Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis the Helge Holden Anniversary Volume

Nils Henrik Risebro and Kristian Seip, Editors Fritz Gesztesy, Harald Hanche-Olsen, Espen R. Jakobsen, Yurii Lyubarskii, Mathematical Physics, and Stochastic Analysis Non-Linear Partial Differential Equations, Series of Congress Reports Series of Congress Reports

Non-Linear Partial Differential Non-Linear Partial Equations, Mathematical Physics, and Stochastic Analysis Differential Equations, The Helge Holden Anniversary Volume Mathematical Physics, Fritz Gesztesy, Harald Hanche-Olsen, Espen R. Jakobsen, Yurii Lyubarskii, Nils Henrik Risebro and Kristian Seip, and Stochastic Analysis Editors The Helge Holden Anniversary Volume This volume is dedicated to Helge Holden on the occasion of his 60th anniversary. It collects contributions by numerous scientists with expertise in non-linear partial differential equations (PDEs), mathematical physics, and stochastic analysis, reflecting to a large degree Helge Holden’s longstanding research interests. Fritz Gesztesy Accordingly, the problems addressed in the contributions deal with a large range of topics, including, in particular, infinite-dimensional analysis, linear and nonlinear Harald Hanche-Olsen PDEs, stochastic analysis, spectral theory, completely integrable systems, random matrix theory, and chaotic dynamics and sestina poetry. They represent to some extent the lectures presented at the conference Non-linear PDEs, Mathematical Physics and Espen R. Jakobsen Stochastic Analysis, held at NTNU, Trondheim, July 4–7, 2016. Yurii Lyubarskii The mathematical tools involved draw from a wide variety of techniques in functional analysis, operator theory, and probability theory. Nils Henrik Risebro

This collection of research papers will be of interest to any active scientist working in Kristian Seip one of the above mentioned areas. Editors

ISBN 978-3-03719-186-6

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EMS Series of Congress Reports

EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature.

Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowron´ski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.) Representations of Algebras and Related Topics, Andrzej Skowron´ski and Kunio Yamagata (eds.) Contributions to Algebraic Geometry. Impanga Lecture Notes, Piotr Pragacz (ed.) Geometry and Arithmetic, Carel Faber, Gavril Farkas and Robin de Jong (eds.) Derived Categories in Algebraic Geometry. Toyko 2011, Yujiro Kawamata (ed.) Advances in Representation Theory of Algebras, David J. Benson, Henning Krause and Andrzej Skowron´ski (eds.) Valuation Theory in Interaction, Antonio Campillo, Franz-Viktor Kuhlmann and Bernard Teissier (eds.) Representation Theory – Current Trends and Perspectives, Henning Krause, Peter Littelmann, Gunter Malle, Karl-Hermann Neeb and Christoph Schweigert (eds.) Functional Analysis and Operator Theory for Quantum Physics. The Pavel Exner Anniversary Volume, Jaroslav Dittrich, Hynek Kovarˇík and Ari Laptev (eds.) Schubert Varieties, Equivariant Cohomology and Characteristic Classes, Jarosław Buczyn´ski, Mateusz Michałek and Elisa Postinghel (eds.) Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis The Helge Holden Anniversary Volume

Fritz Gesztesy Harald Hanche-Olsen Espen R. Jakobsen Yurii Lyubarskii Nils Henrik Risebro Kristian Seip Editors Editors:

Fritz Gesztesy Harald Hanche-Olsen Department of Mathematics Espen R. Jakobsen Baylor University Yurii Lyubarskii Sid Richardson 305I, One Bear Place #97328 Kristian Seip Waco, TX 76798-7328 Department of Mathematical Sciences USA Norwegian University of Science and Technology Email: [email protected] Alfred Getz vei 1 7491 Trondheim Nils Henrik Risebro Norway Department of Mathematics Email: [email protected] University of Oslo [email protected] P.O. Box 1036, Blindern [email protected] N-0316 Oslo [email protected] Norway Email: [email protected]

2010 Mathematics Subject Classification: Primary: 15B52, 35J10, 35L65, 35Q41, 35Q51, 35Q53, 37K10, 42B20, 46N20, 46N30, 46T12, 47B36, 47F05, 60H20, 68N30, 76S05; secondary: 33C45, 35A01, 35A02, 35L80, 37D45, 39A12, 47A10, 47N20, 47N30, 60B20. Key words: Infinite-dimensional analysis, partial differential equations, hyperbolic conservation laws, stochastic analysis, spectral theory, discrete evolution, completely integrable systems, random matrix theory, chaotic dynamics.

ISBN 978-3-03719-186-6

The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

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Typeset by the editors using the authors’ TeX files: Harald Hanche-Olsen, Trondheim, Norway Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Preface

This Festschrift is dedicated to Helge Holden on the occasion of his 60th birthday.

Helge Holden was born to parents Finn Holden (1928–), a textbook author, histo- rian, and high school teacher, and Kirsten nee Wolfhagen (1931–1999), a librarian and psychologist, on September 28, 1956, in Oslo, Norway. Helge grew up with two younger brothers, Steinar, who is professor of economics at the University of Oslo, and Lars, who is Managing Director at the Norwegian Computing Center in Oslo. Interestingly, Helge has a joint paper [58]1 with his two brothers. Helge decided early to study mathematics and physics, and after serving the mandatory one year military service, he entered the University of Oslo in 1976. In the fall of 1980 he started to work on his cand.real. thesis with the late Prof. Raphael Høegh-Krohn (1938–1988) as advisor, working on solvable models in non-relativistic quantum mechanics. In the fall of 1981 he graduated with the cand.real. exam, his grades being so impressive that the result was reported to the King of Norway in the Council of State. Raphael Høegh-Krohn – a charismatic and brilliant mathematical physicist at the University of Oslo – brought Helge in contact with the group around Prof. Sergio Albeverio in Bochum, and the young mathematicians Fabio Martinelli, Werner Kirsch, and, Fritz Gesztesy. Fritz came to Oslo because he had found an improvement in an important paper by Raphael and Sergio (marking the beginning of an intensive collaboration with both of them), and at that occasion Fritz and Helge also started working together, and have continued to do so ever since. In those days, the work centered around so-called point interaction models in nonrelativistic quantum mechanics, and resulted in the book β[1], published by Springer in 1988, translated into Russian, and re-issued in a 2nd edition by the American Mathematical Society with an appendix by Prof. Pavel Exner in 2005. It has established itself as the standard reference in the field, and it is frequently cited in both the physics and mathematics literature. In the early 1980s, Norway was going through rapid and dramatic development. At Christmas of 1969, the first oil was discovered in the North Sea – the Ekofisk field – still one of the largest in the North Sea. The question was how to take advantage

1Numbers refer to the complete list of publications. vi Preface of this to the benefit of Norwegian society. Fortunately, the government decided to build up national competency in Norway, resulting in a dramatic improvement in the standard of living in Norway. Those who visited Norway in the early ’80s and visit now, will see an extreme makeover in Norway – it simply is a different country now. In the early ’80s, the Norwegian oil companies Saga, Hydro, and Statoil needed scientists with a good background in the sciences and in mathematics, and the companies supported mathematical research. Raphael and Helge, together with Nils Henrik Risebro, Lars Holden, Tore Gimse, Kyrre and Frode Bratvedt, and Christian Buchholz, embarked on the project to develop a full scale petroleum reservoir simulator. The approach was based at first on lots of enthusiasm and, retrospectively, less on deep knowledge of reservoir simulation; it was also the start of a life-long collaboration between Helge and Nils Henrik. Unfortunately, Raphael suddenly passed away in 1988, and Helge took over as PhD advisor of Nils Henrik and Lars. The development of the reservoir simulator subsequently changed focus, but it is to this day still used by Schlumberger – the world’s leading oil field services company. At this time there was essentially no activity in partial differential equations in Norway. Together with Prof. Ragnar Winther in Oslo, Helge and Ragnar have, together with their substantial number of students, been instrumental in developing the field of partial differential equations into one of the strongest areasof Norwegian mathematics. The research is no longer restricted to topics relevant to flow in porous media, and a special focus has been to understand the interaction between theoretical results and numerical simulations. There are now strong research groups in partial differential equations at the universities in Oslo, Bergen, and Trondheim, in addition to that at SINTEF. After completion of his PhD, Helge went for the academic year 1985–86 to the Courant Institute of Mathematical Sciences of New York University on a Fulbright scholarship to join the group of Prof. James Glimm. While there, he was offered a permanent position at the Norwegian Institute of Technology (now NTNU – the Norwegian University of Science and Technology), in Trondheim, Norway, and he has remained at NTNU ever since. He spent the spring semester 1989 at the California Institute of Technology, with Prof. Barry Simon, and the academic year 1996/97 at the University of Missouri, Columbia, with Fritz. Helge’s mathematical activity spans several areas. Starting out in mathematical physics, focusing on nonrelativistic quantum mechanics, he later turned his interest to nonlinear partial differential equations. Here the interest initially was concentrated on the study of flow of hydrocarbons in porous media, and the most important mathematical results came in the theory of hyperbolic conservations laws, where, together with Nils Henrik, he wrote the book β[4] that represents the standard presentation of the so-called front-tracking approach to hyperbolic Preface vii conservation laws. With the group of Prof. Bernt Øksendal in Oslo, Helge worked on stochastic differential equations using the so-called white noise approach, and their results were presented in the book β[2]. With Fritz, Helge’s interest turned to completely integrable systems – where the celebrated KdV equation is one of the most recognized examples – and they collected their results in the two-volume treatise β[5, 6], the second volume written jointly with Johanna Michor and Gerald Teschl. Subsequently, Helge’s interest moved to the Camassa–Holm equation, which he had already studied with Fritz in the algebro-geometric setting. Now the problem turned into the study of an evolution equation whose solutions encounter wave breaking, and this study led to further work also on the nonlinear variational wave equation. In this area he worked with Xavier Raynaud and Katrin Grunert. His most recent book discusses so-called operator splitting methods for nonlinear partial differential equations with rough solutions, and is joint with Knut-Andreas Lie, Kenneth H. Karlsen, and Nils Henrik β[7]. A common feature of all of his work is the deep interaction between mathematics and physics. To date, he has co-authored well over 160 publications and 7 books (4 of which are already in 2nd edition, a rare feat). Helge truly enjoys collaboration and to this day has worked with well over 60 collaborators. His work has been extensively cited, according to MathSciNet, almost 2491 times by more than 1631 researchers.2 Helge has received numerous honors, among which we mention the following: – Election to the Norwegian Academy of Science and Letters – Election to the Royal Norwegian Society of Sciences and Letters – Election to the Norwegian Academy of Technological Sciences – Election to the European Academy of Sciences – Fellow, American Mathematical Society – Fellow, Society for Industrial and Applied Mathematics (SIAM) – He gave an invited talk at the European Congress of Mathematics in Stockholm in 2004. At NTNU Helge has been involved in several activities – he has had a staggering number of almost 90 master students and 24 PhD students (3 currently), most of whom are active researchers with successful careers of their own, and he is currently serving on the Board of NTNU. In 1993 he took the initiative to organize an annual Lars Onsager Lecture and Lars Onsager Professorship at NTNU to commemorate the iconic Lars Onsager (1903–76), Nobel Laureate in Chemistry in 1967, whose Alma Mater was the Norwegian Institute of Technology; 25 years later this activity is still going strong.

2MathSciNet, April 1, 2017. No, not an April fools joke! viii Preface

Historically, NTNU has had a strong focus on engineering with a constant source of excellent students coming from the program in Industrial Mathematics, developed together with partners in ECMI – European Consortium of Mathematics in Industry. Helge got involved with this activity through the late Prof. Henrik Martens (1927–93) at NTNU, who was one of the founders of ECMI. Helge served as President of ECMI during the period 2004–06. On the national scene, Helge has been exceptionally active. He chaired several key panels of the Research Council of Norway for extended periods, the most important being the panel that decided on all national individual research grants in mathematics and the natural sciences. In the Scandinavian countries, Helge has chaired evaluations of research and education in mathematics at several universities in Denmark and Sweden. Similarly, on the European scene, Helge has chaired the panel of the European Research Council that awards the prestigious Consolidator Grants in Mathematics. Helge has written numerous articles in Norwegian newspapers and journals on various research political topics and the popularization of mathematics. He has been very actively involved in the Abel Prize in Mathematics since its start in 2003. In particular, he served as Chair of the Abel Board of the Norwegian Academy of Science and Letters, for the period of 2010–14. During his chairmanship several novel activities started, for instance, the competition UngeAbel, the Heidelberg Laureate Forum, and the collaboration with Petroleum Geo-Services. He also took the initiative to edit books about the Abel Laureates, and with Ragni Piene two volumes have been completed ε[7, 12], with a third volume currently in preparation. Helge has always been a strong proponent of international collaboration, and he served as Secretary (2003–06) and Vice President (2007–10) of the European Mathematical Society. Currently he is serving as Secretary of the International Mathematical Union (IMU), with headquarters in Berlin, Germany. He has served on the Board of the Norwegian Academy of Technological Sciences and the Royal Norwegian Society of Sciences and Letters, and during the period 2014–16 he served as President of the latter. While this hints at some account of Helge as the scientist and his substantial service to our profession in the national as well as international arena, we would be amiss not to comment on some personal aspects of our longstanding friendship with him. His kind, yet firm, demeanor, his integrity and sense of fairness, his dedication to science, his intellectual curiosity about the world as a whole, substantially transcending the natural sciences, his deep interest in the arts, and especially, his love of literature, shows him to be the complex and multi-faceted personality we all came to appreciate so much over the years. Several of the editors are collaborators of Helge’s, some for up to 35 years now, all of us view him as a dear and trusted friend. Preface ix

The volume at hand is based to some extent on the conference, Non-linear PDEs, Mathematical Physics and Stochastic Analysis, held at NTNU, Trondheim, July 4–7, 2016 (https://wiki.math.ntnu.no/holden60). The fields represented in the contributions to this volume reflect to a large degree Helge’s longstanding research interests. They center around infinite-dimensional analysis (integrals of probabilistic and oscillatory type), linear and nonlinear partial differential equations (including discrete evolution equations, Ostrovsky–Hunter-type equations, modeling crowd dynamics, porous medium type equations, nonlinear degenerate anisotropic hyperbolic-parabolic equations, Riemann problems for models of polymer flooding, systems of conservation laws, nonlinear dispersive PDEs, compensated compactness and isometric immersions of manifolds), stochastic analysis (optimal control for a system of stochastic Volterra equations), spectral theory (including spectra of leaky surfaces, Hardy–Rellich-type inequalities, dispersion estimate for one-dimensional Schrödinger operators, Schrödinger operators involving the Heisenberg sub-Laplacian), completely integrable systems (including the modified two-component Camassa–Holm system), random matrix theory, and chaotic dynamics and sestina poetry. Finally, we express our sincere gratitude to the staff at the EMS, particularly, Thomas Hintermann, for their help, support, and expertise in producing this volume. We also thank all authors for their contributions and the referees for their invaluable assistance. Happy Birthday, Helge, we hope this volume brings some fond memories and joy! Fritz Gesztesy Harald Hanche-Olsen Espen R. Jacobsen Yurii Lyubarskii Nils Henrik Risebro Kristian Seip

Helge Holden in July 2016 (photo Harald Hanche-Olsen)

Participants at the birthday conference (photo Harald Hanche-Olsen)

Contents

Preface v Fritz Gesztesy, Harald Hanche-Olsen, Espen R. Jacobsen, Yurii Lyubarskii, Nils Henrik Risebro, and Kristian Seip

By such fate of signs 1 Poul G. Hjorth

Optimal control of forward-backward stochastic Volterra equations 3 Nacira Agram, Bernt Øksendal, and Samia Yakhlef

A unified approach to infinite dimensional integrals of probabilistic and oscillatory type with applications to Feynman path integrals 37 Sergio Albeverio and Sonia Mazzucchi

The numbers lead a dance 55 Alan R. Champneys, Poul G. Hjorth, and Harry Man

Compensated compactness in Banach spaces and weak rigidity of isometric immersions of manifolds 73 Gui-Qiang G. Chen and Siran Li

The initial-boundary-value problem for an Ostrovsky–Hunter type equation 97 Giuseppe Maria Coclite, Lorenzo di Ruvo, and Kenneth Hvistendahl Karlsen

Modeling crowd dynamics through hyperbolic – elliptic equations 111 Rinaldo M. Colombo, Maria Gokieli, and Massimiliano D. Rosini

On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type 129 Félix del Teso, Jørgen Endal, and Espen R. Jakobsen

On the spectrum of leaky surfaces with a potential bias 169 Pavel Exner xvi Contents

On the decay of almost periodic solutions of anisotropic degenerate parabolic-hyperbolic equations 183 Hermano Frid

Factorizations and Hardy–Rellich-type inequalities 207 Fritz Gesztesy and Lance Littlejohn

Symmetries and multipeakon solutions for the modified two-component Camassa–Holm system 227 Katrin Grunert and Xavier Raynaud

Vanishing viscosity solutions of Riemann problems for models of polymer flooding 261 Graziano Guerra and Wen Shen

Efficient computation of all speed flows using an entropy stable shock-capturing space-time discontinuous Galerkin method 287 Andreas Hiltebrand and Siddhartha Mishra

Dispersion estimates for spherical Schrödinger equations with critical angular momentum 319 Markus Holzleitner, Aleksey Kostenko, and Gerald Teschl

Sixty years of moments for random matrices 349 Werner Kirsch and Thomas Kriecherbauer

Bound states of Schrödinger type operators with Heisenberg sub-Laplacian 381 Ari Laptev and Andrei Velicu

On Holden’s seven guidelines for scientific computing and development of open-source community software 389 Knut-Andreas Lie

Sharp uniqueness results for discrete evolutions 423 Yurii Lyubarskii and Eugenia Malinnikova

Spatial analyticity of solutions to nonlinear dispersive PDE 437 Sigmund Selberg

Publications by Helge Holden 455

List of Contributors 481 By such fate of signs a sestina for Helge

Poul G. Hjorth

By such fate of signs, that oft hyperbolic turmoil of elegant, brittle equations, is forged in the rugged climate of Norway. Quiet, yet solid, and truly distinguished a singular leader, a trusted companion, riddles surrender, riddles resolve.

Time unfolds in its stedfast resolve, causal in structures, a flow hyperbolic. Gazing in mirrors, we see a companion, experience-laden, ancient equations, less now ‘tween ‘I’ and ‘them’ distinguish. Age comes quietly, even in Norway.

On beautiful fjords, on mountains of Norway, a people so fused by common resolve, to be - there - an equal, and also distinguished to live such a life, at once hyperbolic, but also in service, to ancient equations, that nordic-like sorrow a constant companion.

To be so, to life, a faithful companion, roaming the globe, yet rooted in Norway, to balance in everything weighted equations this takes a special kind of resolve. Never in language, nor deed, hyperbolic, but ever the cool and quiet distinguished. 2 P. G. Hjorth

Thus he himself from others distinguished, remains to his fellows a loyal companion, the laws conserving, the laws hyperbolic, have made him known, not only in Norway as one who could twisted questions resolve into transparently clear blue equations.

His lifelong companion, those eager equations, discovered in Norway, are clearly distinguished, no less hyperbolic than Helge’s resolve. Optimal control of forward-backward stochastic Volterra equations

Nacira Agram, Bernt Øksendal, and Samia Yakhlef

Dedicated to Helge Holden on the occasion of his 60th birthday

Abstract. We study the problem of optimal control of a coupled system of forward-backward stochastic Volterra equations. We use Hida–Malliavin calculus to prove a sufficient and a necessary maximum principle for the optimal control of such systems. Existence and uniqueness of backward stochastic Volterra integral equations are proved. As an application of our methods, we solve a recursive utility optimisation problem in a financial model with memory.

1. Introduction

The purpose of this paper is to establish solution techniques for optimal control of coupled systems of stochastic Volterra equations. Stochastic Volterra equations appear in models for dynamic systems with noise and memory. As a motivating example, consider the following Volterra equation, modeling a stochastic cash flow 푋(푡) = 푋푐(푡) subject to a consumption rate 푐(푡) at time 푡: 푡 푡 푋(푡) = 휉(푡) + ∫ (훼(푡, 푠) − 푐(푠))푋(푠) 푑푠 + ∫ 훽(푡, 푠)푋(푠) 푑퐵(푠) 0 0 푡 (1.1) + ∫ ∫ 휋(푡, 푠, 푒)푋(푠)푁(푑푠,˜ 푑푒), 푡 ∈ [0, 푇] , 0 ℝ 2 2 where 휉∶ [0, 푇] → ℝ and 훼, 훽∶ [0, 푇] → ℝ and 휋∶ [0, 푇] × ℝ0 → ℝ are deterministic functions with 훼, 훽 and 휋 bounded. Here 퐵(푡) = 퐵(푡, 휔) and 푁(푑푡, 푑푒) = 푁(푑푡, 푑푒, 휔) are a Brownian motion and an independent Poisson random measure, respectively, on a complete probability space (Ω, ℱ, 푃). The compensated Poisson random measure 푁˜ is defined by 푁(푑푡,˜ 푑푒) = 푁(푑푡, 푑푒) − 휈(푑푒) 푑푡, where 휈 is the Lévy measure of 푁. We denote by 픽 = {ℱ푡}푡≥0 the right-continuous complete filtration generated by 퐵 and 푁, and we let 픾 ≔ {풢푡}푡≥0 4 N. Agram, B. Øksendal, and S. Yakhlef be a given right-continuous complete subfiltration of 픽, in the sense that

풢푡 ⊆ ℱ푡 for all 푡 ∈ [0, 푇].

The sigma-algebra 풢푡 represents the information available to the consumer at time 푡. Let 풫(픽) be the 휎-algebra of 픽-predictable subsets of Ω × ℝ+, i.e., the 휎-algebra generated by the left continuous 픽-adapted processes. The forward stochastic Volterra integral equation (FSVIE) (1.1) can be written in differential form as 푡 휕훼 푑푋(푡) = 휉′(푡) 푑푡 + (훼(푡, 푡) − 푐(푡))푋(푡) 푑푡 + (∫ (푡, 푠)푋(푠) 푑푠) 푑푡 휕푡 0 푡 휕훽 + 훽(푡, 푡)푋(푡) 푑퐵(푡) + (∫ (푡, 푠)푋(푠) 푑퐵(푠)) 푑푡 휕푡 0 (1.2) + ∫ 휋(푡, 푡, 푒)푋(푡)푁(푑푡,˜ 푑푒) ℝ 푡 휕휋 + (∫ ∫ (푡, 푠, 푒)푋(푠)푁(푑푠,˜ 푑푒)) 푑푡, 푡 ∈ [0, 푇]. 휕푡 ℝ 0 From (1.2) we see that the dynamics of 푋(푡) contains history or memory terms represented by the 푑푠-integrals. Following a suggestion of Duffie and Epstein [5] we now model the total utility of the consumption rate 푐(푡) by a recursive utility process 푌(푡) = 푌푐(푡) defined by the equation 푇 | 푌(푡) = 피 [− ∫ {훾(푠)푌(푠) + ln(푐(푠)푋(푠))} 푑푠 | ℱ푡] ; 푡 ∈ [0, 푇] 푡 | By the martingale representation theorem we see that there exist processes 푍(푡), 퐾(푡, 푒) such that the triple (푌, 푍, 퐾) solves the backward stochastic differential equation (BSDE) 푑푌(푡) = −[훾(푡)푌(푡) + ln(푐(푡)푋(푡)) 푑푡 + 푍(푡) 푑퐵(푡) ⎧ ⎪ + ∫ 퐾(푡, 푒)푁(푑푡,˜ 푑푒); 푡 ∈ [0, 푇] (1.3) ⎨ ⎪ ℝ ⎩ 푌(푇) = 0 We now consider the optimal recursive utility problem to maximise the total recursive utility of the consumption. In other words, we want to find an optimal ∗ consumption rate 푐 ∈ 풰픾 such that sup 푌푐(0) = 푌푐∗(0), (1.4) 푐∈풰픾 Stochastic Volterra equations 5 where 풰픾 is a given set of admissible 픾-adapted consumption processes. This is a problem of optimal control of a coupled system consisting of the forward stochastic Volterra equation (1.1) and the BSDE (1.3). In the following sections we will present solution methods for general optimal control for systems of forward-backward stochastic Volterra equations. Then in the last section we will apply the methods to solve the optimal recursive utility consumption problem above. There has been a lot of research activity recently within stochastic Volterra integral equations (SVIEs) recently, both of forward and backward type. See, e.g., [2, 7, 11, 12, 14, 13, 15, 16, 17, 18]. Perhaps the paper closest to our paper is [13]. However, that paper has a different approach than ours, does not have a sufficient maximum principle, and does not deal with jumps and partial information, as we do.

2. Stochastic maximum principle for FBSVE

This section is an extension to forward-backward systems of the results obtained in [2]. We consider a system governed by a coupled system of controlled forward- backward stochastic Volterra equations (FBSVE) of the form:

푡 푡 푋(푡) = 휉(푡) + ∫ 푏(푡, 푠, 푋(푠), 푢(푠)) 푑푠 + ∫ 휎(푡, 푠, 푋(푠), 푢(푠)) 푑퐵(푠) (2.1) 0 0 푡 + ∫ ∫ 휃(푡, 푠, 푋(푠), 푢(푠), 푒)푁(푑푠,˜ 푑푒), 푡 ∈ [0, 푇], 0 ℝ 푇 푌(푡) = 휂(푋(푇)) + ∫ 푔(푡, 푠, 푋(푠), 푌(푠), 푍(푡, 푠), 퐾(푡, 푠, ⋅ ), 푢(푠)) 푑푠 (2.2) 푡 푇 푇 − ∫ 푍(푡, 푠) 푑퐵(푠) − ∫ ∫ 퐾(푡, 푠, 푒)푁(푑푠,˜ 푑푒), 푡 ∈ [0, 푇] . 푡 푡 ℝ

The quadruple (푋, 푌, 푍, 퐾) is said to be a solution of (2.1)–(2.2) if it satisfies both equations. To the best of our knowledge, results about existence and uniqueness of solutions for such general systems are not known. Conditions under which there exists a unique solution (푌, 푍, 퐾) of (2.2) are studied in Section 3. In the above the functions 휉, 휂 are assumed to be deterministic and 퐶1, while 6 N. Agram, B. Øksendal, and S. Yakhlef the functions 푏(푡, 푠, 푥, 푢)∶ [0, 푇]2 × ℝ × 핌 × Ω → ℝ, 휎(푡, 푠, 푥, 푢)∶ [0, 푇]2 × ℝ × 핌 × Ω → ℝ, 푔(푡, 푠, 푥, 푦, 푧, 푘( ⋅ ), 푢)∶ [0, 푇]2 × ℝ3 × 퐿2(휈) × 핌 × Ω → ℝ, 2 휃(푡, 푠, 푥, 푢, 푒)∶ [0, 푇] × ℝ × 핌 × ℝ0 × Ω → ℝ, are assumed to be continuously differentiable with respect to their first variables, and for all 푡, 푥, 푦, 푧, 푘, 푢, 푒 the processes 푠 ↦ 푏(푡, 푠, 푥, 푢), 푠 ↦ 휎(푡, 푠, 푥, 푢), 푠 ↦ 푔(푡, 푠, 푥, 푦, 푧, 푘( ⋅ ), 푢), 푠 ↦ 휃(푡, 푠, 푥, 푢, 푒) are ℱ푠-measurable for all 푠 ≤ 푡. We assume that 푡 ↦ 푍(푡, 푠) and 푡 ↦ 퐾 (푡, 푠, ⋅ ) are 퐶1 for all 푠, 푒, 휔 and that

푇 푇 휕푍 2 푇 푇 휕퐾 2 피 [∫ ∫ ( (푡, 푠)) 푑푠 푑푡 + ∫ ∫ ∫ ( (푡, 푠, 푒)) 휈(푑푒) 푑푠 푑푡] < ∞. (2.3) 휕푡 휕푡 0 0 0 0 ℝ It is known that (2.3) holds for some linear systems. See [6]. Let 핌 be a given open convex subset of ℝ and let 풰 = 풰픾 be a given family of admissible controls, required to be 픾-predictable, where, as before, 픾 = {풢푡}푡≥0 is a given subfiltration of 픽 = {ℱ푡}푡≥0, in the sense that 풢푡 ⊆ ℱ푡 for all 푡. We associate to the system (2.1)–(2.2) the following performance functional:

푇 퐽(푢) = 피 [∫ 푓(푠, 푋(푠), 푌(푠), 푢(푠)) 푑푠 + 휑(푋(푇)) + 휓(푌(0))] , 0 for given functions

푓∶ [0, 푇] × ℝ2 × 핌 × Ω → ℝ, 휑∶ ℝ → ℝ, 휓∶ ℝ → ℝ.

The functions 휑, 휓 are assumed to be 퐶1, while 푓(푠, 푥, 푦, 푢) is assumed to be 픽- adapted with respect to 푠 and 퐶1 with respect to 푥, 푦, 푢 for each s. We remark here that our performance functional is not of Volterra type. Our optimisation control ∗ problem is to find 푢 ∈ 풰픾 such that sup 퐽(푢) = 퐽(푢∗). (2.4) ᵆ∈풰 Let ℒ be the set of all 픽-adapted stochastic processes, and let ℛ denote the set of all functions 푘∶ ℝ → ℝ. Define the Hamiltonian functional:

ℋ(푡, 푥, 푦, 푧, 푘( ⋅ ), 푣, 푝, 푝( ⋅ ), 푞, 휆, 휆( ⋅ ), 푟( ⋅ ))

≔ 퐻0(푡,푥,푦,푧,푘(⋅),푣,푝,푞,휆,푟(⋅)) + 퐻1(푡, 푥, 푦, 푧, 푘( ⋅ ), 푣, 푝( ⋅ ), 휆( ⋅ )), Stochastic Volterra equations 7 where 3 3 퐻0 ∶ [0, 푇] × ℝ × ℛ × 핌 × ℝ × ℛ → ℝ is defined by

퐻0(푡,푥,푦,푧,푘(⋅),푣,푝,푞,휆,푟(⋅)) ≔ 푓(푡, 푥, 푦, 푣) + 푏(푡, 푡, 푥, 푣)푝 + 휎(푡, 푡, 푥, 푣)푞

+ ∫ 휃(푡, 푡, 푥, 푣)푟(푡, 푒)휈(푑푒) + 푔(푡, 푡, 푥, 푦, 푧, 푘( ⋅ ), 푣)휆 ℝ and 3 퐻1 ∶ [0, 푇] × ℝ × ℛ × 핌 × ℒ × ℒ → ℝ is defined by

퐻1(푡, 푥, 푦, 푧, 푘( ⋅ ), 푣, 푝( ⋅ ), 휆( ⋅ )) 푇 휕푏 푇 휕휎 ≔ ∫ (푠, 푡, 푥, 푣)푝(푠) 푑푠 + ∫ (푠, 푡, 푥, 푣)피[퐷 푝(푠) | ℱ] 푑푠 휕푠 휕푠 푡 푡 푡 푡 푇 휕휃 + ∫ ∫ (푠, 푡, 푥, 푣)피[퐷 푝(푠) | ℱ]휈(푑푒) 푑푠 휕푠 푡,푒 푡 푡 ℝ 푡 휕푔 + ∫ (푠, 푡, 푥, 푦, 푧, 푘( ⋅ ), 푣)휆(푠) 푑푠 휕푠 0 푡 휕푔 휕푍 + ∫ (푠, 푡, 푥, 푦, 푧, 푘( ⋅ ), 푣) (푠, 푡)휆(푠) 푑푠 휕푧 휕푠 0 푡 휕퐾 + ∫ ⟨∇ 푔(푠, 푡, 푥, 푦, 푧, 푘( ⋅ ), 푣), (푠, 푡, ⋅ )⟩휆(푠) 푑푠. 푘 휕푠 0

Here, and in the following, 퐷푡 and 퐷푡,푒 denote the (generalised) Hida–Malliavin derivative at 푡 and at (푡, 푒) with respect to 퐵 and 푁˜, respectively, and ∇푘 denotes the Fréchet derivative with respect to 푘. We refer to the Appendix for more details.

The associated forward-backward system for the adjoint processes 휆(푡), (푝(푡), 푞(푡), 푟(푡, ⋅ )) is

휕ℋ 휕ℋ 푑∇ ℋ 푑휆(푡) ≔ (푡) 푑푡 + (푡) 푑퐵(푡) + ∫ 푘 (푡)푁(푑푡,˜ 푑푒), 0 ≤ 푡 ≤ 푇, 휕푦 휕푧 푑휈 { ℝ (2.5) 휆(0) ≔ 휓′ (푌(0)) , and 8 N. Agram, B. Øksendal, and S. Yakhlef

휕ℋ 푑푝(푡) ≔ − (푡) 푑푡 + 푞(푡) 푑퐵(푡) + ∫ 푟(푡, 푒)푁(푑푡,˜ 푑푒), 0 ≤ 푡 ≤ 푇, 휕푥 ℝ (2.6) 푝(푇) ≔ 휑′(푋(푇)) + 휆(푇)휂′(푋(푇)), where we have used the simplified notation

휕ℋ 휕ℋ (푡) = [ (푡, 푥, 푌(푡), 푍 (푡, .) , 퐾(푡, ⋅ ), 푢(푡), 푝(푡), 푞(푡), 휆(푡), 푟(푡, ⋅ ))] , 휕푥 휕푥 푥=푋(푡)

휕ℋ 휕ℋ and similarly for 휕푦 (푡), 휕푧 (푡).

As in [8] we assume that 퐻 is Fréchet differentiable (퐶1) in the variables 푥, 푦, 푧, 푘, 푢 and that the Fréchet derivative ∇푘퐻 of 퐻 with respect to 푘 ∈ ℛ as a random measure is absolutely continuous with respect to 휈, with Radon–Nikodym derivative 푑∇푘퐻/푑휈. Thus, if ⟨∇푘퐻, ℎ⟩ denotes the action of the linear operator ∇푘퐻 on the function ℎ ∈ ℛ, we have

푑∇푘퐻(휁) ⟨∇푘퐻, ℎ⟩ = ∫ ℎ(휁) 푑∇푘퐻(휁) = ∫ ℎ(휁) 푑휈(휁). ℝ ℝ 푑휈(휁)

The question of existence and uniqueness of the forward-backward system above will not be studied here. It is a subject of future research. See, however our partial result in Section 3.

2.1. A sufficient maximum principle. In this subsection, we prove that under some conditions such as the concavity, a given control ̂푢 which satisfies a maximum condition of the Hamiltonian, is an optimal control for the problem (2.4). From (2.1)–(2.2) we can get the differential forms:

푡 휕푏 푑푋(푡) = 휉′(푡)푑푡 + 푏(푡, 푡, 푋(푡), 푢(푡)) 푑푡 + (∫ (푡, 푠, 푋(푠), 푢(푠)) 푑푠) 푑푡 휕푡 0 푡 휕휎 + 휎(푡, 푡, 푋(푡), 푢(푡)) 푑퐵(푡) + (∫ (푡, 푠, 푋(푠), 푢(푠)) 푑퐵(푠)) 푑푡 휕푡 0 (2.7) + ∫휃(푡, 푡, 푋(푠), 푢(푠), 푒)푁(푑푠,˜ 푑푒) ℝ 푡 휕휃 + (∫ ∫ (푡, 푠, 푋(푠), 푢(푠), 푒)푁(푑푠,˜ 푑푒)) 푑푡, 휕푡 0 ℝ Stochastic Volterra equations 9 and

푑푌(푡) = −푔(푡, 푡, 푋(푡), 푌(푡), 푍(푡, 푡), 퐾(푡, 푡, ⋅ ), 푢(푡)) 푑푡 푇 휕푔 + (∫ (푡, 푠, 푋(푠), 푌(푠), 푍(푡, 푠), 퐾(푡, 푠, ⋅ ), 푢(푠)) 푑푠) 푑푡 휕푡 푡 푇 휕푔 휕푍 + ∫ (푡, 푠, 푋(푠), 푌(푠), 푍(푡, 푠), 퐾(푡, 푠, ⋅ ), 푢(푠)) (푡, 푠) 푑푡 휕푧 휕푡 푡 푇 휕퐾 + ∫ ⟨∇ 푔(푡, 푠, 푋(푠), 푌(푠), 푍(푡, 푠), 퐾(푡, 푠, ⋅ ), 푢(푠)), (푡, 푠, ⋅ )⟩ 푑푡 푘 휕푡 푡 + 푍(푡, 푡) 푑퐵(푡) + ∫ 퐾(푡, 푡, 푒)푁(푑푡,˜ 푑푒) ℝ 푇 휕푍 푇 휕퐾 − (∫ (푡, 푠) 푑퐵(푠)) 푑푡 − (∫ ∫ (푡, 푠, 푒)푁(푑푠,˜ 푑푒)) 푑푡, 휕푡 휕푡 푡 푡 ℝ 푌(푇) = 휂(푋(푇)). (2.8)

We now state and prove a sufficient maximum principle:

Theorem 2.1. Let ̂푢∈ 풰픾, with corresponding solutions 푋(푡)̂ , (푌(푡),̂ 푍(푡,̂ 푠), 퐾̂ (푡, 푠, ⋅ )), 휆(푡)̂ , ( ̂푝(푡), ̂푞(푡), ̂푟 (푡, ⋅ )) of equations (2.7), (2.8), (2.5), and (2.6), respectively. Assume the following:

• (Concavity conditions) The functions

푥 ↦ 휂(푥), 푥 ↦ 휑(푥), 푥 ↦ 휓(푥),

and 푥, 푦, 푧, 푘( ⋅ ), 푢 ↦ ℋ(푡,푥,푦,푧,푘(⋅), 푢,푝,푞,휆,푟), are concave for all 푡, 푝, 푞, 휆, 푟.

• (The maximum condition) ̂ ̂ sup 피 [ℋ(푡, 푋(푡),̂ 푌(푡),̂ 푍̂ (푡) , 푘 (푡, ⋅ ) , 푣, 휆(푡), ̂푝(푡), ̂푞(푡), ̂푟(푡, ⋅ )) ∣ 풢푡 푣∈풰 ̂ ̂ = 피 [ℋ(푡, 푋(푡),̂ 푌(푡),̂ 푍(푡),̂ 푘 (푡, ⋅ ) , ̂푢(푡), 휆(푡), ̂푝(푡), ̂푞(푡), ̂푟(푡, ⋅ )) ∣ 풢푡 , ∀푡 ≥ 0. (2.9)

Then, ̂푢 is an optimal 픾-adapted control. Proof. By considering a suitable increasing family of stopping times converging to 푇, we may assume that all the local martingales appearing in the proof below 10 N. Agram, B. Øksendal, and S. Yakhlef are martingales. In particular, the expectations of the 푑퐵- and 푁(푑푡,˜ 푑푒)-integrals are all 0.

Choose an arbitrary 푢 ∈ 풰픾 and consider

퐽(푢) − 퐽( ̂푢) = 퐼1 + 퐼2 + 퐼3, where

푇 ̂ 퐼1 = 피 [∫ {푓(푡) − 푓(푡)} 푑푡 ] , 퐼2 = 피 [휑 (푋(푇)) − 휑(푋(푇))̂ , 0

퐼3 = 피 [휓 (푌(0)) − 휓(푌(0))̂ , where 푓(푡) = 푓(푡, 푋(푡), 푌(푡), 푢(푡)) and 푓(푡)̂ = 푓(푡, 푋(푡),̂ 푌(푡),̂ ̂푢(푡)). Using a simplified notation

푏(푡, 푡) = 푏(푡, 푡, 푋(푡), 푢(푡)), 푏(푡,̂ 푡) = 푏(푡, 푡, 푋(푡),̂ ̂푢(푡)), 푏(푡, 푠) = 푏(푡, 푠, 푋(푠), 푢(푠)), 휃(푡, 푡, 푒) = 휃(푡, 푡, 푋(푠), 푢(푠), 푒), 휃(푡, 푠, 푒) = 휃(푡, 푠, 푋(푠), 푢(푠), 푒) etc., we get

푇 ̂ 퐼1 = 피[∫ {퐻0(푡) − 퐻̂0(푡) − ̂푝(푡)(푏(푡, 푡) − 푏(푡, 푡)) − ̂푞(푡)(휎(푡, 푡) − ̂휎(푡, 푡)) 0 (2.10) − 휆(푡)̂ (푔(푡, 푡) − ̂푔(푡, 푡)) − ∫ ̂푟(푡, 푒)(휃(푡, 푡, 푒) − 휃(푡,̂ 푡, 푒))휈(푑푒)}푑푡]. ℝ

Using concavity and the Itô formula, we obtain

′ 퐼2 ≤ 피[휑 (푋(푇))̂ (푋(푇) − 푋(푇)̂ ) = 피[ ̂푝(푇)(푋(푇) − 푋(푇)̂ ) − 피[휆(푇)휂̂ ′(푋(푇))̂ (푋(푇) − 푋(푇)̂ ) 푇 푇 = 피[∫ ̂푝(푡)(푑푋(푡) − 푑푋(푡)̂ ) + ∫ (푋(푡) − 푋(푡)̂ )푑 ̂푝(푡) 0 0 푇 + ∫ ̂푞(푡)(휎(푡, 푡) − ̂휎(푡, 푡)) 푑푡 0 푇 + ∫ ∫ ̂푟(푡, 푒)(휃(푡, 푡, 푒) − 휃(푡,̂ 푡, 푒))휈(푑푒) 푑푡] 0 ℝ − 피[휆(푇)휂̂ ′(푋(푇))̂ (푋(푇) − 푋(푇)̂ ) Stochastic Volterra equations 11

푇 푡 휕푏 휕푏̂ = 피[∫ { ̂푝(푡)(푏(푡, 푡) − 푏(푡,̂ 푡) + ∫ ( (푡, 푠) − (푡, 푠)) 푑푠 휕푡 휕푡 0 0 푡 휕휎 휕 ̂휎 + ∫ ( (푡, 푠) − (푡, 푠)) 푑퐵(푠) 휕푡 휕푡 0 푡 휕휃 휕휃̂ + ∫ ∫( (푡, 푠) − (푡, 푠))푁(푑푠,˜ 푑푒)) 휕푡 휕푡 0 ℝ 휕ℋ̂ − (푡)(푋(푡) − 푋(푡)̂ ) + ̂푞(푡)[휎(푡, 푡) − ̂휎(푡, 푡) } 푑푡] 휕푥 푇 + ∫ ∫ ̂푟(푡, 푒)(휃(푡, 푡, 푒) − 휃(푡,̂ 푡, 푒))휈(푑푒) 푑푡 0 ℝ − 피[휆(푇)휂̂ ′(푋(푇)̂ )(푋(푇) − 푋(푇)̂ )]. (2.11)

By the Fubini theorem, we get

푇 푡 휕푏 푇 푇 휕푏 ∫ (∫ (푡, 푠) 푑푠) ̂푝(푡) 푑푡 = ∫ (∫ (푡, 푠) ̂푝(푡) 푑푡) 푑푠 휕푡 휕푡 0 0 0 푠 푇 푇 휕푏 = ∫ (∫ (푠, 푡) ̂푝(푠) 푑푠) 푑푡, (2.12) 휕푠 0 푡 and similarly, by the generalised duality theorems for the Malliavin derivatives [2], we have

푇 푡 휕휎 푇 푡 휕휎 피 [∫ (∫ (푡, 푠) 푑퐵(푠)) ̂푝(푡) 푑푡] = ∫ 피 [∫ (푡, 푠) 푑퐵(푠) ̂푝(푡)] 푑푡 휕푡 휕푡 0 0 0 0 푇 푡 휕휎 = ∫ 피 [∫ (푡, 푠)피[퐷 ̂푝(푡) ∣ ℱ 푑푠] 푑푡 휕푡 푠 푠 0 0 푇 푇 휕휎 = ∫ 피 [∫ (푡, 푠)피[퐷 ̂푝(푡) ∣ ℱ 푑푡] 푑푠 휕푡 푠 푠 0 푠 푇 푇 휕휎 = 피 [∫ ∫ (푠, 푡)피[퐷 ̂푝(푠) ∣ ℱ 푑푠 푑푡] 휕푠 푡 푡 0 푡 (2.13) and

푇 푡 휕휃 피 [∫ (∫ ∫ ( (푡, 푠)) 푁(푑푠,˜ 푑푒)푝(푡)) 푑푡] = 휕푡 0 0 ℝ 12 N. Agram, B. Øksendal, and S. Yakhlef

푇 푡 휕휃 = ∫ 피 [∫ ∫ ( (푡, 푠)) 푁(푑푠,˜ 푑푒)푝(푡)] 푑푡 휕푡 0 0 ℝ 푇 푇 휕휃 = ∫ 피 [∫ ∫ (푡, 푠)피[퐷 푝(푡) | ℱ 휈(푑푒) 푑푡] 푑푠 휕푡 푠,푒 푠 0 푠 ℝ 푇 푇 휕휃 = ∫ 피 [∫ ∫ (푠, 푡)피[퐷 푝(푠) | ℱ 휈(푑푒) 푑푠] 푑푡 (2.14) 휕푠 푡,푒 푡 0 푡 ℝ

Substituting (2.13), (2.14), and (2.12) into (2.11), we get

푇 푇 휕푏 휕푏̂ 퐼 ≤ 피[∫ { ̂푝(푡)(푏(푡, 푡) − 푏(푡,̂ 푡)) + ∫ ̂푝(푠) ( (푠, 푡) − (푠, 푡)) 푑푠 2 휕푠 휕푠 0 푡 푇 휕휎 휕 ̂휎 + ∫ ( (푠, 푡) − (푠, 푡)) 피[퐷 ̂푝(푠) ∣ ℱ] 푑푠 휕푠 휕푠 푡 푡 푡 푇 휕휃 + ∫ ∫ (푠, 푡, 푒)피[퐷 푝(푠) | ℱ 휈(푑푒) 푑푠 휕푠 푡,푒 푡 푡 ℝ 푇 + ∫ ∫( ̂푟(푡, 푒)(휃(푡, 푡, 푒) − 휃(푡,̂ 푡, 푒))휈(푑푒)) 푑푡

0 ℝ 휕ℋ̂ − (푡)(푋(푡) − 푋(푡)̂ ) + ̂푞(푡)(휎(푡, 푡) − ̂휎(푡, 푡))} 푑푡] 휕푥 − 피[휆(푇)휂̂ ′(푋(푇))̂ (푋(푇) − 푋(푇)̂ ) . (2.15)

By the concavity of 휓 and 휂, we obtain

퐼3 = 피[휓(푌(0)) − 휓(푌(0))̂ ≤ 피[휓′(푌(0))̂ (푌(0) − 푌(0)̂ ) = 피[휆(0)(푌(0)̂ − 푌(0))̂ 푇 = 피[휆(푇)̂ (푌(푇) − 푌(푇)̂ ) − 피[∫ (푌(푡) − 푌(푡)̂ )푑휆(푡)̂ 0 푇 푇 휕ℋ̂ + ∫ 휆(푡)̂ (푑푌(푡) − 푑푌(푡)̂ ) + ∫ (푡)(푍(푡, 푠) − 푍(푡,̂ 푠)) 푑푡 휕푧 0 0 푇 푑∇ ℋ + ∫ ∫ 푘 (푡)(퐾(푡, 푠, 푒) − 퐾(푡,̂ 푠, 푒))휈(푑푒) 푑푡] 푑휈 0 ℝ Stochastic Volterra equations 13

≤ 피[휆(푇)휂̂ ′(푋(푇))(푋(푇) − 푋(푇)̂ ) 푇 휕ℋ̂ 푇 − 피[∫ (푡)(푌(푡) − 푌(푡)̂ ) 푑푡 − ∫ 휆(푡)̂ (푔(푡, 푡) − ̂푔(푡, 푡)) 푑푡 휕푦 0 0 푇 푇 휕푔 휕 ̂푔 + ∫ (휆(푡)̂ ∫ ( (푡, 푠) − (푡, 푠)) 푑푠) 푑푡 휕푡 휕푡 0 푡 푇 푇 휕푔 휕푍 휕 ̂푔 휕푍̂ + ∫ 휆(푡)̂ [∫ ( (푡, 푠) (푡, 푠) − (푡, 푠) (푡, 푠)) 푑푠] 푑푡 휕푧 휕푡 휕푧 휕푡 0 푡 푇 푇 휕퐾 휕퐾̂ + ∫ 휆(푡)̂ [∫ (⟨∇ 푔(푡, 푠), (푡, 푠, ⋅ )⟩ − ⟨∇ ̂푔(푡, 푠), (푡, 푠, ⋅ )⟩) 푑푠] 푑푡 푘 휕푡 푘 휕푡 0 푡 푇 푇 휕푍 휕푍̂ + ∫ (휆(푡)̂ ∫ ( (푡, 푠) − (푡, 푠)) 푑퐵(푠)) 푑푡 휕푡 휕푡 0 푡 푇 푇 휕퐾 휕퐾̂ + ∫ (휆(푡)̂ ∫ ∫ ( (푡, 푠, ⋅ ) − (푡, 푠, ⋅ )) 푁(푑푠,˜ 푑푒)) 푑푡 휕푡 휕푡 0 푡 ℝ 푇 휕ℋ̂ + ∫ (푡)(푍(푡, 푠) − 푍(푡,̂ 푠)) 푑푡 휕푧 0 푇 푑∇ ℋ̂ + ∫ ∫ 푘 (푡)(퐾(푡, 푠, 푒) − 퐾(푡,̂ 푠, 푒))휈(푑푒) 푑푡]. (2.16) 푑휈 0 ℝ By the Fubini Theorem, we get

푇 푇 휕푔 푇 푠 휕푔 ∫ (∫ (푡, 푠) 푑푠) 휆(푡)̂ 푑푡 = ∫ (∫ (푡, 푠)휆(푡)̂ 푑푡) 푑푠 휕푡 휕푡 0 푡 0 0 푇 푡 휕푔 = ∫ (∫ (푠, 푡)휆(푠)̂ 푑푠) 푑푡, (2.17) 휕푠 0 0 and 푇 푇 휕푔 휕푍 ∫ 휆(푡)̂ [∫ (푡, 푠) (푡, 푠) 푑푠] 푑푡 휕푧 휕푡 0 푡 푇 푡 휕푔 휕푍 = ∫ (∫ 휆(푠)̂ (푠, 푡) (푠, 푡) 푑푠) 푑푡, 휕푧 휕푠 0 0 푇 푇 휕퐾 ∫ 휆(푡)̂ [∫ ⟨∇ 푔(푡, 푠), (푡, 푠, ⋅ )⟩ 푑푠] 푑푡 푘 휕푡 0 푡 푇 푡 휕퐾 = ∫ (∫ 휆(푠)̂ ⟨∇ 푔(푠, 푡), (푠, 푡, ⋅ )⟩ 푑푠) 푑푡. (2.18) 푘 휕푠 0 0 14 N. Agram, B. Øksendal, and S. Yakhlef

Substituting (2.17)–(2.18) into (2.16), we get

̂ ′ 퐼3 ≤ 피 [휆(푇)휂 (푋(푇)) (푋(푇) − 푋(푇)̂ ) 푇 휕ℋ̂ 푇 − 피 [∫ (푡) (푌(푡) − 푌(푡)̂ ) 푑푡 − ∫ 휆(푡)̂ (푔 (푡, 푡) − ̂푔(푡, 푡)) 푑푡 휕푦 0 0 푇 푡 휕푔 휕 ̂푔 + ∫ ∫ ( (푠, 푡) − (푠, 푡)) 휆(푠)̂ 푑푠 푑푡 휕푠 휕푠 0 0 푇 푡 휕푔 휕푍 휕 ̂푔 휕푍̂ + ∫ (∫ 휆(푠)̂ [ (푠, 푡) (푠, 푡) − (푠, 푡) (푠, 푡)] 푑푠) 푑푡 휕푧 휕푠 휕푧 휕푠 0 0 푇 푡 휕퐾 + ∫ ( ∫ 휆(푠)̂ [⟨∇ 푔(푠, 푡), (푠, 푡, ⋅ )⟩ 푘 휕푠 0 0 휕퐾̂ − ⟨∇ ̂푔(푠, 푡), (푠, 푡, ⋅ )⟩] 푑푠) 푑푡 푘 휕푠 푇 휕ℋ̂ + ∫ (푡) (푍(푡, 푠) − 푍(푡,̂ 푠)) 푑푡]. (2.19) 휕푧 0

Adding (2.10), (2.15), and (2.19), and noting that

푇 휕푏 휕푏̂ 퐻 (푡) − 퐻̂ (푡) = ∫ { (푠, 푡) − (푠, 푡)} ̂푝(푠) 푑푠 1 1 휕푠 휕푠 푡 푇 휕휎 휕 ̂휎 + ∫ { (푠, 푡) − (푠, 푡)} 피[퐷 ̂푝(푠) ∣ ℱ] 푑푠 휕푠 휕푠 푡 푡 푡 푡 휕푔 휕 ̂푔 + ∫ { (푠, 푡) − (푠, 푡)} 휆(푠)̂ 푑푠 휕푠 휕푠 0 푇 푡 휕푔 휕푍 휕 ̂푔 휕푍̂ + ∫ (∫ 휆(푠)̂ [ (푠, 푡) (푠, 푡) − (푠, 푡) (푠, 푡)] 푑푠) 푑푡 + 휕푧 휕푠 휕푧 휕푠 0 0 푇 푡 휕퐾 + ∫ (∫ 휆(푠)̂ [⟨∇ 푔(푠, 푡), (푠, 푡, ⋅ )⟩ 푘 휕푠 0 0 휕퐾̂ − ⟨∇ ̂푔(푠, 푡), (푠, 푡, ⋅ )⟩] 푑푠) 푑푡 푘 휕푠 푇 휕휃 휕휃̂ + ∫ ∫ ( (푠, 푡, 푒) − (푠, 푡, 푒)) 피[퐷 푝(푠)| ℱ]휈(푑푒) 푑푠, 휕푠 휕푠 푡,푒 푡 푡 ℝ we get Stochastic Volterra equations 15

퐽(푢) − 퐽( ̂푢) = 퐼1 + 퐼2 + 퐼3 푇 휕ℋ̂ ≤ 피[∫ {ℋ(푡) − ℋ(푡)̂ − (푡)(푋(푡) − 푋(푡)̂ ) 휕푥 0 휕ℋ̂ 휕ℋ̂ − (푡)(푌(푡) − 푌(푡)̂ ) − (푡)(푍(푡, 푠) − 푍(푡,̂ 푠)) 휕푦 휕푧 푑∇ ℋˆ − ∫ 푘 (푡)(퐾(푡, 푠, 푒) − 퐾(푡,̂ 푠, 푒))휈(푑푒)} 푑푡]. 푑휈 ℝ

By the concavity of ℋ and the maximum condition (2.9), the proof is complete.

2.2. A necessary maximum principle. The concavity condition used in the previous subsection does not always hold in applications. We prove now that if ̂푢∈ 풰픾 is an optimal control for the problem (2.4), then we have the equivalence between being a critical point of 퐽(푢) and a critical point of the conditional Hamiltonian.

We start by defining the derivative processes. For each given 푡 ∈ [0, 푇), let 훼 = 훼(푡) be a bounded 풢푡−measurable random variable, let 휖 ∈ (0, 푇 − 푡], and define

휇(푠) ≔ 훾1[푡,푡+휖](푠), 푠 ∈ [0, 푇]. (2.20)

Assume that ̂푢+ 휖휇 ∈ 풰 for all such 휇, and all sufficiently small nonzero 휖. Then the derivative processes are defined by, writing 푢 for ̂푢 for simplicity from now on,

We see that

푡 휕푏 휕푏 푋′(푡) = ∫ ( (푡, 푠)푋′(푠) + (푡, 푠)휇(푠)) 푑푠 휕푥 휕푢 0 푡 휕휎 휕휎 + ∫ ( (푡, 푠)푋′(푠) + (푡, 푠)휇(푠)) 푑퐵(푠) 휕푥 휕푢 0 푡 휕휃 휕휃 + ∫ ∫ ( (푡, 푠, 푒)푋′(푠) + (푡, 푠, 푒)휇(푠)) 푁(푑푠,˜ 푑푒) 휕푥 휕푢 0 ℝ and

푌′(푡) = 휂′(푋(푇))푋′(푇) 푇 휕푔 휕푔 + ∫ ( (푡, 푠)푋′(푠) + (푡, 푠)푌′(푠) 휕푥 휕푦 푡 휕푔 휕푔 + (푡, 푠)푍′(푡, 푠) + ⟨∇ 푔(푡, 푠), 퐾′(푡, 푠, ⋅ )⟩ + (푡, 푠)휇(푠)) 푑푠 휕푧 푘 휕푢 푇 푇 − ∫ 푍′(푡, 푠) 푑퐵(푠) − ∫ ∫ 퐾′(푡, 푠, 푒)푁(푑푠,˜ 푑푒). 푡 푡 ℝ

Hence

휕푏 휕푏 푑푋′(푡) = [ (푡, 푡)푋′(푡) + (푡, 푡)휇(푡) 휕푥 휕푢 푡 휕2푏 휕2푏 + ∫ ( (푡, 푠)푋′(푠) + (푡, 푠)휇(푠)) 푑푠 휕푡휕푥 휕푡휕푢 0 푡 휕2휎 휕2휎 + ∫ ( (푡, 푠)푋′(푠) + (푡, 푠)휇(푠)) 푑퐵(푠) 휕푡휕푥 휕푡휕푢 0 푡 휕2휃 휕2휃 + ∫ ∫ ( (푡, 푠, 푒)푋′(푠) + (푡, 푠, 푒)휇(푠)) 푁(˜ 푑푠, 푑푒)] 푑푡 휕푡휕푥 휕푡휕푢 0 ℝ 휕휎 휕휎 + ( (푡, 푡)푋′(푡) + (푡, 푡)휇(푡)) 푑퐵(푡) 휕푥 휕푢 휕휃 휕휃 + ∫( (푡, 푡, 푒)푋′(푡) + (푡, 푡, 푒)휇(푡))푁(푑푡,˜ 푑푒), (2.21) 휕푥 휕푢 ℝ and Stochastic Volterra equations 17

푡 푑푌′(푡) = −∇(푔(푡, 푡))(푋′(푡), 푌′(푡), 푍′(푡, 푡), 퐾′(푡, 푡, ⋅ ), 휇(푡)) 푑푡 푇 휕푔 푡 + ∫ ∇ ( (푡, 푠)) (푋′(푠), 푌′(푠), 푍′ (푡, 푠) , 퐾′(푡, 푠, ⋅ ), 휇(푠)) 푑푡 휕푡 푡 푇 휕푔 푡 휕푍 + ∫ ∇ ( (푡, 푠)) (푋′(푠), 푌′(푠), 푍′(푡, 푠), 퐾′(푡, 푠, ⋅ ), 휇(푠)) ( (푡, 푠)) 푑푡 휕푧 휕푡 푡 푇 푡 휕퐾 + ∫ ∇(∇ 푔(푡, 푠))(푋′ (푠) , 푌′(푠), 푍′(푡, 푠), 퐾′(푡, 푠, ⋅ ), 휇(푠)) ( (푡, 푠, ⋅ )) 푑푡 푘 휕푡 푡 + 푍′(푡, 푡) 푑퐵(푡) + ∫ 퐾′(푡, 푡, 푒)푁(푑푡,˜ 푑푒) ℝ 푇 휕푍′ 푇 휕퐾′ − (∫ (푡, 푠) 푑퐵(푠)) 푑푡 − (∫ ∫ (푡, 푠, 푒)푁(푑푡,˜ 푑푒)) 푑푡, (2.22) 휕푡 휕푡 푡 푡 ℝ where we have denoted by ∇ the partial derivatives w.r.t. 푥, 푦, 푧 and 푢 and the 푡 휕 휕 휕 휕 Fréchet derivative w.r.t 푘 such that ∇ = ( 휕푥 , 휕푦 , 휕푧 , ∇푘, 휕ᵆ ) with the second 2 Fréchet derivative ∇푘 ≔ ∇푘∇푘.

Theorem 2.2 (Necessary maximum principle). Let ̂푢∈ 풰픾 with corresponding solutions 푋(푡)̂ , (푌(푡),̂ 푍(푡,̂ 푠), 퐾̂ (푡, 푠, ⋅ )), 휆(푡)̂ , ( ̂푝(푡), ̂푞(푡), ̂푟(푡, ⋅ )) of equations (2.7), (2.8), (2.5), and (2.6), respectively. Then, the following are equivalent:

푇 휕푓 휕푓 휕푓 퐼 = 피[∫ { (푡)푋′(푡) + (푡)푌′(푡) + (푡)휇(푡)} 푑푡], (2.24) 1 휕푥 휕푦 휕푢 0 ′ ′ 퐼2 = 피[휑 (푋(푇))푋 (푇) = 피[푝(푇)푋′(푇) − 피[휆(푇)휂′(푋(푇)푋′(푇)) , ′ ′ 퐼3 = 피[휓 (푌(0))푌 (0) . 18 N. Agram, B. Øksendal, and S. Yakhlef

By the Itô formula

피[푝(푇)푋′(푇) 푇 휕푏 휕푏 = 피[∫ 푝(푡)( (푡, 푡)푋′(푡) + (푡, 푡)휇(푡)) 푑푡 휕푥 휕푢 0 푇 푡 휕2푏 휕2푏 + ∫ 푝(푡){∫ ( (푡, 푠)푋′(푠) + (푡, 푠)휇(푠)) 푑푠} 푑푡 휕푡 휕푥 휕푡 휕푢 0 0 푇 푡 휕2휎 휕2휎 + ∫ 푝(푡){∫ ( (푡, 푠)푋′(푠) + (푡, 푠)휇(푠)) 푑퐵(푠)} 푑푡 휕푡 휕푥 휕푡 휕푢 0 0 푇 푡 휕2휃 휕2휃 + ∫ 푝(푡){∫ ∫( (푡, 푠, 푒)푋′(푠) + (푡, 푠, 푒)휇(푠))푁(푑푠,˜ 푑푒)} 푑푡 휕푡 휕푥 휕푡 휕푢 0 0 ℝ 푇 휕ℋ 푇 휕휎 휕휎 − ∫ 푋′(푡) (푡) 푑푡 + ∫ 푞(푡)( (푡, 푡)푋′(푡) + (푡, 푡)휇(푡)) 푑푡 휕푥 휕푥 휕푢 0 0 푇 휕휃 휕휃 + ∫ (∫( (푡, 푡, 푒)푋′(푡) + (푡, 푡, 푒)휇(푡))푟(푡, 푒)휈(푑푒)) 푑푡]. 휕푥 휕푢 0 ℝ

From (2.12) and (2.13), we have

피[푝(푇)푋′(푇) 푇 휕푏 휕푏 = 피[∫ 푝(푡)( (푡, 푡)푋′(푡) + (푡, 푡)휇(푡)) 푑푡 휕푥 휕푢 0 푇 푇 휕2푏 휕2푏 + ∫ ∫ 푝(푠){( (푠, 푡)푋′(푡) + (푠, 푡)휇(푡)) 푑푠} 푑푡 휕푠 휕푥 휕푠 휕푢 0 푡 푇 휕2휎 휕2휎 푇 + ∫ {( (푠, 푡)푋′(푡) + (푠, 푡)휇(푡)) ∫ 피[퐷 푝(푠) | ℱ 푑푠} 푑푡 휕푠 휕푥 휕푠휕푢 푡 푡 0 푡 푇 푇 휕2휃 휕2휃 + ∫ ∫ ∫( (푠, 푡, 푒)푋′(푡) + (푠, 푡, 푒)휇(푡)) ⋅ 휕푠 휕푥 휕푠 휕푢 0 푡 ℝ

⋅ 피[퐷푡,푒푝(푠) | ℱ푡 휈(푑푒) 푑푡 푇 휕ℋ 푇 휕휎 휕휎 − ∫ (푡)푋′(푡) 푑푡 + ∫ ( (푡)푋′(푡) + (푡)휇(푡)) 푞(푡) 푑푡 휕푥 휕푥 휕푢 0 0 푇 휕휃 휕휃 + ∫ ∫( (푡, 푒)푋′(푡) + (푡, 푒)휇(푡))푟(푡, 푒)휈(푑푒) 푑푡]. 휕푥 휕푢 0 ℝ Stochastic Volterra equations 19

By the Itô formula and (2.21)–(2.22), we get

피[휓′(푌(0))푌′(0) = 피[휆(0)푌′(0) = 피[휆(푇)푌′(푇) + 푇 푡 + 피[∫ 휆(푡){∇푔(푡, 푡)(푋′(푡), 푌′(푡), 푍′(푡, 푡), 퐾′(푡, 푡, ⋅ ), 휇(푡)) 0 푇 휕푔 휕푔 휕푍 휕퐾 − ∫ ∇( (푡, 푠), (푡, 푠) (푡, 푠), ∇ 푔(푡, 푠) (푡, 푠, ⋅ )) ⋅ 휕푡 휕푧 휕푡 푘 휕푡 푡 푡 ⋅ (푋′(푡), 푌′(푡), 푍′(푡, 푡), 퐾′(푡, 푡, ⋅ ), 휇(푡)) } 푑푠 푑푡

푇 푇 휕푍′ + ∫ 휆(푡) ∫ (푡, 푠) 푑퐵(푠)푑푡 휕푡 0 푡 푇 푇 휕퐾′ + ∫ 휆(푡) ∫ ∫ (푡, 푠, 푒)푁(푑푡,˜ 푑푒) 푑푡 휕푡 0 푡 ℝ 푇 휕ℋ 푇 휕ℋ − ∫ (푡)푌′(푡) 푑푡 − ∫ (푡)푍′(푡, 푠) 푑푡 휕푦 휕푧 0 0 푇 푑∇ ℋ − ∫ ∫ 푘 (푡)퐾′(푡, 푠, 푒)휈(푑푒) 푑푡]. 휕휈 0 ℝ

From (2.17)–(2.18) and the Fubini theorem, we have

피[휓′(푌(0))푌′(0) = 피[휆(푇)푌′(푇) 푇 푡 + 피[∫ 휆(푡){∇푔(푡, 푡)(푋′(푡), 푌′(푡), 푍′(푡, 푡), 퐾′(푡, 푡, ⋅ ), 휇(푡)) } 0 푇 푡 휕푔 휕푔 휕푍 휕퐾 + ∫ ∫ 휆(푠){∇( (푠, 푡), (푠, 푡) (푠, 푡), ∇ 푔(푠, 푡) (푠, 푡, ⋅ )) 휕푡 휕푧 휕푡 푘 휕푡 0 0 푡 (푋′(푠), 푌′(푠), 푍′(푡, 푠), 퐾′(푡, 푠, ⋅ ), 휇(푠)) } 푑푠 푑푡

푇 휕ℋ 푇 휕ℋ − ∫ (푡)푌′(푡) 푑푡 − ∫ (푡)푍′(푡, 푠) 푑푡 휕푦 휕푧 0 0 푇 푑∇ ℋ − ∫ ∫ 푘 (푡)퐾′(푡, 푠, 푒)휈(푑푒) 푑푡]. (2.25) 휕휈 0 ℝ 20 N. Agram, B. Øksendal, and S. Yakhlef

Using that

휕ℋ 휕푓 휕푏 휕휎 휕푔 (푡) = (푡) + (푡, 푡)푝(푡) + (푡, 푡)푞(푡) + 휆(푡) (푡, 푡) 휕푥 휕푥 휕푥 휕푥 휕푥 휕휃 푡 휕2푔 + ∫ (푡, 푡, 푒)푟(푡, 푒)휈(푑푒) + ∫ (푠, 푡)휆(푠) 푑푠 휕푥 휕푠 휕푥 0 ℝ 푇 휕2푏 푇 휕2휎 + ∫ (푠, 푡)푝(푠) 푑푠 + ∫ (푠, 푡)피[퐷 푝(푠) | ℱ 푑푠 휕푠 휕푥 휕푠 휕푥 푡 푡 푡 푡 휕2휃 + ∫ (푠, 푡, 푒)피[퐷 푝(푠) | ℱ 휈(푑푒) 푑푠 휕푠 휕푥 푡,푒 푡 ℝ 푡 휕2푔 휕푍 + ∫ (푠, 푡) (푠, 푡)휆(푠) 푑푠 휕푥 휕푧 휕푠 0 푡 휕 휕퐾 + ∫ (∇ 푔(푠, 푡))( (푠, 푡, ⋅ )) 푑푠, (2.26) 휕푥 푘 휕푠 0 and that

푡 휕 ∇ ℋ(푡) = ∇ 푔(푡, 푡)휆(푡) + ∫ ∇ ( 푔(푠, 푡))휆(푠) 푑푠 푘 푘 푘 휕푠 0 푡 휕푔 휕푍 + ∫ ∇ ( (푠, 푡)) (푠, 푡)휆(푠) 푑푠 푘 휕푧 휕푠 0 푡 휕퐾 + ∫ ∇2 푔(푠, 푡) (푠, 푡, ⋅ )휆(푠) 푑푠, (2.27) 푘 휕푠 0

휕ℋ 휕ℋ similarly for (푡) and (푡). Combining (2.24)–(2.25) with (2.23), (2.26)–(2.27) 휕푦 휕푧 and by the definition of 휇, we obtain

푑 푇 휕ℋ 푡+휀 휕ℋ 퐽(푢 + 휖휇) | = 피[∫ (푡)휇(푡) 푑푡] = 피[∫ (푠) 푑푠훼]. 푑휖 | 휕푢 휕푢 휖=0 0 푡

We conclude that 푑 | 퐽(푢 + 휖휇) | = 0 푑휖 휖=0 if and only if 휕ℋ 피[ (푡) | 풢 ] = 0. 휕푢 | 푡 Stochastic Volterra equations 21

3. Existence and uniqueness of solutions of BSVIE

In order to prove the existence the uniqueness of the backward stochastic Volterra integral equations (BSVIE), let us introduce the following BSVIE in the unknown 푌, 푍 and 퐾:

푇 푇 푌(푡) = 휓(푡) + ∫ 푔(푡, 푠, 푌(푠), 푍(푡, 푠), 퐾(푡, 푠, ⋅ )) 푑푠 − ∫ 푍(푡, 푠) 푑퐵(푠) 푡 푡 (3.1) 푇 − ∫ ∫ 퐾(푡, 푠, 푒)푁(푑푠,˜ 푑푒), 푡 ∈ [0, 푇] . 푡 ℝ In this section we prove existence and uniqueness of solutions of (3.1), following the approach by Yong [17] and [18], but now we have jumps. The papers by Wang and Zhang [15] and by Ren [10] studied more general cases of (3.1) and our case can be seen as a particular case of theirs, but we have included this part because it will be more convenient for the reader to have a direct and simple approach. For related results on BSVIE, we refer to Shi and Wang [12, 11].

Let us now introduce the following spaces: 2 2,훽 For any 훽 ≥ 0, let △ ≔ { (푡, 푠) ∈ [0, 푇] ∶ 푡 ≤ 푠 } and 퐻△ [0, 푇] be a space of all processes (푌, 푍, 퐾), such that 푌∶ [0, 푇]×Ω → ℝ is 픽-adapted, and 푍∶ △×Ω → ℝ, 퐾∶ △ × ℝ0 × Ω → ℝ with 푠 → 푍(푡, 푠) and 푠 → 퐾 (푡, 푠, ⋅ ) being 픽-adapted on [푡, 푇], equipped with the norm

푇 푇 2 훽푡 2 훽푠 2 ‖(푌, 푍, 퐾)‖ 2,훽 ≔ 피 ∫ [푒 |푌(푡)| + ∫ 푒 |푍(푡, 푠)| 푑푠 퐻 [0,푇] △ 0 푡 푇 + ∫ ∫ 푒훽푠|퐾(푡, 푠, 푒)|2휈(푑푠, 푑푒)] 푑푡. 푡 ℝ

2,훽 Clearly 퐻△ [0, 푇] is a Hilbert space. It is easy to see that for any 훽 > 0, the norm ‖ ⋅ ‖ 2,훽 is equivalent to ‖ ⋅ ‖퐻2,0[0,푇] obtained from ‖ ⋅ ‖ 2,훽 by taking 훽 퐻△ [0,푇] △ 퐻△ [0,푇] = 0.

We now make the following assumptions: Assumptions (H.1)

• The function 푔∶ [0, 푇]2 × ℝ3 × 퐿2(휈) × Ω → ℝ is such that

푇 푇 2 1. 피[∫ (∫ 푔(푡, 푠, 0, 0, 0) 푑푠) 푑푡] < +∞ 0 푡 22 N. Agram, B. Øksendal, and S. Yakhlef

2. There exists a constant 푐 > 0, such that, for all 푡, 푠 ∈ [0, 푇]

| ′ ′ ′ | |푔(푡,푠,푦,푧,푘(⋅)) − 푔(푡, 푠, 푦 , 푧 , 푘 ( ⋅ ))| 1/2 ≤ 푐(|푦 − 푦′| + |푧 − 푧′| + (∫ |푘(푒) − 푘′(푒)|2휈(푑푒)) ) ℝ for all 푦, 푦′, 푧, 푧′, 푘( ⋅ ), 푘′( ⋅ ) • 휓( ⋅ ) ∈ 퐿2 (Ω, ℝ) ℱ푇 Theorem 3.1. Under the assumptions (H.1), there exists a unique solution (푌, 푍, 퐾) 2,훽 ∈ 퐻△ [0, 푇] of the BSVIE (3.1). 2,훽 Proof. For a given triple of processes (푦( ⋅ ), 푧( ⋅ , ⋅ ), 푘( ⋅ , ⋅ , ⋅ )) ∈ 퐻△ [0, 푇], consider the following simple BSVIE in the unknown triple (푌, 푍, 퐾):

푇 푇 푇 푌(푡) = 휓(푡) + ∫ ̄푔(푡, 푠) 푑푠 − ∫ 푍(푡, 푠) 푑퐵(푠) − ∫ ∫ 퐾(푡, 푠, 푒)푁(푑푠,˜ 푑푒), (3.2) 푡 푡 푡 ℝ where we denote by

̄푔(푡, 푠) = 푔(푡, 푠, 푦(푠), 푧(푡, 푠), 푘(푡, 푠, ⋅ )), for (푡, 푠) ∈ △.

To solve (3.2) for (푌, 푍, 퐾), we introduce the following family of BSDE (paramet- rized by 푡 ∈ [0, 푇]):

푇 휒(푟, 푡) = 휓(푡) + ∫ ̄푔 (푡, 푠) 푑푠 푟 푇 푇 − ∫ 휂(푠, 푡) 푑퐵(푠) − ∫ ∫ 휉(푠, 푡, 푒)푁(푑푠,˜ 푑푒), 푟 ∈ (푡, 푇], 푟 푟 ℝ It is well known that the above BSDE admits a unique adapted solution (휒( ⋅ , 푡), 휂( ⋅ , 푡), 휉( ⋅ , 푡, ⋅ )) and the following estimate holds:

푇 푇 피 [ sup |휒(푟, 푡)|2 + ∫ |휂(푠, 푡)|2 푑푠 + ∫ ∫ |휉(푠, 푡, 푒)|2 휈(푑푒) 푑푠] 푟∈[푡,푇] 푡 푡 ℝ 푇 2 ≤ 퐶피 [|휓(푡)|2 + (∫ ̄푔(푡, 푠) 푑푠) ] . 푡 Now let

푌(푡) = 휒(푡, 푡), 푍(푡, 푠) = 휂(푠, 푡), 퐾 (푡, 푠, ⋅ ) = 휉(푠, 푡, ⋅ ), for all (푡, 푠) ∈ △. Stochastic Volterra equations 23

Then (푌( ⋅ ), 푍( ⋅ , ⋅ ), 퐾( ⋅ , ⋅ )) is an adapted solution to the BSVIE (3.2), and

푇 푇 피[|푌(푡)|2 + ∫ |푍(푡, 푠)|2 푑푠 + ∫ ∫ |퐾(푡, 푠, 푒)|2휈(푑푒) 푑푠] 푡 푡 ℝ 푇 | |2 = 피[|휓(푡) + ∫ ̄푔(푡, 푠) 푑푠| ] 푡 푇 2 ≤ 2피[|휓(푡)|2 + (∫ ̄푔(푡, 푠) 푑푠) ]. 푡 Therefore, by integrating both sides of the inequality above, we get

푇 푇 푇 피 [∫ (|푌(푡)|2 + ∫ |푍(푡, 푠)|2 푑푠 + ∫ ∫ |퐾(푡, 푠, 푒)|2 휈(푑푒) 푑푠) 푑푡] 0 푡 푡 ℝ 푇 푇 2 ≤ 2피 ∫ [|휓(푡)|2 + (∫ ̄푔(푡, 푠) 푑푠) ] 푑푡. 0 푡

Adding and subtracting 푔(푡, 푠, 0, 0, 0) on the left side, then by the Lipschitz assumption, we obtain

푇 푇 푇 피 [∫ (|푌(푡)|2 + ∫ |푍(푡, 푠)|2 푑푠 + ∫ ∫ |퐾(푡, 푠, 푒)|2 휈(푑푒) 푑푠) 푑푡] 0 푡 푡 ℝ 푇 푇 2 ≤ 퐶피 ∫ [|휓 (푡)|2 + (∫ 푔 (푡, 푠, 0, 0, 0) 푑푠) ] 푑푡 0 푡 푇 푇 푇 + 퐶피 [∫ (|푦 (푡)|2 + ∫ |푧(푠)|2 푑푠 + ∫ ∫ |푘(푡, 푠, 푒)|2 휈(푑푒) 푑푠) 푑푡] , 0 푡 푡 ℝ

2,훽 for some constant 퐶. Thus, (푦, 푧, 푘) ↦ (푌, 푍, 퐾) defines a map from 퐻△ [0, 푇] to itself. 2,훽 Now, we want to prove that this mapping is contracting in 퐻△ [0, 푇] under 2,훽 the norm ‖ ⋅ ‖ 2,훽 . We show that if (푦푖, 푧푖, 푘푖) ∈ 퐻△ [0, 푇] for 푖 = 1, 2, and 퐻△ [0,푇] (푌푖, 푍푖, 퐾푖) is the corresponding adapted solution to equation (3.1), then

2 퐶 2 ‖(푌, 푍, 퐾)‖ 2,훽 ≤ ‖(푦, 푧, 푘)‖ 2,훽 . 퐻△ [0,푇] 훽 퐻△ [0,푇]

2,훽 Hence, the mapping (푦, 푧, 푘) ↦ (푌, 푍, 퐾) is contracting on 퐻△ [0, 푇] for large enough 훽 > 0. Then, (푌, 푍, 퐾) is a unique solution for the BSVIE (3.1).

4. Application: Optimal recursive utility consumption

As an illustration of our general results above, we now apply them to solve the optimal recursive utility consumption problem (1.4) described in the Introduction. Our example is related to the examples discussed in [3] and [9], but now the cash flow is modelled by a stochastic Volterra equation and the utility is represented by the recursive utility. As pointed out after (1.2) in the Introduction, the Volterra equation contains history terms and can therefore be viewed as a model for a system with memory. Thus, we assume that the cash flow 푋(푡) = 푋푐(푡) being exposed to a 픾-adapted consumption rate 푐(푡), satisfies the stochastic Volterra equation

푡 푡 푋(푡) = 휉 + ∫ (훼(푡, 푠) − 푐(푠))푋(푠) 푑푠 + ∫ 훽(푡, 푠)푋(푠) 푑퐵(푠) 0 0 푡 (4.1) + ∫ ∫ 휋(푡, 푠, 푒)푋(푠)푁(푑푠,˜ 푑푒), 푡 ∈ [0, 푇], 0 ℝ where we assume for simplicity that 휉 is a (deterministic) constant and 훼, 훽∶ 2 2 [0, 푇] → ℝ and 휋∶ [0, 푇] × ℝ0 → ℝ are deterministic functions with 훼, 훽 and 휋 bounded. The FSVIE (4.1) can be written in its differential form as

푡 휕훼 푑푋(푡) = (훼(푡, 푡) − 푐(푡))푋(푡) 푑푡 + (∫ (푡, 푠)푋(푠) 푑푠) 푑푡 + 휕푡 0 Stochastic Volterra equations 25

푡 휕훽 + 훽(푡, 푡)푋(푡) 푑퐵(푡) + (∫ (푡, 푠)푋(푠) 푑퐵(푠)) 푑푡 휕푡 0 + ∫ 휋(푡, 푡, 푒)푋(푡)푁(푑푡,˜ 푑푒) ℝ 푡 휕휋 + (∫ ∫ (푡, 푠, 푒)푋(푠)푁(푑푠,˜ 푑푒)) 푑푡, 푡 ∈ [0, 푇]. 휕푡 ℝ 0

The recursive utility process 푌(푡) of Duffie and Epstein [5] has the following linear form

푑푌(푡) = −[훾(푡)푌(푡) + ln 푐(푡)푋(푡) 푑푡 + 푍(푡) 푑퐵(푡)

+ ∫ 퐾(푡, 푒)푁(푑푡,˜ 푑푒), 푡 ∈ [0, 푇]. ℝ

Our problem (1.4) is to maximise the performance functional

퐽(푐) ≔ 푌푐(0) over all control processes 푐 ∈ 풰픾, where in this case 풰픾 is the set of all 픾-adapted nonnegative processes.

This problem is a special case of the problem discussed in the previous sections, with 푓 = 0, 휑 = 0, and 휓(푦) = 푦. The Hamiltonian associated to our problem is defined by

푇 휕훼 ℋ(푡, 푠, 푥, 푦, 푝, 푞) = (훼(푡, 푡) − 푐(푡))푝푥 + ∫ (푠, 푡)푥(푠)푝(푠) 푑푠 휕푠 푡 푇 휕훽 + 훽(푡, 푡)푞푥 + ∫ (푠, 푡)푥(푠)피[퐷 푝(푠) | ℱ 푑푠 휕푠 푡 푡 푡 + ∫ 휋(푡, 푡, 푒)푥푟(푡, 푒)휈(푑푒) ℝ 푇 휕휋 + ∫ ∫ (푠, 푡, 푒)푥(푠)피[퐷 푝(푠) | ℱ 휈(푑푒) 푑푠 휕푠 푡,푒 푡 ℝ 푡 + [훾(푡)푦 + ln 푐(푡) + ln 푥 휆.

The corresponding backward-forward system for the adjoint processes (푝, 푞, 푟) 26 N. Agram, B. Øksendal, and S. Yakhlef and 휆 is

푇 휕훼 ⎧푑푝(푡) = −[(훼(푡, 푡) − 푐(푡))푝(푡) + ∫ (푠, 푡)푝(푠) 푑푠 ⎪ 휕푠 ⎪ 푡 ⎪ 푇 휕훽 + 훽(푡, 푡)푞(푡) + ∫ (푠, 푡)피[퐷 푝(푠) | ℱ 푑푠 + ∫ 휋(푡, 푡, 푒)푟(푡, 푒)휈(푑푒) ⎪ 휕푠 푡 푡 ⎪ 푡 ℝ 푇 휕휋 휆(푡) ⎨ + ∫ ∫ (푠, 푡, 푒)피[퐷 푝(푠) | ℱ 휈(푑푠, 푑푒) + ] 푑푡 ⎪ 휕푠 푡,푒 푡 푋(푡) ⎪ ℝ 푡 ⎪ ˜ ⎪ + 푞(푡) 푑퐵(푡) + ∫ 푟(푡, 푒)푁(푑푡, 푑푒), 푡 ∈ [0, 푇], ⎪ ℝ ⎩ 푝(푇) = 0, and 푑휆(푡) = 훾(푡)휆(푡) 푑푡, 푡 ∈ [0, 푇], (4.2) 휆(0) = 1.

The solution of the differential equation (4.2) is

푡 휆(푡) = exp(− ∫ 훾(푠) 푑푠), 푡 ∈ [0, 푇]. 0

Now, maximising the Hamiltonian w.r.t 푐 gives the first order condition

휆(푡) | 푐(푡) = 피[ | 풢 ], 푡 ∈ [0, 푇]. (4.3) 푝(푡)푋(푡) | 푡

Applying Itô’s formula, we get

푑(푝(푡)푋(푡)) = 푝(푡)푑푋(푡) + 푋(푡)푑푝(푡) + 푑[푝(푡)푋(푡)] 푡 휕훼 = 푝(푡){(훼(푡, 푡) − 푐(푡))푋(푡) 푑푡 + (∫ (푡, 푠)푋(푠) 푑푠) 푑푡 휕푡 0 푡 휕훽 + 훽(푡, 푡)푋(푡) 푑퐵(푡) + (∫ (푡, 푠)푋(푠) 푑퐵(푠)) 푑푡 휕푡 0 푡 휕휋 + ∫ 휋(푡, 푡, 푒)푋(푡)푁(푑푡,˜ 푑푒) + (∫ ∫ (푡, 푠, 푒)푋(푠)푁(푑푠,˜ 푑푒)) 푑푡} − 휕푡 ℝ ℝ 0 Stochastic Volterra equations 27

푇 휕훼 − 푋(푡){(훼(푡, 푡) − 푐(푡))푝(푡) 푑푡 + (∫ (푠, 푡)푝(푠) 푑푠) 푑푡 + 훽(푡, 푡)푞(푡) 푑푡 휕푠 푡 푇 휕훽 + (∫ (푠, 푡)피[퐷 푝(푠) | ℱ 푑푠) 푑푡 + ∫ 휋(푡, 푡, 푒)푟(푡, 푒)휈(푑푡, 푑푒) 휕푠 푡 푡 푡 ℝ 푇 휕휋 휆(푡) + (∫ ∫ (푠, 푡, 푒)피[퐷 푝(푠) | ℱ 휈(푑푒) 푑푠) 푑푡 + 푑푡 휕푠 푡,푒 푡 ℝ 푡 푋(푡)

+ 푞(푡) 푑퐵(푡) + ∫ 푟(푡, 푒)푁(푑푡,˜ 푑푒)} ℝ + 훽(푡, 푡)푋(푡)푞(푡) 푑푡 + ∫ 휋(푡, 푡, 푒)푋(푡)푟(푡, 푒)휈(푑푡, 푑푒). ℝ

Collecting the terms, we see that the above reduces to

푡 ⎧ 푝(푡)푋(푡) = 푝(0)푋(0) − ∫ 휆(푠) 푑푠 ⎪ ⎪ 0 ⎪ 푡 ⎪ + ∫ {푝(푠)푋(푠)훽(푠, 푠) − 푋(푠)푞(푠)} 푑퐵(푠) ⎨ 0 ⎪ 푡 ⎪ + ∫ {푝(푠)푋(푠)휋(푠, 푠, 푒) − 푋(푠)푟(푠, 푒)}푁(푑푠,˜ 푑푒), 푡 ∈ [0, 푇], ⎪ ⎪ 0 ⎩푝(푇)푋(푇) = 0.

Therefore, if we define

푢(푡) = 푝(푡)푋(푡), 푣(푡) = 푝(푠)푋(푠)훽(푠, 푠) − 푋(푠)푞(푠), 푤(푡, 푒) = 푝(푠)푋(푠)휋(푠, 푠, 푒) − 푋(푠)푟(푠, 푒), then (푢, 푣, 푤) solves the linear BSDE

푑푢(푡) = −휆(푡) 푑푡 + 푣(푡) 푑퐵(푡) + ∫ 푤(푡, 푒)푁(푑푡,˜ 푑푒), 푡 ∈ [0, 푇], { ℝ 푢(푇) = 0.

The solution of this linear BSDE is

푇 | 푢(푡) = 피[∫ 휆(푠) 푑푠 | ℱ푡] = 푝(푡)푋(푡). 푡 | 28 N. Agram, B. Øksendal, and S. Yakhlef

Combined with (4.3) this gives 푡 | ⎡ exp(− ∫ 훾(푠) 푑푠) | ⎤ ⎢ | ⎥ ∗ ⎢ 0 | ⎥ 푐(푡) = 푐 (푡) = 피 푇 푠 풢푡 . (4.4) ⎢ | | ⎥ ⎢피[∫ exp(− ∫ 훾(푟)푑푟) 푑푠 | ℱ] | ⎥ | 푡 | ⎣ 푡 0 ⎦ In particular, since 휆 > 0 by (4.2) we get that 푝(푡)푋(푡) > 0. Thus we see that 푐(푡) ∗ ∗ is well-defined in (4.3) and 푐 (푡) > 0 for all 푡 ∈ [0, 푇]. Therefore 푐 ∈ 풰픾, and we conclude that 푐∗ is indeed optimal. We have proved Theorem 4.1. The optimal recursive utility consumption rate 푐∗(푡) for the problem (1.4)(with 휉 constant) is given by (4.4).

5. Appendix

5.1. Some basic concepts from Banach space theory. To explain the notation used in this paper, we briefly recall some basic concepts from Banach space theory:

Assume 풳, 풴 are two Banach spaces with norms ‖ ⋅ ‖풳, ‖ ⋅ ‖풴, respectively, and let 퐹∶ 풳 → 풴. • We say that 퐹 has a directional derivative (or Gâteaux derivative) at 푣 ∈ 풳 in the direction 푤 ∈ 풳 if 퐹(푣 + 휀푤) − 퐹(푣) 퐷푤퐹(푣) ≔ lim 휀→0 휀 exists. • We say that 퐹 is Fréchet differentiable at 푣 ∈ 풳 if there exists a continuous linear map 퐴∶ 풳 → 풴 such that ‖퐹(푣 + ℎ) − 퐹(푣) − 퐴(ℎ)‖ lim 풴 = 0. ℎ→0 ‖ℎ‖풳 ℎ∈풳 In this case we call 퐴 the gradient (or Fréchet derivative) of 퐹 at 푣 and we write 퐴 = ∇푣퐹.

• If 퐹 is Fréchet differentiable at 푣 with Fréchet derivative ∇푣퐹, then 퐹 has a directional derivative in all directions 푤 ∈ 풳 and

퐷푤퐹(푣) ≔ ⟨∇푣퐹, 푤⟩ = ∇푣퐹(푤) = ∇푣퐹푤.

In particular, note that if 퐹 is a linear operator, then ∇푣퐹 = 퐹 for all 푣. Stochastic Volterra equations 29

5.2. A brief review of Hida–Malliavin calculus for Lévy processes. For the convenience of the reader, in this section we recall the basic definition and properties of Hida–Malliavin calculus for Lévy processes related to this paper. The following summary is based on [2]. A general reference for this presentation is the book [4]. First, recall the Lévy–Itô decomposition theorem, which states that any Lévy process 푌(푡) with 피[푌2(푡)] < ∞ for all 푡 can be written 푡 푌(푡) = 푎푡 + 푏퐵(푡) + ∫ ∫ 휁푁(푑푠,˜ 푑휁) 0 ℝ with constants 푎 and 푏. In view of this we see that it suffices to deal with Hida– Malliavin calculus for 퐵( ⋅ ) and for

휂( ⋅ ) ≔ ∫∫ 휁푁(푑푠,˜ 푑휁) 0 ℝ separately.

5.3. Hida–Malliavin calculus for B( · ). A natural starting point is the Wiener– 2 Itô chaos expansion theorem, which states that any 퐹 ∈ 퐿 (퐹푇, 푃) can be written ∞

퐹 = ∑ 퐼푛(푓푛) 푛=0 2 푛 for a unique sequence of symmetric deterministic functions 푓푛 ∈ 퐿 (휆 ), where 휆 is Lebesgue measure on [0, 푇] and

푇 푡푛 푡2 퐼푛(푓푛) = 푛! ∫ ∫ ⋯ ∫ 푓푛(푡1, … , 푡푛) 푑퐵(푡1) 푑퐵(푡2) ⋯ 푑퐵(푡푛) 0 0 0

(the 푛-times iterated integral of 푓푛 with respect to 퐵( ⋅ )) for 푛 = 1, 2, … and 퐼0(푓0) = 푓0 when 푓0 is a constant. Moreover, we have the isometry ∞ 피[퐹2] = ‖퐹‖2 = ∑ 푛!‖푓 ‖2 . ‖ ‖퐿2(푝) ‖ 푛‖퐿2(휆푛) 푛=0

Definition 5.1 (Hida–Malliavin derivative 퐷푡 with respect to 퐵( ⋅ )). (퐵) 2 Let 픻1,2 be the space of all 퐹 ∈ 퐿 (ℱ푇, 푃) such that its chaos expansion (4) satisfies ∞ 2 2 ‖퐹‖ (퐵) ≔ ∑ 푛푛!‖푓푛‖ 2 푛 < ∞. 픻1,2 퐿 (휆 ) 푛=1 30 N. Agram, B. Øksendal, and S. Yakhlef

(퐵) For 퐹 ∈ 픻1,2 and 푡 ∈ [0, 푇], we define the Hida–Malliavin derivative or the stochastic gradient of 퐹 at 푡 (with respect to 퐵( ⋅ )), 퐷푡퐹, by ∞

퐷푡퐹 = ∑ 푛퐼푛−1(푓푛( ⋅ , 푡)), 푛=1 where the notation 퐼푛−1(푓푛( ⋅ , 푡)) means that we apply the (푛 − 1)-times iterated integral to the first 푛 − 1 variables 푡1, …, 푡푛−1 of 푓푛(푡1, 푡2, … , 푡푛) and keep the last variable 푡푛 = 푡 as a parameter. One can easily check that

푇 ∞ 2 2 2 피[∫ (퐷푡퐹) 푑푡] = ∑ 푛푛!‖푓푛‖ 2 푛 = ‖퐹‖ (퐵), (5.1) 퐿 (휆 ) 픻1,2 0 푛=1

2 so (푡, 휔) → 퐷푡퐹(휔) belongs to 퐿 (휆 × 푃). 푇 2 Example 5.2. If 퐹 = ∫0 푓(푡) 푑퐵(푡) with 푓 ∈ 퐿 (휆) deterministic, then

퐷푡퐹 = 푓(푡) for a. a. 푡 ∈ [0, 푇].

More generally, if 푢(푠) is Skorohod integrable, 푢(푠) ∈ 픻1,2 for a. a. 푠 and 퐷푡푢(푠) is Skorohod integrable for a. a. 푡, then

푇 푇

퐷푡( ∫ 푢(푠) 훿퐵(푠)) = ∫ 퐷푡푢(푠) 훿퐵(푠) + 푢(푡) for a. a. (푡, 휔), 0 0

푇 where ∫0 휓(푠) 훿퐵(푠) denotes the Skorohod integral of a process 휓 with respect to 퐵( ⋅ ).

Some other basic properties of the Hida–Malliavin derivative 퐷푡 are the following: (i) Chain rule (퐵) 푚 1 Suppose 퐹1, …, 퐹푚 ∈ 픻1,2 and that 휓∶ ℝ → ℝ is 퐶 with bounded partial derivatives. Then, 휓(퐹1, … , 퐹푚) ∈ 픻1,2 and

푚 휕휓 퐷 휓(퐹 , … , 퐹 ) = ∑ (퐹 , … , 퐹 ) 퐷 퐹. 푡 1 푚 휕푥 1 푚 푡 푖 푖=1 푖

(ii) Duality formula 푇 2 (퐵) Suppose 푢(푡) is ℱ푡−adapted with 피[∫0 푢 (푡) 푑푡 < ∞ and let 퐹 ∈ 픻1,2 . Then

푇 푇

피[퐹 ∫ 푢(푡) 푑퐵(푡)] = 피[∫ 푢(푡)퐷푡퐹 푑푡]. (5.2) 0 0 Stochastic Volterra equations 31

(iii) Malliavin derivative and adapted processes If 휑 is an 픽-adapted process, then

퐷푠휑(푡) = 0 for 푠 > 푡.

Remark 5.3. We put 퐷푡휑(푡) = lim 퐷푠휑(푡) (if the limit exists). 푠→푡− Remark 5.4. It was proved in [1] that one can extend the Hida–Malliavin derivative 2 operator 퐷푡 from 픻1,2 to all of 퐿 (ℱ푇, 푃) in such a way that, also denoting the 2 extended operator by 퐷푡, for all 퐹 ∈ 퐿 (ℱ푇, 푃) we have

∗ 2 퐷푡퐹 ∈ (풮) and (푡, 휔) ↦ 피[퐷푡퐹 ∣ ℱ푡] belongs to 퐿 (휆 × 푃) (5.3)

Here (풮)∗ is the Hida space of stochastic distributions. Moreover, the following generalized Clark–Haussmann–Ocone formula was proved:

푇

퐹 = 피[퐹] + ∫ 피[퐷푡퐹 ∣ ℱ푡] 푑퐵(푡) (5.4) 0

2 for all 퐹 ∈ 퐿 (ℱ푇, 푃). See Theorem 3.11 in [1] and also Theorem 6.35 in [4]. We can use this to get the following extension of the duality formula (5.2):

2 Proposition 5.5 (The generalized duality formula). Let 퐹 ∈ 퐿 (ℱ푇, 푃) and let 휑(푡, 휔) ∈ 퐿2(휆 × 푃) be adapted. Then

푇 푇 | | 피[퐹 ∫ 휑(푡) 푑퐵(푡)] = 피[∫ 피[퐷푡퐹 | ℱ푡]휑(푡) 푑푡]. 0 0 Proof. By (5.3) and (5.4) and the Itô isometry we get

푇 푇 푇

피[퐹 ∫ 휑(푡) 푑퐵(푡)] = 피[(피[퐹] + ∫ 피[퐷푡퐹 | ℱ푡 푑퐵(푡)) ∫ 휑(푡) 푑퐵(푡)] 0 0 0 푇

= 피[∫ 피[퐷푡퐹 | ℱ푡 휑(푡) 푑푡]. 0

We will use this extension of the Hida–Malliavin derivative from now on.

5.4. Hida–Malliavin calculus for 푵˜(·). The construction of a stochastic derivative/Hida–Malliavin derivative in the pure jump martingale case follows the same lines as in the Brownian motion case. In this case, the corresponding 2 Wiener–Itô Chaos Expansion Theorem states that any 퐹 ∈ 퐿 (ℱ푇, 푃) (where, in this 32 N. Agram, B. Øksendal, and S. Yakhlef

()˜ 푠 ˜ case, ℱ푡 = ℱ푡 is the 휎−algebra generated by 휂(푠) ≔ ∫0 ∫ℝ 휁푁(푑푟, 푑휁); 0 ≤ 푠 ≤ 푡) can be written as ∞ ˆ2 푛 퐹 = ∑ 퐼푛(푓푛); 푓푛 ∈ 퐿 ((휆 × 휈) ), (5.5) 푛=0

ˆ2 푛 where 퐿 ((휆 × 휈) ) is the space of functions 푓푛(푡1, 휁1, … , 푡푛, 휁푛); 푡푖 ∈ [0, 푇], 휁푖 ∈ ℝ0 2 푛 such that 푓푛 ∈ 퐿 ((휆×휈) ) and 푓푛 is symmetric with respect to the pairs of variables (푡1, 휁1), … , (푡푛, 휁푛). It is important to note that in this case, the 푛−times iterated integral 퐼푛(푓푛) is taken with respect to 푁(푑푡,˜ 푑휁) and not with respect to 푑휂(푡). Thus, we define

퐼푛(푓푛) ≔

푇 푡푛 푡2 푛! ∫ ∫ ∫ ∫ ⋯ ∫ ∫ 푓푛(푡1, 휁1, … , 푡푛, 휁푛)푁(푑푡˜ 1, 푑휁1) ⋯ 푁(푑푡˜ 푛, 푑휁푛) 0 ℝ0 0 ℝ0 0 ℝ0

ˆ2 푛 for 푓푛 ∈ 퐿 ((휆 × 휈) ). The Itô isometry for stochastic integrals with respect to 푁(푑푡,˜ 푑휁) then gives the following isometry for the chaos expansion:

∞ ‖퐹‖2 = ∑ 푛!‖푓 ‖2 . ‖ ‖퐿2(푃) ‖ 푛‖퐿2((휆×휈)푛) 푛=0 As in the Brownian motion case, we use the chaos expansion to define the Malliavin derivative. Note that in this case, there are two parameters 푡, 휁, where 푡 represents time and 휁 ≠ 0 represents a generic jump size.

Definition 5.6 (Hida–Malliavin derivative 퐷푡,휁 with respect to 푁(˜ ⋅ , ⋅ ) [4]). Let ()˜ 2 픻1,2 be the space of all 퐹 ∈ 퐿 (ℱ푇, 푃) such that its chaos expansion (5.5) satisfies

∞ 2 2 ‖퐹‖ (푁)˜ ≔ ∑ 푛푛!‖푓 ‖ < ∞. 픻 푛 퐿2((휆×휈)2) 1,2 푛=1

()˜ For 퐹 ∈ 픻1,2 , we define the Hida–Malliavin derivative 퐷푡,휁퐹 of 퐹 at (푡, 휁) (with respect to 푁(˜ ⋅ )) by ∞

퐷푡,휁퐹 ≔ ∑ 푛퐼푛−1(푓푛( ⋅ , 푡, 휁)), 푛=1 where 퐼푛−1(푓푛( ⋅ , 푡, 휁)) means that we perform the (푛 − 1)−times iterated integral with respect to 푁˜ to the first 푛−1 variable pairs (푡1, 휁1), …, (푡푛, 휁푛), keeping (푡푛, 휁푛) = (푡, 휁) as a parameter. Stochastic Volterra equations 33

In this case, we get the isometry.

푇 ∞ 2 2 2 피[∫ ∫ (퐷푡,휁퐹) 휈(푑휁) 푑푡] = ∑ 푛푛!‖푓푛‖퐿2((휆×휈)푛) = ‖퐹‖ (푁)˜ . 픻1,2 0 ℝ0 푛=0 (Compare with (5.1)). 푇 ˜ Example 5.7. If 퐹 = ∫0 ∫ℝ 푓(푡, 휁)푁(푑푡, 푑휁) for some deterministic 푓(푡, 휁) ∈ 퐿2(휆 × 휈), then 퐷푡,휁퐹 = 푓(푡, 휁) for a. a. (푡, 휁). More generally, if 휓(푠, 휁) is Skorohod integrable with respect to 푁(훿푠,˜ 푑휁), 휓(푠, 휁) ∈ ()˜ 픻1,2 for a. a. 푠, 휁 and 퐷푡,푧휓(푠, 휁) is Skorohod integrable for a. a. (푡, 푧), then

푇 푇

퐷푡,푧(∫ ∫ 휓(푠, 휁)푁(훿푠,˜ 푑휁)) = ∫ ∫ 퐷푡,푧휓(푠, 휁)푁(훿푠,˜ 푑휁) + 푢(푡, 푧) for a. a. 푡, 푧, 0 ℝ 0 ℝ 푇 ˜ where ∫0 ∫ℝ 휓(푠, 휁)푁(훿푠, 푑휁) denotes the Skorohod integral of 휓 with respect to 푁(˜ ⋅ , ⋅ ). (See [4] for a definition of such Skorohod integrals and for more details.)

The properties of 퐷푡,휁 corresponding to those of 퐷푡 are the following: (i) Chain rule [4] ()˜ 푚 Suppose 퐹1, …, 퐹푚 ∈ 픻1,2 and that 휙∶ ℝ → ℝ is continuous and bounded. ()˜ Then, 휙(퐹1, … , 퐹푚) ∈ 픻1,2 and

퐷푡,휁휙(퐹1, … , 퐹푚) = 휙(퐹1 + 퐷푡,휁퐹1, … , 퐹푚 + 퐷푡,휁퐹푚) − 휙(퐹1, … , 퐹푚).

(ii) Duality formula [4] 푇 2 Suppose Ψ(푡, 휁) is ℱ푡-adapted and 피[∫0 ∫ℝ 휓 (푡, 휁)휈(푑휁) 푑푡 < ∞, and let ()˜ 퐹 ∈ 픻1,2 . Then

푇 푇

피[퐹 ∫ ∫ Ψ(푡, 휁)푁(푑푡,˜ 푑휁)] = 피[ ∫ ∫ Ψ(푡, 휁)퐷푡,휁퐹휈(푑휁) 푑푡]. 0 ℝ0 0 ℝ

(iii) Hida–Malliavin derivative and adapted processes [4] If 휑 is an 픽-adapted process, then,

퐷푠,휁휑(푡) = 0 for all 푠 > 푡.

Remark 5.8. We put 퐷푡,휁휑(푡) = lim 퐷푠,휁휑(푡) (if the limit exists). 푠→푡− 34 N. Agram, B. Øksendal, and S. Yakhlef

Remark 5.9. As in Remark 3.2 we note that there is an extension of the Hida– ()˜ 2 Malliavin derivative 퐷푡,휁 from 픻1,2 to 퐿 (ℱ푡 × 푃) such that the following extension of the duality theorem holds:

Proposition 5.10 (Generalized duality formula). Suppose Ψ(푡, 휁) is ℱ푡-adapted 푇 2 2 and 피[∫0 ∫ℝ 휓 (푡, 휁)휈(푑휁) 푑푡 < ∞ and let 퐹 ∈ 퐿 (ℱ푇 × 푃). Then,

푇 푇

피[퐹 ∫ ∫ Ψ(푡, 휁)푁(푑푡,˜ 푑휁)] = 피[ ∫ ∫ Ψ(푡, 휁)피[퐷푡,휁퐹 ∣ ℱ푡]휈(푑휁) 푑푡]. 0 ℝ0 0 ℝ We refer to Theorem 13.26 in [4]. Accordingly, note that from now on we are working with this generalized version of the Hida–Malliavin derivative. We emphasize that this generalized Hida–Malli- avin derivative 퐷푋 exists for all 푋 ∈ 퐿2(푃) as an element of the Hida stochastic distribution space (풮)∗, and it has the property that the conditional expectation 2 피[퐷푋|ℱ푡] belongs to 퐿 (휆 × 푃), where 휆 is Lebesgue measure on [0, 푇]. Therefore, when using this generalized Hida–Malliavin derivative, combined with conditional expectation, no assumptions on Hida–Malliavin differentiability in the classical sense are needed; we can work on the whole space of random variables in 퐿2(푃).

Acknowledgments. Agram and Øksendal carried out their research with support of the Norwegian Research Council, within the research project Challenges in Stochastic Control, Information and Applications (STOCONINF), project number 250768/F20. We want to thank Yaozhong Hu and Yanqing Wang helpful comments. We are also grateful to an anonymous referee for a very valuable and comprehensive report, which helped us to improve the paper considerably.

References

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A unified approach to infinite dimensional integrals of probabilistic and oscillatory type with applications to Feynman path integrals

Sergio Albeverio and Sonia Mazzucchi

Dedicated to Helge Holden on the occasion of his 60th birthday with great admiration and thankfulness

Abstract. An unified approach to infinite dimensional integration in terms of linear continuous functional is presented, including the cases of both probabilistic and oscillatory integrals. Applications to the theory of Feynman path integrals and to the study of high-order heat-type equations are also presented.

1. Introduction

It is a special pleasure to be able to contribute to this conference by presenting a paper on a unified approach to infinite dimensional integrals and their applications. These include many of special interest to Helge, from hydrodynamics to quantum mechanics, quantum field theory and statistical mechanics. Let us start by shortly recalling the origins of the theory of such integrals. N. Wiener in the 1920’s constructed a probability measure 푃 on a space of paths (from the time interval [0, 푇] to ℝ), since then known as Wiener measure, and yielding the full probabilistic description of a time-continuous stochastic process called (mathematical) Brownian motion or Wiener process.1 This is the prototype of an integral on an infinite-dimensional (paths)-space with respect to a positive finite measure (in fact a probability measure after a suitable normalization). We shall recall in section 2 how this measure is connected with the solution of the heat equation (a prototype of a parabolic partial differential equation involving first order differentiation in time and second order differentiation inthespace

1Predecessors of the description of the Brownian motion process can be found in work by Thiele (1880) (in the analysis of astronomical data), Bachelier (in the analysis of stock prices) and Einstein (1905) and Smolukowski (1906) (in the analysis of “physical Brownian motion” of small particles suspended in liquid, see, e.g., [23, 51]) 38 S. Albeverio and S. Mazzucchi variables). Let us here recall that the measure 푃 can be heuristically grasped as an expression of the form

−1 − 1 ∫푡 |훾(푠)|̇ 2 d푠 푃(푑훾) = “푍 푒 2 0 ∏ 푑훾(푠)” (1) 푠∈[0,푇] where ∏푠∈[0,푇] 푑훾(푠) stands for a heuristic “Lebesgue-type measure” on the space 푡 2 1/2 of paths 훾, 푍 for a “normalization constant”, and (∫0 | ̇훾(푠)| d푠) for the kinetic- energy norm (the norm in the space 퐻1,2([0, 푇], ℝ) called Cameron–Martin space). Extensions of measures of this type when [0, 푇] is replaced by a manifold and the paths 훾 are replaced by maps from a manifold to a manifold arise in theories like classical hydrodynamics, in connection with conserved quantities (see, e.g., [4]), in statistical mechanics [6, 11], Euclidean quantum field theory [54, 37, 17], and Euclidean quantum gravity [10]. The corresponding heuristic integrals then look like ∫ 푓(훾)푃(푑훾), (2) Γ Γ being a space of maps, 푓∶ Γ → ℂ a complex-valued function, and 푃(푑훾) a (probability) measure of the heuristic form

푃(푑훾) = “푍−1푒−푆(훾) d훾” (3) with 푆 a suitable functional of 훾 (a generalization of the above “kinetic energy functional”, often called “action functional”).2 Such measures and integrals arise in connection with various evolution equations of mathematical physics which are of first order in time and involve differential or, more generally, pseudodifferential operators in space. They occur both in deterministic problems and in problems involving stochastic terms (like in stochastic differential equations, see [26, 25, 38, 3]). Let us call for brevity such integrals of being of “type I”. Of a different kind are the integrals which arise in connection with hyperbolic problems, including problems of quantum mechanics, wave equation, and acous- tics. A prototype of such equations of the hyperbolic rather than parabolic type is the Schrödinger equation 휕 1 푖 휓(푡, 푥) = − Δ휓(푡, 푥) + 푉(푥)휓(푡, 푥), 푥 ∈ ℝ푑, 푡 ∈ ℝ, (4) 휕푡 2 where 푖 is the imaginary unit, 휓 a complex-valued “wave function” of a time variable 푡 ∈ ℝ and a space variable 푥 ∈ ℝ푑, with 푉 a (usually real-valued) potential term (a function of 푥), with a given initial condition 휓0. Just as 푢, the solution of the 2This is so in quantum field theory. In hydrodynamics 푆 can be related to invariant quantities of the classical equation of motion, see, e.g., [4] for the Euler and Navier–Stokes equations. Infinite dimensional integrals of probabilistic and oscillatory type 39 corresponding “heat equation with sink” obtained by replacing in (4) 푖휕/휕푡 by −휕/휕푡, can be expressed by an integral with respect to the Wiener measure of the form (2) (see section 2), the solution of (4) can be expressed by an infinite dimensional integral of the form (2), with 푃 replaced by an heuristic expression of the form “푍−1푒푖푆(훾) d훾.” (5) This was suggested by Feynman in the 40s [33]. The integral (2) has then the form of an oscillatory integral over the infinite dimensional space Γ (of paths, in the case of the Schrödinger equation). Cameron [24] observed in the 60s that, due to the presence of the imaginary unit 푖 in (5) (푆 being real-valued) the “Feynman measure” (5) in this case cannot be realized as a 휎-additive complex measure with finite total variation over the space of paths Γ. Hence, the integral (2) cannot be defined in the framework of Lebesgue traditional integration theory, but needsan alternative construction. Several efforts have been devoted to the solution ofthis problem, and different approaches have been proposed (see, e.g., [36, 52, 40, 27, 29] as well as [9, 48] and references therein). The main motivation of Feynman to consider (5) was the study of the semiclassical limit of quantum mechanics, namely the study of the asymptotic behavior of the solution 휓 of the Schrödinger equation when the Planck constant ℏ is regarded as a negligible parameter. Indeed when proper physical units are introduced in Eq. (4), the presence of the Planck constant ℏ (hidden in (4)) amounts to having 푆(훾) replaced by 푆(훾)/ℏ in (5). An heuristic application of the “stationary phase method” in the study of the limit of (2) when ℏ ↓ 0 yields, due to the special form of the action functional 푆, that the asymptotic form of quantum mechanical quantities is related to the classical behavior of the underlying physical system. This has been a source of inspiration in quantum physics, and it is a prototype of work in hyperbolic systems (including the relations between wave optics and ray optics, see, e.g., [18, 30, 42]). The implementation of an infinite dimensional version of the stationary phase method in the framework of a rigorous mathematical definition of Feynman integrals has been developed in [7, 2, 53], yielding expansions of integrals (2) around “classical orbits”. We shall call here integrals of the above type (2) and (5) “of type II”. A third class of problems in infinite dimensional integration arise in the study of evolution equations of order larger than 2 in the space variables. A prototype of these is given by the Cauchy problem for a partial differential equation of the form 휕 푢(푡, 푥) = −Δ2푢(푡, 푥) − 푉(푥)푢(푡, 푥), 푡 ∈ [0, +∞), 푥 ∈ ℝ. (6) 휕푡 Such equations have been discussed since the work by Krylov [46] (1960) and Hochberg [41] (1978) (see [49] for a discussion). Solutions in terms of continuous linear functionals which are understood in a suitable spirit as in the work done 40 S. Albeverio and S. Mazzucchi for Feynman path integrals have been recently discussed by the second author [49, 50, 21]. We shall call such realizations of integrals by continuous linear functionals “integrals of type III”. It turns out that a unified approach which covers type I, II, and III canbe developed, and we briefly present this new approach in the rest of this paper, which is structured as follows. In section 2 we introduce integrals of type I and their relations with partial differential equations. In section 3 we introduce integrals of type II in their relations with the Schrödinger equation. In section 4 we present an extension of the integrals of type I and II to integrals of type III covering in particular higher order parabolic (heat-type) equations. It turns out that these extensions are related to studies made for Schrödinger operators with polynomial potentials and related oscillatory integrals with polynomial phase [12, 13].

2. Infinite dimensional integrals and PDEs

Infinite dimensional integration is a powerful tool in the study of dynamical systems. The first example of the deep connections between infinite dimensional probability measures and partial differential equations is the celebrated Feynman– Kac formula (Eq. (8)), providing the representation of the solution of the heat equation with potential 휕 1 푢(푡, 푥) = Δ푢(푡, 푥) − 푉(푥)푢(푡, 푥), 푥 ∈ ℝ푑, 푡 ∈ [0, +∞), {휕푡 2 (7) 푑 푢(0, 푥) = 푢0(푥), 푥 ∈ ℝ , in terms of an integral of the form

− ∫푡 푉(휔(푠)+푥) d푠 푢(푡, 푥) = ∫ 푒 0 푢(0, 휔(푡) + 푥) d푃(휔). (8)

퐶푡 where 푃 is the Wiener probability measure on the Borel 휎-algebra in the Banach 푑 space 퐶푡 of continuous paths 휔∶ [0, 푡] → ℝ starting at the origin, endowed with the sup-norm. The connection between heat equation and Wiener process is just a particular case of a general theory connecting Markov processes with parabolic equations associated to second order elliptic operators (see [28, 34]). Given a Lipschitz map 휎∶ ℝ푑 → 퐿(ℝ푑,ℝ푑) from ℝ푑 to the 푑 × 푑 matrices, a Lipschitz 푑 푑 vector 푏∶ ℝ → ℝ and a 푑−dimensional Wiener process 푊푡, the solution of the Cauchy problem 휕 1 푢(푡, 푥) = Tr[휎(푥)휎∗(푥)퐷2푢(푡, 푥) + ⟨푏(푥), 퐷 푢(푡, 푥)⟩ + 푉(푥)푢(푡, 푥) {휕푡 2 푥 푥 (9) 푢(0, 푥) = 푢0(푥) Infinite dimensional integrals of probabilistic and oscillatory type 41

푥 푥 is related with process 푋 = (푋푡 )푡≥0 solution of the stochastic differential equation

푥 푥 푥 푑푋푡 = 푏(푋푡 ) d푡 + 휎(푋푡 ) d푊푡, { (10) 푋(0) = 푥, 푥 ∈ ℝ푑 by the formula

푡 푥 푥 ∫ 푉(푋푠 ) d푠 푑 푢(푡, 푥) = 피 [푢(0, 푋푡 )푒 0 ] , 푡 ≥ 0, 푥 ∈ ℝ (11)

A representation of this form cannot be proved for the solution to different kinds of PDEs, e.g., of hyperbolic type such as the Schrödinger equation (4) or as parabolic equations associated to high order operators, such as high-order heat-type equations of the form 휕 푢(푡, 푥) = (−1)+1 Δ 푢(푡, 푥) − 푉(푥)푢(푡, 푥), 푡 ∈ [0, +∞), 푥 ∈ ℝ푑. (12) 휕푡 Indeed, in this case it is impossible to prove a representation for the solution of (4) or (12) of the form (11), namely in terms of the expectation with respect to 푑 the probability measure associated to a stochastic process 푋 = (푋푡)푡≥0 on ℝ . In fact equations (4) and (12), unlike Eq. (7) and Eq. (9), do not satisfy a maximum principle, which would be actually deduced from formula (11), at least for 푉 ≡ 0. A deeper understanding of this no-go result can be gained by inspecting one of the (many) proofs of the Feynman–Kac formula and checking the point where the arguments which work in the case of the heat equation fail in the case of equations (4) or (12). We present here an argument based on the proof of formula (8) given in [55]. In the following we shall, for notational simplicity, limit our considerations to the case where 푑 = 1. 2 2 Let us consider the evolution semigroup 푇푡 ∶ 퐿 (ℝ) → 퐿 (ℝ) generated by an 2 2 ∞ 2 operator 퐴∶ 퐷(퐴) ⊂ 퐿 (ℝ) → 퐿 (ℝ) given on 퐶0 functions 푢 ∈ 퐿 (ℝ) by 푑 퐴푢(푥) ≔ 훼 푢(푥), 훼 ∈ ℂ, 푁 ∈ ℕ, 푥 ∈ ℝ, (13) 푑푥 푛 where 훼 ∈ ℂ satisfies the condition Re(훼(푖푦) ) ≤ 0 for all 푦 ∈ ℝ. Let 퐾푡( ⋅ , ⋅ ), 푡 ≥ 0 denote the kernel of 푇푡, namely:

푇푡푢(푥) = ∫ 퐾푡(푥, 푦)푢(푦) d푦 (14) ℝ

In fact 퐾푡 has the form

1 푁 퐾 (푥, 푦) = ∫ 푒푖휉(푥−푦)푒훼(푖휉) 푡 d휉 (15) 푡 2휋 ℝ 42 S. Albeverio and S. Mazzucchi

1 In particular, if 푁 = 2 and 훼 = 2 , 퐾푡 is the fundamental solution of the heat equation: (푥 − 푦)2 퐾 (푥, 푦) = (2휋푡)−1/2 exp (− ) , (16) 푡 2푡

푖 while if 푁 = 2 and 훼 = 2 , 퐾푡 is the fundamental solution of the Schrödinger equation: (푥 − 푦)2 퐾 (푥, 푦) = (2휋푖푡)−1/2 exp (푖 ) . (17) 푡 2푡

By the semigroup property of 푇푡 the Chapman–Kolmogorov equation follows:

∫ 퐾푡(푥, 푦)퐾푠(푦, 푧) d푦 = 퐾푡+푠(푥, 푧). (18) ℝ

Given a continuous bounded function 푉∶ ℝ → ℝ, let us denote (with an abuse of notation) 푉∶ 퐿2(ℝ) → 퐿2(ℝ) the associated multiplication operator defined on ∞ 2 2 the vectors 푢 ∈ 퐶0 (ℝ) by 푉푢(푥) = 푉(푥)푢(푥). Let 퐴 + 푉∶ 퐷(퐴) ⊂ 퐿 (ℝ) → 퐿 (ℝ) 2 2 be the operator sum and 푇푉(푡)∶ 퐿 (ℝ) → 퐿 (ℝ) the associated semigroup, written (퐴+푉)푡 formally as 푇푉(푡) = 푒 . By the Trotter product formula [57], the perturbed semigroup is given by the strong 퐿2(ℝ)-limit

푛 푒(퐴+푉)푡푢 = lim (푒 퐴푡/푛푒푉푡/푛) 푢, 푢 ∈ 퐿2(ℝ). 푛→∞

By taking a subsequence and using (14), the latter is equal for almost every 푥 ∈ ℝ to

푛 푒(퐴+푉)푡푢(푥) = lim (푒 퐴푡/푛푒푉푡/푛) 푢(푥) 푛→∞

푛 푛−1 (푡/푛) ∑푗=1 푉(푥푗) = lim ∫ 푢(푥0)푒 ∏ 퐾푡/푛(푥푗, 푥푗+1) d푥푗 (19) 푛→∞ ℝ푛 푗=0 where 푥푛 ≡ 푥. In the case where 푇푡 is a Markov semigroup and its kernel 퐾푡( ⋅ , ⋅ ) is the density of a probability measure on ℝ, the last line of Eq. (19) can be interpreted as an integral on the space ℝ[0,+∞) with respect to a 휎-additive probability measure, constructed by means of Kolmogorov’s theorem. Indeed Kolmogorov’s existence theorem is the cornerstone for the construction of non-trivial probability measures on infinite dimensional spaces Ω. It was originally proved by Kolmogorov in the case where Ω = ℝ[0,푇] and later generalized by Bochner [20] to the case of projective limit spaces. Fixed an 푥 ∈ ℝ, for any finite set 퐽 = {푡1, 푡2, … , 푡푛} with 퐽 0 < 푡1 < 푡2 < ⋯ < 푡푛 < +∞, let 휇퐽 be the (complex) Borel measure on ℝ , defined Infinite dimensional integrals of probabilistic and oscillatory type 43 by

휇퐽(퐵1 × 퐵2 × ⋯ × 퐵푛) = ∫ ⋯ ∫ 퐾푡푛−푡푛−1(푥푛−1, 푥푛)… 퐵1 퐵푛

⋯ 퐾푡2−푡1(푥1, 푥2)퐾푡1(푥, 푥1) d푥1 ⋯ d푥푛, (20) where 퐵1, 퐵2, … , 퐵푛 are Borel sets in ℝ and 퐾⋅( ⋅ , ⋅ ) is the kernel of the evolution 2 2 semigroup 푇푡 ∶ 퐿 (ℝ) → 퐿 (ℝ) considered above (see Eq. (14) and (15)). By the Chapman–Kolmogorov identity (18), it follows that the family of measures (휇퐽) forms a projective system of (complex) measures [20]. If 퐾푡(푥, 푦) is the kernel of a Markov semigroup, in particular if the degree 푁 of the differential operator 퐴 + defined by (13) is equal to 2 and 훼 ∈ ℝ , then the measures 휇퐽 defined by (20) are probability measures and Kolmogorov’s existence theorem [19] assures the existence of a probability measure 휇 on the space of paths Ω = ℝ[0,푇] endowed by the 휎-algebra 퐴 generated by the cylinder sets of the form

[0,푇] 퐸퐽;퐵1,…,퐵푛 ≔ { 휔 ∈ ℝ ∶ 훾(푡1) ∈ 퐵1, … , 훾(푡푛) ∈ 퐵푛 }

(with 퐽 = {푡1, … , 푡푛} ⊂ [0, 푇] and 퐵1, … 퐵푛 belonging to the Borel 휎-algebra ℬ(ℝ)) satisfying:

휇(퐸퐽;퐵1,…,퐵푛) = 휇퐽(퐵1 × 퐵2 × ⋯ × 퐵푛). (21) In the case where 훼 ∈ ℂ and 푁 = 2 (as in the case of the Schrödinger equation (4) where 훼 = 푖/2 is a purely imaginary constant) or if 푁 > 2 (as in the case of the high- order heat type equation (12)) the kernel 퐾푡(푥, 푦) is not a real positive function which can be interpreted as the density of a probability measure. In these cases formula (20) defines a projective family of either complex or signed (unbounded resp. bounded variation) measures. Kolmogorov’s theorem has been generalized to projective families of complex or signed bounded variation measures by E. Thomas [56]. The following result gives a necessary condition for the existence of the limit of a projective family of complex measures.

Theorem 1. Let {휇퐽} be a projective family of signed or complex bounded Borel measures, on ℝ퐽 labeled by the finite subsets 퐽 of the interval [0, 푇]. A necessary condition for the existence of a (signed or complex) bounded measure 휇 on (Ω, 퐴) satisfying relation (21) is the following uniform bound on the total variation of the measures belonging to the family {휇퐽}:

sup |휇퐽| < +∞, (22) 퐽 where |휇퐽| denotes the total variation of the measure 휇퐽. 44 S. Albeverio and S. Mazzucchi

It {휇퐽} are probability measures then (22) is trivially satisfied. The measure 휇 on (Ω, 퐴) describes a stochastic (in fact Markov, resp. sub-Markov) process 푋푥 = 푥 푥 푥 (푋푡 )푡≥0. The finite dimensional distributions 푃(푋푡1 ∈ 퐵1, … , 푋푡푛 ∈ 퐵푛), 푡1 ≤ ⋯ ≤ 푡푛, 퐵푖 ∈ ℬ(ℝ) for 푖 = 1, … , 푛, are given by

푥 푥 휇(퐸퐽;퐵1,…,퐵푛) = 푃(푋푡1 ∈ 퐵1, … , 푋푡푛 ∈ 퐵푛) = 휇퐽(퐵1 × ⋯ × 퐵푛),

퐽 = {푡1, … , 푡푛} (23) (see, e.g., [19]), and the limit on the right hand side of Eq. (19) can be interpreted in terms of the expectation with respect to the distribution of 푋푥, yielding a probabilistic representation of the form

푡 푥 푥 ∫ 푉(푋푠 ) d푠 푇푡푢(푥) = 피 [푢(푋푡 )푒 0 ]

In the case of complex or signed kernels 퐾, under the assumptions of Theorem 1 the measure 휇 on (Ω, 퐴) satisfying relation (21) exists. In this case it is also possible 푥 푥 to introduce the concept of pseudoprocesses 푋 = (푋푡 )푡≥0, with the family of “finite dimensional distributions” given by the 휇퐽 in the sense that 푋푡(휔) = 휔(푡), 휔 ∈ ℝ[0,푇] and (23) holds again (without of course a probabilistic interpretation). + On the other hand, if 퐾푡(푥, ⋅ ), with 푥 ∈ 퐸 and 푡 ∈ ℝ , are general complex or signed measures, in many interesting cases condition (22) is not satisfied. In particular, in the case where 퐾푡(푥, 푦) is given by (17) or, more generally, by (15) with 푁 > 2, the total variation of the measures 휇퐽 defined by Eq. (20) increases exponentially with the cardinality of the set 퐽 = {푡1, … , 푡푛}. This no-go result was stated for the first time by Cameron [24] in the case 퐾푡(푥, 푦) is the fundamental solution of the Schrödinger equation (4), and by Krylov in the case of the high-order heat type equation (12). The construction of a functional integral representation for the solution of this kind of PDEs cannot be directly based on Lebesgue integration and measure theory. In fact, a generalization of integration theory on infinite dimensional spaces based on the concept of linear (continuous) functional provides an fruitful alternative approach. The integral with respect to a 휎-additive measure 휇 has to be replaced by a linear (continuous) functional 퐿∶ 퐷(퐿) → ℂ defined on a domain 퐷(퐿) which contains the cylinder functions, i.e., the functions 푓∶ Ω → ℂ of the form 푓(휔) ≔ 퐹(휔(푡1), … , 휔(푡푛)) (24) 푛 for some 퐽 = {푡1, … , 푡푛} and a Borel function 퐹 on ℝ . The action of the functional 퐿 on a function 푓 of the form (24) must be given by a (finite dimensional) integral 퐽 on the space ℝ with respect to the measure 휇퐽 defined by (20), namely:

퐿(푓) = ∫ 퐹(푥1, … , 푥푛)퐾푡푛−푡푛−1(푥푛−1, 푥푛) ⋯ 퐾푡2−푡1(푥1, 푥2)퐾푡1(푥, 푥1) d푥1 ⋯ d푥푛. ℝ푛 Infinite dimensional integrals of probabilistic and oscillatory type 45

The following sections shall give an example of a particular implementation of this idea as well its application to the proof of generalized Feynman–Kac formulae for the solution of either the Schrödinger (4) or the high-order heat type equation (12).

3. Oscillatory integrals and Schrödinger equation

An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals has recently been proposed in [15, 14]. It provides a truly infinite dimensional construction of integrals as linear functionals, as much as possible independent of the underlying topological and measure theoretical structure. In fact it includes both the case of probabilistic and oscillatory integrals in infinite dimensions associated respectively either to Markov processes or to Feynman integrals. A particular example of this general theory is given by the infinite dimensional Fresnel integrals, introduced by S. Albev- erio and R. Høegh-Krohn [9, 7] in connection with the mathematical definition of Feynman path integrals and recently generalized by S. Mazzucchi [50] in the proof of a functional integral representation for the solution of the high-order heat type equations. Finite dimensional Fresnel integrals are objects of the following form

푖 ‖푥‖2 푒 2ℏ ∫ 푓(푥) d푥, (25) 푛/2 ℝ푛 (2휋푖ℏ) where ℏ ∈ ℝ ⧵ {0} is a real parameter and 푓∶ ℝ푛 → ℂ a bounded Borel function. They are applied in optics and in the theory of wave diffraction. From a purely mathematical point of view, they have been extensively studied in connection with the theory of Fourier integral operators [42]. Particular interest has been devoted to the study of their asymptotic behavior in the limit the parameter ℏ converges to 0 [30, 42]. The study of integrals of the form (25) in the case where ℝ푛 is replaced by a real separable infinite dimensional Hilbert space ℋ began with the work by Itô [43] and was further developed by S. Albeverio and R. Høegh-Krohn [9]. Their construction is based on a Fourier transform approach. Given a Schwartz test function 푓 ∈ 푆(ℝ푛), the Fresnel integral (25) can be computed in terms of the following Parseval’s identity:

푖 ‖푥‖2 푒 2ℏ − 푖ℏ ‖푥‖2 ∫ 푓(푥) d푥 = ∫ 푒 2 푓(푥)̂ d푥, (26) 푛/2 ℝ푛 (2휋푖ℏ) ℝ푛 46 S. Albeverio and S. Mazzucchi

푓̂being the suitably normalized Fourier transform of 푓. Given a real separable Hilbert space (ℋ, ⟨ , ⟩), let us consider the Banach space ℳ(ℋ) of complex Borel measures on ℋ with finite total variation, endowed with the total variation norm, denoted by ‖휇‖ℳ(ℋ). ℳ(ℋ) is a commutative Banach algebra under convolution, where the unit is the 훿 point measure concentrated at 0. Let ℱ(ℋ) denote the space of complex functions 푓 on ℋ of the form:

푓(푥) = ̂휇(푥) = ∫ 푒푖⟨푥,푦⟩ d휇(푦), 푥 ∈ ℋ (27) ℋ for some 휇 ∈ ℳ(ℋ), 푓 ∈ ℱ(ℋ) being thus the Fourier transform of 휇. By introducing on ℱ(ℋ) the norm ‖푓‖ℱ(ℋ) = ‖휇‖ℳ(ℋ), the map (27) becomes an isometry and ℱ(ℋ) endowed with the norm ‖ ⋅ ‖ℱ(ℋ) becomes a commutative Banach algebra of continuous functions [9]

Definition 2. Let 푓 ∈ ℱ(ℋ). The infinite dimensional Fresnel integral of 푓, 푖 ‖푥‖2 denoted by ∫˜푒 2 푓(푥) d푥, is defined as:

˜ 푖 ‖푥‖2 − 푖ℏ ‖푥‖2 ∫푒 2ℏ 푓(푥) d푥 ≔ ∫ 푒 2 d휇푓(푥), (28) ℋ

푖⟨푥,푦⟩ where 푓(푥) = ∫ℋ 푒 d휇푓(푦), 휇푓 ∈ ℳ(ℋ). Remark 3. The right hand side of (28) is a well defined (absolutely convergent) 푖 ‖푥‖2 Lebesgue integral. Moreover the application 푓 ↦ ∫˜푒 2ℏ 푓(푥) d푥 is a linear continuous functional on ℱ(ℋ).

In [7] the functional defined by (28) has been applied to the construction of a representation for the solution of the Schrödinger equation (4) in the cases where the potential 푉 belongs to ℱ(ℝ푑). 푑 Let ℋ푡 be the Hilbert space of absolutely continuous paths 훾∶ [0, 푡] → ℝ 푡 2 such that 훾(푡) = 0 and ∫0 | ̇훾(푠)| d푠 < ∞, endowed with the inner product 푡 ⟨훾1, 훾2⟩ = ∫0 1̇훾 (푠) ⋅2 ̇훾 (푠) d푠. Let us consider the initial value problem associated to the Schrödinger equation

휕 ℏ2 푖ℏ 휓(푡, 푥) = − Δ휓(푡, 푥) + 푉(푥)휓(푡, 푥), 푥 ∈ ℝ푑, 푡 ∈ ℝ, { 휕푡 2 (29) 푑 휓(0, 푥) = 휓0(푥), 푥 ∈ ℝ ,

ℏ ∈ ℝ+ denoting the reduced Planck constant. Let us assume that the potential 푉∶ ℝ푑 → ℝ is a continuous bounded function belonging to ℱ(ℝ푑), i.e., 푉(푥) = 푖푥푦 푑 ∫ℝ푑 푒 d휇푉(푦) for some complex Borel measure 휇푉 on ℝ . We will also assume Infinite dimensional integrals of probabilistic and oscillatory type 47

2 푑 푑 푖푥푦 that the initial datum 휓0 ∈ 퐿 (ℝ ) belongs to ℱ(ℝ ), i.e., 휓0(푥) = ∫ℝ푑 푒 d휇0(푦). Under this conditions the Hamiltonian operator 퐻∶ 퐷(퐻) ⊂ 퐿2(ℝ푑) → 퐿2(ℝ푑), ∞ 푑 defined on the smooth vectors 휓 ∈ 퐶0 (ℝ ) as ℏ2 퐻휓(푥) = − Δ휓(푥) + 푉(푥)휓(푥), 2

∞ 푑 is essentially self-adjoint on 퐶0 (ℝ ), and its unique self adjoint extension (denoted again 퐻 with an abuse of notation) generates a strongly continuous unitary group 푈(푡) ≡ 푒−(푖/ℏ)퐻푡 on 퐿2(ℝ푑). For any 푡 > 0 the action of the evolution group 푑 2 푑 푈(푡) on the initial vector 휓0 ∈ ℱ(ℝ ) ∩ 퐿 (ℝ ) can be represented by an infinite dimensional Fresnel integral on the Hilbert space ℋ푡. Indeed under the stated 푑 + assumptions, for any 푥 ∈ ℝ and 푡 ∈ ℝ , the map 푓∶ ℋ푡 → ℂ defined as

푖 푡 푓(훾) ≔ 휓 (훾(0) + 푥) exp (− ∫ 푉(훾(푠) + 푥) d푠) , 훾 ∈ ℋ 0 ℏ 푡 0 is an element of the Banach algebra ℱ(ℋ푡). Furthermore its infinite dimensional − 푖 퐻푡 Fresnel integral gives the action on 휓0 of the unitary group 푒 ℏ , namely

− 푖 퐻푡 푖 ‖훾‖2 푒 ℏ 휓(푥) = ∫ 푒 2ℏ 푓(훾) d훾

ℋ푡

푖 ∫푡 |훾(푠)|̇ 2푑푠− 푖 ∫푡 푉(훾(푠)+푥) d푠 = ∫ 푒 2ℏ 0 ℏ 0 휓0(훾(0) + 푥) d훾 훾(푡)=0 The second line shows how the infinite dimensional Fresnel integral provides in this context a rigorous mathematical definition of Feynman path integrals. [33, 9, 48]. For a detailed proof of these results as well as for their applications to the Feynman path integral representation of the solution of the Schrödinger equation, see, e.g., [31, 2, 9, 48]. For other approaches to the mathematical theory of Feynman path integrals see, e.g., [39, 44, 35, 45].

4. Oscillatory integrals with polynomial phase and high-order heat type equations

The definition 2 of infinite dimensional Fresnel integral has been recently generalized to cover the case where the quadratic phase function ‖푥‖2 is replaced by an higher-order polynomial [50]. This new functional, named infinite dimensional Fresnel integral with polynomial phase function can be applied to the construction of a functional integral representation for the solution of a general class of 48 S. Albeverio and S. Mazzucchi high-order heat type equations of the form 휕 휕푝 푢(푡, 푥) = (−푖)푝훼 푢(푡, 푥) + 푉(푥)푢(푡, 푥), 푥 ∈ ℝ, 푡 > 0, {휕푡 휕푥푝 (30) 푢(0, 푥) = 푢0(푥), 푥 ∈ ℝ, where 푝 ∈ ℕ is a positive integer and 훼 ∈ ℂ a complex constant. In the case where 푝 = 2 and 훼 is purely imaginary, Eq. (30) reduces to the Schrödinger equation, while if 푝 is even, namely 푝 = 2푁 and 훼 = (−1)+1 , Eq. (30) is the high-order heat type equation (12). Le (ℬ, ‖ ⋅ ‖) be a real separable Banach space, and let us denote with ℬ∗ its topological dual. Let ℱ(ℬ) denote the space of complex valued functions 푓∶ ℬ∗ → ℂ of the form 푖⟨푥,푦⟩ 푓(푥) = ∫ 푒 d휇푓(푦) (31) ℬ for some complex Borel measure 휇푓 on ℬ. The space ℱ(ℬ) is a Banach algebra of functions, where the product is the pointwise one (푓 ⋅ 푔)(푥) = 푓(푥)푔(푥) and the norm of an element 푓 ∈ ℱ(ℬ) is defined as the total variation of the associated Borel bounded measure (see Eq (31)), namely ‖푓‖ℱ ≔ ‖휇푓‖. Given a continuous function Φ∶ ℬ → ℂ, it is possible to generalize definition 2 to the case where the quadratic phase function ‖푥‖2 is replaced by Φ. Definition 4. Let let Φ∶ ℬ → ℂ be a continuous function such that Re(Φ(푥)) ≤ 0 for all 푥 ∈ ℬ. The infinite dimensional Fresnel integral on ℬ∗ with phase function Φ is the functional 퐿Φ ∶ ℱ(ℬ) → ℂ, given by

Φ(푥) 푖⟨푥,푦⟩ 퐿Φ(푓) ≔ ∫ 푒 d휇푓(푥), 푓 ∈ ℱ(ℬ), 푓(푥) = ∫ 푒 d휇푓(푦). (32) ℬ ℬ

By its definition, it is straightforward to see that the functional 퐿Φ ∶ ℱ(ℬ) → ℂ is linear and continuous in the ℱ(ℬ)-norm. Indeed

Φ |퐿Φ(푓)| ≤ ∫ |푒 | d|휇푓|(푥) ≤ ‖휇푓‖ = ‖푓‖ℱ. ℬ Furthermore the functional 퐿 is normalized, i.e., its value on the constant function ∗ 푓(푥) = 1 ∀푥 ∈ ℬ is equal to 퐿Φ(1) = 1 (푓 being the Fourier transform of the 훿 point measure at 푥 = 0). In particular the functional 퐿Φ generalizes the infinite dimensional Fresnel integrals (definition 2) in the sense that if ℬ ≡ ℋ and Φ(푥) = −푖‖푥‖/2 then ˜ 푖 ‖푥‖2 퐿Φ(푓) = ∫ 푒 2ℏ 푓(푥) d푥 ℋ Infinite dimensional integrals of probabilistic and oscillatory type 49

Let us consider now a particular example of a functional of the form (32) on a suitable Banach space ℬ. Given a positive integer 푝 ∈ ℕ, with 푝 ≥ 2, let us consider the Banach space ℬ푝 of absolutely continuous paths 훾∶ [0, 푡] → ℝ such that 훾(푡) = 0 and weak derivative ̇훾 belonging to 퐿푝([0, 푡]), endowed with the norm

푡 1/푝 푝 ‖훾‖ℬ푝 = (∫ | ̇훾(푠)| d푠) . 0

∗ 1 1 The dual space ℬ푝 is isomorphic to ℬ푞, with 푝 + 푞 = 1, and the pairing between ∗ an element 휂 ∈ ℬ푝 and 훾 ∈ ℬ푝 is given by:

푡

⟨휂, 훾⟩ = ∫ 휂(푠)̇ ̇훾(푠) d푠 휂 ∈ ℬ푞, 훾 ∈ ℬ푝. 0

A function 푓∶ ℬ푞 → ℂ belonging to the Banach algebra ℱ(ℬ푞) has the form

푖 ∫푡 휂(푠)̇ 훾(푠)̇ d푠 푓(휂) = ∫ 푒 0 d휇푓(훾), 휂 ∈ ℬ푞 (33) ℬ푝 for some complex Borel measure 휇푓 on ℬ푝. Let us introduce a homogeneous phase function Φ푝 ∶ ℬ푝 → ℂ of the form

푡 푝 푝 Φ푝(훾) ≔ (−1) 훼 ∫ ̇훾(푠) 푑푠, 훾 ∈ ℬ푝 0 where 훼 ∈ ℂ is a complex constant such that Re(훼) ≤ 0 if 푝 is even and Re(훼) = 0 if 푝 is odd. The corresponding generalized infinite dimensional Fresnel integral

퐿Φ푝 ∶ ℱ(ℬ푝) → ℂ is defined as

푝 푡 푝 (−1) 훼 ∫0 훾(푠)̇ 푑푠 퐿Φ푝(푓) = ∫ 푒 d휇푓(훾), (34) ℬ푝 for 푓 ∈ ℱ(ℬ푝) given by Eq. (33).

Lemma 5. Let 푓∶ ℬ푞 → ℂ be a cylindric function of the following form:

푓(휂) = 퐹(휂(푡1), 휂(푡2), … , 휂(푡푛)), 휂 ∈ ℬ푞,

푛 푛 with 0 ≤ 푡1 < 푡2 < ⋯ < 푡푛 < 푡 and 퐹∶ ℝ → ℂ, 퐹 ∈ ℱ(ℝ ):

푛 푖 ∑ 푦푘푥푘 퐹(푥1, 푥2, … , 푥푛) = ∫ 푒 푘=1 d휈퐹(푦1, … , 푦푛) ℝ푛 50 S. Albeverio and S. Mazzucchi

Then 푓 ∈ ℱ(ℬ푝) and its infinite dimensional Fresnel integral with phase function Φ푝 is given by 푛 퐿 (푓) = ∫ 퐹(푥 , 푥 , … , 푥 ) ∏ 퐺푝 (푥 , 푥 ) d푥 , (35) Φ푝 1 2 푛 푡푘+1−푡푘 푘+1 푘 푘 ℝ푛 푘=1 푝 where 푥푛+1 ≡ 0, 푡푛+1 ≡ 푡 and 퐺푠 is the fundamental solution of the high order 휕 푝 휕푝 heat-type equation 휕푡 푢(푡, 푥) = (−푖) 훼 휕푥푝 푢(푡, 푥) The previous lemma allows the proof of the following functional integral representation of the solution to the high-order heat type equation (30). 2 푖푥푦 Theorem 6. Let 푢0 ∈ ℱ(ℝ) ∩ 퐿 (ℝ) and 푉 ∈ ℱ(ℝ), with 푢0(푥) = ∫ℝ 푒 d휇0(푦) 푖푥푦 and 푉(푥) = ∫ℝ 푒 d휈(푦), where 휇0 and 휈 are bounded complex measures on ℝ. Then the functional 푓푡,푥 ∶ ℬ푞 → ℂ defined by

∫푡 푉(푥+휂(푠)) d푠 푓푡,푥(휂) ≔ 푢0(푥 + 휂(0))푒 0 , 푥 ∈ ℝ, 휂 ∈ ℬ푞, belongs to ℱ(ℬ푞) and its infinite dimensional Fresnel integral with phase function Φ푝 provides a representation for the solution of the Cauchy problem 휕 휕푝 푢(푡, 푥) = (−푖)푝훼 푢(푡, 푥) + 푉(푥)푢(푡, 푥), 푥 ∈ ℝ, 푡 ∈ [0, +∞) {휕푡 휕푥푝 푢(0, 푥) = 푢0(푥), 푥 ∈ ℝ For a detailed proof of these results see [50]. For other approaches to the solution of the high-order heat type equations see [46, 41, 21, 22] and references in [49].

Acknowledgements

The first author first met Helge during his research stay 1977–78 in Oslo.Helge was then a very bright student of Raphael Høegh-Krohn, a very close friend and coworker of the first author. In his thesis Helge solved very important problems in the theory of quantum mechanics with potentials of the point interaction type. From this and related work originated the joint book [5]. Other collaborations in other areas, like quantum (gauge) field theory, like e.g., [8], took place. It has been a great joy to be able to collaborate with him. His open and charming character, joint with his great scientific and organizatorial skills, always made a real pleasure to work with him. Both authors would like to congratulate wholeheartedly Helge for his 60th birthday, express him their sincere admiration and wish him many more years in good health and satisfaction in all his endeavours. They are very grateful to the organizers of the conference for the kind invitation to contribute to this book. Infinite dimensional integrals of probabilistic and oscillatory type 51

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The numbers lead a dance Mathematics of the Sestina

Alan R. Champneys, Poul G. Hjorth, and Harry Man

Dedicated to Helge Holden on the occasion of his 60th birthday

Abstract. Sestinas are poems of 39 lines comprising six verses of six lines each, and a three line final verse or ‘envoi’. The structure of the sestina is built around word repetition rather than strict rhyme. Each verse uses the same set line ending words, but in a permuted order. The form of the permutation is highly specific, and is equivalent to iteration of the tent map. This paper considers for which number 푁 of verses, other than 6, can a sestina-like poem be formed. That is, which 푁 will the prescribed permutation lead to a poem of 푁 verses where no two verses have the same order of their end words. In so doing, a link is found between permutation groups, chaotic dynamics, and Cunningham numbers.

1. Introduction

Sestinas are a form of highly complex poems designed around a particular pattern, see e.g. [6, 13]. Each verse of the poem has six lines and there are 6 verses in total. In addition there is a coda, called an envoi that contains just three lines. For the main poem, the final word of each line is crucial. The collection of sixsuch end-words is invariant from verse to verse, yet the word order is permuted. The permutation from one verse to the next takes a specific form. The idea is similar to that of a riffle shuffle of a pack of cards. The list of words is split intwoandthe words from the second half are meshed with the words from the first half, but in reverse order. Thus, what was the last word is now first, what was the penultimate is now third, etc. This mixing is is sometimes represented in a spiral pattern, as illustrated in Figure 1. Schimel[11] describes the sestina as “like a dance [12], with each stanza representing a reel. Each stanza is based on the stanza directly preceding it. The order of the stanza peels off the lines of the prior stanza, moving ever inwards towards the core: last, first, penultimate, second, antepenultimate, third.” And what of the envoi? It contains all six end words, two per line, with half of the end-words being placed somewhere within the body of the line, and half at 56 A. R. Champneys, P. G. Hjorth, H. Man

Verse 1 Verse 2 one 1 six two 2 one three 3 five four 4 two five 5 four six 6 three start here Figure 1. Illustrating the permutation of the order of the end words when passing from the first to the second verse. Here “one” to “six” represent the end words used in thefirst verse, and the numbers 1 to 6 represent the position within a verse. This spiral illustration, found in several poetry text books e.g. [6, 13], appears somewhat confusing as it is does not represent the actual permutation map. Instead, the arrangement of the second verse is found by following the spiral, beginning with 6. the end. It is like a closing passage of the dance in double time. Some versions of the sestina demand a strict order for the placement of the six words on these lines,

… 5 … 2 … 3 … 4 … 1 … 6 some poets chose to vary the form by using synonyms, or otherwise shake up the form, but the reason behind these alterations has to be implicit in either the imagery or the language e.g., the numbers of a rocket countdown. Such numbers might be fudged if the poem is about someone setting off fireworks in their garden and one firework goes off unexpectedly early. The envoi is a crucial part of each sestina, but because it is not involved directly in the permutation of words from verse to verse, we shall ignore the envoi in the mathematical arguments that follow. Scholars continue to debate on the sestina’s precise origins because of the volume of poems thought to be comprised of sextains (six line verses) between the 11th and 13th centuries. However, the form’s invention is commonly attributed to early 13th century literary giant Arnaut Daniel with his poem (English translation: The Firm Desire That Entered My Heart)1 At the time, his poetic abilities were incontestable, and it set a deliberate challenge to those around him to make the poetry of courtship as tough as possible by forcing the poet to utilise a pattern of repeated words while still keeping their bride to be entertained. It also was incidentally quite pioneering, exhibiting a kind of free verse, years ahead of other

1We shall not give precise references to the poems or poets mentioned in this introduction; this is, after all, primarily an article on mathematics and the text of all the poems can easily be found online. The numbers lead a dance 57 exponents of free verse like Christopher Smart or Walt Whitman, with attention largely focused on content over rhyme. While any challengers would be able to express themselves in a manner that was strictly regimented it also allowed allowed poets to dispense with strict rhyme and meter. This is probably one of the main reasons why in the 20th Century he was championed by the likes of T. S. Eliot and Ezra Pound. Both Eliot and Pound were exponents of an imagistic poetry which was sparse, and that lacked the romance, nostalgia and high rhetoric that had so dominated the poetry of the previous century. Think of poems such as Tennyson’s The Charge of the Light Brigade or Words- worth’s Daffodils and the line, ‘I wandered lonely as a cloud’. Poetry’s horizons opened up under the modernists as they fractured the syntax and rhyme and formal conventions of poetry, and now a poem could be made from found text, and fragments of dialogue, and musical hall songs, or it could be made to look like rain trickling down a page, and most importantly of all, the poem was no longer a piece of text reflecting on an event, but it was the event itself. This shifted the spotlight to both the form a poem could take, and how this interacted directly with its meaning. It was as if someone in an art gallery pointed at a portrait and suddenly made the pronouncement that the frame the portrait was sitting in was just as integral to the artwork’s meaning as the painting itself. This shift in approach is still felt today, and particularly in the workings of the sestina. The sestina structure itself is what frames the poem, and its structure, therefore is subject to the same scrutiny as the poem’s literal meaning. Well composed sestinas can either make the word repetition seem utterly necessary to the unfolding narrative, which is the conventional view, or they can emphasize each endword deliberately to bring the structure more to the fore. Great modern examples of the form include John Ashbery’s The Painter and Paul Muldoon’s extraordinary work Yarrow. There are several modern literary journals such as McSweeneys in San Francisco that have in the past purposefully asked for sestinas to ward off amateur poets – and mention the word ‘sestina’ to anypoetry workshop now, and you can expect a sharp intake of breath from around the room because of its infamous complexity. Mathematically, we can describe the sestina permutation as follows. Let 푚 be the number of 푚-line verses and let 푛 represent the word that is at the end of the 푛th line of verse 푝. Then the position in the (푝 + 1)푠푡 verse is given by the rule

푚 2푛 if 푛 ≤ [ 2 , 푛 ↦ { 푚 (1) 2푚 + 1 − 2푛 if [ 2 < 푛 ≤ 푚, where 푚 is the number of lines in a verse and [⋅] represents the integer part of an 58 A. R. Champneys, P. G. Hjorth, H. Man expression. Thus, for 푚 = 6 we have

1 ↦ 2, 2 ↦ 4, 3 ↦ 6, 4 ↦ 5, 5 ↦ 3, 6 ↦ 1, (2) as constructed in Figure 1.

verse one two three four five six 1st line 1 6 3 5 4 2 2nd line 2 1 6 3 5 4 3rd line 3 5 4 2 1 6 4th line 4 2 1 6 3 5 5th line 5 4 2 1 6 3 6th line 6 3 5 4 2 1

Table 1. Position of the end-words of each line among the six verses of a six-line sestina. Here, the number 1 represents the word that ends the first line of the first verse, 2 represents the word that ends the second line of the first verse, etc. The word in the first line movesto the second line, the word in the second line moves to the fourth line and so on.

For a sestina to work properly, each of the end-words should have a turn at the end of 푛th line of a verse, for each 푛 = 1, 2, … 푚. This indeed occurs if 푚 = 6 as indicated in Table 1 and illustrated in the poems embedded in this article. Here, each word has a turn in each position, and each line within a verse sees each end-word precisely once during the poem. If one were to construct a seventh verse according to the rule (1), then the order of the end-words would be identical to that of the first verse. Thus we find that the permutation forms a cycle of length six. The question we wish to address in this article is for which other verse lengths 푚 does this symmetric, egalitarian distribution of line order among the end-words occur if we use the same basic rule (1) from verse to verse? A simple test shows, for example, that something goes wrong if 푚 = 7, or 푚 = 8, see Tables 2 and 3 respectively. There are also poems known as double sestinas, which have 푚 = 12. For example, one of the first known sestina in English, The Complaint of Lisa by Algernon Charles Swinburne, is actually a double, although the permutation of the end-words does not follow the rule (1) but appears somewhat random. We shall see shortly that there is also a problem applying rule (1) when 푚 = 12. Strictly speaking, we should always choose 푚 to be even in order for their to be an envoi of length 푚/2 with each line containing two end words. But, for purposes of the mathematics, we shall ignore this restriction. Poets who engage on highly regular forms like sestinas tend to do it in part for the challenge, and The numbers lead a dance 59

verse one two three four five six seven 1st line 1 7 4 2 1 7 4 2nd line 2 1 7 4 2 1 7 3rd line 3 6 3 3 3 6 3 4th line 4 2 1 7 4 2 1 5th line 5 5 5 5 5 5 5 6th line 6 3 6 3 6 3 6 7th line 7 4 2 1 7 4 2

Table 2. Similar to Table 1 but for 푚 = 7. Note the fifth line of each verse always ends with the same word. That is, the number 5 is a fixed point of the rule (1). Also the sixth and third lines share the same two words repeatedly; (6 3) is a period-two cycle of (1).

verse one two three four five six seven eight 1st line 1 8 4 2 1 8 4 2 2nd line 2 1 8 4 2 1 8 4 3rd line 3 7 5 6 3 7 5 6 4th line 4 2 1 8 4 2 1 8 5th line 5 6 3 7 5 6 3 7 6th line 6 3 7 5 6 3 7 5 7th line 7 5 6 3 7 5 6 3 8th line 8 4 2 1 8 4 2 1

Table 3. Similar to Table 1 but for 푚 = 8. Note that the pattern repeats at the fifth verse, so that the word that end the first line of the first verse only ever ends the first, second, fourth and eighth lines of any subsequent verse, never the third, fifth, sixth or seventh. In fact, (1, 8, 4, 2) is a period-four cycle of the rule (1), and so is (3, 7, 5, 6). the requirement for a non-integer number of lines in the envoi could provide an opportunity to subvert the form creatively. Of course, the phrase “poetic license” springs to mind.

2. Recent history

Before proceeding, it might be interesting to point out the happenstance that led to this article being written. It started with a chance meeting more than 10 years ago between the first and last author on their regular daily commute from thesame bus stop on the outskirts of Bath to Bristol. Harry at the time was on a placement 60 A. R. Champneys, P. G. Hjorth, H. Man with a publishing house following a successful MA in Creative Writing at Bath Spa University. Alan had recently become the youngest ever head of Department of Engineering Mathematics at Bristol. Harry had written the first two stanzas of a sestina, his opus magnum, with 푚 = 78. Harry initially decided upon the number 78 for the simple fact that it was twice the length (39) of the total number of lines of a conventional sestina including its envoi. A familiar construction technique to most poets in this form is write down on a blank piece of paper a guide to illustrate which words are going to arrive in what position allowing the poem to be grafted onto this template and adjusted. Aware that the total number of lines was now 6,123 it became necessary for Harry to plot out the new larger sestina using a fractionally more adept system than pen and paper! Microsoft Excel provided him with a chart illustrating the word positions for each verse. To his horror, he noticed that already at verse 26, the entire sestina collapsed to the original order and renewed its cycle once more, something that should only happen in a hypothetical extra verse prior to the final envoi. The pleasure in the reading of a sestina for most literary critics is precisely its strategic avoidance of this outcome! With all the carefully chosen words in play already, and all the source material under his belt and the idea firmly in his mind, there was little that could bedone to remedy the problem except for the increasingly large possibility of it all ending up in the wastepaper basket. Not to be outdone by this setback he began to try and establish a means by which to accurately predict the relationship between the number of lines in a verse and the potentially disastrous outcome of word positions prematurely coming back to their original order, and thus to restructure the two original verses while causing minimum damage to the sense of the writing. So should the sestina be grown or shrunk? Unknown to us, a solution was actually available in French, starting with the work in the 1960s of the French Poet Raymond Queneau and his colleague the mathematician Jacques Roubaud [8, 9]. Those results were recently summarised in the excellent article by Michael Saclolo in Notices of the American Mathematical Society [10]. Queneau asked exactly the same question. For what numbers 푚 is an 푚-ina possible? In French a sestina is called a sestine and so Roubard coined the phrase 푞-ines or quenines in honour of Queneau for the admissible 푞-verse poem. This was later formalised by Monique Bringer [2], a student of Roubard, who coined the phase Queneau–Daniell group for the cyclic subgroup of the general linear group of order 푚 generated by the quinine permutation. She was able to provide a partial set of necessary and sufficient conditions for admissible numbers 푚. The complete characterisation of which numbers 푚 are admissible was not actually solved until 2008, in the work of Dumas [5], whose results are reproduced in English in [10]. Dumas’ proof is not however constructive, in the following The numbers lead a dance 61

6 2

3 4 5

Figure 2. The orbit of the end-words for a sestina of length 푚 = 6 sense. It is easy to state necessary and sufficient conditions for a number 푝 to be prime, but there is no simple checkable formula that generates the 푚’th prime number. So it seems to be also with sestina numbers. In what follows we describe an investigation of the generalisation of a sestina, which was arrived at independently of the French group theorists. In so doing so, we uncover an alternative view, establishing a connection with a different branch of mathematics, namely chaotic dynamical systems.

3. Permutation groups

We call 푚 a sestina number if the permutation represented by (1) on the set of 푚 integers has minimal period 푚. Let us recall some basic facts from permutation group theory.

The group of all permutations of 푚 symbols is denoted by 푆푛. Basic theorems [7] tell us that any element of the group has a unique minimal representation in terms of disjoint cycles. Take the permutation (1) with 푚 = 6 as described in (2) and Table 1. A far more compact way of writing this is to look at the orbit of the position of the first end-word after each successive application of the permutation: That is, following the arrows around the circle of (2), we see that the first end-word of the first verse becomes the second end-word of the second verse; the fourth end-word of the third verse, the 5th end-word of the fourth verse; and so on. The circular representation also allows us to find the orbit of any other end-word. For example, to see what happens to the third end-word of the first verse, we start at the number 3 on the clock face and follow the arrows six times. So, in the second 62 A. R. Champneys, P. G. Hjorth, H. Man verse, this word ends the sixth line; it ends the first line of the third verse andso on. This then leads to the more compact notation (1, 2, 4, 5, 3, 6), where the round brackets mean “and repeat”. Now 푚 = 6 is a sestina number because there is representation of the effect of the transformation (1) in terms of a single cycle. If we try the same for 푚 = 7, based on the information in table 2 we see that the permutation is now written (1, 2, 4, 7)(3, 6)(5) which has three separate disjoint cycles. The end-words of lines one, two, four and seven cycle, whereas lines three and six swap end-words between each successive verse, while the fifth line ends with the same word each time. Similarly for 푚 = 8, we have (1, 2, 4, 8)(3, 6, 5, 7) two four-cycles, and for 푚 = 12 we have (1, 2, 4, 8, 9, 7, 11, 3, 6, 12)(5, 10) a 10-cycle and a 2-cycle. This latter case shows the difficulty of trying to construct a “double sestina” using the same transformation (1) as for the standard six-verse transformation.

m cycle representation sestina number ? 1 (1) yes 2 (12) yes 3 (123) yes 4 (124)(3) no 5 (12435) yes 6 (124536) yes 7 (1247)(36)(5) no 8 (1248)(3657) no

Table 4. Cycle structure of the sestina permutation for the first few values of 푚.

Thus, we have established a criterion for 푚 to be a sestina number; namely that the permutation (1) can be expressed as a single cycle of length 푚. Table 4 lists the disjoint cycle representation for the first few 푚. Note that there is no obvious pattern governing which 푚’s lead to a single 푚-cycle and which do not. It is precisely this pattern that we aim to uncover in the rest of this article. The numbers lead a dance 63

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 x Figure 3. Constructing the dynamics of the tent map via the so-called cobwebbing process. Here 푦 is replaced at the next unit of time by its value given by the formula (3). This value is then fed back as the next value of 푦 into the same formula, and so-on. This feedback process is represented as the reflection of the value of the image of agiven 푦-value in the 45∘ line.

4. A connection with chaotic dynamics

The equation (1) can be represented as discrete-time dynamical system acting on the first 푚 integers. A simple re-scaling, letting 푦 = 2푛/(2푚 + 1), shows that repeated iteration of (1) is equivalent to the dynamics of the tent map for 푦 ∈ [0, 1]: 2푦 if 푦 ≤ 1/2, 푦 ↦ { (3) 2 − 2푦 if 1/2 < 푦 ≤ 1. Instead of the integers from 1 to 푚 we now have the points 2푗/(2푚 + 1) with 푗 = 1, …, 푚 distributed between 0 and 1. For any value of 푚 we will call these points sestina points. The dynamics of the map is represented graphically in Fig. 3. To be more precise, this is the tent map with slope 2, which is part of the general family of tent maps 휇푦 if 푦 ≤ 1/2, 푦 ↦ { (4) 휇(1 − 푦) if 1/2 < 푦 ≤ 1, with slope 휇 > 0 [4]. Straightforward analysis shows that if 휇 < 1, then the fixed point 푥 = 0 is the unique attractor of the system. That is, all initial conditions will eventually converge towards 푥 = 0 under repeated iteration of (4). If 휇 = 1, then all points with 푦 ≤ 1/2 are fixed points of this dynamical system. 64 A. R. Champneys, P. G. Hjorth, H. Man

It is when 휇>1that things get interesting. See Figure 3. In fact, among chaotic maps, the tent map is rather special because of the sharp point at 푦 = 1/2. So as 휇 increases through 1, rather than a Feigenbaum period-doubling cascade that is familiar to all who have studied smooth chaotic dynamical systems (see e.g. [4]), the dynamics immediately becomes chaotic. There are still two fixed points, 푦=0and 푦 = 휇/(휇 + 1), but both are unstable. That is, if you choose an initial condition arbitrarily close to one of these points, it moves away under iteration. For 1<휇<√2 then the attractor of the map splits into two non- overlapping sub-intervals of (0, 1), Arbitrary initial conditions are attracted to these two sub-intervals within which there is a chaotic cycling of points. For √2≤휇≤2the separate intervals start to overlap.

Figure 4. Bifurcation diagram showing points on the attractor of the tent map (4) for 0≤휇≤2.

For 휇=2the map is fully chaotic. That is, almost all initial conditions are part of the chaotic set and each region of the chaotic set are visited with equal probability. Starting from some arbitrary 푦-value in the interval (0, 1) and repeatedly iterating the formula (3), we reach a infinite sequence of further 푦-values that never repeat. Nevertheless, this sequence eventually visits arbitrarily close to every 푦-value in the interval [0, 1]. Moreover, no points that start in this interval ever escape. That is, the interval [0, 1] is the unique chaotic attractor of the dynamics. However, embedded within the chaos is a (countable) infinity of unstable periodic orbits with all possible periods. In particular, all rational initial conditions of (3) lie on periodic orbits. To see this, note that if an initial condition 푦 = 푝/푞 The numbers lead a dance 65 for integers 푝 and 푞 then all forward images of this point must be expressible as a fraction 푟/푞 for some integer 푟. Moreover, the map takes the unit interval to itself, hence 0 ≤ 푟 ≤ 푞. Since there are only 푞 + 1 such fractions, this must be a periodic orbit of period at most 푞 + 1. In particular we are interested in the case that 푞 = 푁 for odd 푁 = 2푚 + 1 and 푝 = 2푛 for some 푛 ≤ 푚. The question we seek to address then is: what is the image under repeated iteration of (3) of the specific initial condition 푦 = 2/(2푚 + 1), for each odd integer 2푚+1? If this orbit has minimum period 푚 then we say that 푚 is a sestina number. The only other possibility is that this initial condition lies on a periodic orbit with a lower period 푞. So, it seems we must look at conditions for the existence of periodic orbits of (3) (and hence of (1)) of arbitrary period 푞 ≤ 푚.

5. Conditions for cycles

The example in Table 2 above shows that 푚 = 7 fails to be a sestina number because there exists a fixed point (a 1-cycle) and a 2-cycle. Also, from Table 3, 푚 = 8 fails to be a sestina number because the permutation is decomposed into two disjoint 4-cycles. So in order to characterise which numbers are not sestina numbers, we need to consider conditions for a position 푗 (where 0 < 푗 ≤ 푚) to be part of a period-푞 cycle for 푞 ≤ 푚. Consider first the case of a fixed point. The fixed point for the tent mapisat the intersection between the map and the line 푥 = 푦, and (disregarding the trivial fixed point 푥 = 0 which will not be relevant here) occurs at 푥 = 2/3. If one of the 푚 sestina points, 푥푗 = 2푗/(2푚 + 1), 푗 = 1, … , 푚 happens to coincide with the value 푥 = 2/3, then a 1-cycle will occur, and the number 푚 (if different from 1) will fail as a sestina number. This will happen for all numbers 푚 such that

2푗 2 = 2푚 + 1 3 or 3 ∣ (2푚 + 1) and is obviously the case for 푚 = 7. If we study the condition for 2-cycles, we must find the loci for period-2 points of the tent map. These points are located where the twice repeated tent map intersects the line 푥 = 푦, i.e., at 푥 = 2/5, 2/3, 4/5, see figure 5. For sestina points to coincide with these values, we find that in addition to 3 ∣ (2푚 + 1) that 2푗 2 2푗 2 2푗 4 = or = or = 2푚 + 1 5 2푚 + 1 3 2푚 + 1 5 66 A. R. Champneys, P. G. Hjorth, H. Man

(a) (b) (c) 1 1 1

0 1 0 1 0 1

Figure 5. Location of (a) period-1 (fixpoint), (b) period-2 and (c) period-3 points for the tent map is the abcissa for the intersection between repeated tents and the line 푦 = 푥, see, e.g., [4].

The middle condition gives us 3 ∣ (2푚 +1) (because a period 1 orbit is also a period 2 orbit) but we now also have to exclude

5 ∣ (2푚 + 1) to avoid period-2 orbits, so this condition prevents 푚 (if different from 2) from being a sestina number. For the value 푚 = 7 we have both a 1-cycle and a 2-cycle present, since both 3 and 5 are factors of (2푚 + 1). 3-cycles occur (see figure 5) at the 23 − 1 values 푥 = 2/9, 2/7, 4/9, 4/7, 6/9, 6/7, 8/9, and they will coincide with sestina values if 7 ∣ (2푚 + 1) or 9 ∣ (2푚 + 1). Continuing in this manner, one finds: Proposition 1. The 푞-cycle points are located at 2 2 4 4 2푘 2푞 − 2 2푞 − 2 2푞 푥 = , , , ,…, ,…, , , . 2푞 + 1 2푞 − 1 2푞 + 1 2푞 − 1 2푞 − 1 2푞 + 1 2푞 − 1 2푞 + 1 All in all there are 2푞 − 1 such points. If there is a 푗 such that one of the sestina points 푥 = 2푗/(2푚 + 1) coincides with a 푞-cycle point, then the sestina permutation will contain a 푞-cycle. This happens when ∃푗, 푘 ∈ ℕ, 푗 = 1, …, 푚 and 푘 = 1, …, 2푚−1 such that 2푗 2푘 = 2푚 + 1 2푞 ± 1 or 푘(2푚 + 1) = 푗(2푞 ± 1) Here, ± is taken as “plus or minus“. The necessary existence of at least one 푞-cycle (푞 ≤ 푚) for a sestina permutation over 푚 can be noted in the following The numbers lead a dance 67

Proposition 2. For any odd number 2푚 + 1 there must be a number 푞 ≤ 푚 such that (2푚 + 1) ∣ (2푞 ± 1). We are now in position to give a necessary and sufficient conditions for a number 푚 to be a sestina number. The first sestina point (푗 = 1) must be part of a 푚-cycle which takes it to all the other positions, i.e., the 푚-cycle is not caused by successive 푞-cycles where 푞 is a factor of 푚: Theorem 3. A number 푚 is a sestina number if and only if (2푚 + 1) ∣ (2푚 ± 1) and (2푚 + 1) ∤ (2푞 ± 1) for any 푞 which is a proper factor of 푚. Unfortunately Theorem 3 is not constructive, since in order to check whether an arbitrary 푚 is a sestina number, we have to factor 푚. In particular, it is not clear from the theorem how many sestina numbers there are, even if there are infinitely many or not. The following corollaries establish some more information. Corollary 4. For 푚 to be a sestina number, 2푚 + 1 must be prime. Proof. Suppose that 2푚 + 1 is composite, let 2푚 + 1 = 푟푠. where 푟 and 푠 are both odd and smaller than 2푚 + 1 By the above corollary applied to 푟, there must exist a 푞 ≤ (푟 − 1)/2 such that 푟 divides 2푞 ± 1. Let (2푞 ± 1) = 푟푘. Now we have 푘(2푚+1) = (2푞 ±1)푠 where, by construction, 푘 < 2푞−1 and 푠 < 푚. This is precisely the condition according to Proposition 1 for the existence of a 푞-cycle. But 푞 < 푚 by construction, and hence 푚 cannot be a sestina number.

Corollary 5. Let 2푚 + 1 be a prime number that divides 2푚 ± 1. If 푚 is also a prime, then 푚 is a sestina number. Proof. This follows immediately from Theorem 1, since if 푚 is prime its only factors 푞 are 1 and 푚 itself.

Remarks. 1. Note that Corollary 5 is not a necessary condition for a sestina number. For example, 푚 = 6 and 푚 = 9 are non-prime sestina numbers.

2. The Corollary does not establish how many sestina numbers there are, but at least we have a simple algorithm for finding sestinas of large length. Take a prime 푚 such that 2푚 + 1 is also prime. Test whether (2푚 + 1) is a factor of 2푚 ± 1. If it is, then 푚 is a sestina number. Perhaps this could be the point of departure for a study of the cardinality of sestina numbers. 68 A. R. Champneys, P. G. Hjorth, H. Man

3. When 푚 is itself a prime number, and 2푚 −1 is also prime, then the numbers 2푚 − 1 are the so-called Mersenne primes. More generally Primes of the form 2푚 ± 1 are examples of what are known as Cunningham Primes [1]. Such numbers are named after the British number theorist who in 1925 [3] started what has become known as the Cunningham project of finding factors of numbers of the form 푏푛 ± 1, for 푏 = 2, 3, 5, 6, 7, 10, 11, 12 and large 푛.

6. Discussion

The above description of sestina numbers is in some way less than satisfactory. It relies on the factorisation of large numbers of the form 2푚 ± 1. As is well known such factorisation is a complex computational task. In fact the brute force approach of simply letting the numbers lead a dance, i.e., iterating the tent map 푚 times, provides a far quicker (order 푚) method of deciding whether 푚 is a sestina number (in fact this is the essence of Dumas’ theorem [5, 10]). Using this method it is a straightforward computational task to construct all the sestina numbers less than a certain positive integer. Here, for example, is a list of all sestina numbers up to 푚 = 200:

1, 2, 3, 5, 6, 9, 11, 14, 18, 23, 26, 29, 30, 33, 35, 39, 41, 50, 51, 53, 65, 69, 74, 81, 83, 86, 89, 90, 95, 98, 99, 105, 113, 119, 131, 134, 135, 146, 155, 158, 173, 174, 179, 183 ,186, 189, 191, 194

Also, the characterisation given here does not immediately tell us whether there are infinitely many sestina numbers or not. This question is still open. Finally, we return to the original motivation to this article. How to construct a sestina with 푚 = 78. Here 2푚 + 1 is prime but 푚 isn’t. The permutation (1) splits into three 26-cycles:

(1,2,4,8,16,32,64,29,58,41,75,7,14,28,56,45,67,23,46,65,27,54,49,59,39,78) (3,6,12,24,48,61,35,70,17,34,68,21,42,73,11,22,44,69,19,38,76,5,10,20,40,77) (9,18,36,72,13,26,52,53,51,55,47,63,31,62,33,66,25,50,57,43,71,15,30,60,37,74)

So 푚 = 78 is not a sestina number as defined here; the usual method of making a sestina of this length will not work. Instead, an alternative strategy might be to use the basic sestina permutation (1) 25 times to generate the first 26 verses, then apply something else to perturb the situation so that we do not get locked into a 26 cycle. One example of such a perturbation can be found by noticing that each The numbers lead a dance 69 successive cycle in the above 26-cycles is the image of the previous one under

2푛 if 푛 ≤ [ 푚 ⎧ 3 푚 2푚 푛 ↦ 2푚 + 1 − 3푛 if [ 3 < 푛 ≤ [ 3 (5) ⎨ 2푚 ⎩3푛 − (2푚 + 1) if [ 3 < 푛 ≤ 푚 Hence we can apply this transformation to create the 27th verse, followed by 25 more applications of the basic permutation (1) to create verses 28 to 52, one more application of (5) to create verse 53, finishing off with a final 25 iterations of(1) to complete the sestina. However, this is a mathematical solution. From the point of view of a contemporary poet it might be better for the variation to occur within the poem itself; creatively intentional rather than merely because of the sestina number’s mathematical torsion. Instead, the poet could take the opportunity to to build a relationship between the number 26 and the poem’s content. The obvious example being the number of letters in the alphabet. Harry is still writing his magnum opus.

7. Afterword

This article itself has, in fact, undergone a merry dance. The original chance encounter mentioned in section 2 happened some ten years or so ago. Some of the theory was worked out at the time and presented at the British Applied Mathematics Colloquium, which in the year 2007 was held in Bristol. Harry provided an impromptu performance sestina at the event. Then both authors went back to their day jobs and no article was written. Harry is now fulfilling his then dream ambition to be a published poet and Alan, having completed his stint as Head of Department and other managerial roles, continues to slug it out as a regular engineering mathematics professor. And so it would have remained had it not been for another chance encounter some fifteen years or so before that, coincidentally also in Bristol. Alan, thena finishing PhD at Oxford, got chatting with Poul, then a recently appointed Assistant Prof at the Technical University of Denmark, at a conference on the Dynamics of Numerics and the Numerics of Dynamics. It was quickly established that, in addition to scientific interests in common, each has the same quirky senseof humour and attitude to life. An invitation to Lyngby for the following year ensued, and together they studied the dynamics of chaos amid the beautiful deer park there. A lifelong friendship has become established, but no joint publication has ever result from their collaboration. Until now. A few years ago, chatting over a pint 70 A. R. Champneys, P. G. Hjorth, H. Man of British real ale, Poul expressed the desire to pick up the pathetic half-finished manuscript that had resulted from Alan and Harry’s original collaboration. He soon discovered Saclolo’s article and the preceding work of the French poets and group theorists. We had been scooped. So, once again our desire to publish seems to have been thwarted. Then Poul was invited to speak at the birthday symposium for another longstanding scientific friend, Helge Holden. The story of the sestina mathematics and its links to chaos was once again resurrected, and Poul was even inspired to compose a sestina for Helge (see elsewhere in this Volume). An invitation to write a paper based on his talk has put new impetus into the collaboration; Alan and Harry are now back in touch after their paths were separated. And so the number dance continues.

Acknowledgements. The authors wish to thank Alain Goriely, Philip Holmes and an anonymous referee for enthusiastic comments on earlier versions of this work. Also, special thanks go to Bob Wieman who pointed out to us yet another coincidence; although Cunningham is no known direct ancestor of the first author, his full name was Allan Joseph Champneys Cunningham.

References

[1] J. Brillhart, D.H. Lehmer, J. Selfridge, B. Tuckerman,and S.S. Wagstaff Jr., Factorizations of 푏푛 ± 1, 푏 = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, 3rd ed. Providence, RI: Amer. Math. Soc. (2002). [2] M. Bringer, Sur un probléme de R. Queneau. Mathématiques et Sciences Humaines 27 (1969) 13–20. [3] A.J.C. Cunningham and H.J. Woodall, Factorisation of 푦푛 ± 1, 푦 = 2, 3, 5, 6, 7, 10, 11, 12, up to high powers 푛. London: Hodgson (1925). [4] R. Devaney, An introduction to chaotic dynamical systems. Cambridge, Mass: Addison- Wesley (1989) [5] J.-G. Dumas, Caractérisation des quenines et leur représentation spirale. Mathéma- tiques et Sciences Humaines 184 (2008) 9–23. [6] S. Fry, The Ode Less Travelled: Unlocking the Poet Within London: Arrow (2007) [7] S. MacLane and G. Birkhoff, Algebra 3rd ed. Providence, RI: AMS Chelsea Publishing (1999). [8] R. Queneau, Note complémentaire sur la sextine, Subsidia Pataphysica 1 79–80, (1963). The numbers lead a dance 71

[9] J. Roubaud, Un problème combinatoire posé par poéste lyrque des troubadours. Math- ématiques et Sciences Humaines 27 (1969) 5–12. [10] M.P. Saclolo, How a medieval troubadour became a mathematical figure. Notices of the AMS 58 (2011) 682–687. [11] L. Schimel, Poetic license: some thoughts on sestinas. Writing-World.com published online at www.writing-world.com/poetry/schimel.shtml accessed 20/09/2016 (2001). [12] Sting: Shape of My Heart, on Ten Summoner’s Tales, ©1993 UMG Recordings, Inc. [13] C.B. Whitlow and M. Krysi, Obsession: Sestinas in the Twenty-First Century. Dartmouth College Press (2014).

Compensated compactness in Banach spaces and weak rigidity of isometric immersions of manifolds

Gui-Qiang G. Chen and Siran Li

To Helge Holden on the occasion of his 60th birthday with friendship and affection

Abstract. We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div–curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div–curl lemma can be employed to establish the global weak rigidity of the Gauss–Codazzi–Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.

1. Introduction

In this paper we discuss a unified approach developed recently in[11] towards establishing more general and intrinsic compensated compactness theorems for nonlinear analysis and nonlinear partial differential equations (PDEs), with applications to the weak rigidity of isometric immersions of Riemannian manifolds into Euclidean spaces with lower regularity. Compensated compactness has played an important role in the study of nonlinear PDEs arising from fluid mechanics, calculus of variations, and nonlinear elasticity; cf. [2, 16, 20, 21, 24, 39, 50, 51] and the references cited therein. The div–curl lemma introduced by Murat and Tartar [39, 50] is the cornerstone of the theory, which reads in the simplest and original form as follows: Lemma 1.1 (The div–curl lemma by Murat–Tartar [39, 50]). Let {푣휖}, {푤휖} ⊂ 2 3 3 3 퐿loc(ℝ ;ℝ ) be two sequences of vector fields on ℝ such that 휖 휖 2 푣 ⇀ 푣, 푤 ⇀ 푤 weakly in 퐿loc. 휖 −1 3 휖 Assume that {div 푣 } is pre-compact in 퐻loc (ℝ ; ℝ) and {curl 푤 } is pre-compact in −1 3 3 퐻loc (ℝ ;ℝ ). Then 푣휖 ⋅ 푤휖 → 푣 ⋅ 푤 in the sense of distributions. 74 G.-Q. Chen and S. Li

Two distinctive approaches have been developed in the literature to prove Lemma 1.1: One is via harmonic analysis, and the other is based on the Hodge (de Rham) decomposition theorem; see [16, 23, 33, 40, 45] and the references cited therein. Both approaches depend crucially on the geometry of the Euclidean spaces or Riemannian manifolds; see §2 for a detailed exposition. See also Whitney [54] (Chapter IX, Theorem 17A) for an early version of the div–curl type lemma in the language of his geometric integration theory. One of our crucial observations in [11] is that the div–curl lemma can be reformulated via a functional-analytic approach in generic Banach spaces with two general bounded linear operators in place of div and curl. Indeed, all the necessary properties we need for div and curl in order to conclude the lemma can be extracted as two simple, abstract conditions: One is algebraic, and the other is analytic. Both conditions can be naturally formulated in terms of operator algebras on Banach spaces. This leads to the generalization of the existing versions of the div–curl lemma, and provides the third approach to the theory of compensated compactness. In addition, combining the functional-analytic compensated compactness theorem together with the ellipticity of the Laplace–Beltrami operator, we are ready to obtain a geometrically intrinsic div–curl lemma on Riemannian manifolds. Throughout this paper, the term “intrinsic” means “independent of local coordinates” on Riemannian manifolds. As an application of the new div–curl lemma, we analyze the weak rigidity of isometric immersions of Riemannian manifolds into Euclidean spaces. The problem of isometric immersions/embeddings has been of considerable interest in the development of differential geometry, which has also led to the important developments of new ideas and methods in nonlinear analysis and PDEs (cf. [30, 42, 43, 55]). Moreover, it is well-known in differential geometry that the Gauss–Codazzi–Ricci (GCR) equations are the compatibility conditions for the existence of isometric immersions (see [19, 46]). The GCR equations can be viewed as a first-order nonlinear system of geometric PDEs for the second fundamental forms and normal connections. However, in general, the GCR equations are of no type, neither purely hyperbolic nor purely elliptic. The weak rigidity problem for isometric immersions can be formulated as follows: Given a sequence of isometric immersions of an 푛-dimensional manifold 1,푝 with a 푊loc metric for 푝 > 푛, whose second fundamental forms and normal con- 푝 nections are uniformly bounded in 퐿loc, whether its weak limit is still an isometric 1,푝 immersion with the same 푊loc metric. This rigidity problem has its motivation from both geometric analysis and nonlinear elasticity: The existence of isometric immersions of Riemannian manifolds with lower regularity corresponds naturally to the realization of elastic bodies with lower regularity in the physical space. See Compensated compactness in Banach spaces 75

Ciarlet–Gratie–Mardare [15], Mardare [36], Szopos [49], and the references cited therein. In [11], we have proved that the solvability of the GCR equations in 푊 1,푝 is equivalent to the existence of 푊 2,푝 isometric immersions on Riemannian manifolds. This is done by employing the Cartan formalism, also known as the method of moving frames. We have shown that both the GCR equations and isometric immersions are equivalent to the structural equations of the Cartan formalism. Then, by exploiting the div–curl structure of the GCR equations and the Cartan formalism, we have deduced the global weak rigidity of these geometric PDEs, independent of local coordinates on Riemannian manifolds. Now, in view of the equivalence theorem established above, the weak rigidity of isometric immersions is readily concluded. The rest of this paper is organized as follows: In §2, we first formulate the functional-analytic compensated compactness theorem in Banach spaces and give an outline of its proof, and then deduce a geometrically intrinsic div–curl lemma on Riemannian manifolds as its corollary. Two generalizations of the latter result are also discussed. In §3, we collect some background on differential geometry pertaining to the GCR equations and the Cartan formalism. Finally, in §4, we show the weak rigidity of isometric immersions, together with the weak rigidity of the GCR equations and the Cartan formalism.

2. A Compensated Compactness Theorem in Banach Spaces

In this section we first discuss a functional-analytic compensated compactness theorem. As its consequence, we deduce a geometrically intrinsic div–curl lemma on Riemannian manifolds. To establish the original div–curl lemma, Lemma 1.1, as well as its various generalizations (see [16, 23, 33, 40, 45] and the references cited therein), the following distinctive approaches have been adopted: The first approach, developed by Murat and Tartar in[39, 50], is based on harmonic analysis. It is observed that the first-order differential constraints, namely the pre-compactness of {div 푣휖} and {curl 푤휖} in 퐻−1, lead to the decay properties of {푣휖 ⋅ 푤휖} in the high Fourier frequency region. Coifman–Lions–Meyer–Semmes in [16] extended this lemma by combining the exploitation of this observation with further techniques in harmonic analysis, including Hardy spaces, and commutator estimates of BMO functions and Riesz transforms. The second approach is based on the Hodge decomposition. Robbin–Rogers– 76 G.-Q. Chen and S. Li

Temple in [45] observed that, by writing 푣휖 = ΔΔ−1푣휖 = (grad ∘ div − curl ∘ curl)Δ−1푣휖, (2.1) {푣휖} can be decomposed into a weakly convergent part and a strongly convergent part, and similarly for 푤휖 (also see the exposition in Evans [23]). For this, the advantage can be taken of the first-order differential constraints, the commuta- tivity of the Green operator Δ−1 on ℝ3 with divergence, gradient, and curl, and most crucially, the ellipticity of Δ, so that, for {푣휖 ⋅ 푤휖}, the pairing of the weakly convergent terms pass to the limits via integration by parts, and the pairings of other terms can be dealt with directly. Observe that the Laplace–Beltrami operator Δ defined for differential forms on any oriented closed Riemannian manifold (푀, 푔) is always elliptic, and it has a decomposition similar to (2.1): Δ = d ∘ 훿 + 훿 ∘ d, (2.2) where d is the exterior differential and 훿 is its 퐿2-adjoint (cf. §6 in [53] for the details), so that the div–curl lemma is ready to be generalized to Riemannian manifolds. The third approach, which is the main content of this section, is functional- analytic. As aforementioned, the existing div–curl lemmas are formulated in terms of vector fields or local differential forms on Euclidean spaces(cf. [16, 23, 45, 39, 50]), and some generalizations to Riemannian manifolds are available (cf. [11, 31, 33]). For example, Kozono–Yanagisawa [33] obtained a div–curl lemma using functional-analytic results on 퐿2(ℝ푛), as well as a geometric version, with emphasis on the weak convergence of vector fields up to the boundary of the domain or compact Riemannian manifold, which requires the divergence and curl of the vector fields to be bounded in 퐿2. One of our key observations is that, for the “usual” div–curl lemmas – with the exception of certain end-point cases, e.g., Theorem 2.3, the specific geometry of Euclidean spaces or manifolds plays no essential role. Based on this observation, we have formulated and established a general compensated compactness theorem through bounded linear operators on Banach spaces in [11]. Roughly speaking, it may be stated as follows: If two bounded linear operators 푆 and 푇 between Banach spaces satisfy two conditions: One is algebraic (푆 and 푇 are orthogonal to each other), and the other is analytic (푆 ⊕푇 determines nearly everything), then a result in the spirit of Lemma 1.1 holds, with div and curl replaced by 푆 and 푇, respectively. We now discuss the functional-analytic compensated compactness theorem in Banach spaces, as well as its geometric implications. For some background on functional analysis, we refer to [25]. Let us first explain some notations: In the sequel, ℋ is a Hilbert space over the field 핂 = ℝ or ℂ so that ℋ = ℋ∗, and 푌, 푍 are two Banach spaces over 핂. We use Compensated compactness in Banach spaces 77

ℋ∗ and 푌∗, 푍∗,… to denote the dual Hilbert and Banach spaces, respectively. In what follows, we consider the bounded linear operators:

푆∶ℋ→푌, 푇∶ℋ→푍.

For their adjoint operators, we write

푆† ∶ 푌∗ → ℋ, 푇† ∶ 푍∗ → ℋ.

By ⟨ ⋅ , ⋅ ⟩푌, ⟨ ⋅ , ⋅ ⟩푍, …, we mean the duality pairings on suitable Banach spaces, and notation ⟨ ⋅ , ⋅ ⟩ without subscripts is reserved for the inner product on ℋ. 휖 Furthermore, for any normed vector spaces 푋, 푋1, and 푋2, we write {푠 } ⊂ 푋 for a 휖 sequence {푠 } in 푋 as a subset, and 푋1 ⋐ 푋2 for a compact embedding between the normed vector spaces. We use ‖⋅‖푋 to denote the norm in 푋, write → for the strong convergence of sequences under the norm, and write ⇀ for the weak convergence. 퐵푋 ≔ { 푥 ∈ 푋 ∶ ‖푥‖ ≤ 1 } is the closed unit ball in 푋, and 퐵푋 ≔ { 푥 ∈ 푋 ∶ ‖푥‖ < 1 } is the open unit ball. Moreover, for a linear operator 퐿∶ 푋1 → 푋2, its kernel is written as ker(퐿) ⊂ 푋1, and its range is ran(퐿) ⊂ 푋2. Finally, for 푋1 ⊂ 푋 as a vector ⟂ ∗ subspace, its annihilator is defined as 푋1 ≔ { 푓 ∈ 푋 ∶ 푓(푥) = 0 for all 푥 ∈ 푋1 }. To proceed, we define the following linear operators:

푆 ⊕ 푇∶ ℋ → 푌 ⊕ 푍, (푆 ⊕ 푇)ℎ ≔ (푆ℎ, 푇ℎ); { 푆† ∨ 푇† ∶ (푌 ⊕ 푍)∗ ≅ 푌∗ ⊕ 푍∗ → ℋ, (푆† ∨ 푇†)(푎, 푏) ≔ 푆†푎 + 푇†푏 for ℎ ∈ ℋ, 푎 ∈ 푌∗, and 푏 ∈ 푍∗. The direct sum 푌 ⊕ 푍 is always endowed with the † † † norm: ‖(푦, 푧)‖푌⊕푍 ≔ ‖푦‖푌 + ‖푧‖푍. Also, it is direct to see that (푆 ⊕ 푇) = 푆 ∨ 푇 . Our compensated compactness theorem is formulated in the following: Theorem 2.1 (Theorem 2.1 in [11]). Let ℋ = ℋ∗ be a Hilbert space over 핂, 푌 and 푍 be reflexive Banach spaces over 핂, and 푆∶ ℋ → 푌 and 푇∶ ℋ → 푍 be bounded linear operators satisfying

(Op 1) Orthogonality: 푆 ∘ 푇† = 0, 푇 ∘ 푆† = 0; (2.3)

(Op 2) For some Hilbert space (ℋ; ‖ ⋅ ‖ℋ) such that ℋ embeds compactly into ℋ, ˜ ˜ there exists a constant 퐶 > 0 so that,˜ for all ℎ ∈ ℋ,

‖ℎ‖ℋ ≤ 퐶(‖(푆ℎ, 푇ℎ)‖푌⊕푍 + ‖ℎ‖ℋ) = 퐶(‖푆ℎ‖푌 + ‖푇ℎ‖푍 + ‖ℎ‖ℋ). (2.4) ˜ ˜ Assume that two sequences {푢휖}, {푣휖} ⊂ ℋ satisfy the following conditions: (Seq 1) 푢휖 ⇀ 푢 and 푣휖 ⇀ 푣 in ℋ as 휖 → 0; 78 G.-Q. Chen and S. Li

(Seq 2) {푆푢휖} is pre-compact in 푌, and {푇푣휖} is pre-compact in 푍.

Then ⟨푢휖, 푣휖⟩ → ⟨푢, 푣⟩ as 휖 → 0.

Outline of Proof. We now sketch the main steps of the proof here. The interested readers are referred to [11] for the details. Step 1. Claim: 푆 ⊕ 푇∶ ℋ → 푌 ⊕ 푍 has finite-dimensional kernel and closed range. As 푌 and 푍 are reflexive, ran(푆 ⊕ 푇) is also a reflexive Banach space. This observation guarantees that all the assumptions (Op 1)–(Op 2) and (Seq 1)–(Seq 2) remain valid, provided that 푌 ⊕ 푍 is replaced by ran(푆 ⊕ 푇), i.e., 푆 and 푇 are surjective. Thus, once Step 1 has been established, we can assume that 푆 ⊕ 푇 is Fredholm in the subsequent arguments.

Indeed, to show dim핂 ker(푆 ⊕ 푇) < ∞, by the classical Riesz lemma, it suffices to check that the closed unit ball of ker(푆 ⊕ 푇) is compact in the norm topology of 푌 ⊕ 푍. To this end, let 푗∶ ℋ ↪ ℋ be the compact embedding in (Op 2). Then, ˜ for any ℎ ∈ ℋ such that 푗(ℎ) ∈ 푗[ker(푆 ⊕ 푇)] ∩ 퐵ℋ, the same condition yields ˜

‖ℎ‖ℋ ≤ 퐶(‖푆ℎ‖푌 + ‖푇ℎ‖푍 + ‖푗(ℎ)‖ℋ) ≤ 퐶. (2.5) ˜ Therefore, the unit ball of 푗[ker(푆 ⊕ 푇)] in ℋ is finite-dimensional, and the same conclusion holds for ker(푆 ⊕ 푇) as 푗 is an embedding.˜ To show ran(푆 ⊕ 푇) ⊂ 푌 ⊕ 푍 as a closed subspace, we take any sequence {ℎ휇} ⊂ ℋ such that (푆 ⊕ 푇)ℎ휇 → 푤 in the norm topology of 푌 ⊕ 푍 and argue that 푤 ∈ ran(푆 ⊕ 푇). This follows from the following coercivity estimate: There exists a universal constant 휖0 > 0 such that

‖(푆 ⊕ 푇)ℎ‖푌⊕푍 ≥ 휖0‖푗(ℎ)‖ℋ for all ℎ ∈ ℋ. (2.6) ˜ The estimate in (2.6) is obtained via a contradiction argument, by taking into account of the finite-dimensionality of ker(푆 ⊕ 푇) and the 1-homogeneity of (2.6). Then we decompose ℎ휇 = 푘휇 + 푟휇 for 푘휇 ∈ ker(푆 ⊕ 푇) and 푟휇 ∈ [ker(푆 ⊕ 푇)]⟂. In view of the inequality:

휇1 휇2 휇1 휇2 ‖(푆 ⊕ 푇)(ℎ − ℎ )‖푌⊕푍 ≥ 휖0‖푗(푟 − 푟 )‖ℋ, ˜ we find that {푗(푟휇)} is a Cauchy sequence in ℋ, which converges to some 푗(푟). Then it is direct to check that (푆 ⊕ 푇)푟 = 푤, which˜ leads to the claim in Step 1. Notice in passing that we have obtained the following decomposition of ℋ along 푆 ⊕ 푇: ℋ = ker(푆 ⊕ 푇) ⊕ ran(푆† ∨ 푇†), (2.7) Compensated compactness in Banach spaces 79 where ⊕ is the topological direct sum of the Banach spaces, with the summands being orthogonal with respect to the inner product on ℋ. Moreover, note that only the analytic assumption (Op 2) on 푆 and 푇 has been used in Step 1. Step 2. From now on, 푆⊕푇 is assumed to be surjective and with finite-dimensional kernel. In this step, we decompose each of the two sequences {푣휖} and {푤휖} into three parts: an 푆-free part, a 푇-free part, and a remainder in the finite-dimensional space ker(푆 ⊕ 푇). This is done via the generalized Laplacian. Indeed, we define operator Δ∶/ 푌∗ ⊕ 푍∗ → 푌 ⊕ 푍 as follows:

Δ/ ≔ (푆 ⊕ 푇) ∘ (푆† ∨ 푇†) = 푆푆† ⊕ 푇푇†. (2.8)

Then, thanks to Eq. (2.7) and ker(푆† ∨ 푇†) = [ran(푆 ⊕ 푇)]⟂, we find that Δ/ also has finite-dimensional kernel and closed range and, as in Step 1, Δ/ can be assumed † † to be surjective. Denote by 휋1 ∶ ℋ = ker(푆 ⊕ 푇) ⊕ ran(푆 ∨ 푇 ) → ker(푆 ⊕ 푇) the canonical projection onto the first coordinate, which is a finite-rank (hence compact) operator. Then our decomposition of {푣휖} and {푤휖} are given as follows:

휖 휖 † 휖 † 휖 휖 휖 † 휖 † 휖̃ 푢 = 휋1(푢 ) + 푆 푎 + 푇 푏 , 푣 = 휋1(푣 ) + 푆 ̃푎 + 푇 푏 , { (2.9) † † † † ̃ 푢 = 휋1(푢) + 푆 푎 + 푇 푏, 푣 = 휋1(푣) + 푆 ̃푎+ 푇 푏, for some 푎, ̃푎, 푎휖, ̃푎휖 ∈ 푌∗ and 푏, 푏,̃ 푏휖, 푏휖̃ ∈ 푍∗. Applying the algebraic condition (Op 1) of operators 푆 and 푇, the inner products become:

휖 휖 휖 휖 † 휖 † 휖 † 휖 † 휖̃ ⟨푢 , 푣 ⟩ = ⟨휋1(푢 ), 휋1(푣 )⟩ + ⟨푆 푎 , 푆 ̃푎 ⟩ + ⟨푇 푏 , 푇 푏 ⟩, { (2.10) † † † † ̃ ⟨푢, 푣⟩ = ⟨휋1(푢), 휋1(푣)⟩ + ⟨푆 푎, 푆 ̃푎⟩ + ⟨푇 푏, 푇 푏⟩.

휖 휖 Owing to the compactness of 휋1, ⟨휋1(푢 ), 휋1(푣 )⟩ → ⟨휋1(푢), 휋1(푣)⟩ as 휖 → 0. Therefore, to conclude the theorem, it remains to establish

⟨푆†푎휖, 푆† ̃푎휖⟩ + ⟨푇†푏휖, 푇†푏휖̃ ⟩ → ⟨푆†푎, 푆† ̃푎⟩ + ⟨푇†푏, 푇†푏⟩̃ as 휖 → 0, (2.11) which is the content of the next step. Step 3. To prove the convergence in (2.11), we start with the following two observations:

(i) The left-hand side of (2.11) can be expressed in terms of the generalized Laplacian:

⟨푆†푎휖, 푆† ̃푎휖⟩ + ⟨푇†푏휖, 푇†푏휖̃ ⟩ = ⟨푆푆†푎휖, ̃푎휖⟩ + ⟨푏휖, 푇푇†푏휖̃ ⟩ = ⟨Δ(푎/ 휖, 푏휖̃ ), ( ̃푎휖, 푏휖)⟩ ; (2.12) 푌 푍 푌⊕푍 80 G.-Q. Chen and S. Li

(ii) Multiplying 푆 to 푢휖 and 푇 to 푣휖 in (2.9) and invoking (Op 1), we have

푆푢휖 = 푆푆†푎휖, 푇푣휖 = 푇푇†푏휖̃ , (2.13)

so that Δ(푎/ 휖, 푏휖̃ ) = (푆푢휖, 푇푣휖). (2.14)

Now, as {푆푢휖} ⊂ 푌 and {푇푣휖} ⊂ 푍 are pre-compact by assumption (Seq 2), it suffices to show the boundedness of {( ̃푎휖, 푏휖)} in the norm topology of 푌∗ ⊕ 푍∗ to reach the conclusion. Furthermore, in view of the specific form of the expression involved in (2.11), it is enough to exhibit one particular representative ( ̃푎휖, 푏휖) in the −1 휖 휖 휖 휖 co-set Δ/ {(푆푣 , 푇푢 )} such that ‖( ̃푎 , 푏 )‖푌∗⊕푍∗ ≤ 퐶, where 퐶 > 0 is independent of 휖. As {(푆푣휖, 푇푢휖)} is uniformly bounded in the norm topology of 푌 ⊕ 푍, owing to the weak convergence of {푣휖} and {푤휖} assumed in (Seq 1), the desired result follows from a standard result in functional analysis, which is Claim ♣ in the proof of Theorem 3.1 in [11]. This completes the proof.

With the benefit of hindsight, let us now explain the motivation for Theorem 2.1 and its relations with the earlier versions of the div–curl lemmas. Consider a 3-dimensional oriented closed manifold 푀 (differentiable, or of weaker Sobolev regularity, not necessarily Riemannian). We denote by Ω푞(푀) the space of differential 푞-forms on 푀, by ∗∶ Ω푞(푀) → Ωdim(푀)−푞(푀) the Hodge-star, by d∶ Ω푞(푀) → Ω푞+1(푀) the exterior differential, and by ♯ the tonic operator, i.e., the canonical isomorphism between the co-tangent bundle 푇∗푀 and the tangent bundle 푇푀 by raising indices in the coefficients. It is well-known that div, grad, and curl can be defined intrinsically via the commutative diagram:

d d d Ω0(푀) ⟶ Ω1(푀) ⟶ Ω2(푀) ⟶ Ω3(푀) ↑ ↑ ↑ ↑ ↓↑ Id ↓↑♯ ↓↑♯∘∗ ↓↑∗ grad curl div 퐶∞(푀) ⟶ Γ(푇푀) ⟶ Γ(푇푀) ⟶ 퐶∞(푀) In particular, the Riemannian metric on 푀 plays no role at all. The “orthogonality” of div and curl in the sense of (Op 1) in Theorem 2.1, which follows from the cohomological chain condition d ∘ d = 0, 훿 ∘ 훿 = 0, is a purely algebraic relation. Therefore, it is not surprising that a compensated compactness theorem with greater generality and abstractness is available. Moreover, the Hodge decomposition approach to the div–curl lemma initiated by Robbin–Rogers–Temple in [45] makes use of the Laplacian Δ on flat ℝ3. If we take 푆 = div, 푇 = curl, ℋ = 퐿2(ℝ3;ℝ3), 푌 = 퐻−1(ℝ3; ℝ), and ℋ = 푍 = 퐻−1(ℝ3;ℝ3) with suitable localizations if necessary, the classical div–curl˜ lemma (Lemma 1.1) is immediately recovered. Our generalized Laplacian Δ/ extends the Compensated compactness in Banach spaces 81 flat Laplacian and, more generally, the Laplace–Beltrami operator Δ on manifolds, in view of Eqs. (2.1)–(2.2). The Fredholmness of Δ follows from the Hodge decomposition theorem, cf. §6 in [53]. Before our subsequent development, we remark that the assumption of the re- flexivity of 푌 and 푍 is crucial, since several counterexamples have been constructed for non-reflexive 푌 and 푍 (see [18] and Remark 3.2 in [11]). Now we discuss a geometric consequence of Theorem 2.1. Using the expression for Δ/ = Δ on Riemannian manifolds in terms of 푆 = d and 푇 = 훿 as in Eq. (2.2), we have

Theorem 2.2 (Geometrically intrinsic div–curl lemma A, Theorem 3.3 in [11]). Let 휖 휖 2 푞 ∗ (푀, 푔) be an 푛-dimensional Riemannian manifold. Let {휔 }, {휏 } ⊂ 퐿loc(푀; ⋀ 푇 푀) be two families of differential 푞-forms such that

휖 휖 2 푞 ∗ (i) 휔 ⇀ 휔 and 휏 ⇀ 휏 weakly in 퐿loc(푀; ⋀ 푇 푀);

(ii) There are compact subsets of the corresponding Sobolev spaces, 퐾d and 퐾훿, such that 푞+1 ⎧{d휔휖} ⊂ 퐾 ⋐ 퐻−1 푀; 푇∗푀 , ⎪ d loc ( ⋀ ) ⎨ 푞−1 ⎪ {훿휏휖} ⊂ 퐾 ⋐ 퐻−1 푀; 푇∗푀 . ⎩ 훿 loc ( ⋀ ) Then ⟨휔휖, 휏휖⟩ converges to ⟨휔, 휏⟩ in 풟′(푀), that is,

휖 휖 ∞ ∫ ⟨휔 , 휏 ⟩휓 d푉푔 ⟶ ∫ ⟨휔, 휏⟩휓 d푉푔 for any 휓 ∈ 퐶푐 (푀). 푀 푀 Since the conclusion for the weak continuity in Theorem 2.2 is in the distributional sense, we may assume 푀 to be oriented and closed without loss of generality in the proof. Here and in the sequel, 푊 푘,푝(푀; ⋀푞 푇∗푀) denotes the Sobolev space of differential 푞-forms with 푊 푘,푝–regularity. Then Δ∶ Ω푞(푀) → Ω푞(푀), as well as Δ∶ 푊 푘,푝(푀; ⋀푞 푇∗푀) → 푊 푘−2,푝(푀; ⋀푞 푇∗푀), for 0 ≤ 푞 ≤ 푛 = dim(푀), is elliptic, which is crucial for the verification of condition (Op 2) in Theorem 2.1. As is well-known, the analogous operator on semi-Riemannian manifolds is not elliptic in general, so that Theorem 2.2 may not be extended directly to the semi-Riemannian settings. Next, we state an endpoint case of the above theorem, for which the first- order differential constraints are prescribed in non-reflexive Banach spaces 푊 −1,1, in contrast to condition (Seq 2) in Theorem 2.1. The underlying argument for the proof essentially follows from that in Conti–Dolzmann–Müller [18], which employs a Lipschitz truncation argument and the pre-compactness theorems for 퐿1 82 G.-Q. Chen and S. Li

(e.g., Chacon’s biting lemma, the Dunford–Pettis theorem, etc.) to reduce to the reflexive case.

Theorem 2.3 (Geometrically intrinsic div–curl lemma B, Theorem 6.1 in [11]). 휖 2 푞 ∗ 휖 Let (푀, 푔) be an 푛-dimensional manifold. Let {휔 } ⊂ 퐿loc(푀; ⋀ 푇 푀) and {휏 } ⊂ 2 푞 ∗ 퐿loc(푀; ⋀ 푇 푀) be two families of differential 푞-forms. Suppose that 휖 휖 2 푞 ∗ (i) 휔 ⇀ 휔 and 휏 ⇀ 휏 weakly in 퐿loc(푀; ⋀ 푇 푀) as 휖 → 0;

(ii) There are compact subsets of the corresponding Sobolev spaces, 퐾d and 퐾훿, such that 휖 −1,1 푞+1 ∗ {d휔 } ⊂ 퐾d ⋐ 푊loc (푀; ⋀ 푇 푀), { 휖 −1,1 푞−1 ∗ {훿휏 } ⊂ 퐾훿 ⋐ 푊loc (푀; ⋀ 푇 푀);

(iii) {⟨휔휖, 휏휖⟩} is equi-integrable.

Then ⟨휔휖, 휏휖⟩ converges to ⟨휔, 휏⟩ in 풟′(푀), that is,

휖 휖 ∞ ∫ ⟨휔 , 휏 ⟩휓 d푉푔 ⟶ ∫ ⟨휔, 휏⟩휓 d푉푔 for any 휓 ∈ 퐶푐 (푀). 푀 푀 To conclude this section, we remark that the preceding intrinsic div–curl lemmas (Theorems 2.2–2.3) can be extended to the case of general Hölder exponents, 휖 푟 푞 ∗ 휖 푠 푞 ∗ 1 1 namely that {휔 } ⊂ 퐿loc(푀; ⋀ 푇 푀) and {휏 } ⊂ 퐿loc(푀; ⋀ 푇 푀) with 푟 + 푠 = 1. Since 퐿푟 is not a Hilbert space unless 푟 = 2, such generalizations cannot be directly deduced from Theorem 2.1. Nevertheless, they follow from similar arguments, with slight modifications in light of Eq. (2.2). We refer to Theorems 3.7, Theorem 6.2, and Appendix in [11] for the details; also see §5 in [33].

3. Isometric Immersions of Riemannian Manifolds and the Gauss–Codazzi–Ricci (GCR) Equations

In this section, we briefly discuss the geometric preliminaries for the isometric immersions of Riemannian manifolds. We restrain ourselves to the constructions directly related to our subsequent development. We refer to the classical texts [19, 22, 46] for more detailed treatments on differential geometry, to Han–Hong [30], as well as the classical papers by Nash [42, 43], for isometric immersions. From now on, let 푀 be an 푛-dimensional Riemannian manifold, and let 푔 be a Riemannian metric on 푀. Motivated by the applications in nonlinear elasticity (cf. 1,푝 2,푝 [2, 13, 15, 36]), we consider the metrics of weaker regularity: 푔 ∈ 푊loc .A 푊loc 푛+푘 map 푓∶ (푀, 푔) → (ℝ , 푔0) for the Euclidean metric 푔0 is an immersion if the Compensated compactness in Banach spaces 83 differential d푓푃 is injective for each 푃 ∈ 푀, and it is an embedding if 푓 itself is also injective. Moreover, 푓 is isometric if

d푓 ⊗ d푓 = 푔, (3.1) that is, for any 푃 ∈ 푀 and vector fields 푋, 푌 on 푀,

d푓푃(푋) ⋅ d푓푃(푌) = 푔푃(푋, 푌), (3.2) where 푔푃 denotes the metric evaluated at 푃, and ⋅ is the Euclidean dot product on ℝ푛+푘. Notice that Eq. (3.2) makes sense in the distributional sense when 푝∗ = 푛푝/(푛 − 푝) ≥ 2, and that 푔 has a continuous representative when 푝 > 푛, in view of the Sobolev embeddings 푊 2,푝(ℝ푛) ↪ 푊 1,푞(ℝ푛) for 1 ≤ 푞 ≤ 푝∗ and 푊 1,푝(ℝ푛) ↪ 퐶0(ℝ푛) for 푝 > 푛. In differential geometry, the description of an isometric immersion is equivalent to the determination of how ℝ푛+푘 – viewed as its own tangent spaces – can be split into the immersion-independent and immersion-dependent geometry of 푀. More precisely, for each point 푃 ∈ 푀, we have the vector space direct sum

푛+푘 ⟂ ℝ = 푇푃푀 ⊕ 푇푃푀 , (3.3)

⟂ 푛+푘 where 푇푃푀 is the tangent space of 푀 at 푃, and 푇푃푀 is its complement in ℝ , interpreted as the normal space. To study the isometric immersions, two approaches have been employed from the PDE point of view. One is to deal directly with Eq. (3.2), which is a first-order, nonlinear, generally under-determined PDE; the other is to derive a PDE system by taking two more derivatives and solves for the compatibility conditions. The former approach has been employed by Nash [42, 43] to establish the existence of 퐶1 and 퐶푘 isometric embeddings for large enough co-dimensions; also see Günther [29] for a simplification. For the latter approach, the compatibility conditions read schematically as follows:

0 = Curvature of ℝ푛+푘

⎧Curvature in (tangential, tangential) direction = 0, ⟺ Curvature in (tangential, normal) direction = 0, (♠) ⎨ ⎩Curvature in (normal, normal) direction = 0. To continue, we use Latin letters 푋, 푌, 푍, 푊, … to denote the tangential vector fields in Γ(푇푀), write Greek letters 휉, 휂, 휁, … for the normal vector fields in Γ(푇푀⟂), and identify vector fields with first-order differential operators. Then 푋푌 and 푋휉 are vector fields in Γ(푇푀) and Γ(푇푀⟂), respectively. Let us consider the well- known geometric quantities: 84 G.-Q. Chen and S. Li

• Immersion-independent quantities. Taking one derivative in 푔 leads to the Levi–Civita connection (or covariant derivative) ∇∶ Γ(푇푀) × Γ(푇푀) → Γ(푇푀), and taking one further derivative gives us the Riemann curvature tensor 푅∶ Γ(푇푀) × Γ(푇푀) × Γ(푇푀) × Γ(푇푀) → ℝ. • Immersion-dependent quantities. For given ∇, 푅 as above, consider the 푛+푘 isometric immersion 푓∶ (푀, 푔) ↪ (ℝ , 푔0). We define the second fundamental form 퐵∶ Γ(푇푀) × Γ(푇푀) → Γ(푇푀⟂) by

퐵(푋, 푌) ≔ 푋푌 − ∇푋푌, (3.4) and the normal connection ∇⟂ ∶ Γ(푇푀) × Γ(푇푀⟂) → Γ(푇푀⟂) by

⟂ ⟂ ∇푋휉 ≔ projection of 푋휉 (at each point) onto 푇푀 . (3.5)

Then the right-hand sides of the schematic equations in (♠) can be expressed via the quantities (푔, ∇, 푅, 퐵, ∇⟂), resulting in the Gauss, Codazzi, and Ricci equations in (3.6), (3.7), and (3.8), respectively, below; see also Theorems 2.1–2.2 in [11] and §6 in [19].

2,푝 푛+푘 Proposition 3.1. Suppose that 푓 ∈ 푊loc (푀, 푔; ℝ , 푔0) for 푝 > 푛 is an isometric immersion. Then the following compatibility conditions are satisfied in 풟′:

⟨퐵(푋, 푊), 퐵(푌, 푍)⟩ − ⟨퐵(푌, 푊), 퐵(푋, 푍)⟩ = 푅(푋, 푌, 푍, 푊), (3.6) 푋퐵(푌, 푍, 휂) − 푌퐵(푋, 푍, 휂) ⟂ = ([푋, 푌], 푍, 휂) − 퐵(푋, ∇푌푍, 휂) − 퐵(푋, 푍, ∇푌휂) ⟂ + 퐵(푌, ∇푋푍, 휂) + 퐵(푌, 푍, ∇푋휂), (3.7) ⟂ ⟂ 푋⟨∇푌휉, 휂⟩ − 푌⟨∇푋휉, 휂⟩ ⟂ ⟂ ⟂ ⟂ ⟂ = ⟨∇[푋,푌]휉, 휂⟩ − ⟨∇푋휉, ∇푌휂⟩ + ⟨∇푌휉, ∇푋휂⟩ ⟂ ⟂ + 퐵(푋휉 − ∇푋휉, 푌, 휂) − 퐵(푋휂 − ∇푋휂, 푌, 휉), (3.8) where 푋, 푌, 푍, 푊 ∈ Γ(푇푀), 휂, 휉 ∈ Γ(푇푀⟂), and [푋, 푌] = 푋푌 − 푌푋 is the Lie bracket. Here and in the sequel, we have used ⟨ ⋅ , ⋅ ⟩ to denote all the inner products induced by metrics, and 퐵(푌, 푍, 휂) ≔ ⟨퐵(푌, 푍), 휂⟩. From the PDE perspectives, we view the immersion-dependent quantities (퐵, ∇⟂) as to be solved, and the immersion-independent quantities (푔, ∇, 푅) as being fixed. Indeed, in the isometric immersion problem, metric 푔 is prescribed, so are all the immersion-independent quantities; thus, the immersion-dependent geometry determines the whole of the isometric immersion. Therefore, in the sequel, the GCR equations are always considered as a first-order nonlinear PDE system for (퐵, ∇⟂). Compensated compactness in Banach spaces 85

Proposition 3.1 says that the GCR equations form a necessary condition for the existence of isometric immersions. The converse is known as the “realization problem” in elasticity: Given (퐵, ∇⟂) satisfying the GCR equations, construct an isometric immersion (i.e., design an elastic body) whose immersion-dependent ⟂ ∞ 1,푝 geometry is prescribed by (퐵, ∇ ). This problem for both 퐶 and 푊loc metrics has been answered in the affirmative, globally on simply-connected manifolds; see Tenenblat [52] for the former, and Mardare [36, 37] and Szopos [49] for the latter. In [11], we adapt the geometric arguments in [52] to re-prove the realization 1,푝 theorem in 푊loc regularity, which simplifies the proofs in [36, 37, 49]. Moreover, this method sheds light on the weak rigidity problem of isometric immersions, which is the main content of §4. Finally, we briefly sketch the main tool in[52]– the Cartan formalism – which serves as a bridge between the geometric problem of isometric immersions and the PDEs (GCR equations). In full generality, consider a vector bundle 퐸 over 푀 of 푘 푘 fibre ℝ , trivialized on a local chart 푈 ⊂ 푀, i.e., 퐸|푈 ≅ 푈×ℝ as a diffeomorphism. 푖 1 Let {휕푖} ⊂ Γ(푇푈) be an orthonormal frame, and let {휔 } ⊂ Ω (푈) be its dual (co- frame). Then we choose {휂푛+1, … , 휂푛+푘} ⊂ Γ(퐸) as an orthonormal basis for fibre ℝ푘, and set 푖 휔푗 (휕푘) ≔ ⟨∇휕푘휕푗, 휕푖⟩, (3.9) 푖 훼 휔훼(휕푗) = −휔푖 (휕푗) ≔ ⟨퐵(휕푖, 휕푗), 휂훼⟩, (3.10) 휔훼(휕 ) ≔ ⟨∇퐸 휂 , 휂 ⟩, (3.11) 훽 푗 휕푗 훼 훽 where ∇퐸 is the bundle connection: ∇퐸 = ∇⟂ for 퐸 = 푇푀⟂ = the normal bundle. 1,푝 All these constructions make sense in distributions for 푔 ∈ 푊loc . Moreover, here and in the sequel, the following index convention is adopted: 1≤푖,푗≤푛; 1≤푎,푏,푐,푑,푒≤푛+푘; 푛+1≤훼,훽,훾≤푛+푘. In this setting, the GCR equations on bundle 퐸 are equivalent to the following two systems, known as the first and second structural equations of the Cartan formalism (cf. [52, 46, 47]):

푖 푗 푖 d휔 = ∑ 휔 ∧ 휔푗 , (3.12) 푗 푎 푐 푎 d휔푏 = − ∑ 휔푏 ∧ 휔푐 (3.13) 푐 for each 푖, 푎, 푏. These equations can be represented compactly as first-order nonlinear Lie algebra-valued PDEs. Denoting by 픰픬(푛 + 푘) the Lie algebra of antisymmetric (푛 + 푘) × (푛 + 푘) matrices, we can write (3.12)–(3.13) as d푤 = 푤 ∧ 푊, d푊 + 푊 ∧ 푊 = 0, (3.14) 86 G.-Q. Chen and S. Li

푎 1 where 푊 = {푤푏 } ∈ Ω (푈; 픰픬(푛 + 푘)) which is known as the connection one-forms, 푤 = (휔1, … , 휔푛, 0, … , 0)⊤ ∈ Ω1(푈; ℝ푛+푘), and ∧ operates by the wedge product on the factor of differential forms and matrix multiplication on the factor ofthe matrix Lie algebra, with respect to the factorization:

1 Ω1 푈; 픰픬(푛 + 푘) ≅ Γ 푇∗푈 ⊗ 픰픬(푛 + 푘) . (3.15) ( ) (⋀ ) In other words, the structural equations recast in (3.14) are also intrinsic, i.e., independent of the choice of local moving frames/coordinates {휕푖} and {휂훼}.

4. Weak Rigidity of Isometric Immersions

Finally, we discuss the weak rigidity of isometric immersions with weaker regular- 2,푝 ity in 푊loc as in the previous sections. The rigidity problem of isometric immersions concerns the following: If {푓휖} is 푛+푘 a sequence of isometric immersions of a manifold 푀 into (ℝ , 푔0), which converges to a map 푓∶ 푀 → ℝ푛+푘 in a certain topology, is 푓 still an isometric immersion? This problem has a history of celebrated results. Nash in [42] showed that the 퐶1 isometric immersions are not rigid. In particular, any 퐶∞ short (i.e., distance- shrinking) immersion is 퐶0-close to a 퐶1 isometric immersion; see also [5] for a recent computer visualisation. In the same sense, Borisov in [4] proved that 퐶1,훼 isometric immersions are not rigid for 훼 > 0 below a certain value, and this value has been improved in [17]. On the other hand, the 퐶1,훼 isometric immersions for large enough 훼 are classically known to be rigid; cf. [43] and the references therein. More recently, deep connections have been established between the transition phenomenon from the non-rigidity to rigidity of the 퐶1,훼 isometric immersions and Onsager’s conjecture (concerning the dissipative weak solutions to the Euler equations in fluid dynamics). We refer the readers to[7] and the references cited therein for such developments. Our focus is on the weak rigidity problem motivated by applications. In this case, for the sequence of isometric immersions {푓휖} that is weakly and locally 2,푝 2,푝 convergent in 푊 for 푝 > 푛 = dim(푀), we ask if the weak limit 푓 is still a 푊loc isometric immersion. Indeed, we answer the question in the affirmative, thanks to the locally uniform 퐿푝 bounds on the immersion-dependent geometry. This is in the spirit of the works by Langer [34] and the recent generalization by Breuning [3]. Our result can be formulated as follows:

Theorem 4.1 (Corollary 5.2 in [11]). Let 푀 be an 푛-dimensional simply-connected 1,푝 휖 Riemannian manifold with 푊loc metric 푔 for 푝 > 푛. Suppose that {푓 } is a family Compensated compactness in Banach spaces 87 of isometric immersions of 푀 into ℝ푛+푘 with Euclidean metric, uniformly bounded 2,푝 푛+푘 in 푊loc (푀; ℝ ), whose second fundamental forms and normal connections are {퐵휖} and {∇⟂,휖}, respectively. Then, after passing to subsequences, {푓휖} converges 2,푝 푛+푘 to 푓 weakly in 푊loc which is still an isometric immersion 푓∶ (푀, 푔) → ℝ . Moreover, the corresponding second fundamental form 퐵 is a weak limit of {퐵휖}, and the corresponding normal connection ∇⟂ is a weak limit of {∇⟂,휖}, both taken in the 푝 weak topology in 퐿loc.

Outline of Proof. We sketch the proof in three steps. For the details, we refer to §5 (Step 1), §4.2–§4.3 (Step 2), and §4.4 (Step 3) in [11].

2,푝 Step 1. We show the equivalence between the existence of 푊loc isometric immer- 1,푝 sions and the existence of 푊loc solutions of the GCR equations in distributions (Proposition 3.1). Then the weak rigidity of isometric immersions is translated to the weak rigidity of the GCR equations. Indeed, at the end of §3, it is remarked that the GCR equations are equivalent to the structural equations (3.14) of the Cartan formalism. Hence, by Proposition 3.1, Eq. (3.14) is a necessary condition for the existence of isometric immersions. Conversely, we follow the arguments in [52] to transform Eq. (3.14) into first- order nonlinear PDEs on Lie groups. More precisely, the isometric immersion 푓 satisfies the following equations (formulated as initial value problems) for 퐴 ∈ 1,푝 푊loc (푈; 푂(푛 + 푘)), where 푂(푛 + 푘) is the group of (푛 + 푘) × (푛 + 푘) symmetric matrices:

푊 = d퐴 ⋅ 퐴⊤, d푓 = 푤 ⋅ 퐴. (4.1)

The above two equations are known as the Pfaff and the Poincaré systems. In the smooth case, they can be solved by the Frobenius theorem, by checking that the solution distribution is involutive. For the weak regularity case, we apply the theorems due to Mardare [36, 37] for the existence of solutions to Eq. (4.1). Then 1,푝 0 d푓 ∈ 푊loc ↪ 퐶loc for 푛 > 푝, and it is non-degenerate and distance-preserving, thanks to the Poincaré system and the definition of 푊. This implies that 푓 is indeed an isometric immersion.

Step 2. The GCR equations in Proposition 3.1 are reformulated to manifest the div–curl structures, which admits the application of the intrinsic div–curl lemma, Theorem 2.2. For this purpose, let us fix the tangential vector field 푍 and normal vector fields (퐵) (∇⟂) (휉, 휂), and define the 2-tensor fields 푉푍,휂 , 푉휉,휂 ∶ Γ(푇푀) × Γ(푇푀) → Γ(푇푀) and 88 G.-Q. Chen and S. Li

(퐵) (∇⟂) 1-forms Ω푍,휂,Ω휉,휂 as follows:

(퐵) 푉푍,휂 (푋, 푌) ≔ 퐵(푋, 푍, 휂)푌 − 퐵(푌, 푍, 휂)푋, ⟂ (∇ ) ⟂ ⟂ 푉휉,휂 (푋, 푌) ≔ ⟨∇푌휉, 휂⟩푋 − ⟨∇푋휉, 휂⟩푌, (퐵) Ω푍,휂 ≔ −퐵(•, 푍, 휂), ⟂ (∇ ) ⟂ Ω휉,휂 ≔ ⟨∇• 휉, 휂⟩. For simplicity, we often drop the indices in both Ω and 푉 from now on. To wit, these Ω’s are nothing but the contractions of (퐵, ∇⟂), and the 푉’s are obtained by applying Ω to the 2-Grassmannian (i.e., the space of 2-planes) in 푇푀 and polarized in the anti-symmetric fashion. Recall that the divergence can be defined intrinsically on manifolds by div 푋 ≔ ∗(ℒ푋 d푉푔), where ℒ denotes the Lie derivative, and the following well-known identities hold on manifolds:

ℒ푋 = d ∘ 휄푋 + 휄푋 ∘ d for 푋 ∈ Γ(푇푀), { d훼(푋, 푌) = 푋훼(푌) − 푌훼(푋) − 훼([푋, 푌]) for 훼 ∈ Ω1(푀), 푋, 푌 ∈ Γ(푇푀).

Thus, the divergence of 푉’s and the generalized curl (i.e., d) of Ω’s can be expressed as

(퐵) div (푉푍,휂 (푋, 푌)) = 푌퐵(푋, 푍, 휂) − 푋퐵(푌, 푍, 휂) + 퐵(푋, 푍, 휂) div 푌 − 퐵(푌, 푍, 휂) div 푋, (4.2)

⟂ (∇ ) ⟂ ⟂ div (푉휉,휂 (푋, 푌)) = −푌⟨∇푋휉, 휂⟩ + 푋⟨∇푌휉, 휂⟩ ⟂ ⟂ + ⟨∇푌휉, 휂⟩ div 푋 − ⟨∇푋휉, 휂⟩ div 푌, (4.3) (퐵) d(Ω푍,휂)(푋, 푌) = 푌퐵(푋, 푍, 휂) − 푋퐵(푌, 푍, 휂) + 퐵([푋, 푌], 푍, 휂), (4.4) ⟂ (∇ ) ⟂ ⟂ ⟂ d(Ω휉,휂 )(푋, 푌) = −푌⟨∇푋휉, 휂⟩ + 푋⟨∇푌휉, 휂⟩ − ⟨∇[푋,푌]휉, 휂⟩, (4.5)

⟂ ⟂ where the terms 퐵(푋, 푍, 휂) div 푌, 퐵(푌, 푍, 휂) div 푋, ⟨∇푌휉, 휂⟩ div 푋, ⟨∇푋휉, 휂⟩ div 푌, ⟂ ⟂ 퐵([푋, 푌], 푍, 휂), and ⟨∇[푋,푌]휉, 휂⟩ are linear in (퐵, ∇ ), while the other terms on the right-hand sides of the above four equations involve first-order derivatives of (퐵, ∇⟂). Moreover, for further development, it is crucial to observe that

(퐵) (퐵) div (푉푍,휂 (푋, 푌)) = d(Ω푍,휂)(푋, 푌) + [linear terms], (4.6) (∇⟂) (∇⟂) div (푉휉,휂 (푋, 푌)) = d(Ω휉,휂 )(푋, 푌) + [linear terms]. Next, using the tensor fields 푉 and Ω introduced above, we can reformulate the GCR system as the following equations with emphasis on the pairings of 푉’s Compensated compactness in Banach spaces 89 and Ω’s:

(퐵) (퐵) ∑⟨푉푍,휂 (푋, 푌), Ω푊,휂⟩ = 푅(푋, 푌, 푍, 푊), (4.7) 휂 (퐵) (∇⟂) (퐵) d(Ω푍,휂)(푋, 푌) + ∑⟨푉휂,훽 (푋, 푌), Ω푍,훽⟩ 훽

+ 퐵(푌, ∇푋푍, 휂) − 퐵(푋, ∇푌푍, 휂) = 0, (4.8)

(∇⟂) (∇⟂) (∇⟂) (퐵) (퐵) d(Ω휉,휂 )(푋, 푌) + ∑⟨푉휂,훽 (푋, 푌), Ω휉,훽 ⟩ = ∑⟨푉푍,휉 (푋, 푌), Ω푍,휂⟩, (4.9) 훽 푍 where all the summations are at most countable and locally finite. Therefore, we have transformed the GCR equations in Proposition 3.1 into Eqs. (4.7)–(4.9), expressed in terms of the tensor fields 푉 and Ω. Furthermore, the divergence of 푉 roughly equals to the generalized curl of the corresponding Ω, which involves the derivatives of solutions (퐵, ∇⟂) up to the first order. Step 3. Now we are at the stage of applying the geometrically intrinsic div– curl lemma (Theorem 2.2) to conclude the weak rigidity of isometric immersions. Let {퐵휖, ∇⟂,휖} be the second fundamental forms and normal connections associated to the sequence of isometric immersions {푓휖}. As {푓휖} is uniformly 2,푝 (퐵휖) (퐵휖) (∇⟂,휖) (∇⟂,휖) bounded in 푊loc , the tensor fields {푉 ,Ω , 푉 ,Ω } are uniformly 푝 bounded in 퐿loc, so that they are pre-compact in the weak topology. In view of Eqs. (4.8)–(4.9) and (4.6), the Cauchy–Schwarz inequality immediately yields (퐵휖) (퐵휖) (∇⟂,휖) (∇⟂,휖) 푝/2 that {div 푉 , dΩ , div 푉 , dΩ } are uniformly bounded in 퐿loc , which ′ −1,푝 ′ compactly embeds into 푊loc for some 1 < 푝 < 2. On the other hand, they are −1,푝 uniformly bounded in 푊loc for 푝 > 푛 ≥ 2. Thus, by interpolation, we find that

(퐵휖) (퐵휖) (∇⟂,휖) (∇⟂,휖) −1 {div 푉 , dΩ , div 푉 , dΩ } are pre-compact in 퐻loc , which is precisely the desired first-order differential constraints for the geometrically intrinsic div–curl lemma, Theorem 2.2. Therefore, applying Theorem 2.2, we obtain the following subsequent convergence results in 풟′(푀):

(퐵휖) (퐵휖) (퐵) (퐵) ⟨푉푊,휂 (푋, 푌), Ω푍,휂 ⟩ ⟶ ⟨푉푊,휂(푋, 푌), Ω푍,휂⟩,

(∇⟂,휖) (∇⟂,휖) (∇⟂) (∇⟂) ⟨푉휂,훽 (푋, 푌), Ω휉,훽 ⟩ ⟶ ⟨푉휂,훽 (푋, 푌), Ω휉,훽 ⟩, (퐵휖) (퐵휖) (퐵) (퐵) ⟨푉푍,휉 (푋, 푌), Ω푍,휂 ⟩ ⟶ ⟨푉푍,휉 (푋, 푌), Ω푍,휂⟩,

(∇⟂,휖) (퐵휖) (∇⟂) (퐵) ⟨푉휂,훽 (푋, 푌), Ω푍,훽 ⟩ ⟶ ⟨푉휂,훽 (푋, 푌), Ω푍,훽⟩, 90 G.-Q. Chen and S. Li so that we can pass to the limits in Eqs. (4.7)–(4.9). As shown in Step 2, these equations are equivalent to the GCR equations, which leads to the weak continuity of the GCR equations. Finally, by Step 1, we know that the existence of solutions of the GCR equations in 풟′(푀) are equivalent to the existence of isometric im- 2,푝 푛+푘 mersions in 푊loc (푀; ℝ ). Thus, the assertion is proved on the local trivialized chart 푈 ⊂ 푀 as in Step 1. When 푀 is simply-connected, we can pass from the local to global by a standard monodromy argument.

We now make three comments on our main theorem, Theorem 4.1. First of all, in Step 1, we have given a geometrically intrinsic proof of the realization theorem. This can be summarized as follows: Corollary 4.2 (Theorem 5.2 in [11]). Let (푀, 푔) be an 푛-dimensional, simply- 1,푝 푘 connected Riemannian manifold with metric 푔 ∈ 푊loc for 푝 > 푛, and let (퐸, 푀, ℝ ) 1,푝 퐸 푝 be a vector bundle over 푀. Assume that 퐸 has a 푊loc metric 푔 and an 퐿loc connec- 퐸 퐸 퐸 푝 tion ∇ such that ∇ is compatible with 푔 . Moreover, suppose that there is an 퐿loc tensor field 푆∶ Γ(퐸) × Γ(푇푀) → Γ(푇푀) satisfying

⟨푋, 푆휂(푌)⟩ − ⟨푆휂(푋), 푌⟩ = 0,

푝 and a corresponding 퐿loc tensor field 퐵∶ Γ(푇푀) × Γ(푇푀) → Γ(퐸) defined by

⟨퐵(푋, 푌), 휂⟩ = −⟨푆휂(푋), 푌⟩.

Then the following are equivalent: (i) The GCR equations as in Proposition 3.1 with 푅⟂ replaced by 푅퐸, the Riemann curvature operator on the bundle; (ii) The Cartan formalism;

2,푝 푛+푘 (iii) The existence of a global isometric immersion 푓 ∈ 푊loc (푀; ℝ ) such that the induced normal bundle 푇(푓푀)⟂, normal connection ∇⟂, and second fundamental form can be identified with 퐸, ∇퐸, and 퐵, respectively. In (i)–(ii), the equalities are taken in the distributional sense and, in (iii), the isometric 2,푝 immersion 푓 ∈ 푊loc is unique a.e., modulo the Euclidean group of rigid motions ℝ푛+푘 ⋊ 푂(푛 + 푘). In view of the above corollary, for the purpose of weak rigidity, it is more natural to investigate the Cartan formalism. In particular, the GCR equations are recast into the compact identity d푊 = 푊 ∧ 푊, which is the second structural equation (as in well-known in geometry, the first structural equation expresses the torsion-free property of the Levi–Civita connection). However, notice that the connection Compensated compactness in Banach spaces 91

1-form 푊 consists of only (퐵, ∇⟂) so that, for the sequence of isometric immersions 휖 2,푝 푛+푘 휖 {푓 } uniformly bounded in 푊loc (푀; ℝ ), the corresponding {푊 } is uniformly 푝 휖 bounded in 퐿loc(푀; 픰픬(푛 + 푘)). Let 푊 be a weak limit of {푊 }. Then, via similar arguments as in Steps 2–3 in the proof of Theorem 4.1, we can pass the limits in

d푊 휖 = 푊 휖 ∧ 푊 휖 ⟹ d푊 = 푊 ∧ 푊. (4.10)

Therefore, applying Corollary 4.2, we obtain a simplified proof of Theorem 4.1. Second, for the most physically relevant case of the isometric immersing/embedding of a 2-dimensional manifold (i.e., a surface) into ℝ3, we can also establish the weak rigidity of the GCR equations in the critical case 푝 = 푛 = 2 (where the Ricci equation is trivial). This is because, on the right-hand side of the Gauss equation (3.6), we have the Gauss curvature 푅(푋, 푌, 푍, 푊), which is a fixed 퐿1 function in the setting of isometric immersions, thanks to the Cauchy–Schwarz inequality. Therefore, it is equi-integrable. Then we can apply the critical case of the div–curl lemma (Theorem 2.3).

Corollary 4.3 (Theorem 6.3 in [11]). Let 푀 be a 2-dimensional, simply-connected 1 휖 2 surface, and let 푔 be a metric in 퐻loc. If {푓 } is a family of 퐻loc isometric immersions of 푀 into ℝ3 such that the corresponding second fundamental forms {퐵휖} are uniformly 2 휖 2 퐿 -bounded. Then, after passing to a subsequence, {푓 } converges to 푓 weakly in 퐻loc which is still an isometric immersion 푓∶ (푀, 푔) → ℝ3. Moreover, the corresponding 휖 2 second fundamental form 퐵 is a limit point of {퐵 } in the 퐿loc topology. Finally, it is easy to derive a slightly more general version of the weak rigidity theorem, Theorem 4.1, by allowing the metrics to be unfixed and strongly conver- 1,푝 gent in 푊loc for 푝 > 푛. Such scenarios naturally arise in the regularization of a singular metric into smooth ones; cf. §7.2 in [11]. We remark in passing that the analogies of Theorem 4.1 and Corollaries 4.2–4.3 for isometric immersions into semi-Euclidean spaces of semi-Riemannian submanifolds (i.e., the metrics are non-degenerate, but may no longer be positive- definite; see O’Neill [44]) are also valid. For the possibly degenerate hypersurfaces, using the machinery of rigging fields (cf. [48, 35, 38]), a counterpart of the Cartan formalism can be established, which leads to the weak rigidity, provided that the 푝 rigging fields are uniformly 퐿loc bounded. For a rigorous formulation and the proof of these results, see our forthcoming paper [12]. As discussed above, we have established the weak rigidity of isometric immersions (Theorem 4.1) in [11]. It would be interesting to explore its relation with the rigidity/non-rigidity results in stronger topologies (see the discussion at the beginning of §4), to extend it to the larger framework of the h-principle laid down by Gromov [28], and to examine what the possible implications are in fluid 92 G.-Q. Chen and S. Li dynamics, in view of the connections between isometric immersions and Euler equations (see [1, 13] and [7]).

Acknowledgement. Gui-Qiang Chen’s research was supported in part by the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), the UK EPSRC Award to the EPSRC Centre for Doctoral Training in PDEs (EP/L015811/1), and the Royal Society–Wolfson Research Merit Award (UK). Siran Li’s research was supported in part by the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE (EP/E035027/1).

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The initial-boundary-value problem for an Ostrovsky–Hunter type equation

Giuseppe Maria Coclite, Lorenzo di Ruvo, and Kenneth Hvistendahl Karlsen

To Helge Holden on his 60th birthday with friendship and great admiration

Abstract. We consider an Ostrovsky–Hunter type equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.

1. Introduction

The nonlinear evolution equation

3 휕푥(휕푡푢 + 푎 휕푥푓(푢) + 푏훽 휕푥푥푥푢) = 훾푢, 푎, 푏, 훽, 훾 ∈ ℝ, (1.1) generalizes 1 휕 (휕 푢 + 휕 푢2 − 훽 휕3 푢) = 훾푢, (1.2) 푥 푡 2 푥 푥푥푥 which was derived by Ostrovsky [30] to model small-amplitude long waves in a rotating fluid of a finite depth. In addition, (1.2) generalizes the Korteweg–de Vries equation (that corresponds to 훾 = 0) by the additional term induced by the Coriolis force. Mathematical properties of the Ostrovsky equation (1.1) were studied recently in many works, including the local and global well-posedness in the energy space [18, 23, 26, 37], stability of solitary waves [21, 24, 27], and convergence of solutions in the limit of the Korteweg–de Vries equation [22, 27]. We will consider the limit of no high-frequency dispersion (훽 = 0), in which case (1.2) reads 1 휕 (휕 푢 + 휕 푢2) = 훾푢. (1.3) 푥 푡 2 푥 In this form the equation (1.3) is known under different names, such as the reduced Ostrovsky equation [31, 35], the Ostrovsky–Hunter equation [2], the short-wave 98 G. M. Coclite, L. di Ruvo, and K. H. Karlsen equation [19], and the Vakhnenko equation [28, 32]. It is deduced by considering two asymptotic expansions of the shallow water equations: first, with respect to the rotation frequency, and then with respect to the amplitude of the waves (see [17, 20]). By integrating (1.3) in 푥, we obtain the system formulation of problem (1.3)

휕푡푢 + 푢 휕푥푢 = 훾푃, 휕푥푃 = 푢. (1.4) In [11, 17], the authors proved the well-posedness of the entropy solution of the homogeneous initial boundary value problem, and the Cauchy problem for (1.4). In [3], the authors improved the results of [11, 17], proving the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem, and the uniqueness of the entropy solution for the Cauchy problem using an Oleinik type estimate from [4]. Moreover, the convergence of the solutions of (1.2) to the entropy solutions of (1.4) is proven in [5]. 1 3 When 푎 = − 6 , 푓(푢) = 푢 , and 푏 = 1,(1.1) reads 1 휕 (휕 푢 − 휕 푢3 − 훽 휕3 푢) = 훾푢. (1.5) 푥 푡 6 푥 푥푥푥 This equation is known as the regularized short pulse equation, and it was derived by Costanzino, Manukian and Jones [16] in the context of the nonlinear Maxwell equations with high-frequency dispersion. Mathematical properties of the regularized short pulse equation (1.5) were studied recently, including the local and global well-posedness in the energy space [16, 33], and stability of solitary waves [16]. In the limit of no high-frequency dispersion (훽 = 0), (1.5) reads 1 휕 (휕 푢 − 휕 푢3) = 훾푢. (1.6) 푥 푡 6 푥 This equation, which is termed short pulse equation, was introduced recently by Schäfer and Wayne [34] as a model equation describing the propagation of ultra-short light pulses in silica optical fibers. The well-posedness of the entropy solution for the initial-boundary-value problem and the Cauchy problem for (1.6) is proved in [6, 7]. Moreover, the convergence of the solutions of (1.5) to the entropy ones of (1.6) is studied in in [8]. As 훾 → 0 in (1.3), (1.5), we obtain the following scalar conservation law

푢2 푢3 휕 푢 + 휕 푓(푢) = 0, 푓(푢) = , − . (1.7) 푡 푥 2 6 In [9], the convergence of the solutions of (1.3), (1.6) to the entropy ones of (1.7) is proved. In the same paper the the authors studied also the limits 훽, 훾 → 0 for (1.2) and (1.5). An Ostrovsky–Hunter type equation 99

Finally, in [15], the authors prove the convergence of a finite difference scheme to the unique entropy solution of (1.3), (1.6) on a bounded domain with periodic boundary conditions. That result also provides an existence proof for periodic entropy solutions for (1.3) and (1.6). In this paper, we choose 푎 = 1, 푏 = 0 in (1.1) and we study the following initial boundary value problem

휕 ( 휕 푢 + 휕 푓(푢)) = 훾푢, 푡 > 0, 0 < 푥 < 1, ⎧ 푥 푡 푥 ⎪ 푢(푡, 0) = 훼(푡), 푡 > 0, ⎪ 휕 푓(푢(푡, 0)) = −훼′(푡), 푡 > 0, (1.8) ⎨ 푥 ⎪ 푢(푡, 1) = 훽(푡), 푡 > 0, ⎪ ⎩ 푢(0, 푥) = 푢0(푥), 0 < 푥 < 1, where we assume that1

∞ ∞ 푢0 ∈ 퐿 (0, 1), 훼, 훽 ∈ 퐿 (0, ∞) ∩ 퐵푉loc(0, ∞). (1.9)

Moreover, the flux 푓 is assumed to be smooth and genuinely nonlinear, in the sense that

푓 ∈ 퐶2(ℝ), |{ 푢 ∈ ℝ ∶ 푓″(푢) = 0 }| = 0. (1.10)

One of the main differences between our results and the previous ones (seein particular [15]) is that here we do not assume that the initial datum 푢0 has zero mean. We use the same approach used for the Camassa–Holm equation [13, 38]: we give the definition of entropy solution for (1.8) using the system formulation stated in (1.4). Indeed in what follows we develop all the theory using

푥 ⎧ 휕 푢 + 휕 푓(푢) = 훾 ∫ 푢(푡, 푦) d푦, 푡 > 0, 0 < 푥 < 1, ⎪ 푡 푥 ⎪ 0 푢(푡, 0) = 훼(푡), 푡 > 0, (1.11) ⎨ ⎪ 푢(푡, 1) = 훽(푡), 푡 > 0, ⎪ ⎩ 푢(0, 푥) = 푢0(푥), 0 < 푥 < 1,

1 Since 훼 ∈ 퐵푉loc(0, ∞), it is differentiable a.e., and so the boundary condition on 휕푥푓(ᵆ(푡, 0)) make sense. 100 G. M. Coclite, L. di Ruvo, and K. H. Karlsen or 휕 푢 + 휕 푓(푢) = 훾푃, 푡 > 0, 0 < 푥 < 1, ⎧ 푡 푥 ⎪ 휕 푃 = 푢, 푡 > 0, 0 < 푥 < 1, ⎪ 푥 ⎪ 푃(푡, 0) = 0, 푡 > 0, (1.12) ⎨ 푢(푡, 0) = 훼(푡), 푡 > 0, ⎪ ⎪ 푢(푡, 1) = 훽(푡), 푡 > 0, ⎪ ⎩ 푢(0, 푥) = 푢0(푥), 0 < 푥 < 1. Here we have a second unknown 푃 satisfying a homogeneous Dirichlet boundary condition at 푥 = 0, which follows from the boundary condition on 휕푥푓(푢(푡, 0)) in (1.8). The equivalence between (1.11) or (1.12) and (1.8) holds only for smooth solutions. Due to the regularizing effect of the 푃 equation in (1.12) we have that

푢 ∈ 퐿∞((0, 푇) × (0, 1)) ⟹ 푃 ∈ 퐿∞(0, 푇; 푊 1,∞(0, 1)). (1.13)

Therefore, if a map 푢 ∈ 퐿∞((0, 푇) × (0, 1)), 푇 > 0, satisfies, for every convex map 휂 ∈ 퐶2(ℝ),

ᵆ ′ ′ ′ 휕푡휂(푢) + 휕푥푞(푢) − 훾휂 (푢)푃 ≤ 0, 푞(푢) = ∫ 푓 (휉)휂 (휉) d휉, (1.14) in the sense of distributions, then [14, Theorem 1.1] provides the existence of 휏 휏 strong traces 푢0, 푢1 on the boundaries 푥 = 0, 1, respectively. We look for entropy solution of (1.8), or (1.11), or (1.12), based on the following definition:

Definition 1.1. We say that 푢 ∈ 퐿∞((0, 푇) × (0, 1)), where 푇 > 0, is an entropy solution of the initial-boundary value problem (1.8) and (1.9) if

i) 푢 is a distributional solution of (1.12) or equivalently of (1.4), namely for every test function 휙 ∈ 퐶∞(ℝ × (0, 1)) with compact support

∞ 1 1

∫ ∫ (푢 휕푡휙 + 푓(푢) 휕푥휙 + 훾푃휙) d푡 d푥 + ∫ 푢0(푥)휙(0, 푥) d푥 = 0; 0 0 0

ii) for every convex function 휂 ∈ 퐶2(ℝ) with corresponding 푞 defined by 푞′ = 푓′휂′ the entropy inequality (1.14) holds in the sense of distributions in (0, ∞)×(0, 1), namely for every nonnegative test function 휙 ∈ 퐶∞(ℝ × (0, 1)) with compact support An Ostrovsky–Hunter type equation 101

∞ 1 ′ ∫ ∫ (휂(푢) 휕푡휙 + 푞(푢) 휕푥휙 + 훾푃휂 (푢)휙) d푡 d푥 0 0 1

+ ∫ 휂(푢0(푥))휙(0, 푥) d푥 ≤ 0; 0 iii) for every convex function 휂 ∈ 퐶2(ℝ) with corresponding 푞 defined by 푞′ = 푓′휂′, the boundary entropy condition 휏 ′ 휏 푞(푢0(푡)) − 푞(훼(푡)) − 휂 (훼(푡))(푓(푢0(푡)) − 푓(훼(푡))) 휏 ′ 휏 (1.15) ≤ 0 ≤ 푞(푢1(푡)) − 푞(훽(푡)) − 휂 (훽(푡))(푓(푢1(푡)) − 푓(훽(푡))). holds for a.e. 푡 ∈ (0, ∞). The main result of this paper is the following theorem. Theorem 1.2. Assume (1.8), (1.9), and (1.10). The initial-boundary value problem (1.8) and (1.9) possesses a unique entropy solution 푢 in the sense of Definition 1.1. Moreover, if 푢1 and 푢2 are two entropy solutions of (1.8) and (1.9) in the sense of Definition 1.1, the following inequality holds

훾푡 ‖푢1(푡, ⋅) − 푢2(푡, ⋅)‖퐿1(0,1) ≤ 푒 ‖푢1(0, ⋅) − 푢2(0, ⋅)‖퐿1(0,1), (1.16) for almost every 푡 > 0. The paper is organized as follows. In Section 2, we prove some a priori estimates on a vanishing viscosity approximation of (1.12). Those and a compensated compactness based argument will us allow to select a converging subsequence within the vanishing viscosity approximations and prove our main result in Section 3.

2. Vanishing viscosity approximation

Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (1.12). Fix a small number 휖 > 0, and let 푢휖 = 푢휖(푡, 푥) be the unique classical solution of the following mixed problem 휕 푢 + 휕 푓(푢 ) = 훾푃 + 휖 휕2 푢 , 푡 > 0, 0 < 푥 < 1, ⎧ 푡 휖 푥 휖 휖 푥푥 휖 ⎪ 휕 푃 = 푢 , 푡 > 0, 0 < 푥 < 1, ⎪ 푥 휖 휖 ⎪ 푢 (푡, 0) = 훼 (푡), 푡 > 0, 휖 휖 (2.1) ⎨ 푢 (푡, 1) = 훽 (푡), 푡 > 0, ⎪ 휖 휖 ⎪ 푃 (푡, 0) = 0, 푡 > 0, ⎪ 휖 ⎩ 푢휖(0, 푥) = 푢휖,0(푥), 0 < 푥 < 1, 102 G. M. Coclite, L. di Ruvo, and K. H. Karlsen

∞ where 푢휖,0, 훼휖, 훽휖 are 퐶 approximation of 푢0, 훼, 훽 such that

푢휖,0(0) = 훼휖(0), 푢휖,0(1) = 훽휖(0),

‖푢 ‖ 2 ≤ ‖푢 ‖ 2 , ‖푢 ‖ ∞ ≤ ‖푢 ‖ ∞ , 휖,0 퐿 (0,1) 0 퐿 (0,1) 휖,0 퐿 (0,1) 0 퐿 (0,1) (2.2) ‖훼휖‖퐿∞(0,∞) ≤ ‖훼‖퐿∞(0,∞), ‖훽휖‖퐿∞(0,∞) ≤ ‖훽‖퐿∞(0,∞), ′ ′ ‖훼휖‖퐿1(0,푇) ≤ 푇푉(0,푇)(훼), ‖훽휖‖퐿1(0,푇) ≤ 푇푉(0,푇)(훽), 푇 > 0, for every 휖 > 0. Clearly (2.1) is equivalent to the integro-differential problem 푥 ⎧ 휕 푢 + 휕 푓(푢 ) = 훾 ∫ 푢 (푡, 푦) d푦 + 휖 휕2 푢 , 푡 > 0, 0 < 푥 < 1, ⎪ 푡 휖 푥 휖 휖 푥푥 휖 ⎪ 0 푢 (푡, 0) = 훼 (푡), 푡 > 0, (2.3) ⎨ 휖 휖 ⎪ 푢 (푡, 1) = 훽 (푡), 푡 > 0, ⎪ 휖 휖 ⎩ 푢휖(0, 푥) = 푢휖,0(푥), 0 < 푥 < 1. The existence of such solutions can be easily obtained by fixing a small number 훿 > 0 and considering the further approximation of (2.1) (see [10, 12, 17]) 휕 푢 + 휕 푓(푢 ) = 훾푃 + 휖 휕2 푢 , 푡 > 0, 0 < 푥 < 1, ⎧ 푡 휖,훿 푥 휖,훿 휖,훿 푥푥 휖,훿 ⎪ −훿 휕2 푃 + 휕 푃 = 푢 , 푡 > 0, 0 < 푥 < 1, ⎪ 푥푥 휖,훿 푥 휖,훿 휖,훿 ⎪ 푢휖(푡, 0) = 훼휖(푡), 푡 > 0, ⎨ 푢 (푡, 1) = 훽 (푡), 푡 > 0, ⎪ 휖 휖 ⎪ 푃 (푡, 0) = 휕 푃 (푡, 0) = 0, 푡 > 0, ⎪ 휖,훿 푥 휖,훿 ⎩ 푢휖,훿(0, 푥) = 푢휖,0(푥), 0 < 푥 < 1, and then sending 훿 → 0. Let us prove some a priori estimates on 푢휖. Lemma 2.1. For every 푡 ∈ (0, ∞),

훾푡 ‖푢휖(푡, ⋅)‖퐿∞(0,1) ≤ 휅푒 , (2.4) where 휅 = ‖푢0‖퐿∞(0,1) + ‖훼‖퐿∞(0,∞) + ‖훽‖퐿∞(0,∞) . In particular,

훾푡 ‖푃휖(푡, ⋅)‖퐿∞(0,1), ‖휕푥푃휖(푡, ⋅)‖퐿∞(0,1) ≤ 휅푒 , 푡 ≥ 0. (2.5) Proof. Due to (2.3), 푥 ′ 2 휕푡푢휖 + 푓 (푢휖) 휕푥푢휖 − 휖 휕푥푥푢휖 ≤ 훾 ∫ |푢휖(푡, 푦)| d푦 ≤ 훾‖푢휖(푡, ⋅)‖퐿∞(0,1) . 0 An Ostrovsky–Hunter type equation 103

Since the map ℱ(푡) ≔ 휅푒훾푡, 푡 ∈ (0, ∞), solves the equation 푑ℱ = 훾ℱ(푡) d푡 and (see (2.2))

max {푢휖(0, 푥), 훼휖(푡), 훽휖(푡), 0} ≤ ℱ(푡), (푡, 푥) ∈ (0, ∞) × (0, 1), the comparison principle for parabolic equations implies that

푢휖(푡, 푥) ≤ ℱ(푡), (푡, 푥) ∈ (0, ∞) × (0, 1).

In a similar way we can prove that

푢휖(푡, 푥) ≥ −ℱ(푡), (푡, 푥) ∈ (0, ∞) × (0, 1).

This concludes the proof of (2.4). Finally, (2.5) follows from the identities

푥

푃휖(푡, 푥) = ∫ 푢휖(푡, 푦) d푦, 휕푥푃휖 = 푢휖, 0 and (2.4). Lemma 2.2. We have that

∞ 2 {푢휖} is uniformly bounded in 퐿 (0, 푇; 퐿 (0, 1)), 푇 > 0, 휖>0 (2.6) {√휖 휕 푢 } is uniformly bounded in 퐿2((0, 푇) × (0, 1)), 푇 > 0. 푥 휖 휖>0 Proof. Consider the functions

푣휖(푡, 푥) = 푢휖(푡, 푥) − 푤휖(푡, 푥),

푤휖(푡, 푥) = (1 − 푥)훼휖(푡) + 푥훽휖(푡).

We have that

2 휕푡푣휖 + 휕푥푓(푢휖) = 훾푃휖 + 휖 휕푥푥푣휖 − 휕푡푤휖,

푣휖(푡, 0) = 푣휖(푡, 1) = 0, 훾푡 ‖푣휖(푡, ⋅)‖퐿∞(0,1) ≤ 휅(푒 + 1), ′ ′ 휕푡푤휖(푡, 푥) = (1 − 푥)훼휖(푡) − 푥훽휖(푡),

휕푥푤휖(푡, 푥) = 훽휖(푡) − 훼휖(푡). 104 G. M. Coclite, L. di Ruvo, and K. H. Karlsen

Therefore, thanks to the Hölder inequality

푑 1 푣2 1 ∫ 휖 d푥 = ∫ 푣 휕 푣 d푥 d푡 2 휖 푡 휖 0 0 1 1 2 = 휖 ∫ 푣휖 휕푥푥푣휖 d푥 − ∫ 푣휖 휕푥푓(푢휖) d푥 0 0 1 푥 1

+ 훾 ∫ ∫ 푣휖(푡, 푥)푢휖(푡, 푦) d푥 d푦 − ∫ 휕푡푤휖푣휖 d푥 0 0 0 1 1 1 ′ ′ + 훾 (∫ |푢휖| d푥)(∫ |푣휖| d푥) + (|훼휖(푡)| + |훽휖(푡)|) ∫ |푣휖| d푥 0 0 0 1 1 ᵆ휖(푡,푥) 2 ′ ≤ −휖 ∫ (휕푥푣휖) d푥 − ∫ 휕푥 (∫ 휉푓 (휉) d휉) d푥 0 0 0 1

+ 훽휖(푡)푓(훽휖(푡)) − 훼휖(푡)푓(훼휖(푡)) − (훽휖(푡) − 훼휖(푡)) ∫ 푓(푢휖) d푥 0 1 1 1 ′ ′ + 훾 (∫ |푢휖| d푥)(∫ |푣휖| d푥) + (|훼휖(푡)| + |훽휖(푡)|) ∫ |푣휖| d푥 0 0 0 1 훽휖(푡) 2 ′ ≤ −휖 ∫ (휕푥푣휖) d푥 − ∫ 휉푓 (휉) d휉 0 훼휖(푡)

+ 훽휖(푡)푓(훽휖(푡)) − 훼휖(푡)푓(훼휖(푡))

− (훽휖(푡) − 훼휖(푡))‖푓(푢휖(푡, ⋅))‖퐿∞(0,1)

+ 훾‖푢휖(푡, ⋅)‖퐿∞(0,1)‖푣휖(푡, ⋅)‖퐿∞(0,1) ′ ′ + (|훼휖(푡)| + |훽휖(푡)|)‖푣휖(푡, ⋅)‖퐿∞(0,1). Integrating over (0, 푡) and using Lemma 2.1,(1.9), and (2.2) we obtain that

∞ 2 {푣휖} is uniformly bounded in 퐿 (0, 푇; 퐿 (0, 1)), 푇 > 0 휖>0 (2.7) {√휖 휕 푣 } is uniformly bounded in 퐿2((0, 푇) × (0, 1)), 푇 > 0. 푥 휖 휖>0

Since, thanks to (1.9) and (2.2), {푤휖}휖>0 and {휕푥푤휖}휖>0 are uniformly bounded in 퐿∞((0, 푇) × (0, 1)), 푇 > 0,(2.7) implies (2.6).

3. Proof of the main result

This section is devoted to the proof of Theorem 1.2. We begin with the following result. An Ostrovsky–Hunter type equation 105

Lemma 3.1. There exists a function 푢 ∈ 퐿∞((0, 푇) × (0, 1)) that is a distributional solution of (1.12) and satisfies (1.14) in the sense of distributions for every convex entropy 휂 ∈ 퐶2(ℝ).

We construct a solution by passing to the limit in a sequence {푢휖}휖>0 of viscosity approximations (2.1). We use the compensated compactness method [36].

Theorem 3.2 (Tartar). Let {푣휈}휈>0 be a family of functions defined on (0, ∞)×(0, 1). ∞ Assume that {푣휈}휈>0 is uniformly bounded in 퐿loc((0, ∞) × (0, 1)) and the family {휕 휂(푣 ) + 휕 푞(푣 )} 푡 휈 푥 휈 휈>0

−1 2 ′ ′ ′ is compact in 퐻loc ((0, ∞) × (0, 1)) for every convex 휂 ∈ 퐶 (ℝ), where 푞 = 휂 푓 . ∞ Then a sequence {휈푛}푛∈ℕ ⊂ (0, ∞), 휈푛 → 0, and a map 푣 ∈ 퐿loc((0, ∞) × (0, 1)), exist such that

푝 푣휈푛 → 푣 a.e. and in 퐿loc((0, ∞) × (0, 1)), 1 ≤ 푝 < ∞. The following compact embedding result of Murat [29] is useful. Theorem 3.3 (Murat). Let Ω be a bounded open subset of ℝ , 푁 ≥ 2. Suppose that −1,∞ the sequence {ℒ푛}푛∈ℕ of distributions is bounded in 푊 (Ω). In addition, suppose that ℒ푛 = ℒ1,푛 + ℒ2,푛, where {ℒ } lies in a compact subset of 퐻−1(Ω) and {ℒ } lies in a bounded 1,푛 푛∈ℕ loc 2,푛 푛∈ℕ 1 −1 subset of 퐿loc(Ω). Then {ℒ푛}푛∈ℕ lies in a compact subset of 퐻loc (Ω).

Lemma 3.4. There exists a subsequence {푢휖푘}푘∈ℕ of {푢휖}휖>0 and a limit function 푢 ∈ 퐿∞((0, 푇) × (0, 1)), 푇 > 0, such that

푝 푢휖푘 → 푢 a.e. and in 퐿 ((0, 푇) × (0, 1)), 1 ≤ 푝 < ∞, 푇 > 0. (3.1) Moreover, we have

푝 1,푝 푃휖푘 → 푈 a.e. and in 퐿 ((0, 푇); 푊 (0, 1)), 1 ≤ 푝 < ∞, 푇 > 0. (3.2) where 푥 푈(푡, 푥) = ∫ 푢(푡, 푦) d푦, 푡 > 0, 0 < 푥 < 1. (3.3) 0 Proof. Let 휂 ∶ ℝ → ℝ be any convex 퐶2 entropy function, and let 푞∶ ℝ → ℝ be the corresponding entropy flux defined by

푞′ = 푓′휂′. 106 G. M. Coclite, L. di Ruvo, and K. H. Karlsen

′ By multiplying the first equation in (2.1) with 휂 (푢휖) and using the chain rule, we get 2 ″ 2 ′ 휕푡휂(푢휖) + 휕푥푞(푢휖) = 휖⏟⎵⎵⏟⎵⎵⏟ 휕푥푥휂(푢휖) ⏟⎵⎵⎵⎵⎵⎵⎵⎵⏟⎵⎵⎵⎵⎵⎵⎵⎵⏟− 휖휂 (푢휖) (휕푥푢휖) + 훾휂 (푢휖)푃휖, 1 2 ≕ ℒ휖 ≕ ℒ휖 1 2 where ℒ휖, ℒ휖 are distributions. By Lemmas 2.1 and 2.2, for any 푇 > 0,

1 −1 ℒ휖 → 0 in 퐻 ((0, 푇) × (0, 1)) as 휖 → 0, 2 1 (3.4) {ℒ휖}휖>0 is uniformly bounded in 퐿 ((0, 푇) × (0, 1)). Therefore, Murat’s lemma [29] implies that

{휕 휂(푢 ) + 휕 푞(푢 )} 푡 휖 푥 휖 휖>0 −1 lies in a compact subset of 퐻loc ((0, 푇) × (0, 1)), 푇 > 0. (3.5) The 퐿∞ bound stated in Lemma 2.1, (3.5), and Tartar’s compensated compactness method [36] give the existence of a subsequence {푢휖푘}푘∈ℕ and a limit function 푢 ∈ 퐿∞((0, 푇) × (0, 1)), 푇 > 0, such that (3.1) holds. Finally, thanks to the Hölder inequality and the identities

푥 ∫ 푃휖푘(푡, 푥) = 푢휖푘(푡, 푦) d푦, 휕푥푃휖푘 = 푢휖푘, 0 (3.2) follows from (3.1).

We are now ready for the proof of Theorem 1.2. Proof of Theorem 1.2 Since, thanks to Lemma 3.4, 푢 ∈ 퐿∞((0, 푇)×(0, 1)), 푇 > 0 is a distributional solution of the problem (see (3.3))

휕 푢 + 휕 푓(푢) = 훾푈, (푡, 푥) ∈ (0, ∞) × (0, 1), ⎧ 푡 푥 ⎪ 푢(0, 푥) = 푢 (푥), 푥 ∈ (0, 1), 0 (3.6) ⎨ 푢(푡, 0) = 훼(푡), 푡 > 0, ⎪ ⎩ 푢(푡, 1) = 훽(푡), 푡 > 0, that satisfies the entropy inequalities (1.14),[14, Theorem 1.1] tells us that the 휏 휏 limit 푢 admits strong boundary traces 푢0, 푢1 at (0, ∞) × {푥 = 0}, (0, ∞) × {푥 = 1}, respectively. Since, arguing as in [14, Section 3.1] (indeed our solution is obtained as the vanishing viscosity limit of (3.6)), [14, Lemma 3.2] and the boundedness of the source term 푈 (cf. (1.13)) imply (1.15). An Ostrovsky–Hunter type equation 107

Finally, we have to prove the uniqueness and the stability of the entropy solution to (1.8), and (1.9). To this end, let 푢1, 푢2 be two entropy solutions. It is enough to prove that (1.16) holds. Since 푢1 and 푢2 are entropy solutions of (3.6), we can use [1, Corollary 2.6], to assemble the inequality

‖푢1(푡, ⋅) − 푢2(푡, ⋅)‖퐿1(0,1) ≤ 훾‖푈1 − 푈2‖퐿1((0,푡)×(0,1)), (3.7) for 푡 ∈ (0, ∞), where 푈1 and 푈2 are defined as in (3.3). Moreover, (3.3) says that 푥

푈1(푡, 푥) − 푈2(푡, 푥) = ∫ (푢1(푡, 푦) − 푢2(푡, 푦)) d푦, 0 so ‖푈1(푡, ⋅) − 푈2(푡, ⋅)‖퐿1(0,1) ≤ ‖푢1(푡, ⋅) − 푢2(푡, ⋅)‖퐿1(0,1). Hence, by (3.7),

‖푢1(푡, ⋅) − 푢2(푡, ⋅)‖퐿1(0,1) ≤ 훾‖푢1 − 푢2‖퐿1((0,푡)×(0,1)) 푡

= 훾 ∫ ‖푢1(푠, ⋅) − 푢2(푠, ⋅)‖퐿1(0,1)푑푠. 0 Therefore, (1.16) follows from Gronwall’s lemma.

Acknowledgments. The authors thank the anonymous referee for the useful remarks. G. M. Coclite and L. di Ruvo are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This work was supported by the Research Council of Norway via grant no. 250674/F20.

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Modeling crowd dynamics through hyperbolic – elliptic equations

Rinaldo M. Colombo1, Maria Gokieli2, and Massimiliano D. Rosini3

This work is dedicated to Helge Holden, with gratitude and esteem.

Abstract. Inspired by the works of Hughes [22, 23], we formalize and prove the well- posedness of a hyperbolic–elliptic system whose solutions describe the dynamics of a moving crowd. The resulting model is here shown to be well-posed and the time of evacuation from a bounded environment is proved to be finite. This model also provides a microscopic description of the individuals’ behaviors.

1. Introduction

We consider the problem of describing how pedestrians exit an environment. From a macroscopic point of view, we identify the crowd through the pedestrians’ density, say 휌 = 휌(푡, 푥), and assume that the crowd behavior is well described by the continuity equation

+ 휕푡휌 + ∇ ⋅ (휌 푉(푥, 휌)) = 0 , (푡, 푥) ∈ ℝ × Ω , (1.1) where Ω ⊂ ℝ2 is the environment available to pedestrians, 푉 = 푉(푥, 휌) ∈ ℝ2 is the velocity of the individual at 푥, given the presence of the density 휌. At present, a fully rigorous derivation of (1.1) on the basis of a microscopic to macroscopic limit is apparently unavailable. The case of 1-dimensional space is considered in [16], see also [11, Section 4] and [12] for vehicular traffic modeling. Several choices for the velocity function are available in the literature, see for instance [5, 6, 9, 10, 15, 22, 23, 24, 30] for velocities depending nonlocally on the density, and [25, Section 4.1] for velocities depending locally on the density. Here, we posit the following (local with respect to the density) assumption:

푉(푥, 휌) = 푣(휌) 푤(푥) (1.2) where 푣 = 푣(휌) is a smooth non-increasing scalar function, motivated by the common attitude of moving faster when the density is lower. A key role is played 112 R. M. Colombo, M. Gokieli, and M. D. Rosini by 푤 = 푤(푥): this vector identifies the route followed by the individual at 푥. It is reasonable to assume that the individual at 푥 follows the shortest path from 푥 towards the nearest exit. This naturally suggests to choose 푤 parallel to ∇휙, the potential 휙 being the solution to the eikonal equation on Ω. Extending the results in [2, 17, 18] obtained in the 1-dimensional space to the 2-dimensional space we consider the following elliptic regularization of the eikonal equation:

2 ‖∇휙‖ − 훿 Δ휙 = 1 , 푥 ∈ Ω , where 훿 is strictly positive. This is a standard, so called “viscous”, approximation of the eikonal equation, that essentially dates back to the classical existence result in [14, Section IV]. In the present modeling setting, 훿 represents the difficulty with which pedestrians choose to behave differently from their neighbors. Clearly, the resulting vector field ∇휙 depends only on Ω, namely only on the geometry of the environment available to the pedestrians, i.e., on the positions of the exits, on the possible presence of obstacles, and so on. We assume that the boundary 휕Ω is partitioned in walls, say Γ푤, exits, say Γ푒, and corners, say Γ푐; namely 휕Ω = Γ푤 ∪ Γ푒 ∪ Γ푐, the set Γ푒, Γ푤, Γ푐 being two by two disjoint. Γ푐 is a discrete subset of 휕Ω. Also Γ푒 and Γ푤 are subsets of 휕Ω, and they are open in the topology they inherit from 휕Ω. It is then natural to choose 휙 as solution to the elliptic equation 2 ⎧ ‖∇휙‖ − 훿 Δ휙 = 1 푥 ∈ Ω ∇휙(휉) ⋅ 휈(휉) = 0 휉 ∈ Γ (1.3) ⎨ 푤 ⎩ 휙(휉) = 0 휉 ∈ Γ푒 , 휈(휉) being the outward unit normal to 휕Ω at 휉. To select the direction 푤(푥) followed by the pedestrian at 푥 we set 푤 = 풩(−∇휙) , (1.4) the map 풩 being a regularized normalization, that is 푥 풩(푥) = , (1.5) 2 √휃2 + ‖푥‖ for a fixed strictly positive parameter 휃. Finally, the evolution of the crowd density 휌 is then found solving the following scalar conservation law:

+ 휕푡휌 + ∇ ⋅ (휌 푣(휌) 푤(푥)) = 0 (푡, 푥) ∈ ℝ × Ω { 휌(0, 푥) = 휌표(푥) 푥 ∈ Ω (1.6) 휌(푡, 휉) = 0 (푡, 휉) ∈ ℝ+ × 휕Ω , where 휌표 is the initial crowd distribution. In other words, for a given domain Ω, from (1.3) we obtain the vector field ∇휙, that is used in (1.4) to define 푤 and then Modeling crowd dynamics through hyperbolic – elliptic equations 113 from (1.6) we obtain how the pedestrians’ density 휌 evolves in time starting from the initial density 휌표. Remark that the boundary condition 휌(푡, 휉) = 0 has to be understood in the sense of conservation laws, see [4, 13] and Definition 2.3 below. Indeed, the choice in (1.6) allows a positive outflow from Ω through Γ푒 thanks to the definition of 푤, as proved in (E.2) of Proposition 2.2. We prove below that the model consisting of (1.3)–(1.4)–(1.6) is well posed, i.e., it admits a unique solution which is a continuous function of the initial data. Moreover, we also ensure that the evacuation time is finite. Remark that the model (1.3)–(1.4)–(1.6) is completely defined by the physical domain Ω, by the function 푣 = 푣(휌) and by the initial datum 휌표, apart from the regularizing parameters 훿 and 휃. The next two sections are devoted to the detailed formulation of the problem, to the statement of the well-posedness result and of further qualitative properties of the model (1.3)–(1.4)–(1.6). All technical details are gathered in Section 4.

2. Well-Posedness

Throughout, we denote ℝ+ = [0, +∞[. For 푥 ∈ ℝ2 and 푟 > 0, 퐵(푥, 푟) stands for the open disk centered at 푥 with radius 푟. For any measurable subset 푆 of ℝ2, we denote by |푆| its 2-dimensional Lebesgue measure. Recall that two (non-empty) 2 subsets 퐴1, 퐴2 of ℝ are separate whenever 퐴1 ∩ 퐴2 = ∅ = 퐴1 ∩ 퐴2. A key role is played by the geometry of the domain Ω. Here we collect the conditions necessary in the sequel, see Figure 1. (Ω.1) Ω ⊂ ℝ2 is non-empty, open, bounded and connected.

(Ω.2) The boundary 휕Ω admits the disjoint decomposition 휕Ω = Γ푤 ∪ Γ푒 ∪ Γ푐, where Γ푤 and Γ푒 are separate and are finite union of open 1-dimensional ퟑ,휸 manifolds of class 퐂 , for a given 훾 ∈ ]0, 1[; Γ푒 is non-empty; Γ푐 is a discrete finite set; and Γ푤 ∩ Γ푒 ⊆ Γ푐 ⊆ Γ푤.

(Ω.3) For any 푥 ∈ Γ푐, there exists an 휖 > 0 such that the intersection 퐵(푥, 휖) ∩ Ω is exactly a quadrant of the disk 퐵(푥, 휖). The requirement (Ω.1) is clear. In (Ω.2), the term open has to be understood with respect to the topology inherited by 휕Ω. Again concerning (Ω.2), introduce the connected components of Γ푤,Γ푒 and Γ푐, i.e.,

푛푤 푛푒 푛푐 Γ = Γ푖 ,Γ = Γ푖 , and Γ = {퐽 }. 푤 ⋃ 푤 푒 ⋃ 푒 푐 ⋃ 푖 푖=1 푖=1 푖=1 114 R. M. Colombo, M. Gokieli, and M. D. Rosini

3 Γe Γ1 2 w Γ1 Γe 2 6 w Γe Γw

4 2 Γ Γw w 5 Γw

Γ1 2 e 3 Γw 4 Γw 1 Γw Γe 3 Γw

Figure 1. Two examples of sets Ω with the notation used in (Ω.2) and in (Ω.3).

푖 Each of the Γ푒 is an exit, while the 퐽푖 are points where the regularity of 휕Ω is 푖 푖 ퟑ,휸 allowed to be lower. Condition (Ω.2) implies that each Γ푤 and each Γ푒 is a 퐂 manifold. Since Γ푐 ⊆ Γ푤, along the boundary 휕Ω, between two different exits there is always a wall or, in other words, there can not be two exits separated only by a corner point. Condition (Ω.2) also implies that 푛푒 ≥ 1, so that there is at least one exit. Moreover, apart from the trivial case where 휕Ω = Γ푒, the set Γ푐 may not be empty. Note also that any corner point 퐽푖 in Γ푐 is either a doorjamb, if 퐽푖 ∈ Γ푒, or a wall corner, if 퐽푖 ∈ (Γ푤 ⧵ Γ푒). Condition (Ω.3) says that the angles between each door and the walls are right and convex, and additionally that these contain straight segments. This is a technical assumption, related to the subtle mixed boundary conditions: Dirichlet and Neumann conditions meet at the doorjamb points. Condition (Ω.3) ensures the regularity of solutions in a neighborhood of these points, a property that might not hold for general angles. Throughout, by solution to (1.3) we mean generalized solution in the sense of the following definition (see [20, Chapters 8 and 13]). Definition 2.1. Let Ω satisfy (Ω.1). A function 휙 ∈ 퐇ퟏ(Ω; ℝ) is a generalized solution to (1.3) if trΓ푒 휙 = 0 and

2 훿 ∫ ∇휙(푥) ⋅ ∇휂(푥) d푥 + ∫ (‖∇휙(푥)‖ − 1) 휂(푥) d푥 = 0 Ω Ω

ퟏ for any 휂 ∈ 퐇 (Ω; ℝ) such that trΓ푒 휂 = 0.

Above, trΓ푒 휂 denotes the trace of 휂 on Γ푒. We refer to [19, Chapter 5.5] for the definition and properties of the trace operator. Note that no generalized solution to (1.3) can vanish a.e. on Ω. The next proposition provides the basic existence result for the solutions to (1.3), together with some qualitative properties. Modeling crowd dynamics through hyperbolic – elliptic equations 115

Proposition 2.2 (Elliptic Problem). Let Ω satisfy (Ω.1), (Ω.2), (Ω.3). Fix 훿 > 0. Then, problem (1.3) admits a unique generalized solution 휙 ∈ 퐂ퟑ(Ω; ℝ) with the properties: (E.1) For a.e. 푥 ∈ Ω, ∇휙(푥) ≠ 0.

(E.2) For all 휉 ∈ Γ푒, −∇휙(휉) ⋅ 휈(휉) > 0. |Ω| max 휙 |Ω| max 휙 (E.3) exp (− 휕Ω ) ≤ − ∫ ∇휙(휉) ⋅ 휈(휉) d휉 ≤ exp ( 휕Ω ). 훿 훿 훿 훿 Γ푒 The proof of the above proposition is postponed to Section 4. Here, we note that properties (E.1), (E.2) and (E.3) have clear consequences on the properties of the solutions to the full system (1.3)–(1.4)–(1.6). Indeed, setting 푤 as in (1.4), property (E.1) implies that 푤 vanishes only on a set of measure 0; (E.2) ensures that 푤 is non zero and points outwards along exits; (E.3) can be used to provide bounds on the evacuation time. In the hyperbolic problem (1.6), we use the following assumptions, which are standard in the framework of conservation laws:

ퟐ (C.1) 푣 ∈ 퐂 ([0, 푅max]; [0, 푉max]) is weakly decreasing, 푣(0) = 푉max, and 푣(푅max) = 0. ∞ (C.2) 휌표 ∈ (퐁퐕 ∩ 퐋 )(Ω; [0, 푅max]).

Above, 푅max, respectively 푉max, is the maximal density, respectively speed, possibly reached by the pedestrians. We recall also the definition of entropy solution to (1.6), which originates in [32], see also [4, p. 1028]. Here, we refer to [13, Definition 2.1]. Definition 2.3. Let the conditions (Ω.1), (Ω.2), (C.1) and (C.2) hold. Let 푤 ∈ ퟐ ∞ 퐂 (Ω; 퐵(0, 1)). A function 휌 ∈ (퐋 ∩ 퐁퐕)([0, 푇] × Ω; [0, 푅max]) is an entropy solution to the initial – boundary value problem (1.6) if for any test function ퟐ 2 + 휁 ∈ 퐂퐜(]−∞, 푇[ × ℝ ;ℝ ) and for any 푘 ∈ [0, 푅max] 푇

∫ ∫ {|휌(푡, 푥) − 푘| 휕푡휁(푡, 푥)} d푥 d푡 0 Ω 푇 +∫ ∫{ sign (휌(푡, 푥) − 푘) (휌(푡, 푥) 푣 (휌(푡, 푥)) − 푘 푣(푘)) 푤(푥) ⋅ ∇휁(푡, 푥)} d푥 d푡 0 Ω

+ ∫ |휌표(푥) − 푘| 휁(0, 푥) d푥 Ω 푇

+ ∫ ∫ {tr휕Ω 휌(푡, 휉) 푣(tr휕Ω 휌(푡, 휉)) − 푘 푣(푘)} 푤(휉) ⋅ 휈(휉) 휁(푡, 휉) d휉 d푡 ≥ 0. 0 휕Ω 116 R. M. Colombo, M. Gokieli, and M. D. Rosini

As above, tr휕Ω 푢 stands for the operator trace at 휕Ω applied to the 퐁퐕 function 푢, see for instance [19, § 5.5] or [13, Appendix]. Note that if the solution has bounded total variation in time, it has a trace at 푡 = 0+.

Proposition 2.4 (Hyperbolic Problem). Let the conditions (Ω.1), (Ω.2) and (C.1) hold. Let 푤 ∈ 퐂ퟐ(Ω; 퐵(0, 1)). Then, problem (1.6) generates the map

+ ퟏ ퟏ 풮∶ ℝ × (퐋 ∩ 퐁퐕)(Ω; [0, 푅max]) → (퐋 ∩ 퐁퐕)(Ω; [0, 푅max]) 푡 , 휌 ↦ 풮푡휌 with the following properties:

(H.1) 풮 is a semigroup.

(H.2) 풮 is Lipschitz continuous with respect to the 퐋ퟏ-norm, more precisely for any 푠, 푡 ∈ [0, 푇]

‖풮 휌 − 풮 휌 ‖ ≤ [ sup TV(풮 휌 )] |푡 − 푠| . ‖ 푡 표 푠 표‖퐋∞(Ω;ℝ) 휏 표 휏∈[푠,푡]

(H.3) For any 푡 ∈ [0, 푇]

‖풮 휌 ‖ ≤ ‖휌 ‖ exp(퐶 푡) , ‖ 푡 표‖퐋∞(Ω;ℝ) ‖ 표‖퐋∞(Ω;ℝ) 1

TV (풮푡휌표) ≤ 퐶2 (1 + 푡 + TV(휌표)) exp(퐶2 푡) ,

′ where the constants 퐶1, 퐶2 depend only on ‖푣 ‖ ퟐ,∞ , ‖푤‖ ퟐ,∞ 2 , 퐖 ([0,푅max];ℝ) 퐖 (Ω;ℝ ) and 푅max.

ퟏ (H.4) For any 휌표 ∈ (퐋 ∩퐁퐕)(Ω; [0, 푅max]), the orbit 푡 ↦ 풮푡휌표 is the unique solution to (1.6) in the sense of Definition 2.3.

The proof of the above proposition is deferred to Section 4, where it is shown that the above statements follow from [13, Theorem 2.7]. We now give the definition of solution to(1.3)–(1.4)–(1.6).

Definition 2.5. Let the assumptions (Ω.1), (Ω.2), (Ω.3), (C.1) and (C.2) hold. ퟏ ∞ The pair of functions (휙, 휌) ∈ 퐇 (Ω; ℝ) × (퐋 ∩ 퐁퐕)([0, 푇] × Ω; [0, 푅max]) solves the problem (1.3)–(1.4)–(1.6) if 휙 is a generalized solution to (1.3) in the sense of Definition 2.1 and 휌 is an entropy solution to (1.6) in the sense of Definition 2.3 with 푤 given by (1.4).

The next theorem ensures the well-posedness of the hyperbolic–elliptic model (1.3)–(1.4)–(1.6). Modeling crowd dynamics through hyperbolic – elliptic equations 117

Theorem 2.6 (Mixed Problem). Assume the conditions (Ω.1), (Ω.2), (Ω.3), (C.1), and (C.2) hold. For any 훿, 휃 > 0, the hyperbolic–elliptic problem (1.3)–(1.4)–(1.6) generates a map

+ ퟏ ퟏ ℳ∶ ℝ × (퐋 ∩ 퐁퐕)(Ω; [0, 푅max]) → (퐋 ∩ 퐁퐕)(Ω; [0, 푅max]) 푡 , 휌 ↦ ℳ푡휌 with the following properties: (M.1) ℳ is a semigroup. (M.2) ℳ is Lipschitz continuous with respect to the 퐋ퟏ-norm, more precisely for any 푠, 푡 ∈ [0, 푇]

‖ℳ 휌 − ℳ 휌 ‖ ≤ [ sup TV(ℳ 휌 )] |푡 − 푠| . ‖ 푡 표 푠 표‖퐋∞(Ω;ℝ) 휏 표 휏∈[푠,푡]

(M.3) For any 푡 ∈ [0, 푇] we have that (휙, 휌) = ℳ푡휌표 satisfies

‖휌‖ ≤ ‖휌 ‖ exp(퐶 푡) , ‖ ‖퐋∞(Ω;ℝ) ‖ 표‖퐋∞(Ω;ℝ) 1

TV (휌) ≤ 퐶2 (1 + 푡 + TV(휌표)) exp(퐶2 푡) ,

where the constant 퐶1 > 0 depends on ‖푞‖ ퟏ,∞ and ‖푤‖ ퟏ,∞ 2 , 퐖 ([0,푅max];ℝ) 퐖 (Ω;ℝ )

while the constant 퐶2 depends on ‖푞‖ ퟐ,∞ and ‖푤‖ ퟐ,∞ 2 , where 퐖 ([0,푅max];ℝ) 퐖 (Ω;ℝ ) as usual we set 푞(휌) = 휌 푣(휌).

ퟏ (M.4) For all 휌표 ∈ (퐋 ∩퐁퐕)(Ω; [0, 푅max]), the orbit 푡 ↦ ℳ푡휌표 is the unique solution to (1.3)–(1.4)–(1.6) in the sense of Definition 2.5. The above result is a direct consequence of Proposition 2.2 and Proposition 2.4.

3. Qualitative Properties

Here, we aim at further qualitative properties of the solutions to (1.3)–(1.4)–(1.6) that have a relevant meaning in the present setting. Introduce for ̂푥∈ Ω the path 푝푥̂ followed by those pedestrians that are at ̂푥 at time 푡 = 0, i.e., the map 푝푥̂ is defined for 푡 ≥ 0 as the solution to the Cauchy problem ̇푥= 푤(푥), { where 푤 = 풩(−∇휙) . (3.1) 푥(0) = ̂푥, Above, 풩 is defined in (1.5) and 휙 is the solution to (1.3). 118 R. M. Colombo, M. Gokieli, and M. D. Rosini

The terminology here is inspired by that of fluid mechanics. Given a solution 휌 = 휌(푡, 푥) to the continuity equation (1.1)–(1.2), the solutions to the ordinary differential equation ̇푥= 푣 (휌(푡, 푥)) 푤(푥) are often referred to as the particle’s paths, see, e.g., [7, Chapter 1, § 1.1.ii]. Therefore, 푣(휌) being a scalar, we refer to the solution to (3.1) as to the pedestrians’ trajectories. Proposition 3.1 (Pedestrians’ Trajectories). Let Ω satisfy (Ω.1), (Ω.2), (Ω.3) and call 휙 the solution to (1.3) provided by Proposition 2.2. Then:

2 (Q.1) For any ̂푥∈ Ω, there exists a unique globally defined path 푝푥̂ ∶ 퐼푥̂ → ℝ solving (3.1), 퐼푥̂ being a suitable non trivial real interval. (Q.2) Any two paths either coincide or do not intersect, in the sense that for any ̂푥, 푦̂ ∈ Ω

either ̂푥∈ 푝푦̂(퐼푦̂) and 푝푥̂(퐼푥̂) ⊆ 푝푦̂(퐼푦̂) 푝푥̂(퐼푥̂) ∩ 푝푦̂(퐼푦̂) ≠ ∅ ⟹ { or 푦̂ ∈ 푝푥̂(퐼푥̂) and 푝푦̂(퐼푦̂) ⊆ 푝푥̂(퐼푥̂).

(Q.3) There exist a subset Ω̂ ⊂ Ω with |Ω̂ | = 0 and a map 푇∶ Ω ⧵ Ω̂ → ℝ+ such that 퐼푥̂ = [0, 푇푥̂] and 푝푥̂(푇푥̂) ∈ Γ푒 for all 푥 ∈ Ω ⧵ Ω̂ .

The proof is deferred to Section 4. In other words, 푇푥̂ is the time that the pedestrian leaving from point ̂푥 needs to reach the exit. Property (Q.3) ensures that this time is finite for a.e. initial position .̂푥 Figure 2 shows that the set Ω̂ may not be avoided under the present assumptions.

Γe Ωˆ

Γe

Figure 2. An example in which the set Ω̂ in Proposition 3.1 is necessarily non empty. In the room above, due to the presence of the two exits Γ푒, the vector field 푤 vanishes along the dotted segment Ω̂ .

4. Technical Details

We choose the following notation to denote a vector orthogonal to a given vector in ℝ2: 푣 −푣 if 푣 = [ 1] , then 푣⟂ = [ 2] . 푣2 푣1 Modeling crowd dynamics through hyperbolic – elliptic equations 119

We frequently use the boundedness and Lipschitz continuity of the map 풩 as defined in (1.5), namely

‖푁(푥)‖ ≤ 1 for all 푥 ∈ ℝ2 , 1 (4.1) ‖푁(푥 ) − 푁(푥 )‖ ≤ ‖푥 − 푥 ‖ for all 푥 , 푥 ∈ ℝ2 . 1 2 휃 1 2 1 2 The Hopf–Cole transformation (see e.g. [19, Chapter 4.4.1])

푢 = 푒−휙/훿 (4.2) transforms generalized solutions to (1.3) into generalized solutions to the linear problem 푢 = 훿2 Δ푢 푥 ∈ Ω ⎪⎧ ∇푢(휉) ⋅ 휈(휉) = 0 휉 ∈ Γ푤 (4.3) ⎪⎨ ⎩ 푢(휉) = 1 휉 ∈ Γ푒 , whose precise definition (see e.g. [20, Chapter 8]) is here below. Definition 4.1. A function 푢 ∈ 퐇ퟏ(Ω; ℝ) is a generalized solution to (4.3) on Ω if trΓ푒 푢 ≡ 1 and

훿2 ∫ ∇푢(푥) ⋅ ∇휂(푥) d푥 + ∫ 푢(푥) 휂(푥) d푥 = 0 (4.4) Ω Ω for any 휂 ∈ 퐇ퟏ(Ω; ℝ) such that tr 휂 ≡ 0. Γ푒 The next Lemma collects various information on (4.3). Lemma 4.2. Fix a positive 훿 and let Ω satisfy (Ω.1) and (Ω.2). Then, (u.1) Problem (4.3) admits a unique generalized solution 푢∈(퐇ퟏ ∩ 퐂∞)(Ω; ℝ) in ퟑ the sense of Definition 4.1. Moreover, 푢 ∈ 퐂 (Ω ⧵ Γ푐; ℝ). (u.2) There exists a positive 휛 dependent only on Ω such that 푢(푥) ∈ ]휛, 1[ for all 푥 ∈ Ω, so that 푢(푥) ∈ [휛, 1] also for all 푥 ∈ Ω.

(u.3) The solution 푢 to (4.3) satisfies ∇푢(휉) ⋅ 휈(휉) > 0 for all 휉 ∈ Γ푒. (u.4) The set { 푥 ∈ Ω ∶ ∇푢(푥) = 0 } of critical points of 푢 has measure 0. If in addition Ω satisfies (Ω.3), then: (u.5) 푢 ∈ 퐂ퟑ(Ω; ℝ). (u.6) If ̄푥∈ Ω is a critical point of 푢, then the Hessian matrix 퐷2푢( ̄푥) has at least one positive eigenvalue. 120 R. M. Colombo, M. Gokieli, and M. D. Rosini

Proof. Consider the different items above separately. ⋆ (u.1): we use the Lax–Milgram lemma, see [19, Section 6.2.1]. Introduce the ퟏ Hilbert space 퐻 = { 휂 ∈ 퐇 (Ω; ℝ) ∶ trΓ푒 휂 = 0 a.e. on Γ푒 } endowed with the usual scalar product and the coercive bilinear form

푎(푢, 휂) = 훿2 ∫ ∇푢(푥) ⋅ ∇휂(푥) d푥 + ∫ 푢(푥) 휂(푥) d푥 . Ω Ω Note that 퐻 is a closed subspace of 퐇ퟏ(Ω; ℝ) by the Trace Theorem [19, Chapter 5.5, Theorem 1]. Indeed, if 푢푘 is a sequence in 퐻 converging to 푢 in 퐇ퟏ(Ω; ℝ), then ‖ 푘 ‖ ‖ 푘 ‖ ‖푢‖ ퟐ = ‖푢 − 푢‖ ퟐ ≤ 퐶 ‖푢 − 푢‖ ퟏ → 0 , 퐋 (Γ푒;ℝ) 퐋 (Γ푒;ℝ) 퐇 (Ω;ℝ) for a constant 퐶 depending only on Ω, so that 푢 ∈ 퐻. A function 푢 ∈ 퐇ퟏ(Ω; ℝ) is a generalized solution to (4.3) if and only if 푣 = 푢−1 ∈ 퐻 and 푎(푣, 휂) = − ∫Ω 휂(푥) d푥 for all 휂 ∈ 퐻. The map 휂 ↦ ∫Ω 휂(푥) d푥 is a linear functional over 퐻. By the Lax– Milgram lemma, we infer the existence and uniqueness of a generalized solution 푢 to (4.3) such that 푢 ∈ 퐻 ⊂ 퐇ퟏ(Ω; ℝ). Moreover, 푢 ∈ 퐂∞(Ω; ℝ) by [19, Theorem 3 in Chapter 6.3 and Theorem 6 in Section 5.6.3]. By (Ω.1) and (Ω.2), the results ퟑ in [1, Theorem 9.3] ensure that 푢 ∈ 퐂 (Ω ⧵ Γ푐; ℝ). ퟏ ⋆ (u.2): note that, due to the boundary conditions along Γ푒 and Γ푤, no 퐇 solution to (4.3) can be constant. The function 휂 = (푢 − 1)+, where (푣)+ = max(푣, 0), is in 퐇ퟏ(Ω; ℝ) and inserting it in (4.4) we get

2 2 훿2 ∫ ‖∇(푢 − 1)+‖ + ∫ |(푢 − 1)+| + ∫(푢 − 1)+ = 0 . Ω Ω Ω This leads to (푢 − 1)+ ≡ 0 a.e. in Ω, and, by the continuity of 푢 on Ω, 푢(푥) ≤ 1 for all 푥 ∈ Ω. The map 푢 satisfies (4.3) in the strong sense everywhere in Ω. Hence, by the maximum principle [28, Chapter 2, Theorem 6], 푢(푥) < 1 for all 푥 ∈ Ω. We show now that 푢 > 0. As 푢 is continuous in Ω, it attains its minimum.

Assume, by contradiction, that minΩ 푢 = −푚 for some 푚 ≥ 0. Then, by applying the maximum principle to −푢, we know that there exists 휉 ∈ 휕Ω such that 푢(휉) = −푚. We apply now Hopf’s Lemma, more precisely its extension from [26], to domains satisfying the cone condition (instead of the ball condition as in the original work by Hopf, see e.g. [28, Theorem 8 in Chapter 2]), which implies that the normal derivative of 푢 at 휉 is positive, contradicting (4.3). ⋆ (u.3): is an immediate consequence of (u.2), due to the boundary conditions in (4.3). ⋆ (u.4): denote by 퐷2푢 the Hessian matrix of 푢 and note that { 푥 ∈ Ω ∶ ∇푢(푥) = 0 } = { 푥 ∈ Ω ∶ ∇푢(푥) = 0 and det 퐷2푢(푥) = 0 } ∪ { 푥 ∈ Ω ∶ ∇푢(푥) = 0 and det 퐷2푢(푥) ≠ 0 } . Modeling crowd dynamics through hyperbolic – elliptic equations 121

The former set has 2-dimensional measure zero by Sard Theorem [29] applied to ∇푢. The latter set consists of isolated points all belonging to the compact set Ω, hence it is finite. Therefore, |{ 푥 ∈ Ω ∶ ∇푢(푥) = 0 }| = 0. ퟑ ⋆ (u.5): we verify that 푢 is 퐂 at the points in Γ푐 under condition (Ω.3). To this aim, we adapt the arguments in [31, Proof of Theorem 3.1], there applied to Poisson equation. Fix 푥표 ∈ Γ푐∩Γ푒, i.e., 푥표 is a doorjamb. Let 휖 be as in (Ω.3), call ℓ = 휖/2 and choose 푥1 ∈ Γ푒 ∩ 퐵(푥표, ℓ) with 푥1 ≠ 푥표. Let 휈 be a unit vector such that 휈 ⋅ (푥1 − 푥표) = 0 and pointing outward Ω at 푥1. Define 푥2 = 푥1 − ℓ휈 and 푥3 = 푥표 − ℓ휈. Call 푅 the open rectangle with vertexes 푥표, 푥1, 푥2, 푥3, denote by 푥푖 푥푗 the open segment

2 푥푖 푥푗 = { 푥 ∈ ℝ ∶ 푥 = (1 − 휃)푥푖 + 휃푥푗, 휃 ∈ ]0, 1[ },

′ and by 픖 the symmetry about the straight line including 푥표푥3 and 푅 = 픖(푅). ′ Define the rectangle ℛ = 푅 ∪ 푥표푥3 ∪ 푅 and consider the problem

x0 x1 2 ⎧−훿 Δ푤(푥) + 푤(푥) = 0 푥 ∈ ℛ R′ ⎪ R Ω 푤(휉) = 1 휉 ∈ 푥표푥1 ∪ 픖(푥표푥1)

⎨ 푤(휉) = 푢(휉) 휉 ∈ 푥1푥2 ∪ 푥2푥3 ⎪ x2 x3 ⎩ 푤(휉) = 푤 (픖(휉)) 휉 ∈ 픖(푥1푥2 ∪ 푥2푥3). Note that the boundary condition is of class 퐂∞ by the regularity of 푢 proved above. The Lax–Milgram lemma ensures that the function 푤 exists, is unique and is in 퐂∞(ℛ; ℝ). By construction, 푤 is symmetric with respect to the straight line 푥표 + ℝ 휈, in the sense that 푤(푥) = 푤 (픖(푥)) for all 푥 ∈ ℛ.

This in turn implies that

∇푤(휉) ⋅ 휈(휉) = 0 for all 푥 ∈ 푥표푥3 .

∞ ∞ Due to the 퐂 regularity of the boundary of ℛ at 푥표, 푤 is of class 퐂 in a neigh- ∞ borhood of 푥표. By uniqueness, 푤 = 푢 on ℛ. Hence, 푢 is of class 퐂 also in a neighborhood of 푥표 restricted to Ω. If 푥표 ∈ (Γ푐 ⧵Γ푒), to prove the regularity of 푢 at 푥표 we proceed as above, simply replacing the Dirichlet condition on 푥표푥1 by a homogeneous Neumann one, applying again the Lax–Milgram lemma and concluding by symmetry and uniqueness. ⋆ (u.6): the characteristic equation det (퐷2푢( ̄푥) − 휆퐼) = 0 in the case of a 2-dimensional problem is a quadratic equation with real solutions 휆1( ̄푥), 휆2( ̄푥) satisfying

2 휆1( ̄푥) 휆2( ̄푥) = det 퐷 푢( ̄푥) , 휆1( ̄푥) + 휆2( ̄푥) = Δ푢( ̄푥) . 122 R. M. Colombo, M. Gokieli, and M. D. Rosini

Note that by the 퐂ퟐ regularity of 푢 proved at (u.1), the equation 푢 = 훿2Δ푢 is −2 satisfied in whole Ω. By (u.2), 휆1( ̄푥) + 휆2( ̄푥) = 훿 푢( ̄푥) > 0, so that at least one of the eigenvalues has to be (strictly) positive. Proof of Proposition 2.2. By (4.2) and straightforward computations it is clear that (1.3) has a solution if and only if (4.3) has a solution which is positive a.e. in Ω. Point (u.1) in Lemma 4.2 ensures the existence and uniqueness of a solution to (4.3). Moreover, by (u.2) in Lemma 4.2 this solution is strictly positive a.e. in Ω. This allows to define 휙 = −훿 ln 푢. The remaining regularity statements and (E.1) follow again from Lemma 4.2 by (4.2). So as to obtain (E.2), note first that −∇휙 ⋅ 휈 = (훿/푢) ∇푢⋅휈 > 0 everywhere on Γ푒 by (4.2) and (u.3) in Lemma 4.2. Then, integrate (4.3) on Ω, use Green’s theorem and again Lemma 4.2 to obtain (E.3).

Proof of Proposition 2.4. The present proof follows from [13, Theorem 2.7]. Indeed, referring to the notation therein, we define 푞(휌) = 휌 푣(휌) and verify the necessary assumptions. 2 ퟑ,휸 (Ω3,γ) Ω is a bounded open subset of ℝ with piecewise 퐂 boundary 휕Ω by (Ω.1) and (Ω.2). (F) This condition is immediate since in the present case we have 퐹 ≡ 0. (f) In our case 푓(푡, 푥, 휌) = 휌 푣(휌) 푤(푥). By (C.1) and the assumption that 푤 is in (퐂ퟐ ∩ 퐖ퟏ,∞) (ℝ; 퐵(0, 1)), we have that 푓 is of class 퐂ퟐ and moreover

′ 휕휌푓(푡, 푥, 휌) = 푞 (휌) 푤(푥) , 2 ″ 휕휌휌푓(푡, 푥, 휌) = 푞 (휌) 푤(푥) , ′ 휕휌∇ ⋅ 푓(푡, 푥, 휌) = 푞 (휌 )∇ ⋅ 푤(푥)

∞ + are all functions of class 퐋 on ℝ × Ω × [0, 푅max].

‖풮 휌 ‖ ≤ (‖휌 ‖ + 푐 푡) exp(푐 푡) by [13, Formula (2.5)] ‖ 푡 표‖퐋∞(Ω;ℝ) ‖ 표‖퐋∞(Ω;ℝ) 2 1

TV(풮푡휌표) ≤ (풜1 + 풜2 푡 + 풜3 TV(휌표)) exp(풜4 푡) by [13, Formula (6.44)] where, with reference to [13, Formula (5.1)] and [13, § 6], the constants 푐1, 푐2, 풜1, …, 풜4 are estimated as follows:

′ 푐1 = 1 + ‖푞 ‖ ∞ ‖∇ ⋅ 푤‖ ∞ 퐋 ([0,푅max];ℝ) 퐋 (Ω;ℝ)

≤ 1 + ‖푞‖ ퟏ,∞ ‖푤‖ ퟏ,∞ , 퐖 ([0,푅max];ℝ) 퐖 (Ω;ℝ) Modeling crowd dynamics through hyperbolic – elliptic equations 123

푐2 = 0 ,

풜1 = 풪(1) ‖퐷푓‖ ∞ 푛×(1+푛) 퐋 (Ω×[0,푅max];ℝ )

≤ 풪(1) ‖푞‖ ퟏ,∞ ‖푤‖ ퟏ,∞ 푛 , 퐖 ([0,푅max];ℝ) 퐖 (Ω;ℝ )

풜2 = 풪(1) ‖퐷푓‖ ퟏ,∞ 푛×(1+푛) 퐖 (Ω×[0,푅max];ℝ )

≤ 풪(1) ‖푞‖ ퟐ,∞ ‖푤‖ ퟐ,∞ 푛 , 퐖 ([0,푅max];ℝ) 퐖 (Ω;ℝ )

′ 풜3 = 풪(1) + ‖푞 ‖ ∞ ‖푤‖ ∞ 푛 퐋 ([0,푅max];ℝ) 퐋 (Ω;ℝ )

≤ 풪(1) + ‖푞‖ ퟏ,∞ ‖푤‖ ∞ 푛 , 퐖 ([0,푅max];ℝ) 퐋 (Ω;ℝ )

풜4 = 풪(1) [1 + ‖퐷푓‖ ퟏ,∞ 푛×(1+푛) ] 퐖 (Ω×[0,푅max];ℝ )

≤ 풪(1) [1 + ‖푞‖ ퟐ,∞ ‖푤‖ ퟐ,∞ 푛 ] 퐖 ([0,푅max];ℝ) 퐖 (Ω;ℝ ) and the above norms of 푞 are bounded by (C.1) and by the adopted assumption on 푤. For technical reasons, below we fix an arbitrary open subset Ω′ of ℝ2 containing Ω and extend the unique generalized solution 휙 ∈ 퐂ퟑ(Ω; ℝ) of (1.3) given in ˜ ퟑ 2 ˜ ˜ Proposition 2.2 introducing a map 휙 ∈ 퐂퐜 (ℝ ;ℝ) such that 휙 ≡ 휙 in Ω and 휙 ≡ 0 in ℝ2 ⧵ Ω′. This is possible thanks to the regularity of 휙 and to the following result.

Lemma 4.3 ([20, Lemma 6.37]). Let Ω satisfy (Ω.1), (Ω.2), (Ω.3). For any open subset Ω′ of ℝ2 such that Ω ⊂ Ω′, there exists a constant 퐶 such that for any 푓 ∈ ퟑ ˜ ퟑ 2 퐂 (Ω; ℝ), there exists a map 푓 ∈ 퐂퐜(ℝ ; ℝ) with

˜ 푓(푥) for all 푥 ∈ Ω ‖ ˜‖ 푓(푥) = { 2 ′ and ‖푓 ‖ ≤ 퐶 ‖푓‖ ퟑ . 0 for all 푥 ∈ ℝ ⧵ Ω 퐂ퟑ(ℝ2;ℝ) 퐂 (Ω;ℝ)

Proof of Proposition 3.1. First, apply Lemma 4.3 and extend 휙 to a 휙˜ ∈ 퐂ퟑ(ℝ2; ℝ). Define ˜푤(푥) = 풩 (−∇휙(푥)˜ ). By (4.1), Lemma 4.3 and Proposition 2.2, ˜푤∈ 퐂ퟎ,ퟏ(ℝ2;ℝ2). Hence, for any fixed ̂푥∈ ℝ2, the Cauchy problem

̇푥= ˜푤(푥) { (4.5) 푥(0) = ̂푥

2 admits a unique solution ˜푝푥̂ ∶ ℝ → ℝ . Define

+ 푇푥̂ = sup { 푡 ∈ ℝ ∶ ˜푝푥̂([0, 푡]) ⊂ Ω } and 푝푥̂(푡) = ˜푝푥̂(푡) for 푡 ∈ [0, 푇푥̂]. 124 R. M. Colombo, M. Gokieli, and M. D. Rosini

By construction, the map 푝푥̂ solves (3.1). By the standard theory of ordinary differential equations, (Q.1) and (Q.2) are proved. We consider now (Q.3). Note that (4.5) is dissipative in Ω, in the sense that 휙˜ is a (strict) Lyapunov function for (4.5) in Ω, i.e., 휙˜ decreases along the path 푡 → 푝푥̂(푡) as long as 푝푥̂(푡) ∈ Ω. In fact, as long as 푝푥̂(푡) ∈ Ω

−1/2 d d 2 2 휙˜(푝 (푡)) = 휙 (푝 (푡)) = − (휃2 + ‖∇휙 (푝 (푡))‖ ) ‖∇휙 (푝 (푡))‖ , d푡 푥̂ d푡 푥̂ 푥̂ 푥̂ which is strictly negative whenever ̂푥 is not a critical point. By LaSalle’s Prin- ciple [21, Theorem 9.22, see also Lemma 9.21 and Theorem 14.17], as 푡 goes to infinity, every bounded path 푝푥̂ that remains in Ω is attracted towards the set of equilibria, i.e., of critical points of (4.5). More precisely, setting

there exists (푡 ) such that 휔( ̂푥) = { 푥 ∈ ℝ2 ∶ 푛 푛∈ℕ } lim푛→+∞ 푡푛 = +∞ and lim푛→+∞ 푝푥̂(푡푛) = 푥 ˜ 2 퐸퐷 = { 푥 ∈ 퐷 ∶ ∇휙(푥) = 0 } for 퐷 ⊆ ℝ we proved that if 푥 ∈ 휔( ̂푥) ∩ Ω for a ̂푥∈ Ω, then ∇휙(푥) = 0. Note that for any ̂푥∈ Ω, the path 푥̃푝 ̂ exiting ̂푥 does not intersect Γ푤. Indeed, by the boundary condition imposed along Γ푤 in (1.3)

Γ푤 = { 푥 ∈ Γ푤 ∶ ∇휙(푥) = 0 } ∪ { 푥 ∈ Γ푤 ∶ ∇휙(푥) ≠ 0 and ∇휙(푥) ⋅ 휈(푥) = 0 } . The former set above is clearly invariant, both positively and negatively, with respect to (4.5), hence it can not be reached by a path 푡 → 푝푥̂(푡) starting in Ω. The latter consists of trajectories solving (4.5) that are entirely contained in Γ푤, since 푤 is parallel to Γ푤. As a consequence, for any ̂푥∈ Ω, either the path 푡 → 푝푥̂(푡) crosses Γ푒, or it stays in Ω and approaches a point in the set 퐸Ω, namely 휔( ̂푥) ⊆ 퐸Ω. It remains to determine the behavior of the system near the critical points in 퐸Ω. We proceed by linearisation around ,̄푥 with ∇휙( ̄푥) = 0. Denote by 퐴( ̄푥) the first order total derivative of 푁(−∇휙) computed at ̄푥∈ 퐸Ω. By direct computations, 1 퐴( ̄푥) = 퐷푁 (−∇휙( ̄푥)) = − 퐷2휙( ̄푥) , (4.6) 휃 thanks to ∇휙( ̄푥) = 0. Recall the map 푢 given by (4.2). Due to (4.3) and (4.6) we have 1 훿 퐴( ̄푥) = 퐷2푢( ̄푥) , 휃 푢( ̄푥) proving that 퐴( ̄푥) is symmetric and diagonalizable. By (u.6) in Lemma 4.2, 퐴( ̄푢) has at least one strictly positive eigenvalue, say 휆2 > 0. Consider now two cases, depending on the value attained by the other eigenvalue 휆1: Modeling crowd dynamics through hyperbolic – elliptic equations 125

⋆ 휆1 ≠ 0: Then, by the Hartman–Grobman theorem, see e.g. [21, Theorem 9.35], depending on the sign of 휆1, ̄푥 is either a source or a saddle. In both cases, it is an isolated point of 퐸Ω, so that ̄푥∈ 휔( ̂푥) implies { ̄푥} = 휔( ̂푥), by the connectedness of 휔( ̂푥). This is possible only if 휆1 < 0, i.e., ̄푥 is a saddle, and ̂푥 belongs to the stable manifold consisting of two trajectories entering ,̄푥 which is a set of measure zero.

⋆ 휆1 = 0: Then, ̄푥 is not necessarily an isolated point of ℰΩ. We use here the result of Palmer [27] about the local central manifold, which is an invariant 1-dimensional set containing all possible critical points in a neighborhood of .̄푥 This result can be seen as a generalization of the Hartman–Grobman theorem, and gives the instability of the central manifold, see also [3, § 4], [8, § 9.2-9.3], and [21, Theorem 10.14]. Let 퐵 be the change of coordinates matrix such that 퐵퐴( ̄푥)퐵−1 is diagonal, with 퐴( ̄푥) given in (4.6). By means of the linear change of variables 푦(푡) = 퐵 (푝푥̂(푡) − ̄푥), the differential equation in (4.5) can be written as 푦̇ = 푓 (푦 , 푦 ) { 1 1 1 2 (4.7) 푦2̇ = 휆2 푦2 + 푓2(푦1, 푦2), where 푓 ∈ 퐂ퟐ(ℝ2;ℝ2) is bounded, see Lemma 4.3, and satisfies 푓(0) = 0. The dependence of 퐵, 푓 and 휆2 upon ̄푥 is here neglected. We obtain from [27] that there exist a Lipschitz continuous function ℎ and a homeomorphism 퐻∶ ℝ+ ×ℝ2 → ℝ2, such that the graph of ℎ is the local central manifold and the map 푧(푡) = 퐻 (푡, 푦(푡)), with 퐻(푡, 0) = 0, solves ̇푧 = 푓 (푧 , ℎ(푡, 푧 )) { 1 1 1 1 (4.8) 2̇푧 = 휆2 푧2 , provided 푦 solves (4.7). As a matter of fact, ℎ can be proved to be also 퐂ퟐ, see [3, Proposition 4.1] or [21, Theorem 10.14]. Then, by continuity of 퐻, there exists 푟0 > 0 such that if ‖푦(푡)‖ < 푟0, then |푧2(푡)| = |퐻2 (푡, 푦(푡))| < |푧2(0)|. Solving the second equation in (4.8), we obtain that for 푦(0) such that 푧2(0) = 퐻2 (0, 푦(0)) ≠ 0, there exists 푡∗ > 0 such that ‖푦(푡)‖ > 푟0 for all 푡 > 푡∗. Going back to the original 푥-variable, for any neighborhood 푂 of ̄푥 with 푂 ⊆ ℝ2, introduce

푊 = { 푥 ∈ 푂 ∶ 퐻2 (0, 퐵(푥 − ̄푥)) = 0 } .

We have obtained that if ̂푥∈ 푂 ⧵ 푊, then 푝푥̂(푡) is outside 풪 for all 푡 > 푡∗. Thus, ̄푥 can be attractive only for the points lying on 푊, which is clearly a 1-dimensional manifold and has 2-dimensional Lebesgue measure equal to 0. Moreover, 푊 as a whole is repulsive. Therefore, 휔( ̂푥) ∩ 푊 is non-empty only if the path passing through ̂푥 lies inside 푊. Hence the 1-dimensional Lebesgue measure of 휔( ̂푥) ∩ 푊 is 0. 126 R. M. Colombo, M. Gokieli, and M. D. Rosini

+ Finally, for almost all ,̂푥 the path 푝푥̂(ℝ ) given by (4.5) is not attracted by 퐸Ω, hence it has to reach the exit Γ푒, i.e., there exists a positive finite time 푇푥̂ such that 푝푥̂(푇푥̂) ∈ Γ푒.

Acknowledgments. The authors were supported by the INDAM–GNAMPA project Leggi di conservazione nella modellizzazione di dinamiche di aggregazione. MDR acknowledges support from Università degli Studi di Ferrara Project 2017 “FIR: Modelli macroscopici per il traffico veicolare o pedonale”, from INdAM– GNAMPA Project 2017 “Equazioni iperboliche con termini nonlocali: teoria e modelli” and from the National Science Centre, Poland, Project “Mathematics of multi-scale approaches in life and social sciences” No. 2017/25/B/ST1/00051.

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[28] Murray H Protter and Hans F Weinberger, Maximum principles in differential equations, Springer, 1984. [29] Arthur Sard et al., The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc 48 (1942), no. 12, 883–890. [30] M. Twarogowska, P. Goatin, and R. Duvigneau, Macroscopic modeling and simulations of room evacuation, Applied Mathematical Modelling 38 (2014), no. 24, 5781 – 5795. [31] Evgenii Alekseevich Volkov, Differentiability properties of solutions of boundary value problems for the Laplace and Poisson equations on a rectangle, Trudy Matematicheskogo Instituta im. VA Steklova 77 (1965), 89–112. [32] A. I. Vol′pert, Spaces 퐵푉 and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302. MR 0216338 (35 #7172) On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type

Félix del Teso, Jørgen Endal, and Espen R. Jakobsen

Dedicated to Helge Holden, who never stops inspiring us, on the occasion of his 60th birthday

Abstract. We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of theform

휕푡푢 − 퐴휑(푢) = 0.

These equations are possibly degenerate nonlinear diffusion equations with a general non- decreasing continuous nonlinearity 휑, and the largest class of linear symmetric nonlocal diffusion operators 퐴 considered so far. The operators are defined from a bilinear energy form ℰ and may be degenerate and have some 푥-dependence. The fractional Laplacian, symmetric finite differences, and any generator of symmetric pure jump Lévy processes are included. The main results are (i) an Oleĭnik type uniqueness result for energy solutions; (ii) an existence (and uniqueness) result for distributional solutions with finite energy; and (iii) equivalence between the two notions of solution, and as a consequence, new well- posedness results for both notions of solutions. We also obtain quantitative energy and related 퐿푝-estimates for distributional solutions. Our uniqueness results are given for a class of functions defined from test functions by completion in a certain topology. We study rigorously several cases where this space coincides with standard function spaces. In particular, for operators comparable to fractional Laplacians, we show that this space is a parabolic homogeneous fractional Sobolev space.

1. Introduction

In this paper we study uniqueness and existence of solutions with finite energy of the following two related Cauchy problems of nonlocal porous medium type,

휆 휕푡푢 − 퐴 [휑(푢)] = 0 in 푄푇 ≔ ℝ × (0, 푇), (1.1)

푢(푥, 0) = 푢0(푥) on ℝ , (1.2) 130 F. del Teso, J. Endal, and E. R. Jakobsen and

휇 휕푡푢 − ℒ [휑(푢)] = 0 in 푄푇, (1.3)

푢(푥, 0) = 푢0(푥) on ℝ , (1.4) where 푢 = 푢(푥, 푡) is the solution, 푇 > 0, 퐴휆 and ℒ휇 are nonlocal (convection-) diffusion operators, the nonlinearity 휑 is any continuous non-decreasing function, 1 ∞ and 푢0 ∈ 퐿 ∩ 퐿 . The problems are nonlinear degenerate parabolic, and include the fractional porous medium equations [28], where ℒ휇 = −(−Δ)훼/2 and 휑(푢) = 푢|푢|푚−1 for 훼 ∈ (0, 2) and 푚 > 0. Included are also Stefan problems, filtration equations, and generalized porous medium equations. See the introductions of [28, 26, 22] for more information. Both problems are connected to a bilinear energy form defined as 1 ℰ [푓, 푔] ≔ ∬ (푓(푦) − 푓(푥))(푔(푦) − 푔(푥)) Λ(d푥, d푦), (1.5) 휆 2 ℝ푁×ℝ푁⧵퐷 where 퐷 ≔ { (푥, 푥) ∶ 푥 ∈ ℝ } is the diagonal and Λ is a nonnegative Radon 휆 measure on ℝ × ℝ ⧵ 퐷. The operator 퐴 is the generator of ℰ휆 defined by

휆 ℰ휆[푓, 푔] = − ∫ 푓퐴 [푔] d푥 (1.6) ℝ푁 (see Corollary 1.3.1 in [31]), while ℒ휇 = 퐴휆 for the special case where Λ = 휇(푥 + d푦) d푥. In general 퐴휆 is symmetric, 푥-dependent, and has no closed expression, while ℒ휇 is an 푥-independent operator with integral representation

휇 ℒ [휙](푥) = ∫ (휙(푥 + 푧) − 휙(푥) − 푧 ⋅ 퐷휙(푥)ퟏ|푧|≤1) 휇(d푧), (1.7) ℝ푁⧵{0} where 퐷 is the gradient, ퟏ|푧|≤1 an indicator function, and 휇 a symmetric (even) non-positive Lévy measure satisfying ∫ |푧|2 ∧ 1 휇(d푧) < ∞. The operator ℒ휇 is nonnegative and symmetric, and the fractional Laplacian is an example. A first warning is that 퐴휆 is not a pure diffusion operator in general: Under density and symmetry assumptions on Λ, 퐴휆 will have an integral representation like (1.7) with 푥-dependent 휇 plus an additional drift term! A second warning is that the 푥-dependence in 퐴휆 is restricted, e.g. −푎(푥)(−Δ)훼/2 is not covered! We refer to Section 2.1 for precise assumptions and to Section 2.4 for a discussion and examples of 퐴휆. This work was inspired by the two recent papers [26] and [22] (see also [23]), which contain well-posedness results for energy (or weak) solutions of (1.1)–(1.2) and distributional (or very weak) solutions of (1.3)–(1.4) respectively. These very Nonlocal equations of porous medium type 131 general results require different techniques and formulations. The uniqueness argument of [22] is based on a complicated resolvent approximation procedure of Brézis and Crandall [18], while in [26] it is based on an easier and more direct argument by Oleĭnik et al. [34]. The first part of this paper is devoted to Oleĭnik type uniqueness arguments for (1.1)–(1.2). We try to push this argument as far as possible, and in the process we extend some of the results and arguments of [26]. E.g., we remove absolute continuity, symmetry, and comparability assumptions. We also discuss the appli- cability and limitations of the method. Our uniqueness results are given for a class of functions defined from test functions by completion in a certain topology. We study rigorously several cases where this space coincides with standard function spaces. In particular, for operators (globally) comparable to fractional Laplacians, we show that this space is a parabolic homogeneous fractional Sobolev space. In an appendix we also provide rigorous definitions and results of these spaces, some of which we were not able to find in the literature. In the second part of the paper we study the equivalence between energy and distributional formulations in the setting of (1.3)–(1.4). A main result is a new existence result for distributional solutions with finite energy. This existence result and the uniqueness result of [22] is then transported from distributional solutions to energy solutions by equivalence, while the Oleĭnik uniqueness results of the first part is transported in the other direction. These result are all either new, orfor the Oleĭnik results, represent a much simpler approach to obtaining uniqueness compared to [22]. At the end, we give several new quantitative energy and related 퐿푝-estimates for distributional solutions. The type of bilinear form defined in (1.5) plays a central role in probability theory. It is associated with a Dirichlet form and a corresponding symmetric Markov process, see e.g. [31] for a general theory. The type of “nonlocal” bilinear form we consider here is similar to those studied in e.g. [37, 5]. In the linear case (휑(푢) = 푢), equations (1.1) and (1.3) are (at least formally) Kolmogorov equations for the transition probability densities of the corresponding Markov processes (see e.g. Section 3.5.3 in [4]). Let us now give a brief summary of previous works on (1.1)–(1.2) and (1.3)–(1.4). We focus first on the 푥-dependent equation (1.1). In the linear case there is a large amount of literature. Some of the main trends in the more PDE oriented community are described in the two surveys [33, 39] (along with extensions to other types of nonlinear equations). When 휑 is nonlinear, we are not aware of any other result than the ones presented in [26]. There the authors consider operators 퐴휆 where the densities of the measures are comparable to the density of the fractional Laplacian. Existence and uniqueness is discussed in the first part, but the main 132 F. del Teso, J. Endal, and E. R. Jakobsen focus of the paper is to prove continuity/regularity and long time asymptotics for energy solutions. There is a vast literature on special cases of (1.3)–(1.4). In the linear fractional 훼/2 case 휕푡푢 + (− Δ) 푢 = 0 for 훼 ∈ (0, 2), we have well-posedness even for measure data and solutions growing at infinity [6, 13]. If we replace (− Δ)훼/2 by an operator ℒ whose measure has integrable density, well-posedness results can be found in [17]. In the case of the fractional porous medium equation (see above), existence, uniqueness and a priori estimates are proven for (strong) 퐿1-energy solutions in [27, 28]. We also mention that there are results for that equation in weighted 퐿1-spaces [14], with logarithmic diffusion (휑(푢) = log(1+푢))[29], singular or ultra fast diffusions [11], weighted equations with measure data [32], and problems on bounded domains [12, 15, 16]. There are other ways to investigate these equations: In [10, 19, 40, 9, 42], the authors consider a so-called porous medium equation with fractional pressure, and in [3] they consider bounded diffusion operators that can be represented by non-singular integral operators on the form (1.7). Finally, we mention that in the presence of (nonlinear) convection in (1.3)–(1.4), additional entropy conditions are needed to have uniqueness [1, 20, 21]; a counterexample for uniqueness of distributional solutions is given in [2].

Outline. In Section 2 we state the assumptions and present and discuss our main results. The main uniqueness result is proven in Section 3. Properties such as equivalence of distributional and energy solutions, existence of distributional solutions with finite energy, and energy and 퐿푝-estimates are finally proven in Section 4. In Appendices A, B, and C we give rigorous results on the Sobolev spaces used in this paper along with the proofs of characterizations of the uniqueness function class in terms of common function spaces.

Notation. We use the same notation as in [22] except for the ones we explicitly mention here: The (Borel) measure 휇 is said to be even if 휇(퐵) = 휇(−퐵) for all Borel sets 퐵. We say that the (Borel) measure Λ(d푥, d푦) is symmetric if Λ(d푥, d푦) = Λ(d푦, d푥).A kernel 휆(푥, d푦) on ℝ × ℬ(ℝ ⧵ {푥}) satisfies: (i) 퐵 ↦ 휆(푥, 퐵) is a positive measure on ℬ(ℝ ⧵ {푥}) for each fixed 푥 ∈ ℝ ; and (ii) 푥 ↦ 휆(푥, 퐵) is a Borel measurable function for every 퐵 ∈ ℬ(ℝ ⧵ {푥}). An operator 퐿 is symmetric 2 on 퐿 if (푢, 퐿푣)퐿2 = (퐿푢, 푣)퐿2. From the bilinear form ℰ휆 defined in (1.5) we define a seminorm (the energy) and a space,

|푓|2 ≔ ℰ [푓, 푓], 퐸휆 휆

퐸휆(ℝ ) ≔ { 푓∶ ℝ → ℝ ∶ 푓 is measurable and |푓|퐸휆 < ∞ }, Nonlocal equations of porous medium type 133 and the related parabolic (energy) seminorm and space,

푇 |푓|2 ≔ ∫ |푓(⋅, 푡)|2 d푡, 푇,퐸휆 퐸휆 0 2 퐿 (0, 푇; 퐸휆(ℝ )) ≔ { 푓∶ 푄푇 → ℝ ∶ 푓 is measurable and |푓|푇,퐸휆 < ∞ }. The Cauchy–Schwartz inequality holds in this setting (cf. Lemma 3.1):

| 푇 | | | ∫ ℰ휆[푓(⋅, 푡), 푔(⋅, 푡)] d푡 ≤ |푓|푇,퐸휆|푔|푇,퐸휆. | 0 |

2. Main results

In this section we give the assumptions, main results, and a discussion of these. There are two sections with results. Section 2.2 contains a sequence of uniqueness results for energy solutions of (1.1)–(1.2), while Section 2.3 contains results about (1.3)–(1.4). There we prove the equivalence of energy and distributional solutions with finite energy, the existence of the latter type of solutions, and transport uniqueness and existence results between the two formulations. The results we obtain are either new or represent a much more efficient way to obtain such results compared to previous arguments.

2.1. Assumptions. We start with the bilinear form ℰ휆 defined in (1.5). To have a more practical formulation of the assumptions, we first rewrite (1.5): We assume that Λ has as kernel 휆̃ ≥ 0 with respect to d푥, Λ(d푥, d푦) = 휆(푥,̃ d푦) d푥, change variables 푦 → 푥 + 푧, and set 휆(푥, d푧) ≔ 휆(푥,̃ 푥 + d푧) to obtain

1 ℰ [푓, 푔] = ∫ ∫ (푓(푥 + 푧) − 푓(푥))(푔(푥 + 푧) − 푔(푥)) 휆(푥, d푧) d푥. (2.1) 휆 2 ℝ푁 |푧|>0

Our assumptions on ℰ휆 can then be formulated as follows:

̃ (Aλ0) Λ has as kernel 휆 ≥ 0 on ℝ × ℬ(ℝ ⧵ {푥}),

Λ(d푥, d푦) = 휆(푥,̃ d푦) d푥.

̃ (Aλ1) The translated kernel 휆(푥, d푧) ≔ 휆(푥, 푥 + d푧) satisfies

2 1 (i) Σ휆(푥) ≔ ∫ |푧| 휆(푥, d푧) ∈ 퐿loc(ℝ ); and 0<|푧|≤1 134 F. del Teso, J. Endal, and E. R. Jakobsen

1 (ii) Π휆(푥) ≔ ∫ 휆(푥, d푧) ∈ 퐿loc(ℝ ). |푧|>1

(Aλ2) Λ is symmetric,

∫ ∫ Λ(d푥, d푦) = ∫ ∫ Λ(d푥, d푦) for all Borel 퐴 ×퐵 ⊂ ℝ ×ℝ ⧵퐷. 퐴 퐵 퐵 퐴

In some results, we need to strengthen assumption (Aλ1).

(A’λ1) Assumption (Aλ1) holds and in addition

∞ (i) Π휆 ∈ 퐿 (ℝ ); and (ii) 휆(푥, d푧) is locally shift-bounded: For some constant 퐶 > 0,

휆(푥 + ℎ, 퐵) ≤ 퐶휆(푥, 퐵) for all 푥, ℎ ∈ ℝ with |ℎ| ≤ 1 and Borel 퐵 ⊂ 퐵(0, 1) ⧵ {0}.

(A”λ1) Assumption (Aλ1) holds and in addition 푐 d푧 푚휇 (d푧) ≤ 휆(푥, d푧) ≤ 푀휇 (d푧) where 휇 (d푧) = ,훼 , 훼 훼 훼 |푧|+훼

for some 0 < 푚 ≤ 푀, 훼 ∈ (0, 2), and every 푥 ∈ ℝ .

The remaining assumptions we will use in this paper are given below.

(Aμ) 휇 ≥ 0 is an even Radon measure on ℝ ⧵ {0} satisfying

∫ |푧|2 휇(d푧) + ∫ 1 휇(d푧) < ∞. |푧|≤1 |푧|>1

(Aφ) 휑 ∶ ℝ → ℝ is continuous and non-decreasing.

1 ∞ (Au0 ) 푢0 ∈ 퐿 (ℝ ) ∩ 퐿 (ℝ ).

∞ Remark 2.1. (a) By (Aλ0) and (Aλ1), ℰ휆[⋅, ⋅] is well-defined on 퐶c (ℝ ), nonnegative and symmetric,

∞ ℰ휆[푓, 푓] ≥ 0 and ℰ휆[푓, 푔] = ℰ휆[푔, 푓] for 푓, 푔 ∈ 퐶c (ℝ ).

∞ Moreover, by Example 1.2.4 in [31], (ℰ휆, 퐶c (ℝ )) is a closable Markovian form on 퐿2(ℝ ) and its closure a regular Dirichlet form. Nonlocal equations of porous medium type 135

(b) It is easy to check that (Aλ1” ) ⇒ (Aλ1’ ) ⇒ (Aλ1), see also the remarks on locally 휆 shift-bounded kernels in Section 2.4. Assumption (Aλ1” ) implies that 퐴 is 훼/2 comparable to −(−Δ) , and local shift-boundedness in (Aλ1’ ) is used to show that functions with finite energy can be approximated by test functions (cf. Theorem 2.6).

휇 2 ∞ (c) By (Aμ), the operator ℒ defined by (1.7) is well-defined on 퐶 (ℝ ) ∩ 퐿 (ℝ ), non-positive and symmetric. The generator of any symmetric pure jump Lévy process is included, like, e.g., the fractional Laplacian and symmetric finite difference operators.

(d) If 휆(푥, d푧) = 휇(d푧), then (Aμ) implies (A휆0), (A휆1’), and (A휆2). The first two are trivial, while the third follows by, e.g., Lemma 6.4 in [22].

(e) Without loss of generality we can assume 휑(0) = 0 (by adding a constant).

2.2. Uniqueness results for energy solutions. In this section we give several uniqueness results for energy (or weak) solutions of (1.1)–(1.2). These results follow from an extension of the Oleĭnik argument.

1 Definition 2.2 (Energy solutions). A function 푢 ∈ 퐿loc(푄푇) is an energy solution of (1.1)–(1.2) if

2 (i) 휑(푢) ∈ 퐿 (0, 푇; 퐸휆(ℝ )); and

∞ (ii) for all 휓 ∈ 퐶c (ℝ × [0, 푇)),

푇

∫ (∫ 푢휕푡휓 d푥 − ℰ휆[휑(푢), 휓]) d푡 + ∫ 푢0(푥)휓(푥, 0) d푥 = 0. 0 ℝ푁 ℝ푁

Remark 2.3.

(a) The integrals in (ii) are well-defined by (Aλ0), (Aλ1’ ), (Au0 ), and the regularity of 푢 and 휑(푢). From (ii) it follows that the initial condition 푢0 is assumed in the distributional sense (푢0 is a weak initial trace):

∞ ess lim ∫ 푢(푥, 푡)휓(푥, 푡) d푥 = ∫ 푢0(푥)휓(푥, 0) d푥 ∀휓 ∈ 퐶c (ℝ × [0, 푇)). 푡→0+ ℝ푁 ℝ푁

2 (b) By the support of the test functions, we could take 퐿loc([0, 푇); 퐸휆(ℝ )) in (i).

To state the uniqueness results, we will introduce spaces in which the Oleĭnik argument works. A particular requirement is that test functions are dense in these 136 F. del Teso, J. Endal, and E. R. Jakobsen spaces with respect to the weakest convergence that can be used in the proof. This is encoded in the following space:

∞ 2 푋 ≔ { 푓 ∈ 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휆(ℝ )) ∶

∞ there exists {휓푛}푛∈ℕ ⊂ 퐶c (ℝ × [0, 푇)) such that

|휓푛 − 푓|푇,퐸휆 → 0 as 푛 → ∞ and

1 ∬ 휓푛휙 d푥 d푡 → ∬ 푓휙 d푥 d푡 for all 휙 ∈ 퐿 (푄푇) as 푛 →∞ }. 푄푇 푄푇 Below we show that limits can be avoided to get more useful characterizations of such spaces if we (i) go to subspaces, e.g.

2 2 ∞ 2 푋 ∩ 퐿 (푄푇) = 퐿 (푄푇) ∩ 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휆(ℝ )); (2.2) or (ii) restrict the operator by assuming (A”λ1) which implies

∞ 2 푋 = 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휇훼(ℝ )). (2.3) We refer to Theorem 2.6 below for precise statements. Our most general uniqueness result applies to energy solutions in the following class of functions:

1 ∞ 풰풞 ≔{ 푢 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∶ 휑(푢) ∈ 푋 }.

Theorem 2.4 (Uniqueness 1). Assume (Aφ), (Aλ0), (Aλ1), and (Au0 ). Then there is at most one energy solution 푢 of (1.1)–(1.2) in 풰풞. A proof can be found in Section 3. Remark 2.5. A similar but less general uniqueness result is given by Theorem 1.1 in [26]. They assume that 휆(푥, d푧) is absolutely continuous with a density comparable to the Lévy measure of the 훼-stable process, and hence 퐴휆 is comparable to 훼/2 −(−Δ) . In this case (A’λ1) is satisfied in view of the discussion in Section 2.4. Note that in general the uniqueness class 풰풞 is smaller than the natural existence class 1 ∞ 2 ℰ풞 ≔ { 푢 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∶ 휑(푢) ∈ 퐿 (0, 푇; 퐸휆(ℝ )) }. This is an intrinsic problem with the Oleĭnik argument when it is extended to such general settings as we consider here, and it is also observed in [26]. However, the two classes may coincide under additional assumptions, e.g., if 휑(푢) also belongs to 퐿2 ∩ 퐿∞ or if 퐴휆 is comparable to −(−Δ)훼/2. This is a consequence of the following result. Nonlocal equations of porous medium type 137

Theorem 2.6. Assume (Aλ0) and (Aλ2).

(a) If (A’λ1) holds, then (2.2) holds.

(b) If (A”λ1) holds, then (2.3) holds. The proofs are given in Appendices A and C respectively. See also Section 2.4 for a possible alternative based on recurrence. By Theorem 2.4 and Theorem 2.6, we now have:

Corollary 2.7 (Uniqueness 2). Assume (Aλ0), (Aλ2), (Aφ), and (Au0 ) hold.

(a) If (Aλ1’ ) holds, then there is at most one energy solution 푢 of (1.1)–(1.2) such 2 that 푢 ∈ ℰ풞 and 휑(푢) ∈ 퐿 (푄푇).

(b) If (Aλ1” ) holds, then there is at most one energy solution 푢 of (1.1)–(1.2) such that 푢 ∈ ℰ풞.

Remark 2.8. When the operator 퐴휆 is comparable to the fractional Laplacian 훼/2 −(−Δ) for 훼 ∈ (0, 2) (i.e. (Aλ1” ) holds), the uniqueness and existence classes coincide, and if if 푁 > 훼 they satisfy

1 ∞ 2 훼/2 풰풞 = ℰ풞 = { 푢 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∶ 휑(푢) ∈ 퐿 (0, 푇; 퐻̇ (ℝ )) }. (2.4)

The latter space is often used in the porous medium setting [45, 28], see also [26]. See Appendix B for rigorous definitions and properties of the homogeneous fractional Sobolev spaces 퐻̇ 훼/2(ℝ ) and 퐿2(0, 푇; 퐻̇ 훼/2(ℝ )), some of these we were not able to find in the literature. ∞ ∞ Note that if (Aφ) holds and 푢 ∈ 퐿 (푄푇), then 휑(푢) ∈ 퐿 (푄푇). Now let 훽 ∈ (0, 1] and assume 휑 is locally 훽-Hölder continuous at 0:

|휑(푠) − 휑(0)| sup < ∞ for all 푅 > 0. (2.5) 훽 |푠|<푅 |푠|

∞ Then, since 휑(0) = 0 and 푢 ∈ 퐿 (푄푇),

2훽 2 푢 ∈ 퐿 (푄푇) ⟹ 휑(푢) ∈ 퐿 (푄푇).

1 ∞ 2훽 1 By interpolation, functions 푢 ∈ 퐿 ∩ 퐿 belongs to 퐿 for 훽 ∈ [ 2 , 1]. This leads us to our next result:

Corollary 2.9 (Uniqueness 3). Assume (Aφ), (Aλ0), (Aλ1’ ), (Aλ2), and (Au0 ) hold. If 1 in addition (2.5) holds for some 훽 ∈ [ 2 , 1], then there is at most one energy solution 푢 of (1.1)–(1.2) such that 푢 ∈ ℰ풞. 138 F. del Teso, J. Endal, and E. R. Jakobsen

Now we specialize to the case 휆(푥, d푧) = 휇(d푧) and 퐴휆 = ℒ휇. Equation (1.1) then becomes equation (1.3). From all the above uniqueness results and Remark 2.1 (d) we obtain the following uniqueness results for (1.3)–(1.4).

Corollary 2.10 (Uniqueness 4). Assume (Aφ), (Aμ), and (Au0 ) hold.

(a) There is at most one energy solution 푢 of (1.3)–(1.4) in 풰풞.

(b) There is at most one energy solution 푢 of (1.3)–(1.4) such that 푢 ∈ ℰ풞 and 2 휑(푢) ∈ 퐿 (푄푇).

1 (c) If in addition (2.5) holds for some 훽 ∈ [ 2 , 1], then there is at most one energy solution 푢 of (1.3)–(1.4) such that 푢 ∈ ℰ풞.

2.3. Equivalence with distributional solutions and consequences. In this section we study the connection between distributional (or very weak) solutions and energy (or weak) solutions. We focus on the simpler case where 퐴휆 = ℒ휇, and hence the measure 휆(푥, d푧) = 휇(d푧) is independent of 푥. In other words, we consider the Cauchy problem (1.3)–(1.4). In general, 퐴휆 will have an additional drift/convection term compared to ℒ휇, see Section 2.4. This gives rise to a nonlinear convection term in the equation and the possibility that solutions develop shocks (cf. e.g. [20] and references therein). Whether this happens or not here is not known and another reason to avoid this case now. We state an equivalence result for the two solution concepts, existence and uniqueness results for distributional solutions with finite energy, and then transport these results from distributional solutions to energy solutions. The uniqueness results of the previous section are transported in the opposite direction, and the different uniqueness results are then compared. We also give quantitative energy and related 퐿푝-estimates for distributional solutions.

1 Definition 2.11 (Distributional solutions). A function 푢 ∈ 퐿loc(푄푇) is a distributional solution of (1.3)–(1.4) if

휇 ∬ (푢휕푡휓 + 휑(푢)ℒ [휓]) d푥 d푡 + ∫ 푢0(푥)휓(푥, 0) d푥 = 0 푁 푄푇 ℝ ∞ ∀휓 ∈ 퐶c (ℝ × [0, 푇)).

The integral is well-defined under the assumptions (Aφ), (Aμ), and (Au0 ) if also 휑(푢) ∈ 퐿∞ (which is the case when 푢 ∈ 퐿∞). This weaker notion of solutions does not require finite energy, but when the energy is finite, the two notionsof solutions will be equivalent. Nonlocal equations of porous medium type 139

∞ Theorem 2.12 (Equivalent notions of solutions). Assume (Aφ), (Aμ), 푢0 ∈ 퐿 (ℝ ), ∞ and 푢 ∈ 퐿 (푄푇). Then the following statements are equivalent: (a) 푢 is an energy solution of (1.3)–(1.4). (b) 푢 is a distributional solution of (1.3)–(1.4) such that

2 휑(푢) ∈ 퐿 (0, 푇; 퐸휇(ℝ )).

We prove this result in Section 4.1. In the setting of this paper, it turns out that there always exists distributional solutions with finite energy.

Theorem 2.13 (Existence 1). Assume (Aφ), (Aμ), and (Au0 ). Then there exists a distributional solution 푢 of (1.3)–(1.4) satisfying 1 ∞ 1 (i) 푢 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∩ 퐶([0, 푇]; 퐿loc(ℝ )) and 2 (ii) 휑(푢) ∈ 퐿 (0, 푇; 퐸휇(ℝ )). This is one of the main results of this paper and will be proven at the end of Section 4.2. For such solutions we have a new uniqueness result by equivalence, Theorem 2.12, and the uniqueness result for energy solutions in Corollary 2.10.

Corollary 2.14 (Uniqueness 5). Assume that (Aφ), (Aμ), and (Au0 ) hold. (a) There is at most one distributional solution 푢 of (1.3)–(1.4) in 풰풞. (b) There is at most one distributional solution 푢 of (1.3)–(1.4) such that 푢 ∈ ℰ풞 2 and 휑(푢) ∈ 퐿 (푄푇). 1 (c) If in addition (2.5) holds for some 훽 ∈ [ 2 , 1 , then there is at most one distributional solution 푢 of (1.3)–(1.4) such that 푢 ∈ ℰ풞. Note that we have uniqueness in a smaller class than we have existence for by Theorem 2.13. This uniqueness result should also be compared to our recent general uniqueness result from [22].

Theorem 2.15 (Uniqueness 6, Theorem 2.8 in [22]). Assume (Aφ), (Aμ), and (Au0 ). Then there is at most one distributional solution 푢 of (1.3)–(1.4) satisfying

1 ∞ 1 푢 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∩ 퐶([0, 푇]; 퐿loc(ℝ )). In particular, any solution from Theorem 2.13 is unique. This result is more general than Corollary 2.14, but the proof is also more complicated. When Corollary 2.14 applies, a greatly simplified uniqueness argument is available (as we have seen). In view of the equivalence in Theorem 2.12, we can also transport results in the other direction: from distributional solutions to energy solutions. First we obtain a new existence result as an immediate consequence of Theorem 2.13. 140 F. del Teso, J. Endal, and E. R. Jakobsen

Corollary 2.16 (Existence 2). Assume (Aφ), (Aμ), and (Au0 ). Then there exists an energy solution 푢 of (1.3)–(1.4) satisfying

1 ∞ 1 푢 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∩ 퐶([0, 푇]; 퐿loc(ℝ )). In the case of 푥-independent operators, this existence result is much more general than the result given in Theorem 1.1 in [26]. Uniqueness results for energy solutions of (1.3)–(1.4) are given in Corollary 2.10. These results hold for a smaller class of functions than the above existence results. However, a (more) general uniqueness result can be obtained from the result for distributional solutions in Theorem 2.15 and equivalence.

Corollary 2.17 (Uniqueness 6). Assume (Aφ), (Aμ), and (Au0 ). Then there is at most one energy solution 푢 of (1.3)–(1.4) satisfying

1 ∞ 1 푢 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∩ 퐶([0, 푇]; 퐿loc(ℝ )). The proof is immediate. The solutions of Corollary 2.16 are therefore unique, and this result is stronger than the Oleĭnik type result Corollary 2.10. In view of the well-posedness of both energy and distributional solutions and the equivalence between the two notions of solutions, we now have a full equivalence result under assumptions (Aφ), (Aμ), and (Au0 ).

Corollary 2.18 (Equivalent notions of solutions 2). Assume (Aφ), (Aμ), (Au0 ), 1 ∞ 1 and 푢 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∩ 퐶([0, 푇]; 퐿loc(ℝ )). Then 푢 is an energy solution of (1.3)–(1.4) if and only if it is a distributional solution. We end this section by new quantitative energy and related 퐿푝-estimates for the unique distributional solution 푢 provided by Theorems 2.13 and 2.15. This type of estimate is widely used for different local and nonlocal equations of porous medium type, see the discussion in Section 2.4. All proofs are given in Section 4.2. 푤 Now, define Φ ∶ ℝ → ℝ by Φ(푤) = ∫0 휑(휉) d휉. Then we have:

Theorem 2.19 (Energy inequality). Assume (Aφ), (Aμ), and (Au0 ). Then the distributional solution 푢 of (1.3)–(1.4) satisfies

∫ Φ(푢(푥, 휏)) d푥 + |휑(푢)|2 ≤ ∫ Φ(푢 ) d푥 for 휏 ∈ (0, 푇]. 휏,퐸휇 0 ℝ푁 ℝ푁 Since Φ ≥ 0, we immediately have a quantitative bound on the energy.

Corollary 2.20. Assume (Aφ), (Aμ), and (Au0 ). Then the distributional solution 푢 of (1.3)–(1.4) satisfies

2 |휑(푢)| ≤ ‖휑(푢 )‖ ∞ 푁 ‖푢 ‖ 1 푁 < ∞. 푇,퐸휇 0 퐿 (ℝ ) 0 퐿 (ℝ ) Nonlocal equations of porous medium type 141

There is also a second type of energy inequality that implies 퐿푝-bounds.

푝 Theorem 2.21 (퐿 -bounds). Assume (Aφ), (Aμ), and (Au0 ). Then the distributional solution 푢 of (1.3)–(1.4) satisfies, for 0 < 휏 ≤ 푇,

푝 푝 ∫ |푢(푥, 휏)| d푥 ≤ ∫ |푢0(푥)| d푥 for all 푝 ∈ [1, ∞), ℝ푁 ℝ푁 and in the case 푝 = ∞,

‖푢(⋅, 휏)‖퐿∞(ℝ푁) ≤ ‖푢0‖퐿∞(ℝ푁).

2.4. Remarks.

Locally shift-bounded kernels. Let 휇(d푧) be a nonnegative locally finite Borel measure on ℝ ⧵ {0} and 푗(푥, 푧) a measurable function satisfying

0 < 푚 ≤ 푗(푥, 푧) ≤ 푀 < ∞.

Then the kernel 휆(푥, d푧) = 푗(푥, 푧)휇(d푧) is not only locally, but also globally, shift-bounded in the sense that for all 푥, ℎ ∈ ℝ and Borel 퐵 ⊂ ℝ ⧵ {0}, 휆(푥 + ℎ, 퐵) 푀 ≤ . 휆(푥, 퐵) 푚 +훼 Examples of 휇 are Lévy measures of Lévy processes, e.g. 휇(d푧) = (푐,훼 /|푧| ) d푧 for the 훼-stable process (훼 ∈ (0, 2)) with the fractional Laplacian as generator. The latter case corresponds exactly to assumption (A”λ1).

Recurrence and alternative characterization of 푿. In Theorem 2.6 (a) approximation by test functions is obtained by an additional assumption on the function class. Alternatively, as in part (b), we can keep the original function class, 휆 but restrict the bilinear form ℰ휆 (and hence the generator 퐴 ). In the elliptic setting such results are given in Theorem 3.2 in [38] under the assumptions that (Aλ0), ∞ (Aλ1’ ) and (Aλ2) hold and the closure of (ℰ휆, 퐶c (ℝ )) is recurrent. A condition ensuring recurrence for symmetric Lévy processes is given in Section 37 in [36]. E.g. the fractional Laplacian −(−Δ)훼/2 for 훼 ∈ (0, 2) is recurrent if 푁 ≤ 훼 – which is a rather restrictive assumption! Similar results are true in our parabolic setting. Assuming recurrence, or rather, assuming existence of the sequence of cut-off functions mentioned in Lemma 3.1 in [38], we get

∞ 2 푋 = 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휆(ℝ )). 142 F. del Teso, J. Endal, and E. R. Jakobsen

The proof is an easy modification of the proof of Theorem 3.2 in[38] if we assume (Aλ0),(Aλ1’ ), and (Aλ2) hold (as in Theorem 2.6 (a)) and, in addition, ∬ |푥 − 2 푇 푦| Λ(d푥, d푦) < ∞. Note that the latter condition implies ∫0 ℰ휆[휙푛, 휙푛] d푡 → 0 for 1 any 휙푛 ∈ 퐶c (푄푇) such that 휙푛 → 1 푎.푒. and ‖퐷휙푛‖퐿∞ → 0. However, this extra condition excludes all Lévy processes and all 푥-independent generators.

Integral representations of the operators 푨흀. In general the operator 퐴휆 is abstractly defined from ℰ휆 by formula (1.6). However explicit integral representation formulas exist under additional assumptions on the kernel 휆(푥, d푧) (cf. (2.1)). We follow [37] and assume (Aλ0) and (Aλ1) hold and 휆(푥, d푧) = 휆(푥,̃ 푥 + d푧) = 푗(푥, 푥 + 푧) d푧, where 푗 ≥ 0 is a symmetric measurable function on ℝ × ℝ ⧵ {푥} such that

∫ |푧| |푗(푥, 푥 + 푧) − 푗(푥, 푥 − 푧)| d푧 < ∞. |푧|≤1 Symmetric here means that 푗(푥, 푦) = 푗(푦, 푥). Note that we now have Λ(d푥, d푦) = 푗(푥, 푦) d푥 d푦 in (1.5). By Theorem 2.2 in [37], it then follows that

휆 퐴 [휙](푥) ≔ ∫ (휙(푥 + 푧) − 휙(푥) − 푧 ⋅ 퐷휙(푥)ퟏ|푧|≤1) 푗(푥, 푥 + 푧) d푧 |푧|>0 1 + ∫ 푧ퟏ (푗(푥, 푥 + 푧) − 푗(푥, 푥 − 푧)) d푧 ⋅ 퐷휙(푥) 2 |푧|≤1 |푧|>0

2 for 휙 ∈ 퐶c (ℝ ). Compare with (1.7) and note that the second integral is like a drift term that vanishes if 푗(푥, 푥 + 푧) = 푗(푥, 푥 − 푧). Under slightly stronger 휆 2 assumptions, this 퐴 coincides on 퐶c (ℝ ) with the generator of the closure of ∞ 2 (ℰ휆, 퐶c (ℝ )) in 퐿 (ℝ ) – see Proposition 2.5 in [37]. Let us simplify and assume that

푗(푥, 푦) = 푗1(푥, 푦)휇(푥 − 푦) for 푗1 symmetric, 푗1(푥, 푥 + 푧) = 푗1(푥, 푥 − 푧), and 휇 even. This 푗 is symmetric and +훼 푗(푥, 푥 + 푧) = 푗(푥, 푥 − 푧). Taking 푗1(푥, 푦) = 푎(푥) + 푎(푦) and 휇(푧) = 푐,훼 /|푧| , the Lévy density of the fractional Laplacian, we get an 푥-depending fractional Laplace like operator:

휆 훼/2 퐴1[휙](푥) = − 푎(푥)(−Δ) 휙(푥) 푐,훼 + ∫ (휙(푥 + 푧) − 휙(푥) − 푧 ⋅ 퐷휙(푥)ퟏ|푧|≤1)푎(푥 + 푧) +훼 d푧. |푧|>0 |푧| Nonlocal equations of porous medium type 143

From this example we also learn that our class of operators does not include the simplest and most natural 푥-depending fractional Laplace operator,

−푎(푥)(−Δ)훼/2휙(푥), since it only satisfies the symmetry assumption on 푗 (or (Aλ2)) if 푎 is constant!

On 푳풑-estimates. If 휑(푢) = 푢|푢|푚−1 and ℒ휇 = −(−Δ)휍/2, then by [28] the estimate corresponding to Theorem 2.21 takes the form

휏 2 푝 | 휍/4 (푝+푚−1)/2| 푝 ∫ |푢(푥, 휏)| d푥 +∫ ∫ |(−Δ) |푢| | d푥 d푡 ≤ ∫ |푢0(푥)| d푥. (2.6) ℝ푁 0 ℝ푁 ℝ푁 Note the additional energy term. A closer look at our proof, see Corollary 4.12 and the proof of Theorem 2.21, reveals that we could also have an 퐿푝-estimate with some energy. However, this energy is only a limit and hard to characterize under our weak assumptions. Such 퐿푝 type decay estimates are an essential tool for nonlinear diffusion equations of porous medium type. They imply that |푢|(푝+푚−1)/2 belongs to some Sobolev space. This estimate and the Nash–Gagliardo-Nirenberg inequality can be used in a Moser iteration argument to obtain an 퐿1 − 퐿∞ smoothing effect and then existence of energy solutions with initial data merely in 퐿1 [44, 45, 28, 29, 26]. The other main application of the 퐿푝-energy estimates is as key steps in Sobolev or Simon type compactness arguments. Such arguments are used in [44, 10, 9, 40, 41, 42] to prove existence of energy solutions through the resolution of a sequence of smooth approximate problems and passing to the limit in view of compactness.

3. Proof of uniqueness for energy solutions

In this section we prove Theorem 2.4. We start by some preliminary results.

2 Lemma 3.1 (Cauchy–Schwartz). Assume (Aλ1). If 푓, 푔 ∈ 퐿 (0, 푇; 퐸휆(ℝ )), then

| 푇 | | | ∫ ℰ휆[푓(⋅, 푡), 푔(⋅, 푡) d푡 ≤ |푓|푇,퐸휆|푔|푇,퐸휆. | 0 | The proof is as for the classical Cauchy–Schwartz inequality, and we omit it.

2 푇 Lemma 3.2. Assume (Aλ1). If 푓 ∈ 퐿 (0, 푇; 퐸휆(ℝ )) and 푔(푥, 푡) = ∫푡 푓(푥, 푠) d푠, then |푔|2 ≤ 1 푇2|푓|2 . 푇,퐸휆 2 푇,퐸휆 144 F. del Teso, J. Endal, and E. R. Jakobsen

Proof. By Jensen’s inequality and Tonelli’s lemma,

푇 1 푇 |푔|2 ≤ ∫ ∬(푇 − 푡) ∫ |푓(푥 + 푧, 푠) − 푓(푥, 푠)|2 d푠 휆(푥, d푧) d푥 d푡 푇,퐸휆 2 0 푡 푇 푇 = ∫ (푇 − 푡)( ∫ |푓(⋅, 푠)|2 d푠) d푡, 퐸휆 0 푡 and the result follows.

Since an energy solution has some regularity, the weak formulation of the equation will hold also with less regular test functions. We will now formulate such a type of result in the relevant setting for the Oleĭnik argument.

1 1 Lemma 3.3. Let 푢 be an energy solution of (1.1)–(1.2). If 푢 ∈ 퐿 (푄푇), 푢0 ∈ 퐿 (ℝ ), 2 and 휑(푢) ∈ 퐿 (0, 푇; 퐸휆(ℝ )), then for any 휙 ∈ 푋,

푇 푇

∫ (− ∫ 푢휙 d푥 − ℰ휆 [휑(푢), ∫ 휙(⋅, 푠) d푠]) d푡 0 ℝ푁 푡 푇

+ ∫ 푢0(푥) (∫ 휙(푥, 푠) d푠) d푥 = 0. ℝ푁 0

푇 In other words, we may take 휓(푥, 푡) = ∫푡 휙(푥, 푠) d푠 in Definition 2.2 for 휙 ∈ 푋. Note that the integrals are well-defined: see Lemma 3.2. From the proof below it follows that the choice of space 푋 is (close to) optimal.

∞ Proof. From the definition of 푋 there is 퐶c (ℝ × [0, 푇)) ∋ 휙푛 → 휙 ∈ 푋 for the convergence in 푋 as 푛 → ∞. Let

푇 푇

휓(푥, 푡) ≔ ∫ 휙(푥, 푠) d푠 and 휓푛(푥, 푡) ≔ ∫ 휙푛(푥, 푠) d푠. 푡 푡

∞ Observe that 휓푛 ∈ 퐶c (ℝ ×[0, 푇)) since 휙푛 is. By the Cauchy–Schwartz inequality, Lemma 3.2, and the convergence in 푋, we see that

푇 푇2 ∫ ℰ휆[휑(푢), (휓푛 − 휓)] d푡 ≤ |휑(푢)| |휙푛 − 휙| → 0 as 푛 → ∞. 푇,퐸휆 2 푇,퐸휆 0

1 Since 푢 ∈ 퐿 (푄푇) and 휙푛 converges in 푋, we also have

∬ 푢(휕푡휓푛 − 휕푡휓) d푥 d푡 = − ∬ 푢(휙푛 − 휙) d푥 d푡 → 0 as 푛 → ∞. 푄푇 푄푇 Nonlocal equations of porous medium type 145

In a similar way, ∫ℝ푁 푢0(푥)(휓푛 − 휓)(푥, 0) d푥 → 0. The result now follows from taking 휓 = 휓푛 in the definition of energy solutions (Definition 2.2), and using the above estimates to pass to the limit.

Remark 3.4. A closer inspection of the proof reveals that strong |⋅|푇,퐸휆 convergence cannot be replaced by the corresponding weak convergence. The reason is that the weak convergence property for the test functions 휙푛 is lost when they are integrated in time to yield the 휓푛’s. Note that for the proof of Lemma 3.3, the definition of 푋 is essential in the sense ∞ that we take those functions which can be approximated by 퐶c -functions. This lemma is crucial in the Oleĭnik argument below because we want to take

푇 휓(푥, 푡) = ∫ (휑(푢) − 휑(푣))(푥, 푠) d푠 푡 as a test function. By Lemma 3.3, we need that 휑(푢), 휑(푣) ∈ 푋 for this to be possible, and this explains this strange assumption and space.

Proof of Theorem 2.4 (Uniqueness 1). Assume there are two different energy solutions 푢 and 푣 of (1.1) with the same initial data (1.2). Let 푈 = 푢 − 푣 and Φ = 휑(푢) − 휑(푣), and note that the proof is complete if we can show that 푈 = 0 a.e. in 푄푇. To show that, we subtract the energy formulation of the equations for 푢 and 푣 (Definition 2.2). Since the initial data are the same, this leads to

푇 ∞ ∫ (∫ 푈휕푡휓 d푥 − ℰ휆[Φ, 휓]) d푡 = 0 for all 휓 ∈ 퐶c (ℝ × [0, 푇)). (3.1) 0 ℝ푁

Now we adapt the classical argument of Oleĭnik et al. [34] and seek to take

∫푇 Φ(푥, 푠) d푠 0 ≤ 푡 < 푇 휁(푥, 푡) = { 푡 0 푡 ≥ 푇, as a test function. Since Φ ∈ 푋 (by the definition of 풰풞), this can be done by Lemma 3.3, and hence

푇

∫ (− ∫ 푈Φ d푥 − ℰ휆[Φ, 휁]) d푡 = 0. (3.2) 0 ℝ푁 146 F. del Teso, J. Endal, and E. R. Jakobsen

푇| | Since ∫0 |ℰ휆[Φ, 휁]| d푡 < ∞ by Lemma 3.2, we have by Fubini’s theorem 푇

∫ ℰ휆[Φ, 휁] d푡 0 1 푇 = ∫ ∫ ∫ (Φ(푥 + 푧, 푡) − Φ(푥, 푡))(휁(푥 + 푧) − 휁(푥)) 휆(푥, d푧) d푥 d푡 2 0 ℝ푁 |푧|>0 1 푇 = ∫ ∫ ∫ (Φ(푥 + 푧, 푡) − Φ(푥, 푡))× 2 ℝ푁 |푧|>0 0 푇 × ∫ (Φ(푥 + 푧, 푠) − Φ(푥, 푠)) d푠 d푡 휆(푥, d푧) d푥. 푡 Then by the identity for 퐹 ∈ 퐿1((0, 푇)), 2 푇 푇 푇 푇 1 푇 ∫ 퐹(푡) (∫ 퐹(푠) d푠) d푡 = ∫ ∫ 퐹(푡)퐹(푠) d푠 d푡 = (∫ 퐹(푡) d푡) 2 0 푡 0 푡 0 푇 푇 푇 푠 (which follows easily since ∫0 ∫푡 … d푠 d푡 = ∫0 ∫0 … d푡 d푠), 2 푇 1 푇 ∫ ℰ [Φ, 휁] d푡 = ∫ ∫ (∫ (Φ(푥 + 푧, 푡) − Φ(푥, 푡)) d푡) 휆(푥, d푧) d푥 ≥ 0. 휆 4 0 ℝ푁 |푧|>0 0 Returning to (3.2), we then find that 푇 ∫ ∫ 푈Φ d푥 d푡 ≤ 0. 0 ℝ푁

Since 휑 is non-decreasing by (Aφ), 푈Φ ≥ 0 a.e., and it then follows that 푈Φ = 0 a.e. in 푄푇. This means that at a.e. point, either 푈 = 0 or Φ = 0, and hence since 푈 = 0 implies Φ = 0 by definition,

Φ = 0 a.e. in 푄푇. Then by equation (3.1), 푇 ∞ ∫ ∫ 푈휕푡휓 d푥 d푡 = 0 for all 휓 ∈ 퐶c (ℝ × [0, 푇)). 0 ℝ푁 푇 ∞ ∞ Since 휓(푥, 푡) ≔ ∫푡 휙(푥, 푠) d푠 ∈ 퐶c (ℝ × [0, 푇)) for arbitrary 휙 ∈ 퐶c (푄푇), 푇 ∞ − ∫ ∫ 푈휙 d푥 d푡 = 0 for all 휙 ∈ 퐶c (푄푇), 0 ℝ푁 and hence 푈 = 0 a.e. in 푄푇 by du Bois-Reymond’s lemma. Nonlocal equations of porous medium type 147

4. Distributional solutions with finite energy

Our main focus in this section is to prove Theorems 2.12, 2.13, 2.19, and 2.21. First, we prove the equivalence of notions of solutions. Second, we consider an approximate problem of (1.3)–(1.4). The energy and 퐿푝-estimates are then shown to hold for the solution of that problem. A compactness result will give us convergence of solutions of the approximate problem, and we thus obtain existence of some limit solution of the full problem satisfying the same estimates. We recall that (i) ℒ휇[휓] is well-defined for 휓 ∈ 퐶2(ℝ ) ∩ 퐿∞(ℝ ); (ii) ℒ휇[휓] is bounded in 퐿1/퐿∞ for 휓 ∈ 푊 2,1/푊 2,∞; and (iii) ℒ휇 is symmetric for e.g. functions 2,1 2,∞ in 푊 /푊 (see Lemma 3.5 in [22]). Note also that for 휇 replaced by 휇푟 ≔ ∞ 1 ∞ 1 ∞ 휇ퟏ|푧|>푟, (i)–(iii) holds when we only assume that 휓 is in 퐿 , 퐿 /퐿 , and 퐿 /퐿 (see Remark 3.6 (b) in [22]).

4.1. Equivalent notions of solutions. We establish the relation between the (푥-independent) bilinear form and our Lévy operator, as a consequence, we get equivalence of energy and distributional solutions under certain conditions.

∞ ∞ Proposition 4.1. Assume (Aμ). For any 휓 ∈ 퐶c (ℝ ) and 푣 ∈ 퐿 (ℝ ) ∩ 퐸휇(ℝ ), we have

∫ 푣(푥)ℒ휇[휓](푥) d푥 ℝ푁 1 = − ∫ ∫ (푣(푥 + 푧) − 푣(푥))(휓(푥 + 푧) − 휓(푥)) 휇(d푧) d푥 = −ℰ [푣, 휓]. 2 휇 ℝ푁 |푧|>0

Remark 4.2. The result holds as long as both sides make sense.

Lemma 4.3. Assume that 휈 ≥ 0 is an even Radon measure with 휈(ℝ ) < ∞, and 1 1 푝 푞 1 ≤ 푝, 푞 ≤ ∞ with 푝 + 푞 = 1. For any 푓 ∈ 퐿 (ℝ ) and 푔 ∈ 퐿 (ℝ ), we have

휈 ∫ 푔(푥)ℒ [푓](푥) d푥 = −ℰ휈[푓, 푔]. ℝ푁

This proof is postponed to Appendix D.

Proof of Proposition 4.1. Replace 휈 by 휇푟 = 휇ퟏ|푧|>푟 in Lemma 4.3, and let 푔 = 푣 and 푓 = 휓. Then the result follows by Lebesgue’s dominated convergence theorem + as 푟 → 0 since ퟏ|푧|>푟 ≤ 1. 148 F. del Teso, J. Endal, and E. R. Jakobsen

Proof of Theorem 2.12 (Equivalent notions of solutions).

(푎) ⟹ (푏) In Definition 2.2, we have that |휑(푢)|푇,퐸휇 < ∞, and then we can use Proposition 4.1 to obtain (note that 휑(푢) ∈ 퐿∞(ℝ )) 푇 휇 ∞ ∫ ∫ 푢휕푡휓+휑(푢)ℒ [휓] d푥 d푡+∫ 푢0(푥)휓(푥, 0) d푥 = 0 ∀휓 ∈ 퐶c (ℝ ×[0, 푇)). 0 ℝ푁 ℝ푁

(푏) ⟹ (푎) We write Definition 2.11 in the following way

푇 휇 ∫ ( ∫ 푢휕푡휓 d푥 + ∫ 휑(푢)ℒ [휓] d푥) d푡 0 ℝ푁 ℝ푁

∞ + ∫ 푢0(푥)휓(푥, 0) d푥 = 0 ∀휓 ∈ 퐶c (ℝ × [0, 푇)). ℝ푁

By the assumptions, |휑(푢)|푇,퐸휇 < ∞, and hence we can use Proposition 4.1 in the other direction to get energy solutions.

4.2. The approximate problem of (1.3)–(1.4). By using a priori and existence results for a simplified version of (1.3)–(1.4), we can take the limit of a sequence of solutions of such problems, and then conclude that some limit solution of the full problem exists and enjoys the energy and 퐿푝-estimates. Let 휔푛 be a family of mollifiers defined by

휔푛(휎) ≔ 푛 휔 (푛휎) (4.1)

∞ for fixed 0 ≤ 휔 ∈ 퐶c (ℝ ) with supp 휔 ⊆ 퐵(0, 1), 휔(휎) = 휔(−휎), ∫ 휔 = 1, and define

휑푛(푥) ≔ 휑 ∗ 휔푛(푥) − 휑 ∗ 휔푛(0) where 휔푛 is given by (4.1) with 푁 = 1. (4.2) Now, consider the following approximation of (1.3)–(1.4) where the measure 휇 is replaced by 휇푟 = 휇ퟏ|푧|>푟 and the nonlinear diffusion flux 휑 is replaced by 휑푛:

휇푟 휕푡푢푟,푛 − ℒ [휑푛(푢푟,푛)] = 0 in 푄푇, (4.3)

푢푟,푛(푥, 0) = 푢0(푥) on ℝ , (4.4) with

휇푟 ℒ [휓](푥) = ∫ (휓(푥 + 푧) − 휓(푥)) 휇푟(d푧). |푧|>0 ∞ Note that 휑푛 ∈ 퐶 (ℝ) (and hence, locally Lipschitz), 휑푛(0) = 0, and 휑푛 → 휑 locally uniformly on ℝ by (Aφ), the properties of mollifiers, and Remark 2.1 (f). Furthermore, recall that for any 푟 > 0, the operator ℒ휇푟[휓] is well-defined for merely bounded 휓. Nonlocal equations of porous medium type 149

Remark 4.4. Since (4.3)–(4.4) is just a special case of (1.3)–(1.4), existence, uniqueness, (uniform) 퐿1-, 퐿∞-bounds, and time regularity holds for (4.3)–(4.4) by Theo- rem 2.10 in [22] or by [24, 25] through limit procedures and compactness results for entropy or numerical solutions.

Theorem 4.5 (Existence and uniqueness, Theorem 2.8 in [22]). Assume (Aφ),

(Aμ), and (Au0 ). Then there exists a unique distributional solution 푢푟,푛 of (4.3)–(4.4) satisfying 1 ∞ 1 푢푟,푛 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∩ 퐶([0, 푇]; 퐿loc(ℝ )).

Now we first prove that (4.3) holds a.e., and then we deduce energy and clean 퐿푝-estimates (the latter by a Stroock–Varopoulos type result) from the rather general inequality in Proposition 4.7 below.

Lemma 4.6. Assume (Aφ), (Aμ), and (Au0 ). Then the distributional solution 푢푟,푛 of (4.3)–(4.4) with initial data 푢0 satisfies

1 ∞ 휇푟 휕푡푢푟,푛 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) and 휕푡푢푟,푛 = ℒ [휑푛(푢푟,푛)] a.e. in 푄푇.

Proof. By the definition of distributional solutions for (4.3)–(4.4) and the symmetry of ℒ휇푟,

휇푟 휇푟 − ∬ 푢푟,푛휕푡휓 d푥 d푡 = ∬ 휑푛(푢푟,푛)ℒ [휓] d푥 d푡 = ∬ ℒ [휑푛(푢푟,푛)]휓 d푥 d푡. 푄푇 푄푇 푄푇

휇푟 1,∞ Hence ℒ [휑푛(푢푟,푛)] is the weak time derivative of 푢푟,푛. Since 휑푛 ∈ 푊 (ℝ), 1 ∞ 휇푟 1 ∞ 휑푛(푢푟,푛) ∈ 퐿 ∩ 퐿 , and thus we get that 푔 ≔ ℒ [휑푛(푢푟,푛)] ∈ 퐿 (푄푇) ∩ 퐿 (푄푇). 1 Assume also 푢푟,푛 ∈ 퐶 . Then 휕푡푢푟,푛 = 푔, and we can use the fundamental theorem of calculus to see that

1 ‖푢푟,푛( ⋅ , ⋅ + ℎ) − 푢푟,푛 ‖ 1 ‖ − 푔‖ ≤ ∫ ‖푔( ⋅ , ⋅ + 푠ℎ) − 푔‖퐿 (푄푇) d푠. ‖ ℎ ‖ 1 퐿 (푄푇) 0

By an approximation argument in 퐿1, this inequality holds also without the 퐶1 assumption. Taking the limit as ℎ → 0+ (the right-hand side goes to zero by Lebesgue’s dominated convergence theorem, since translations in 퐿1 are continuous), we obtain that

푢푟,푛(푥, 푡 + ℎ) − 푢푟,푛(푥, 푡) 1 lim = 푔(푥, 푡) in 퐿 (푄푇), ℎ→0+ ℎ and hence, 휕푡푢푟,푛 exists and equals 푔 a.e. in 푄푇. 150 F. del Teso, J. Endal, and E. R. Jakobsen

∞ To prove the next result, we need to define cut-off functions: Consider 풳 ∈ 퐶c (ℝ ) such that 풳 ≥ 0, 풳 = 1 when |푥| ≤ 1, and 풳 = 0 when |푥| > 2, and define ⋅ 풳 (⋅) ≔ 풳 ( ) ∈ 퐶∞(ℝ ) for 푅 > 0. (4.5) 푅 푅 c

1,∞ Proposition 4.7. Assume (Aφ), (Aμ), and (Au0 ), and 0 < 휏 ≤ 푇. Let Ψ ∈ 푊loc (ℝ) with Ψ(0) = 0. Then the distributional solution 푢푟,푛 of (4.3)–(4.4) satisfies

휏 ′ 휇푟 ∫ Ψ(푢푟,푛(푥, 휏)) d푥 − ∫ ∫ Ψ (푢푟,푛(푥, 푡))ℒ [휑푛(푢푟,푛(푥, 푡)) d푥 d푡 ℝ푁 0 ℝ푁

= ∫ Ψ(푢0(푥)) d푥. ℝ푁 Remark 4.8. On page 1256 in [28], a similar result as the above is obtained for Ψ(푢) nonnegative, non-decreasing and convex.

1,∞ 1,∞ Proof. Observe that we may assume Ψ ∈ 푊 (ℝ) since Ψ ∈ 푊loc (ℝ) and ∞ 휇푟 푢푟,푛, 푢0 ∈ 퐿 . By Lemma 4.6, 휕푡푢푟,푛 = ℒ [휑푛(푢푟,푛)] a.e. in 푄푇. Multiply this ′ a.e.-equation by Ψ (푢푟,푛(푥, 푡))풳푅(푥) (where 풳푅 is defined in (4.5)) and integrate (in 푥) over ℝ to get

′ 휇푟 ′ ∫ 휕푡푢푟,푛Ψ (푢푟,푛)풳푅 d푥 = ∫ ℒ [휑푛(푢푟,푛)]Ψ (푢푟,푛)풳푅 d푥. ℝ푁 ℝ푁 By Lemma 4.6 and the Sobolev chain rule given by Theorem 2.1.11 in [46], the left-hand side equals ∫ℝ푁 휕푡Ψ(푢푟,푛)풳푅 d푥. Note that the function 풳푅 converges pointwise to 1, is bounded by 1, and is integrable. Hence we can move the time derivative outside the integral on the left-hand side by Lebesgue’s dominated 1 convergence theorem and Lemma 4.6, since |휕푡Ψ(푢푟,푛)풳푅| ∈ 퐿 (ℝ ):

d ∫ Ψ(푢 )풳 d푥 = ∫ ℒ휇푟[휑 (푢 )]Ψ′(푢 )풳 d푥. d푡 푟,푛 푅 푛 푟,푛 푟,푛 푅 ℝ푁 ℝ푁

∞ We integrate in time from 푡 = 0 to 푡 = 휏, using that 풳푅 ∈ 퐶c (ℝ ) and 푢푟,푛 ∈ 1 퐶([0, 푇]; 퐿loc(ℝ )) (cf. Theorem 4.5) to obtain

∫ Ψ(푢푟,푛(푥, 휏))풳푅(푥) d푥 − ∫ Ψ(푢0(푥))풳푅(푥) d푥 ℝ푁 ℝ푁 휏 (4.6) 휇푟 ′ = ∫ ∫ ℒ [휑푛(푢푟,푛(⋅, 푡))](푥)Ψ (푢푟,푛(푥, 푡))풳푅(푥) d푥 d푡. 0 ℝ푁 Nonlocal equations of porous medium type 151

1,∞ ′ 1 Since Ψ ∈ 푊 (ℝ) and Ψ(0) = 0, |Ψ(푤)| ≤ ‖Ψ (푤)‖퐿∞|푤| ∈ 퐿 for 푤 = 푢푟,푛, 푢0. 휇푟 Moreover, since 푢푟,푛, 휑푛(푢푟,푛) and hence also ℒ [휑푛(푢푟,푛)] is integrable, we get 휇푟 ′ 1 |ℒ [휑푛(푢푟,푛)]Ψ (푢푟,푛)풳푅| ∈ 퐿 (ℝ × (0, 휏)). Then Lebesgue’s dominated convergence theorem can be used on both sides of (4.6) as 푅 → ∞ to complete the proof.

푤 Corollary 4.9 (Energy estimate). Let Φ푛(푤) ≔ ∫0 휑푛(휉) d휉. Under the assumptions of Proposition 4.7,

2 ∫ Φ푛(푢푟,푛(푥, 휏)) d푥 + |휑푛(푢푟,푛)| = ∫ Φ푛(푢0(푥)) d푥. 휏,퐸휇푟 ℝ푁 ℝ푁 In particular,

|휑 (푢 )| ≤ ∫ Φ (푢 (푥)) d푥 ≤ ‖휑 (푢 )‖ ∞ 푁 ‖푢 ‖ 1 푁 < ∞. 푛 푟,푛 휏,퐸휇푟 푛 0 푛 0 퐿 (ℝ ) 0 퐿 (ℝ ) ℝ푁

1 ′ Proof. We observe that Φ푛 ∶ ℝ → ℝ is 퐶 and Φ푛(0) = 0. Moreover, Φ푛(푤) = ∞ 휑푛(푤) which is bounded when 푤 ≕ 푢푟,푛, 푢0 ∈ 퐿 by (Aφ) and (4.2). Hence, Φ푛 is Lipschitz, and thus, we can replace Ψ by Φ푛 in Proposition 4.7 to get

휏 휇푟 ∫ Φ푛(푢푟,푛(푥, 휏)) d푥 − ∫ ∫ 휑푛(푢푟,푛(푥, 푡))ℒ [휑푛(푢푟,푛(⋅, 푡)) (푥) d푥 d푡 ℝ푁 0 ℝ푁

= ∫ Φ푛(푢0(푥)) d푥. ℝ푁

휇푟 Since (4.2) hold and ℒ [휑(푢푟,푛)] is integrable, we conclude the first part by Lemma 4.3 (take 푓 = 휑푛(푢푟,푛) = 푔). For the last part, we use that Φ푛(푢0) = ′ |Φ푛(푢0)| ≤ ‖Φ푛(푢0)‖퐿∞|푢0|, and hence, since Φ푛 ≥ 0,

2 |휑푛(푢푟,푛)| ≤ ∫ Φ푛(푢0(푥)) d푥 ≤ ‖휑푛(푢0)‖퐿∞‖푢0‖퐿1 휏,퐸휇푟 ℝ푁 which completes the proof.

1 Lemma 4.10 (General Stroock–Varopoulos). Assume (Aλ1), 푄, 푅, 푆 ∈ 퐶 (ℝ), ′ 2 ′ ′ (푆 ) ≤ 푄 푅 , and |푄(휓)|푇,퐸휆, |푅(휓)|푇,퐸휆 < ∞ for some 휓 ∶ 푄푇 → ℝ. Then

푇

∫ ℰ휆[푄(휓(⋅, 푡)), 푅(휓(⋅, 푡)) d푡 ≥ |푆(휓)| . 푇,퐸휆 0 152 F. del Teso, J. Endal, and E. R. Jakobsen

Proof. Assume without loss of generality that 푏 > 푎. By the Fundamental theorem of calculus, Cauchy–Schwartz’ inequality, and 푄′푅′ ≥ (푆′)2, we obtain

푏 푏 2 2 (푄(푏) − 푄(푎))(푅(푏) − 푅(푎)) = ∫ (√푄′(푡)) d푡 ∫ (√푅′(푡)) d푡 푎 푎 푏 2 푏 2 2 ≥ (∫ √푄′(푡)푅′(푡) d푡) ≥ (∫ 푆′(푡) d푡) = (푆(푏) − 푆(푎)) . 푎 푎

By the definition of ℰ휆 and | ⋅ |푇,퐸휆, the result follows.

Remark 4.11.

(a) See Proposition 4.11 in [17] for a similar result.

(b) Observe that the same lemma holds for a nonnegative even Radon measure 푝 휈 with 휈(ℝ ) < ∞ under the simplified assumption 푄(휓) ∈ 퐿 (푄푇) and 푞 1 1 푅(휓) ∈ 퐿 (푄푇) with 1 ≤ 푝, 푞 ≤ ∞ and 푝 + 푞 = 1.

푝 푝 푤 ″ ′ Corollary 4.12 (퐿 -bound). Let Λ(휉) = |휉| and Ξ푛(푤) = ∫0 √Λ (휉)휑푛(휉) d휉. Under the assumptions of Proposition 4.7 and 푝 ∈ (1, ∞),

푝 2 푝 ∫ |푢푟,푛(푥, 휏)| d푥 + |Ξ푛(푢푟,푛)| d푡 ≤ ∫ |푢0(푥)| d푥 휏,퐸휇푟 ℝ푁 ℝ푁 In particular,

푝 푝 ∫ |푢푟,푛(푥, 휏)| d푥 ≤ ∫ |푢0(푥)| d푥 < ∞. ℝ푁 ℝ푁

2 Remark 4.13. The above result also ensures that |Ξ푛(푢푟,푛)| is uniformly bound- 휏,퐸휇푟 ed in 푟 and 푛.

푝 Proof. Observe that 푢푟,푛 ∈ 퐿 (푄푇) for 푝 ∈ (1, ∞) by standard interpolation in 퐿푝-spaces. 1,∞ Case 1: 푝 ∈ [2, ∞). The function Λ is convex, Λ ∈ 푊loc (ℝ), and Λ(0) = 0. That is, we can replace Ψ by Λ in Proposition 4.7 to get

휏 ′ 휇푟 ∫ Λ(푢푟,푛(푥, 휏)) d푥 − ∫ ∫ Λ (푢푟,푛(푥, 푡))ℒ [휑푛(푢푟,푛(⋅, 푡)) (푥) d푥 d푡 ℝ푁 0 ℝ푁 (4.7)

= ∫ Λ(푢0(푥)) d푥 ℝ푁 Nonlocal equations of porous medium type 153

′ 푝−2 ″ 푝−2 ′ 1,∞ Note that Λ (휉) = 푝|휉| 휉 and Λ (휉) = 푝(푝 − 1)|휉| . Since Λ ∈ 푊loc (ℝ), 2 ∞ ′ 2 푢푟,푛 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇), and (4.2) holds, 푔 ≔ Λ (푢푟,푛(⋅, 푡)) ∈ 퐿 (ℝ ), and 2 푓 ≔ 휑푛(푢푟,푛(⋅, 푡)) ∈ 퐿 (ℝ ). By Lemma 4.3,

휏 휏 ′ 휇푟 ′ − ∫ ∫ Λ (푢푟,푛)ℒ [휑푛(푢푟,푛) d푥 d푡 = ∫ ℰ휇푟[Λ (푢푟,푛), 휑푛(푢푟,푛) d푡. 0 ℝ푁 0 ′ Then by Lemma 4.10 and Remark 4.11 (b) (take 푄 ≔ Λ and 푅 ≔ 휑푛), 휏 ′ 2 ∫ ℰ휇 [Λ (푢푟,푛(⋅, 푡)), 휑푛(푢푟,푛(⋅, 푡)) d푡 ≥ |Ξ푛(푢푟,푛)| ≥ 0, 푟 휏,퐸휇 0 푟

′ 2 ″ ′ since Ξ푛 satisfies (Ξ푛) ≤ Λ 휑푛. Hence the corollary follows by (4.7). Case 2: 푝 ∈ (1, 2). We follow the idea of the proof of Corollary 5.12 in [9]. For each 훿 > 0, consider the function Λ훿 such that

푝−2 ′ ″ 2 2 2 푝−2 Λ훿(0) = Λ훿(0) = 0 and Λ훿(휉) = 푝(푝 − 1) ((훿 + 휉 ) − 훿 ) .

″ 푝−2 Note that 0 ≤ Λ훿(휉) ≤ 푝(푝 − 1)|휉| , and then, | 휉 | | 휉 | ′ | ″ | 푝−1 | ′ | 푝 |Λ훿(휉)| = |∫ Λ훿(푠) d푠| ≤ 푝|휉| and |Λ훿(휉)| ≤ |∫ Λ훿(푠) d푠| ≤ |휉| . | 0 | | 0 |

′ ∞ 1 Since 푔 ≔ Λ훿(푢푟,푛(⋅, 푡)) ∈ 퐿 (ℝ ) and 푓 ≔ 휑푛(푢푟,푛(⋅, 푡)) ∈ 퐿 (ℝ ), we get – by following the calculations in Case 1 – that

2 ∫ Λ훿(푢푟,푛(푥, 휏)) d푥 + |Ξ푛,훿(푢푟,푛)| ≤ ∫ Λ훿(푢0(푥)) d푥 (4.8) 휏,퐸휇 ℝ푁 푟 ℝ푁 with ᵆ푟,푛 ″ ′ Ξ푛,훿(푢푟,푛) = ∫ √Λ훿(휉)휑푛(휉) d휉 ≥ 0. 0 ″ ″ ′ By a direct argument, using Λ훿,Λ , 휑푛 ≥ 0 and Cauchy–Schwartz’s inequality, we obtain

ᵆ푟,푛 | | ″ ″ ′ |Ξ푛,훿(푢푟,푛) − Ξ푛(푢푟,푛)| ≤ ∫ √|Λ훿(휉) − Λ (휉)|√휑푛(휉) d휉 0

ᵆ푟,푛 ᵆ푟,푛 ″ ″ ′ ≤ ∫ |Λ훿(휉) − Λ (휉)| d휉 ∫ 휑푛(휉) d휉 (4.9) √ 0 √ 0

ᵆ푟,푛 1/2 ″ ″ ≤ ‖휑 (푢 )‖ ∞ ∫ |Λ (휉) − Λ (휉)| d휉. 푛 푟,푛 퐿 (푄푇) 훿 √ 0 154 F. del Teso, J. Endal, and E. R. Jakobsen

Since the integrand in the last inequality is dominated by 2푝(푝 − 1)|휉|푝−2 which 푝−2 integrates to 2푝|푢푟,푛| 푢푟,푛, we use Lebesgue’s dominated convergence theorem + + to conclude that Ξ푛,훿 → Ξ푛 as 훿 → 0 . Taking the limit as 훿 → 0 in (4.8), by using Fatou’s lemma on the left-hand side and Lebesgue’s dominated convergence 푝 theorem (|Λ훿(푢0(푥))| ≤ |푢0(푥)| ) on the right-hand side, the corollary follows. Remark 4.14. Observe that by (4.9),

1/2 (푝−1)/2 |Ξ (푢 )| ≤ 푝‖휑 (푢 )‖ ∞ ‖푢 ‖ ∞ < ∞, 푛,훿 푟,푛 푛 푟,푛 퐿 (푄푇) 푟,푛 퐿 (푄푇) and similarly for Ξ푛(푢푟,푛). Hence, both are well-defined for all 푝 ∈ (1, ∞). The existence of a distributional solution of (1.3)–(1.4) with finite energy (cf. The- orem 2.13) will follow from the following compactness theorem:

Theorem 4.15 (Compactness). Assume (Aφ), (Aμ), and (Au0 ). Let {푢푟,푛}푟,푛∈ℕ be a sequence of distributional solutions of (4.3)–(4.4). Then there exists a subsequence 1 {푢푟푗,푛푗}푗∈ℕ and a 푢 ∈ 퐶([0, 푇]; 퐿loc(ℝ )) such that

1 푢푟푗,푛푗 → 푢 in 퐶([0, 푇]; 퐿loc(ℝ )) as 푗 → ∞.

1 ∞ 1 Moreover, 푢 ∈ 퐿 (푄푇) ∩ 퐿 (푄푇) ∩ 퐶([0, 푇]; 퐿loc(ℝ )) is a distributional solution of (1.3)–(1.4).

Remark 4.16. We have that ‖푢‖퐿1/퐿∞ ≤ ‖푢0‖퐿1/퐿∞ by Fatou’s lemma and Remark 4.4 (the limit of a uniformly bounded sequence is uniformly bounded by the same bound).

1 ∞ Proof. Observe that the sequence {푢푟푗,푛푗}푗∈ℕ enjoy 퐿 -, 퐿 -bounds, and time regularity by Remark 4.4, and that these bounds are independent of 푗 (see Section 4 in [22]). ∞ Moreover, for any 휓 ∈ 퐶c (ℝ ),

휇 (ℒ휇 − ℒ 푟푗 )[휓](푥) = ∫ (휓(푥 + 푧) − 휓(푥) − 푧 ⋅ 퐷휓(푥))휇(d푧),

|푧|≤푟푗

휇 푟푗 휇 1 + and hence, ℒ [휓] → ℒ [휓] in 퐿 (ℝ ) as 푟푗 → 0 by Lebesgue’s dominated convergence theorem. We also have,

2 2 sup ∫ min{|푧| , 1} d휇푟푗(푧) ≤ ∫ min{|푧| , 1} d휇(푧) < ∞, 푟푗>0 |푧|>0 |푧|>0 and 휑푛푗 → 휑 locally uniformly as 푛푗 → ∞ by (4.2). Thus, we are in the setting of Theorem 2.12 in [22] and the result follows. Nonlocal equations of porous medium type 155

We are now ready to prove Theorems 2.13, 2.19, and 2.21. Proof of Theorem 2.13 (Existence 1). In light of Theorem 4.15, it only remains to 2 prove that the limit 푢 is such that 휑(푢) ∈ 퐿 (0, 푇; 퐸휇(ℝ )). Recall that Φ푛푗(푤) = ∫푤 휑 (휉) d휉 and Φ(푤) = ∫푤 휑(휉) d휉. Now, 0 푛푗 0 | | | | ∫ Φ(푢0) d푥 − ∫ Φ푛푗(푢0) d푥 ≤ ∫ |Φ(푢0) − Φ푛푗(푢0)| d푥 | ℝ푁 ℝ푁 | ℝ푁 ᵆ0

≤ ∫ ∫ |휑(휉) − 휑푛푗(휉)| d휉 d푥 ℝ푁 0

1 ≤ ‖푢0‖퐿 sup |휑(휉) − 휑푛푗(휉)|, |휉|≤‖ᵆ0‖퐿∞ and since 휑 → 휑 locally uniformly, lim ∫ Φ (푢 ) d푥 = ∫ Φ(푢 ) d푥. 푛푗 푛푗→∞ ℝ푁 푛푗 0 ℝ푁 0 Observe also that by Theorem 4.15 (and Remark 4.4), (4.2), and the proof of

Theorem 2.6 in [22], we can take a further subsequence to get that 휑푛푗(푢푟푗,푛푗) → 휑(푢) a.e. in 푄푇 as 푗 → ∞. 2 2 For any 푅 ≥ 푟푗 > 0, |휑푛 (푢푟 ,푛 )|퐸 ≤ |휑푛 (푢푟 ,푛 )|퐸 , and thus, by the second 푗 푗 푗 휇푅 푗 푗 푗 휇푟 part of Corollary 4.9, 푗 휏 2 ∫ |휑푛 (푢푟 ,푛 )| d푡 ≤ ∫ Φ푛 (푢0) d푥. 푗 푗 푗 퐸휇 푗 0 푅 ℝ푁 Taking the limit as 푗 → ∞, we obtain, by Fatou’s lemma, the above calculations, and the estimate Φ(푢0) ≤ ‖휑(푢0)‖퐿∞|푢0|, that 휏 2 ∫ |휑(푢)|퐸 d푡 ≤ ∫ Φ(푢0) d푥 ≤ ‖휑(푢0)‖퐿∞‖푢0‖퐿1. 휇푅 0 ℝ푁 Another application of Fatou’s lemma, as 푅 → 0+, and the choice 휏 = 푇 yield

∞ 푁 1 푁 |휑(푢)|푇,퐸휇 ≤ ‖휑(푢)‖퐿 (ℝ )‖푢0‖퐿 (ℝ ) < ∞. The proof is complete.

By Theorem 2.15, we know that any subsequence of {푢푟,푛}푟,푛∈ℕ converges to the same limit, and hence, the whole sequence converges since it is bounded by Remark 4.4. Let us then continue with the proof of the energy and 퐿푝-estimates for the distributional solution of (1.3)–(1.4). Proof of Theorem 2.19 (Energy inequality). By Remark 4.16,

|Φ(푢(푥, 푡)) − Φ푛(푢푟,푛(푥, 푡))|

≤ sup |휑(휉)| |푢(푥, 푡) − 푢푟,푛(푥, 푡)| + ‖푢0‖퐿∞ sup |휑(휉) − 휑푛(휉)|. |휉|≤2‖ᵆ0‖퐿∞ |휉|≤‖ᵆ0‖퐿∞ 156 F. del Teso, J. Endal, and E. R. Jakobsen

Since 휑푛 → 휑 locally uniformly and we can find a subsequence of {푢푟,푛}푟,푛 such that 푢푟푗,푛푗 → 푢 a.e. in 푄푇 as 푗 → ∞ by Theorem 4.15, Φ푛푗(푢푟푗,푟푗(푥, 푡)) → Φ(푢(푥, 푡)) pointwise a.e. in 푄푇 as 푗 → ∞. The conclusion then follows by Corollary 4.9, Fatou’s lemma, and the proof of Theorem 2.13. Proof of Theorem 2.21 (퐿푝-bounds). By Fatou’s lemma and Theorem 4.15, we can take the limit as 푗 → ∞ (since 푢푟푗,푛푗 → 푢 a.e. in 푄푇 as 푗 → ∞ by considering a further subsequence in Theorem 4.15) in the second estimate in Corollary 4.12 to obtain the result. The cases 푝 = 1 and 푝 = ∞ are explained in Remark 4.16.

Acknowledgments. E. R. Jakobsen was supported by the Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway. F. del Teso was supported by the FPU grant AP2010-1843 and the grants MTM2011-24696 and MTM2014-52240-P from the Ministry of Education, Culture and Sports, Spain, the BERC 2014-2017 program from the Basque Government, and BCAM Severo Ochoa excellence accredita- tion SEV-2013-0323 from Spanish Ministry of Economy and Competitiveness (MINECO). We would like to thank Stefano Lisini and Giampiero Palatucci for fruitful discussions on homogeneous fractional Sobolev spaces.

A. Proof of Theorem 2.6 (a)

2 2 ∞ 2 Obviously 푋 ∩ 퐿 (푄푇) ⊂ 퐿 (0, 푇; 퐸휆(ℝ )) ∩ 퐿 (푄푇) ∩ 퐿 (푄푇), and we must show the opposite inclusion: Any

2 ∞ 2 푓 ∈ 퐿 (0, 푇; 퐸휆(ℝ )) ∩ 퐿 (푄푇) ∩ 퐿 (푄푇) 2 belongs to 푋 ∩퐿 (푄푇). To do so, we must prove that 푓 can be suitably approximated ∞ by functions in 퐶c (ℝ × [0, 푇)). We will now explain how to build such an approximation. +1 Let 훿 > 0 and 푔훿 ∶ ℝ → ℝ be defined by

푔훿(푥, 푡) ≔ 푓(푥, 푡)ퟏ[2훿,푇−3훿](푡) and mollify 푔훿 to get

퐺훿(푥, 푡) ≔ 푔훿 ∗푥,푡 휌훿(푥, 푡) = ∬ 푔훿(푦, 푠)휌훿(푥 − 푦, 푡 − 푠) d푦 d푠 ℝ푁+1 −−1 ∞ +1 where 휌훿 is defined by 휌훿(휎, 휏) ≔ 훿 휌(휎/훿, 휏/훿) for a fixed 0 ≤ 휌 ∈ 퐶c (ℝ ) satisfying supp 휌 ⊆ 퐵(0, 1) × [−1, 1], 휌(휎, 휏) = 휌(−휎, −휏), and ∬ 휌 = 1. Note that ∞ +1 |푔훿| ≤ |푓| and 퐺훿 ∈ 퐶 (ℝ ) with support in ℝ × [훿, 푇 − 2훿]. Nonlocal equations of porous medium type 157

2 ∞ Lemma A.1. Assume (Aλ0), (Aλ1’ ), (Aλ2), and 푓 ∈ 퐿 (0, 푇; 퐸휆(ℝ )) ∩ 퐿 (푄푇) ∩ 2 퐿 (푄푇). +1 (a) 퐺훿 ∈ 퐶0(ℝ ). 2 2 2 (b) |퐺훿| ≤ 퐶|푓| + 4‖푓‖ 2 ‖Π휆‖ ∞ 푁 for some constant 퐶 ≥ 0. 푇,퐸휆 푇,퐸휆 퐿 (푄푇) 퐿 (ℝ ) Remark A.2. If 휆 is globally shift-bounded, that is, we replace the statement “퐵 ⊂

퐵(0, 1) ⧵ {0}” with “for all 퐵 ∈ ℝ ⧵ {0}” in (A’λ1), then in (b) we get

2 2 |퐺훿| ≤ 퐶|푓| . 푇,퐸휆 푇,퐸휆

∞ In this case, we do not have to assume Π휆 ∈ 퐿 (ℝ ) and (Aλ2).

2 +1 Proof. (a) Since the Fourier transforms of 푔훿 and 휌훿 are both in 퐿 (ℝ ), the properties of the Fourier transform and Hölder’s inequality yield

1 +1 ℱ(퐺훿) = ℱ(푔훿 ∗ 휌훿) = ℱ(푔훿)ℱ(휌훿) ∈ 퐿 (ℝ ). (A.1)

The result then follows by the Riemann–Lebesgue lemma which gives that 퐺훿 = −1 +1 ℱ (ℱ(퐺훿)) ∈ 퐶0(ℝ ). (b) The proof is a straightforward adaptation of the proof of Lemma 2.2 in [38] and the estimate |푔훿| ≤ |푓|.

Next, we recall a useful truncation from [38]: Let 푇훿 ∶ ℝ → ℝ be defined by 1 1 푇 (푥) ≔ min{max{− , 푥 − min{max{−훿, 푥}, 훿}}, } for all 푥 ∈ ℝ. 훿 훿 훿 Observe that for all 푥, 푦 ∈ ℝ

+ |푇훿(푥)| ≤ |푥|, |푇훿(푥) − 푇훿(푦)| ≤ |푥 − 푦|, and 푇훿(푥) → 푥 as 훿 → 0 . (A.2)

∞ We can now define a 퐶c -approximation of 푓:

푤훿(푥, 푡) ≔ 푇훿[퐺훿] ∗푥,푡 휌훿(푥, 푡). (A.3)

2 ∞ Lemma A.3. Assume (Aλ0), (Aλ1’ ), (Aλ2), and 푓 ∈ 퐿 (0, 푇; 퐸휆(ℝ )) ∩ 퐿 (푄푇) ∩ 2 퐿 (푄푇). Then:

∞ +1 (a) 푤훿 ∈ 퐶c (ℝ ) and supp 푤훿 ⊂ ℝ × [0, 푇 − 훿].

+ 2 (b) ‖푤훿 − 푓‖퐿 (푄푇) → 0 as 훿 → 0 .

2 ∞ (c) For some 퐾 ≥ 0, ‖푤훿‖퐿 (푄푇) + ‖푤훿‖퐿 (푄푇) + |푤훿|푇,퐸휆 ≤ 퐾 for all 훿 > 0. 158 F. del Teso, J. Endal, and E. R. Jakobsen

Remark A.4. If 푓 ∈ 퐿푝 for some 푝 ∈ [1, ∞), similar arguments show that (b) can + 푝 be replaced by ‖푤훿 − 푓‖퐿 (푄푇) → 0 as 훿 → 0 . Moreover, if the measure 휆 is globally shift-bounded, then we can relax assumption (Aλ1’ ) as in Remark A.2, and replace the previous uniform bound of |푤훿|푇,퐸휆 by 퐶|푓|푇,퐸휆 in (c).

Proof. (a) Since 퐺훿 vanishes at infinity, 푇훿[퐺훿] has compact support, and therefore ∞ +1 푤훿 ≔ 푇훿[퐺훿] ∗푥,푡 휌훿 ∈ 퐶c (ℝ ). Moreover, supp 푇훿[퐺훿] ⊂ ℝ × [훿, 푇 − 훿] and ∞ hence supp 푤훿 ⊂ ℝ × [0, 푇 − 훿]. As a consequence, 푤훿 ⊂ 퐶c (ℝ × [0, 푇)). 2 2 2 2 (b) Note that |푇훿[퐺훿]| ≤ |퐺훿| by (A.2), ‖퐺훿‖퐿2 ≤ ‖푔훿‖퐿2, and |푔훿| ≤ |푓| , and thus, all these functions are in 퐿2. Hence,

2 2 2 2 ‖푤훿 − 푓‖퐿 (푄푇) ≤ ‖푤훿 − 퐺훿‖퐿 (푄푇) + ‖퐺훿 − 푔훿‖퐿 (푄푇) + ‖푔훿 − 푓‖퐿 (푄푇) and by the properties of mollifiers and (A.2),

‖푤훿 − 퐺훿‖퐿2 ≤ ‖푇훿[퐺훿] ∗푥,푡 휌훿 − 푇훿[푔훿] ∗푥,푡 휌훿‖퐿2 + ‖푇훿[푔훿] ∗푥,푡 휌훿 − 퐺훿‖퐿2

≤ ‖푇훿[퐺훿] − 푇훿[푔훿]‖퐿2 + ‖푇훿[푔훿] − 푔훿‖퐿2

≤ ‖퐺훿 − 푔훿‖퐿2 + ‖푇훿[푔훿] − 푔훿‖퐿2.

2 2 Finally, we can use Lebesgue’s dominated convergence theorem (|푇훿[푔훿]| ≤ |푔훿| 2 2 by (A.2) and |푔훿| ≤ |푓| ) and the properties of mollifiers to conclude. (c) According to Lemma A.1 (b),

2 2 2 |푤훿| ≤ 퐶|푇훿[퐺훿]| + 4‖푇훿[퐺훿]‖ 2 ‖Π휆‖ ∞ 푁 , 푇,퐸휆 푇,퐸휆 퐿 (푄푇) 퐿 (ℝ ) and then by (A.2),

2 2 2 2 |푇훿[퐺훿]| ≤ |퐺훿| and ‖푇훿[퐺훿]‖ 2 ≤ ‖퐺훿‖ 2 . 푇,퐸휆 푇,퐸휆 퐿 (푄푇) 퐿 (푄푇)

So, by another application of Lemma A.1, the properties of mollifiers, |푔훿| ≤ |푓|, and |푔훿(푥, 푡) − 푔훿(푦, 푡)| ≤ |푓(푥, 푡) − 푓(푦, 푡)|, we have 2 2 2 |푤훿| ≤ 퐶(|푓| + ‖푓‖ 2 ‖Π휆‖ ∞ 푁 ). 푇,퐸휆 푇,퐸휆 퐿 (푄푇) 퐿 (ℝ )

2 2 Note also that by part (b), ‖푤훿‖퐿 (푄푇) ≤ ‖푓‖퐿 (푄푇), and moreover, by the properties of mollifiers, (A.2), and the definition of 푔훿,

∞ ∞ 푁+1 ∞ 푁+1 ∞ ‖푤훿‖퐿 (푄푇) ≤ ‖푇훿[퐺훿]‖퐿 (ℝ ) ≤ ‖퐺훿‖퐿 (ℝ ) ≤ ‖푓‖퐿 (푄푇). This completes the proof.

∞ To prove Theorem 2.6 (a), we will define from {푤훿}훿>0 a 퐶c -sequence that converges also in | ⋅ |푇,퐸휆. Nonlocal equations of porous medium type 159

Proof of Theorem 2.6 (a). This proof is an adaptation the proof of Theorem 2.4 2 2 in [38]. Note that by standard arguments 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휆(ℝ )) is a Hilbert 푇 space with inner product ⟨ ⋅ , ⋅ ⟩ 2 + ∫ ℰ [ ⋅ , ⋅ ] d푡. By Lemma A.3 (c) and the 퐿 (푄푇) 0 휆

Banach–Saks theorem, there is a subsequence {푤훿푘}푘∈ℕ such that the Césaro mean ̃ 2 2 of this subsequence converges to some function 푓 ∈ 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휆(ℝ )):

푛 2 푛 2 ‖ 1 ‖ | 1 | ‖ ∑ 푤 − 푓‖̃ + | ∑ 푤 − 푓|̃ → 0 as 푛 → ∞. 푛 훿푘 푛 훿푘 ‖ 푘=1 ‖ 2 | 푘=1 | 퐿 (푄푇) 푇,퐸휆

Let 휙 = 1 ∑푛 푤 . Then 휙 ∈ 퐶∞(ℝ × [0, 푇)) and is uniformly bounded in 푛 푛 푘=1 훿푘 푛 c ∞ 퐿 (푄푇) since 푤훿 is (cf. Lemma A.3 (c)). By the Banach–Alaoglu theorem, we can take a further subsequence (also denoted 휙푛) such that

∗ ∞ 휙푛 ⇀ 푓 in 퐿 (푄푇) as 푛 → ∞.

2 + Since 푤훿 → 푓 in 퐿 (푄푇) as 훿 → 0 by Lemma A.3 (b), any subsequence and 2 any Césaro mean of this subsequence converges to 푓 ∈ 퐿 (푄푇). All three notions of convergence implies distributional convergence, and hence, the result follows since by uniqueness of limits, 푓 = 푓̃ = 푓 in 풟′(ℝ × [0, 푇)) and then a.e.

휶 푵 2 휶 푵 B. On the spaces 푯̇ 2 (ℝ ) and 푳 (0, 푻; 푯̇ 2 (ℝ ))

In the first part of this section we prove the equivalence between three different 훼 definitions of the homogeneous Sobolev space 퐻̇ 2 (ℝ ) when 푁 > 훼. These results are well-known, but we were unable to find proofs that directly apply to our 2 훼 setting. Then in the second part, we define the parabolic space 퐿 (0, 푇; 퐻̇ 2 (ℝ )) and show some of its properties. Note that we do not define this space as a Bochner space, but rather as an iterated 퐿2–퐻̇ 훼/2 space. Our discussion heavily relies on [8], [7], [38], and [35]. In the next section we use these results to prove Theorem 2.6 (b).

Proposition B.1. Assume 훼 ∈ (0, 2) and 푁 > 훼. Let 푓 ∈ 풮′(ℝ ), a tempered distribution, ℱ{푓} its Fourier transform, and

2 |푓| 훼 ≔ ∫ |휉|훼|ℱ{푓}(휉)|2 d휉 < ∞. 퐻̇ 2 (ℝ푁) ℝ푁 The following definitions of 퐻̇ 훼/2(ℝ ) are equivalent: ̇ 훼/2 ′ 1 (a) 퐻1 (ℝ ) ≔ { 푓 ∈ 풮 (ℝ ) ∶ ℱ{푓} ∈ 퐿loc(ℝ ) and |푓|퐻̇ 훼/2(ℝ푁) < ∞ }, 160 F. del Teso, J. Endal, and E. R. Jakobsen

|⋅| ̇훼/2 푁 훼/2 ∞ 퐻 (ℝ ) (b) 퐻̇2 (ℝ ) ≔ 퐶c (ℝ ) , and ̇ 훼/2 2/(−훼) (c) 퐻3 (ℝ ) ≔ { 푓 ∈ 퐿 (ℝ ) ∶ |푓|퐻̇ 훼/2(ℝ푁) < ∞ }.

훼/2 Proof. 1) By Propositions 1.34 and 1.37 and Theorem 1.38 in [7], 퐻̇1 (ℝ ) is a Hilbert space with the norm

2 훼 2 |푓(푥 + 푧) − 푓(푥)| |푓|퐻̇ 훼/2(ℝ푁) = ∫ |휉| |ℱ{푓}(휉)| d휉 = 푐,훼 ∫ ∫ +훼 d푧 d푥, ℝ푁 ℝ푁 |푧|>0 |푧|

훼/2 2/(−훼) and 퐻̇1 (ℝ ) is continuously embedded in 퐿 (ℝ ) with

‖푓‖퐿2푁/(푁−훼)(ℝ푁) ≤ 퐶|푓|퐻̇ 훼/2(ℝ푁). (B.1)

2 We also note that −훼 ∈ (2, ∞) as long as 푁 > 훼. 훼/2 훼/2 훼/2 2) 퐻̇2 (ℝ ) ⊂ 퐻̇1 (ℝ ): For any 푓 ∈ 퐻̇2 (ℝ ), it is clear that 푓 is a tempered 1 distribution and |푓|퐻̇ 훼/2(ℝ ) < ∞. Furthermore, ℱ{푓} ∈ 퐿loc(ℝ ) since for any compact 퐾 ⊂ ℝ , we use Hölder’s inequality to get

1/2 1/2 ∫ |ℱ{푓}| d휉 ≤ (∫ |휉|훼|ℱ{푓}|2 d휉) (∫ |휉|−훼 d휉) < ∞. (B.2) 퐾 ℝ푁 퐾

훼/2 훼/2 3) 퐻̇1 (ℝ ) ⊂ 퐻̇2 (ℝ ): Due to Remark A.4, we can proceed as in Section A (or 훼/2 ∞ Theorem 2.4 in [38]): For any 푓 ∈ 퐻̇1 (ℝ ), we construct an 퐶c -approximation 푤훿 satisfying

훿→0+ ‖푤훿 − 푓‖퐿2푁/(푁−훼)(ℝ푁) ⟶ 0 and |푤훿|퐻̇ 훼/2(ℝ푁) ≤ 퐶|푓|퐻̇ 훼/2(ℝ푁). (B.3)

훼/2 Hence since 퐻̇1 (ℝ ) is a Hilbert space, the Banach–Saks theorem ensures the ̃ ̇ 훼/2 existence of a subsequence {푤훿푘}푘∈ℕ and 푓 ∈ 퐻1 (ℝ ) such that

푛 2 | 1 | | ∑ 푤 − 푓|̃ → 0 as 푛 → ∞. |푛 훿푘 | 푘=1 퐻̇ 훼/2(ℝ푁)

By (B.1), these Césaro means converge to 푓̃in 퐿2/(−훼) . But by (B.3), they also converge to 푓 in 퐿2/(−훼) , and hence 푓 = 푓̃a.e. 훼/2 훼/2 2/(−훼) 4) 퐻̇3 (ℝ ) ⊂ 퐻̇1 (ℝ ): Since 푓 ∈ 퐿 (ℝ ), it is a tempered distribu- 1 tion, and ℱ{푓} ∈ 퐿loc(ℝ ) by (B.2). 훼/2 훼/2 5) 퐻̇1 (ℝ ) ⊂ 퐻̇3 (ℝ ): This is just a consequence of (B.1). Nonlocal equations of porous medium type 161

We now define and analyze the parabolic space 퐿2(0, 푇; 퐻̇ 훼/2(ℝ )). In the proof we will use the following iterated 퐿푝-space [8]:

퐿2(0, 푇; 퐿푞(ℝ )) = { 푓∶ ℝ × (0, 푇) → ℝ ∶ 푓 is measurable 푇 2 and ∫ ‖푓(⋅, 푡)‖퐿푞 d푡 < ∞ }, 0 for some 푞 ∈ (1, ∞). Note that this space is not a priori a Bochner space. 푐 d푧 Lemma B.2. Let 훼 ∈ (0, 2), 푁 > 훼, and 휇 (d푧) ≔ ,훼 . Then the space 훼 |푧|+훼

2 ̇ 훼/2 2 2/(−훼) 2 퐿 (0, 푇; 퐻 (ℝ )) ≔ 퐿 (0, 푇; 퐿 (ℝ )) ∩ 퐿 (0, 푇; 퐸휇훼(ℝ )) is a Hilbert space with inner product

푇 푇 훼 ⟨휓, 휙⟩ ≔ ∫ ∫ |휉| ℱ{휓(⋅, 푡)}(휉)ℱ{휙(⋅, 푡)}(휉) d휉 d푡 = 푐,훼 ∫ ℰ휇훼[휓, 휙] d푡. 0 ℝ푁 0

√⟨⋅,⋅⟩ 2 훼/2 ∞ Moreover, 퐿 (0, 푇; 퐻̇ (ℝ )) = 퐶c (ℝ × [0, 푇)) . Proof. 1) Embedding. Since 푓 ∈ 퐿2(0, 푇; 퐻̇ 훼/2(ℝ )), we have as a consequence of properties of iterated 퐿푝-spaces [8] and Fubini’s theorem that

2 2 ‖푓(⋅, 푡)‖퐿2푁/(푁−훼)(ℝ푁) ∈ 퐿 (0, 푇) and |푓(⋅, 푡)|퐻̇ 훼/2(ℝ푁) ∈ 퐿 (0, 푇).

2/(−훼) It follows that for a.e. 푡 ∈ (0, 푇), 푓(⋅, 푡) ∈ 퐿 (ℝ ) ∩ 퐸휇훼(ℝ ), and then 훼/2 푓(⋅, 푡) ∈ 퐻̇1 (ℝ ) by Proposition B.1 (c). By (B.1), we can then conclude that

푇 푇 ∫ ‖푓(⋅, 푡)‖2 d푡 ≤ 퐶 ∫ |푓(⋅, 푡)|2 d푡. (B.4) 퐿2푁/(푁−훼)(ℝ푁) 퐻̇ 훼/2(ℝ푁) 0 0

푇 2 2 2) Inner product space. Obviously ⟨ ⋅ , ⋅ ⟩ = ∫ | ⋅ | 훼/2 푁 d푡 = | ⋅ | defines 0 퐻̇ (ℝ ) 푇,퐸휇훼 푇 2 (the square of) a seminorm. By (B.4), ∫0 |푓( ⋅ , 푡)|퐻̇ 훼/2(ℝ푁) d푡 = 0 implies 푓 = 0 a.e. in 푄푇, and hence the seminorm is a full norm. Now it is easy to check that the space is an inner product space. 2 훼/2 3) Completeness. Let {푓푛}푛∈ℕ be a Cauchy sequence in 퐿 (0, 푇; 퐻̇ (ℝ )). By definition and (B.4), it follows that

푇 푇 ∫ ‖푓 ( ⋅ , 푡) − 푓 ( ⋅ , 푡)‖2 d푡 ≤ ∫ |푓 ( ⋅ , 푡) − 푓 ( ⋅ , 푡)|2 d푡 → 0 푛 푚 퐿2푁/(푁−훼)(ℝ푁) 푛 푚 퐻̇ 훼/2(ℝ푁) 0 0 162 F. del Teso, J. Endal, and E. R. Jakobsen as 푛, 푚 → ∞. Hence the sequence is Cauchy also in 퐿2(0, 푇; 퐿2/(−훼) (ℝ )). By [8], this space is complete, and sequences convergent in norm contain pointwise a.e. convergent subsequences. Therefore there is 푓 ∈ 퐿2(0, 푇; 퐿2/(−훼) (ℝ )) such that 푇

∫ ‖푓푛( ⋅ , 푡) − 푓( ⋅ , 푡)‖퐿2푁/(푁−훼)(ℝ푁) → 0 as 푛 → ∞, 0 and a subsequence 푓푛푘 → 푓 a.e. in 푄푇 as 푛 → ∞. By Fatou’s lemma,

2 훼/2 which goes to zero as 푚 → ∞. Hence 푓푛 → 푓 in 퐿 (0, 푇; 퐻̇ (ℝ )), and then by the triangle inequality 푓 ∈ 퐿2(0, 푇; 퐻̇ 훼/2(ℝ )). 2 훼/2 ∞ 4) Density. For any 푓 ∈ 퐿 (0, 푇; 퐻̇ (ℝ )), the 퐶c (ℝ × [0, 푇)) functions 푤훿 defined in (A.3) in Appendix A satisfy

훿→0+ ‖푤 − 푓‖ 2 2푁/(푁−훼) 푁 ⟶ 0 and |푤 | ≤ 퐶|푓| . (B.5) 훿 퐿 (0,푇;퐿 (ℝ )) 훿 푇,퐸휇훼 푇,퐸휇훼 This follows from Remark A.4, and the fact that the iterated 퐿푝-spaces have similar properties as the usual 퐿푝-spaces with respect to mollifications (by [8], inequalities for convolutions, continuity of translations, dominated convergence etc. are similar). Since 퐿2(0, 푇; 퐻̇ 훼/2(ℝ )) is a Hilbert space, the Banach–Saks theorem ̃ 2 ̇ 훼/2 implies there is a subsequence {푤훿푘}푘∈ℕ and 푓 ∈ 퐿 (0, 푇; 퐻 (ℝ )) such that

푛 2 | 1 | | ∑ 푤 − 푓|̃ → 0 as 푛 → ∞. 푛 훿푘 | 푘=1 | 푇,퐸휇훼 By (B.4), these Césaro means converge to 푓̃ in 퐿2(0, 푇; 퐿2/(−훼) (ℝ )). But by (B.5), they also converge to 푓 in this space, and hence 푓 = 푓̃a.e.

C. Proof of Theorem 2.6 (b)

1) By (Aλ1” ), the measure 휆 is globally shift-bounded and the (semi)norms on 퐸휆 훼/2 ̇ 훼/2 훼/2 and 퐻 are comparable: 푚|푓|퐻̇ ≤ |푓|퐸휆 ≤ 푀|푓|퐻̇ . The latter gives

훼/2 퐸휆(ℝ ) = { 푓 ∶ 푓 is measurable and |푓|퐻̇ < ∞ } = 퐸휇훼(ℝ ). Nonlocal equations of porous medium type 163

2) For 푁 ≤ 훼, the fractional Laplacian (and hence also 퐴휆) is recurrent. In this case an easy modification of the proof of Theorem 3.1 in[38] yields

∞ 2 푋 = 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휇훼(ℝ )). See the discussion on recurrence in Section 2.4 for more details. ∞ 2 3) For 푁 > 훼, we always have 푋 ⊂ 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휇훼(ℝ )). To prove the ∞ ∞ 2 reverse inclusion, we must show that 퐶c is dense in 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휇훼(ℝ )). This will follow from Lemma B.2 if we can show that ∞ 2 2 ̇ 훼/2 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휇훼(ℝ )) ⊂ 퐿 (0, 푇; 퐻 (ℝ )). ∞ 2 To prove this, we must show that any 푔 ∈ 퐿 (푄푇) ∩ 퐿 (0, 푇; 퐸휇훼(ℝ )) also belongs to 퐿2(0, 푇; 퐿2/(−훼) (ℝ )). Let ℎ(푡) ≔ ‖푔( ⋅ , 푡)‖2 for 푡 ∈ (0, 푇). By Section 252P in [30] and 퐿2푁/(푁−훼)(ℝ푁) measurability of 푔 on ℝ × (0, 푇), ℎ is a measurable function on (0, 푇). We need to prove that ℎ belongs to 퐿1(0, 푇). As a consequence of Fubini’s theorem,

∞ 2 ‖푔( ⋅ , 푡)‖퐿∞(ℝ푁) ∈ 퐿 (0, 푇) and |푔( ⋅ , 푡)|퐻̇ 훼/2(ℝ푁) ∈ 퐿 (0, 푇), ∞ and then 푔( ⋅ , 푡) ∈ 퐿 (ℝ )∩퐸휇훼(ℝ ) for a.e. 푡 ∈ (0, 푇). Hence for such 푡, 푔( ⋅ , 푡) is 훼/2 a tempered distribution and belongs to 퐻̇1 (ℝ ) by the argument of Step 4) in the 2 proof of Proposition B.1. Then by the embedding (B.1), ℎ(푡) ≤ 퐶|푔( ⋅ , 푡)|퐻̇ 훼/2(ℝ푁) 2 1 1 for a.e. 푡, and hence since |푔( ⋅ , 푡)|퐻̇ 훼/2(ℝ푁) ∈ 퐿 (0, 푇), it follows that ℎ ∈ 퐿 (0, 푇) and 푔 ∈ 퐿2(0, 푇; 퐿2/(−훼) (ℝ )). The proof is complete.

D. Proof of Lemma 4.3

1) Assume 1 < 푝, 푞 < ∞. Consider 푓푛(푥) ≔ (푓 ∗ 휔푛)(푥) and 푔푛(푥) ≔ (푔 ∗ 휔푛)(푥) for 휔푛 defined by (4.1). By a direct computation,

(푓푛푔푛)(푥 + 푧) − (푓푛푔푛)(푥) − 푧 ⋅ 퐷(푓푛푔푛)(푥)

= 푓푛(푥)(푔푛(푥 + 푧) − 푔푛(푥) − 푧 ⋅ 퐷푔푛(푥))

+ 푔푛(푥) (푓푛(푥 + 푧) − 푓푛(푥) − 푧 ⋅ 퐷푓푛(푥))

+ (푓푛(푥 + 푧) − 푓푛(푥))(푔푛(푥 + 푧) − 푔푛(푥)) Integrate the above equality against 휈(d푧) d푥 to get

휈 ∫ ℒ [푓푛푔푛](푥) d푥 ℝ푁

휈 휈 = ∫ 푓푛(푥)ℒ [푔푛](푥) d푥 + ∫ 푔푛(푥)ℒ [푓푛](푥) d푥 + 2ℰ휈[푓푛, 푔푛]. ℝ푁 ℝ푁 164 F. del Teso, J. Endal, and E. R. Jakobsen

The three terms on the right-hand side are well-defined by Hölder’s inequality 휈 since the measure 휈 is finite. By Fubini’s theorem, ∫ℝ푁 ℒ [푓푛푔푛](푥) d푥 = 0 and thus, we obtain

휈 휈 0 = ∫ 푓푛(푥)ℒ [푔푛](푥) d푥 + ∫ 푔푛(푥)ℒ [푓푛](푥) d푥 + 2ℰ휈[푓푛, 푔푛]. (D.1) ℝ푁 ℝ푁 By standard estimates for mollifiers, Tonelli’s lemma, and Hölder’s inequality,

≤ 2휈(ℝ )(‖푔‖퐿푞(ℝ푁)‖푓푛 − 푓‖퐿푝(ℝ푁) + ‖푓‖퐿푝(ℝ푁)‖푔푛 − 푔‖퐿푞(ℝ푁)), and

|2ℰ휈[푓푛, 푔푛] − 2ℰ휈[푓, 푔]|

≤ 4휈(ℝ ) (‖푔‖퐿푞(ℝ푁)‖푓푛 − 푓‖퐿푝(ℝ푁) + ‖푓‖퐿푝(ℝ푁)‖푔푛 − 푔‖퐿푞(ℝ푁)) .

휈 Note that a similar argument holds for ∫ℝ푁 푔푛ℒ [푓푛] d푥. Taking the limit as 푛 → ∞ and using the properties of mollifiers, we obtain (D.1) for 푓, 푔 replacing 푓푛, < 푔푛 respectively. Since ℒ휈 is symmetric, we obtain

휈 ∫ 푔(푥)ℒ [푓](푥) d푥 = −ℰ휈[푓, 푔]. ℝ푁

2) Assume 푝 = 1, 푞 = ∞. Again we mollify, 푓푛(푥) ≔ (푓 ∗ 휔푛)(푥) and 푔푚(푥) ≔ (푔 ∗ 휔푚)(푥), and we obtain (D.1) as above. We deduce (almost as before) that

≤ 2휈(ℝ )‖푔‖퐿∞(ℝ푁)‖푓푛 − 푓‖퐿1(ℝ푁)

+ ∫ ∫ |푓푛(푥)| |(푔푚(푥 + 푧) − 푔(푥 + 푧)) − (푔푚(푥) − 푔(푥))| 휈(d푧) d푥, ℝ푁 ℝ푁 and

|2ℰ휈[푓푛, 푔푚] − 2ℰ휈[푓, 푔]|

≤ ∫ ∫ |푓푛(푥 + 푧) − 푓푛(푥)| ℝ푁 ℝ푁

|(푔푚(푥 + 푧) − 푔(푥 + 푧)) − (푔푚(푥) − 푔(푥))|휈(d푧) d푥

+ 4휈(ℝ )‖푔‖퐿∞(ℝ푁)‖푓푛 − 푓‖퐿1(ℝ푁) Nonlocal equations of porous medium type 165

Note that

|(푔푚(푥 + 푧) − 푔(푥 + 푧)) − (푔푚(푥) − 푔(푥))| ≤ 4‖푔‖퐿∞(ℝ푁)

1 and |푓푛(푥)| ∈ 퐿 (ℝ ). Hence, for fixed 푛, we may send 푚 → ∞ by Lebesgue’s dominated convergence theorem to obtain (D.1) for 푓푛, 푔. Then we send 푛 → ∞ to obtain the same for 푓, 푔. Again, we use the symmetry of ℒ휈 to complete the proof.

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On the spectrum of leaky surfaces with a potential bias

Pavel Exner

To my friend Helge Holden on the occasion of his 60th birthday

Abstract. We discuss operators of the type 퐻 = −Δ + 푉(푥) − 훼훿(푥 − Σ) with an attractive interaction, 훼 > 0, in 퐿2(ℝ3), where Σ is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and 푉 is a potential bias being a positive constant 푉0 in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete 2 one with a particular attention to the critical case, 푉0 = 훼 . We show that 휎disc(퐻) is then empty if the bias is supported in the ‘exterior’ region, while in the opposite case isolated eigenvalues may exist.

1. Introduction

An anniversary is usually an opportunity to look back at the achievements of the jubilee. In Helge’s case the picture is impressive as he contributed significantly to several different areas of mathematical physics. Nevertheless, one of his works made a much larger impact than any other, namely the monograph [1] first published in 1988. It is a collective work but Helge’s hand is unmistakably present in the exposition, and I add that it makes me proud to be a part of the second edition of this book. This also motivates me to choose a problem from this area as a topic of this paper. I am going to discuss operators of the type

퐻 = −Δ + 푉(푥) − 훼훿(푥 − Σ) , 훼 > 0 , (1) in 퐿2(ℝ3), where the 훿-potential is supported by an infinite surface Σ dividing the space into two regions, of which one is supposed to be convex, and 푉 is a potential bias being a positive constant in one of the regions and zero in the other. The question to be addressed concerns spectral properties of such operators, in particular, how they depend on the geometry of Σ. We will observe similarities with recent results obtained in the two-dimensional analogue of the present problem 170 P. Exner

[6], especially a peculiar asymmetry. On the other hand, however, it is not likely to have these results extended to higher dimensions, cf. a remark at the end of Section 5.

2. Statement of the problem and the results

Let us begin with formulating the problem described above in proper terms.

2.1. Assumptions. As we shall see, the geometry of Σ will be decisive for spectral properties of the operator (1). We focus our attention on the following class of surfaces: (a) Σ is topologically equivalent to a plane dividing ℝ3 into two regions such that one of them is convex. The trivial case of two halfspaces is excluded.

1 (b) Σ may contain at most finite families 풞 = {퐶푗} of finite 퐶 curves, which are either closed or have regular ends, and 풫 = {푃푗} of points such that outside the set 풞 ∪ 풫 the surface is 퐶2 smooth admitting a parametrization with a uniformly elliptic metric tensor. To distinguish the two regions we shall refer to the convex one as ‘interior’ and denote it Ωint, the other will be ‘exterior’, denoted as Ωext. The curve finiteness in assumption (b) refers to the Hausdorff distance, i.e. to the metric inherited from the ambient three-dimensional Euclidean space. The assumption implies, in particular, that Σ is 퐶2 smooth outside a compact. Note also that the curves indicating the non-smooth parts of Σ may in general touch or cross; the regular ends mean that they can be prolonged locally without losing the 퐶1 property. Since Σ is by assumption topologically equivalent to a plane, we can use an atlas consisting of a single chart, in other words, a map Σ∶ ℝ2 → ℝ3 provided we accept the licence that the fundamental forms and quantities derived from them may not exist at the points of 풞 ∪ 풫, nevertheless, geodesic distances remain well defined across these singularities. Furthermore, by assumption (b) the principal curvatures 푘1, 푘2 of Σ are well defined outside a compact. We will suppose that (c) Σ is asymptotically planar, that is, the principal curvature vanish as the geodesic distance from a fixed point tends to infinity. Equivalently, one can require that both the Gauss and mean curvatures given by 1 퐾 = 푘1푘2 and 푀 = 2 (푘1 +푘2), respectively, vanish asymptotically. We also assume that (d) there is a 푐 > 0 such that |Σ(푠) − Σ(푡)| ≥ 푐|푠 − 푡| holds for any 푠, 푡 ∈ ℝ2. On the spectrum of leaky surfaces with a potential bias 171

This ensures, in particular, that there are no cusps at the points of 풞 ∪ 풫 where Σ is not smooth; in view of assumption (a) such a constant must satisfy 푐 < 1. Given a bounded potential 푉, one can demonstrate in the same way as in [3, Sec. 4] that under the stated assumptions the quadratic form 푞 = 푞훼,Σ,푉 defined by

2 2 1/2 푞[휓] ≔ ‖∇휓‖ + (휓, 푉휓) − 훼 ∫ |휓(Σ(푠))| 푔 (푠) d푠1 d푠2 , (2) ℝ2 where 푠 = (푠1, 푠2) are the coordinates used to parametrize Σ and 푔 = det(푔푖푗) is the appropriate squared Jacobian defined by means of the metric tensor (푔푖푗), with the domain 퐻1(ℝ3), is closed and below bounded. Thus it is associated with a unique self-adjoint operator which we identify with 퐻 = 퐻훼,Σ,푉 of (1) above. In fact such a claim is valid for a much wider class of potentials; however, we focus here our attention on a particular case. By hypothesis (a) above the surface Σ splits ℝ3 into two regions, and we assume that

(e) 푉(푥) = 푉0 > 0 in one of these regions and 푉(푥) = 0 in the other.

2.2. An auxiliary problem. In the trivial case we have excluded the problem is solved easily by separation of variables. It is useful to look at the transverse part which we will need in the following. It is given by the operator

d2 ℎ = − − 훼훿(푥) + 푉(푥) , (3) d푥2 where 푉(푥) = 푉0 for 푥 > 0 and 푉(푥) = 0 otherwise, associated with the quadratic form 휙 ↦ ‖휙′‖2 − 훼|휙(0)|2 + (휙, 푉휙) defined on 퐻1(ℝ). Properties of this operator are easily found; we adopt without proof from [6] the following simple results.

Lemma 2.1.

(i) 휎ess(ℎ) = [0, ∞). 2 (ii) The operator ℎ has no eigenvalues for 푉0 ≥ 훼 .

2 2 훼 −푉0 2 (iii) The operator ℎ has a unique eigenvalue 휇 = − ( 2훼 ) for 푉0 < 훼 .

2 2 (iv) If 푉0 = 훼 the equation ℎ휓 = 0 has a bounded weak solution 휓 ∉ 퐿 (ℝ). 2 2 (v) For 푉0 > 0 and any 휑 ∈ 퐶 (ℝ+) ∩ 퐿 (ℝ+) we have

∞ ′ 2 2 2 ∫ (|휑 | + 푉0|휑| )(푥) d푥 ≥ √푉0 |휑(0)| . 0 172 P. Exner

Relations between the coupling constant and the potential bias will play an important role in the following. For the sake of brevity, we shall call the case (iv) of the lemma critical, and similarly we shall use the terms subcritical for case (iii) and 2 supercritical for the situation where 푉0 > 훼 .

2.3. The results. Let us look first at the essential spectrum. As usual inthe Schrödinger operator theory it is determined by the behavior of the interaction at large distances. In view of assumption (c) we expect that asymptotically the situation approaches the trivial case with separated variables mentioned above, and indeed, we have the following result.

Theorem 2.2. 휎ess(퐻) = [휇, ∞) holds under the assumptions (a)–(e), where 휇 ≔ 1 −2 2 2 2 − 4 훼 (훼 − 푉0) for 푉0 < 훼 and 휇 ≔ 0 otherwise. The question about the existence of the discrete spectrum is more involved and the potential bias makes the answer distinctively asymmetric. In particular, a critical or supercritical potential supported in the exterior region prevents negative eigenvalues from existence.

Theorem 2.3. Under the stated assumptions, suppose that 푉(푥) = 0 holds in Ωint 2 and 푉(푥) = 푉0 ≥ 훼 in Ωext, then 휎(퐻) = 휎ess(퐻) = [0, ∞) . On the other hand, the operator (1) may have isolated eigenvalues in the (sub-) critical regime as we are going to illustrate on examples. In Section 5 we discuss the case of a conical surface and show that the discrete spectrum of 퐻 is nonempty provided 푉0 is small enough. The most interesting, though, is the critical case, 2 푉0 = 훼 with the bias in the interior region. In Section 6 we discuss another example, this time with Σ being a ‘rooftop’ surface, and show that for suitable values of parameters we have here 휎disc(퐻) ≠ ∅.

3. Proof of Theorem 2.2

The argument splits into two parts. First we shall demonstrate the implication

휈 ∈ 휎ess(퐻) if 휈 ≥ 휇 . (4)

To this goal, one has to find for any fixed 휈 ≥ 휇 and any 휀 > 0 an infinite- dimensional subspace ℒ ⊂ Dom(퐻) such that ‖(퐻 − 휈)휓‖ < 휀 holds for any 휓 ∈ ℒ. We denote 휁 ≔ 휈 − 휇 ≥ 0 and choose a pair of functions of unit 퐿2 norm, 2 2 ‖ ‖ 1 2 푓 ∈ 퐶0(ℝ ) with the property that ‖(−Δ − 휁)푓‖ < 4 휀, and 푔 ∈ 퐶0(ℝ) such that 1 ‖푔‖ = 1 and ‖(ℎ − 휇)푔‖ < 4 휀. Such functions can always be found in view of the On the spectrum of leaky surfaces with a potential bias 173 fact that the essential spectrum of the two-dimensional Laplacian is [0, ∞) and of the properties of the operator ℎ stated in Lemma 2.1. (푗) In the next step we choose a sequence of surface points 푎푗 = Σ(푠 ) such that |푠(푗)| → ∞ as 푗 → ∞. With each of them we associate the Cartesian system of coordinates (푥(푗), 푦(푗)) where 푥(푗) are the Cartesian coordinates in the tangential plane (푗) to Σ at the point 푎푗 and 푦 is the distance from this tangential plane. By assumption (b), the points 푎푗 can be always chosen in such a way that this coordinate choice (푗) (푗) (푗) (푗) makes sense. This allows us to define the functions 휓푗(푥 , 푦 ) = 푓(푥 )푔(푦 ). By construction, each of them has a compact support of the diameter independent of 푗, hence in view of assumption (d) one can pick them so that supp 휓푗 ∩Σ is simply connected. Using then a straightforward telescopic estimate in combination with the requirement ‖푓‖ = ‖푔‖ = 1 we get the inequality

‖(퐻 − 휈)휓푗‖ ≤ ‖(−Δ푥 − 휁)푓‖ + ‖(ℎ − 휇)푔‖ + 푉0‖휓푗|풜푗‖ + ‖(훿Σ푗 − 훿Σ)휓푗‖, (5) where Σ푗 is the tangential plane at 푎푗 and 풜푗 is the part of the function support 2 squeezed between Σ and Σ푗; the last term is understood as the 퐿 -norm over the two surface segments contained in the border of 풜푗. The first two terms on the 1 right-hand side of (5) are by construction bound by 2 휀. Furthermore, in view of the assumptions (a)–(c) in combination with the smoothness of the functions 푓, 푔, which are the same for all the 휓푗, the other two terms tend to zero as 푗 → ∞, hence ‖(퐻 − 휈)휓푗‖ < 휀 holds for all 푗 large enough. In addition, one can always choose ′ the points 푎푗 in such a way that supp 휓푗 ∩ supp 휓푗′ = ∅ holds for 푗 ≠ 푗 , which means that Weyl’s criterion hypothesis is satisfied. To complete the proof of the theorem, we have to demonstrate the opposite implication, in other words, to check the validity of the relation

휎ess(퐻) ∩ (−∞, 휇) = ∅ , (6) which is equivalent to inf 휎ess(퐻) ≥ 휇. While in the first part of the proof we have extended to the present case the argument used in the two-dimensional situation, now we choose a different approach because the localization estimates employed in [6] become more involved here. We will need an auxiliary result which is a sort of modification of Proposition 2.5 in[7].

Lemma 3.1. Let ℎ denote the operator (3) acting on the interval (−푑, 푑) with Neumann boundary conditions at the endpoints, associated with the form 휙 ↦ ′ 2 2 1 2 ‖휙 ‖ − 훼|휙(0)| + (휙, 푉휙) defined on 퐻 (−푑, 푑). If 푉0 ≤ 훼 , there are positive 푐0, 푑0 −1 such that for all 푑 > 푑0 we have inf 휎(ℎ ) ≥ 휇 − 푐0푑 . If, on the other hand, 2 푉0 > 훼 holds, then ℎ ≥ 0 for all 푑 large enough.

Proof. We observe that the ground-state eigenfunction of ℎ corresponding the 174 P. Exner eigenvalue 휇푑 < 0 is of the form

휓(푥) = 푐1휒(0,푑) cosh 휅1(푥 − 푑) + 푐2휒(−푑,0) cosh 휅1(푥 + 푑) , where 휅1 ≔ √−휇푑 and 휅2 ≔ √푉0 − 휇푑. Since the function has to be continuous at 푥 = 0 and satisfy 휓′(0+) − 휓′(0−) = −훼휓(0), we get the spectral condition

휅1 tanh 휅1푑 + 휅2 tanh 휅2푑 = 훼 . (7)

As a function of −휇푑, the left-hand side is increasing from √푉0 tanh √푉0푑, behav- ing asymptotically as 2√−휇 + 풪((−휇)−1/2), the equation (7) has a unique solution 2 for any fixed 푑 > 0 provided 푉0 ≤ 훼 . Since the left-hand side is monotonous also with respect to 푑, we have 휇푑 < 휇∞ where 휇∞ = 휇 of Lemma 2.1(iii). To get a lower bound we have to estimate the left-hand side of (7) from below. We will do that using the rough bound tanh 푥 > 1 − 2e−2푥 > 1 − 푥−1. Writing the solution of the appropriate estimating condition in the form ̃휇푑 = 휇 − 훿, we find −1 −2 after a short computation that ̃휇푑 = 휇 − 2푑 + 풪(푑 ) holds as 푑 → ∞, which together with the inequality 휇푑 > ̃휇푑 yields the result. 2 If 푉0 > 훼 no solution exists for a sufficiently small 푑. The condition (7) can be then modified replacing one or both hyperbolic tangents by the trigonometric one, however, we will not need it; it is enough to note that the lowest eigenvalue of ℎ — which certainly exists as ℎ as a Sturm–Liouville operator on a finite interval has a purely discrete spectrum — is positive for 푑 large enough.

3 Consider now the neighborhood Ω푑 ≔ { 푥 ∈ ℝ ∶ dist(푥, Σ) < 푑 } of the surface. Furthermore, fix a point 푥0 ∈ Σ and divide Σ into two parts, Σ푅 consisting of the 푐 points the geodesic distance from 푥0 is larger than 푅 and Σ푅 = Σ ⧵ Σ푅. Note that by assumptions (a) and (b), outside a compact Σ has a well defined normal, and the points of Ω푑 can be written as 푥Σ + 푛푥Σ푢 with |푢| < 푑, where 푥Σ is the point satisfying dist(푥, Σ) = dist(푥, 푥Σ); for 푑 small enough this part of Ω푑 does not intersect itself, i.e., the point 푥Σ is unique. Consequently, for 푅 sufficiently large and 푑 sufficiently small we may define Ω푑,푅 ≔ { 푥Σ + 푛푥Σ푢 ∶ 푥Σ ∈ Σ푅, |푢| < 푑 } 푐 푐 3 and Ω푑,푅 ≔ Ω푑 ⧵ Ω푑,푅. We also introduce the Ω푑 ≔ ℝ ⧵ Ω푑 consisting of two 3 푐 푐 connected components, so together we have ℝ = Ω푑,푅 ∩ Ω푑,푅 ∩ Ω푑. Using these notions we employ a bracketing argument. Changing the domain of 퐻 by additional Neumann conditions imposed at the boundaries of the three domains, we obtain a lower bound to our operator,

퐻 ≥ 퐻 ⊕ 퐻 푐 ⊕ 퐻 푐 . Ω푑,푅 Ω푑,푅 Ω푑 For the proof of (6) only the first part on right-hand side is relevant, because 퐻 푐 corresponds to a compact region and its essential spectrum is thus void, and Ω푑,푅 On the spectrum of leaky surfaces with a potential bias 175 inf 휎 (퐻 푐 ) = 0 holds obviously. To analyze the first part, which we for brevity ess Ω푑 denote as 퐻푑,푅 we employ a geometric argument similar to that used in [4], the difference being the constant potential 푉0 to one side of Σ푅. The ‘pierced layer’ 3 Ω푑,푅 can be regarded as a submanifold in ℝ equipped with the metric tensor (퐺 ) 0 퐺 = ( 휇휈 ) , 퐺 = (훿휍 − 푢ℎ 휍)(훿휌 − 푢ℎ 휌)푔 , (8) 푖푗 0 1 휇휈 휇 휇 휍 휍 휌휈 referring to the curvilinear coordinates (푠1, 푠2, 푢), where 푔휌휈 is the metric tensor 휍 of Σ푅 and ℎ휇 is the corresponding Weingarten tensor; we conventionally use the Greek notation for the range (1, 2) of the indices and the Latin for (1, 2, 3), and we employ the Einstein summation convention. In particular, the volume element of 1/2 2 Ω푑 is given by 푑Ω ≔ 퐺 d 푠 d푢 with

2 2 2 퐺 ≔ det(퐺푖푗) = 푔 [(1 − 푢푘1)(1 − 푢푘2)] = 푔(1 − 2푀푢 + 퐾푢 ) ; (9) for brevity we use the shorthand 휉(푠, 푢) ≡ 1−2푀(푠)푢 +퐾(푠)푢2. Next we introduce

−1 휚푅 ≔ (max{‖푘1‖∞, ‖푘2‖∞}) . Σ푅

By assumption (c) we have 휚푅 → ∞ as 푅 → ∞, and as self-intersections of Ω푑,푅 are avoided as long 푑 < 휚푅; we see that the layer halfwidth 푑 can be in fact chosen arbitrarily big provided 푅 is sufficiently large. At the same time, the transverse component of the Jacobian satisfies the inequalities 퐶−(푑, 푅) ≤ 휉(푠, 푢) ≤ 퐶+(푑, 푅), −1 2 where 퐶±(푑, 푅) ≔ (1 ± 푑휚푅 ) , hence for a fixed 푑 and 푅 large the Jacobian is essentially given by the surface part; recall that the metric tensor 푔휇휈 is assumed to be uniformly elliptic, 푐 − 훿휇휈 ≤ 푔휇휈 ≤ 푐+훿휇휈 with positive 푐±. Now we use these geometric notions to asses the spectral threshold of 퐻푑,푅. Passing to the curvilinear coordinates (푠, 푢), we write the corresponding quadratic form as 푖푗 2 (휕푖휓, 퐺 휕푗휓)퐺 + (휓, 푉휓)퐺 − 훼 ∫ |휓(푠, 0)| dΣ |푠|>푅 flat 2 flat defined on 퐻1(Ω푑,푅, dΩ), where ( ⋅ , ⋅ )퐺 means the scalar product in 퐿 (Ω푑,푅, dΩ), 1/2 flat the symbol dΣ stands for 푔 (푠) d푠, and Ω푑,푅 ≔ { 푞 ∶ |푠| > 푅, |푢| < 푑 }. Using the diagonal form (8) of the metric tensor together with (9) and the above mentioned bound to the factor 휉(푠, 푢), we find

2 2 2 (휓, 퐻푑,푅휓)퐺 ≥ ∫ (|휕ᵆ휓| + 푉|휓| ) dΩ − 훼 ∫ |휓(푠, 0)| dΣ flat Ω푑,푅 |푠|>푅

− 2 2 2 ≥ 휉푅 ∫ (|휕ᵆ휓| + 푉|휓| ) dΣ d푢 − 훼 ∫ |휓(푠, 0)| dΣ, flat Ω푑,푅 |푠|>푅 176 P. Exner

− + where 휉 ≔ infΩ 휉(푠, 푢). Introducing similarly 휉 ≔ sup 휉(푠, 푢) and using 푅 푑,푅 푅 Ω푑,푅 − Lemma 3.1 with the coupling constant 훼푅 = 훼/휉푅 , we get − 휉푅 −1 2 (휓, 퐻푑,푅휓)퐺 ≥ + (휇 − 푐0푑 ) ‖휓‖퐺 휉푅 provided 푑 is large enough. It is clear from the above discussion that to any 휀 > 0 −1 1 one can choose 푅 and 푑 sufficiently large to get 푐0푑 < 2 휀, and at the same time, − + 휚푅 > 푑 and 휉푅 /휉푅 < 휀/2휇. Consequently, inf 휎ess(퐻) > 휇 − 휀 which concludes the proof. Remarks 3.2. (a) The proof did not employ the part of assumption (a) speaking about convexity of one of the regions to which the surface divides the space and, in fact, neither the fact that Σ is simply connected. (b) The bound in Lemma 3.1 is a rough one but it suffices for the present purpose; in reality the error caused by the Neumann boundary is exponentially small as 푑 → ∞, similarly as in [7]. Note that we use such an estimate in a different way than in the said work: there we made the error small by choosing a large 훼 while here the coupling constant is fixed but we choose a large 푑 which we are allowed to do being far enough in the asymptotic region.

4. Proof of Theorem 2.3

In view of Theorem 2.2 it is sufficient to check that 푞[휓] ≥ 0 holds for any 휓 ∈ 2 3 퐶0(ℝ ). The contribution from Ωint to the quadratic form is non-negative and may be neglected; this yields the estimate 푞[휓] ≥ 푞ext[휓], where

2 2 2 1/2 2 푞ext[휓] ≔ ∫ (|∇휓| + 푉0|휓| ) (푥) d푥 − 훼 ∫ |휓(Σ(푠))| 푔 (푠) d 푠 . (10) 2 Ωext ℝ To estimate the quantity from below, we note that by assumption (b) there is a family of open connected subsets Σ푙, 푙 = 1, … , 푁, of Σ which are mutually disjoint and such that Σ = ⋃ Σ and Σ| is 퐶2 smooth for any 푙. It may happen that 푁 = 1 푙 푙 Σ푙 if Σ ⧵ (풞 ∪ 풫) is connected, on the other hand, 푁 ≥ 2 has to hold, for instance, when one of the curves of the family 풞 is closed, or more generally, if 풞 contains a loop. The number of the Σ푙’s can be made larger if we divide a smooth part of the surface by an additional boundary, but by assumption (b) the partition can be always chosen to have a finite number of elements, and moreover, only one of the Σ푙’s is not precompact in Σ. Next we associate the sets Ω푙 ≔ { 푥Σ + 푛푥Σ푢 ∶ 푥Σ ∈ Σ푙, 푢 < 0 } ⊂ Ωext with the Σ푙’s, where the negative sign refers to the fact that the normal vector points conventionally into the interior domain. Since the On the spectrum of leaky surfaces with a potential bias 177 latter is convex by assumption (a), no two halflines {푥Σ ∈ Σ푙, 푢 < 0} emerging from different points of Σ can intersect, and consequently, the sets Ω푙 are mutually disjoint, if 풞 ∪ 풫 ≠ ∅ the closure ⋃ Ω may be a proper subset of the exterior 푙 푙 domain. This yields

2 2 2 1/2 2 푞ext[휓] ≥ ∑ ∫ (|∇휓| + 푉0|휓| ) (푥) d푥 − 훼 ∫ |휓(Σ(푠))| 푔 (푠) d 푠 , 2 푙 Ω푙 ℝ and passing in the first integral to the curvilinear coordinates in analogy with the previous proof, we get

0 푖푗 2 1/2 2 푞ext[휓] ≥ ∑ ∫ ∫ (((휕푖휓) 퐺 (휕푗휓) + 푉0|휓| ) 퐺 )(푠, 푢) d 푠 d푢 푙 푀푙 −∞ − 훼 ∫ |휓(Σ(푠))|2 푔1/2(푠) d2푠 , ℝ2 where 푀푙 is the pull-back of the surface component Σ푙 by the map Σ. Neglecting 휇휈 the non-negative term (휕휇휓) 퐺 (휕휈휓) and using (10), we arrive at the estimate

0 2 2 1/2 푞ext[휓] ≥ ∑ ∫ ∫ (|휕ᵆ휓| + 푉0|휓| )(푠, 푢) 푔 (푠) (1 − 푢푘1(푠)) 푙 푀푙 −∞

2 2 1/2 2 × (1 − 푢푘2(푠))d 푠 d푢 − 훼 ∫ |휓(Σ(푠))| 푔 (푠) d 푠 . ℝ2

However, 1 − 푢푘휇(푠) ≥ 1 holds for 휇 = 1, 2 and 푢 < 0 because both the principal curvatures are non-negative in view of the convexity assumption. Furthermore, the difference between ⋃ 푀 and ℝ2 is a zero measure set, hence we finally find 푙 푙 0 2 2 2 1/2 2 푞[휓] ≥ ∫ {∫ (|휕ᵆ휓| + 푉0|휓| )(푠, 푢) d푢 − 훼|휓(푠, 0)| } 푔 (푠) d 푠 , ℝ2 −∞ where, with the abuse of notation, we have employed the symbol 휓(푠, 0) for 휓(Σ(푠)), but 훼 ≤ √푉0 holds by assumption, and consequently, the expression in the curly brackets is positive by Lemma 2.1(v). This concludes the proof.

5. Example of a conical surface

Consider now the the situation where Σ = 풞휃 is a circular conical surface of an opening angle 2휃 ∈ (0, 휋), in other words

3 2 2 풞휃 = { (푥, 푦, 푧) ∈ ℝ ∶ 푧 = cot 휃 √푥 + 푦 } . 178 P. Exner

This surface satisfies the assumptions (a)–(e) with 풫 consisting of a single point, the tip of the cone, hence if the potential bias is supported in the exterior of this cone, the spectrum of the corresponding operator is by Theorems 2.2 and 2.3 purely essential, 휎(퐻) = [휇, ∞). Let us look now what happens in the opposite case when the bias is in the interior. In view of the symmetry it is useful to employ the cylindrical coordinates relative to the axis of 풞휃. Since our potential is independent of the azimuthal angle, the operator 퐻 now commutes with the corresponding component of the angular momentum operator, −푖휕휑, and allows thus for a partial-wave decomposition, 퐻 = ⨁ 퐻(푚). Writing the wave function in the standard way through its 푚∈ℤ reduced components,

휔 (푟, 푧) 휓(푟, 휑, 푧) = ∑ 푚 e푖푚휑 , 푚∈ℤ √2휋푟 we can rewrite the quadratic form (2) as the sum 푞[휓] = ∑푚∈ℤ 푞푚[휓], where

2 2 4푚 − 1 2 푞푚[휓] ≔ ‖∇휔푚‖퐿2(ℝ2 ) + ∫ 2 |휔푚(푟, 푧)| d푟 d푧 (11) + 2 4푟 ℝ+

2 2 + ∫ 푉(푟, 푧) |휔 (푟, 푧)| d푟 d푧 − 훼 ‖휔 | ‖ 2 2 , 푚 푚 Γ휃 퐿 (ℝ+) 2 ℝ+

2 where ℝ+ is the halfplane { (푟, 푧) ∶ 푟 > 0, 푧 ∈ ℝ } and Γ휃 is the halfline 푧 = 푟 cot 휃. We note first that it is sufficient to focus on the component with vanishing angular momentum in the partial-wave decomposition.

Proposition 5.1. Let 푉(푥) = 푉0 휒Ωint(푥) with 푉0 > 0, then 푞푚[휓] ≥ 휇 holds for any nonzero 푚 ∈ ℤ and all 휓 ∈ 퐻1(ℝ3).

Proof. If 푚 ≠ 0, the second term on the right-hand side of (11) is non-negative, and one estimates the form from below by neglecting it. Following the paper [2] 2 where the case 푉0 = 0 is treated, we introduce in the halfplane ℝ+ another pair of orthogonal coordinates, 푠 measured along Γ휃 and 푡 in the perpendicular direction; the axes of the (푡, 푠) system are rotated with respect to those of (푟, 푧) around the point 푟 = 푧 = 0 by the angle 휃. In these coordinates we have

∞ ∞ ∞ 2 2 2 푞푚[휓] ≥ ‖∇휔푚‖ 2 2 + 푉0 ∫ d푠 ∫ |휔푚(푠, 푡)| d푡 − 훼 ∫ |휔푚(푠, 0)| d푠 , 퐿 (ℝ+) 0 −푠 tan 휃 0 where with the abuse of notation we write 휔푚(푠, 푡) for the function value in the 1 3 rotated coordinates. If 휓 ∈ 퐻 (ℝ ) and 푚 ≠ 0, the reduced wave function 휔푚 On the spectrum of leaky surfaces with a potential bias 179

1 2 belongs to 퐻 (ℝ+). By ˜휔푚 we denote its extension to the whole plane by the zero value in the other halfplane, { (푠, 푡) ∶ 푠 ∈ ℝ, 푡 < −푠 tan 휃 }, which naturally belongs to 퐻1(ℝ2). Such function form a subspace in 퐻1(ℝ2), however; hence we have

inf 푞푚[휓] 1 2 휔푚∈퐻 (ℝ+)

2 2 2 = inf {‖∇˜휔푚‖ 2 2 + 푉0 ∫ |˜휔푚(푠, 푡)| d푠 d푡 − 훼 ∫ |˜휔푚(푠, 0)| d푠} 1 2 퐿 (ℝ+) 휔푚∈퐻 (ℝ+) ℝ2 ℝ

2 2 2 ≥ inf {‖∇휚‖ 2 2 + 푉0 ∫ |휚(푠, 푡)| d푠 d푡 − 훼 ∫ |휚(푠, 0)| d푠} . 휚∈퐻1(ℝ2) 퐿 (ℝ+) ℝ2 ℝ Noting finally that form on the right-hand side of the last inequality isasso- ciated with the self-adjoint operator in 퐿2(ℝ2) which has separated variables, 2 −휕푠 ⊗ 퐼푡 + 퐼푠 ⊗ ℎ, where ℎ is the operator (3) with the variable 푥 replaced by −푡, we obtain the desired claim from Lemma 2.1. On the other hand, the component with zero momentum can give rise to a non- trivial discrete spectrum, at least as long the potential bias is weak enough. 2 Proposition 5.2. To any integer 푁 there is a number 푣 ∈ (0, 훼 ) such that we have #휎disc(퐻) ≥ 푁 for 0 ≤ 푉0 < 푣 . Proof. Because the potential bias we consider is a bounded perturbation, { 퐻 ∶ 훼,풞휃,푉 푉0 ≥ 0 } is a type (A) holomorphic family in the sense of [8]. This means, in particular, that the eigenvalues, if they exist, are continuous functions of 푉0. The same is by Theorem 2.2 true for the essential spectrum threshold. Since the bias-free operator 퐻 has by [2] an infinite number of isolated eigenvalues accumulating 훼,풞휃,0 1 2 at − 4 훼 , the continuity implies the result. Before proceeding further, let us note than in the higher-dimensional analogue of this problem the geometrically induced discrete spectrum is void in the absence of the bias, cf. [9], hence one does not expect a counterpart of Proposition 5.2 to hold either.

6. Example of a rooftop surface

In the previous example we left open the question whether the discrete spectrum could survive up to the critical value of the potential. To demonstrate that this is possible, consider now another example in which the surface Σ = ℛ퐿,휃 is defined through its cuts Γ푧 at the fixed value of the coordinate 푧, i.e. Σ = { Γ푧 ∶ 푧 ≥ 0 }. We ∞ suppose that each Γ푧 is a 퐶 loop in the (푥, 푦)-plane, being a border of a convex region, and consisting of 180 P. Exner

1 (i) two line segments { (푥, ±푧 tan 휃) ∶ |푥| ≤ 2 퐿 }, and (ii) two arcs connecting the loose ‘right’ and ‘left’ ends of the segment, respectively. We suppose in addition that that these arcs corresponding to different values of 푧 are mutually homothetic.

We can regard ℛ퐿,휃 as coming from cutting the cone of the previous example into two halves and inserting in between a wedge-shaped strip of height 퐿, modulo a smoothing in the vicinity of the interface lines. It is easy to check that such a surface satisfies assumptions (a)–(d) of Section 2 with the set 풞 consisting of 1 the segment { (푥, 0, 0) ∶ |푥| ≤ 2 퐿 } and 풫 = ∅. We have the following result concerning the critical operator 퐻 = 퐻 2: 훼,ℛ퐿,휃,훼

Proposition 6.1. 휎푑푖푠푐(퐻) is nonempty provided 퐿 is sufficiently large, and moreover, the number of negative eigenvalues can be made larger than any fixed integer by choosing 휃 small enough.

Proof. We employ the result from the two-dimensional case [6], where an attractive 훿 interaction supported by a broken line of the opening angle 2휃 ∈ (0, 휋) gives rise at least one bound states, and to a larger number for 휃 small. Let 휙 = 휙(푦, 푧) be an eigenfunction of the two-dimensional problem corresponding to an eigenvalue ∞ 휆 < 0. We choose a function 푔 ∈ 퐶0 (−1, 1) and use

1/2 휓휀 ∶ 휓휀(푥, 푦, 푧) = 휀 푔(휀푥)휙(푦, 푧) as a trial function. Since the variables are separated, it is straightforward to find the value of the quadratic form (2), namely

′ 2 2 푞[휓휀] = (휀‖푔 ‖ + 휆) ‖휓‖ .

The expression in the bracket can be made negative by choosing 휀 small enough, 1 and since the support of 휓휀 lies within the layer { (푥, 푦, 푧) ∶ |푥| ≤ 2 퐿 }, it is sufficient to choose 퐿 > 2휀. Moreover, the argument applies to any eigenvalue 휆 of the two-dimensional problem, which concludes the argument.

Note that the result will not change if the surface Σ is deformed outside the support of the trial function which means, in particular, that convexity assumption may be weakened.

Acknowledgments. The research was supported by the Czech Science Foun- dation within the project 17-01706S. On the spectrum of leaky surfaces with a potential bias 181

References

[1] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed. with appendix by P. Exner, AMS Chelsea, Rhode Island, 2005. [2] J. Behrndt, P. Exner, V. Lotoreichik, Schrödinger operators with 훿-interactions supported on conical surfaces, J. Phys. A: Math. Theor. 47 (2014), 355202 (16pp) [3] J.F. Brasche, P. Exner, Yu.A. Kuperin, P. Šeba, Schrödinger operators with singular interactions, J. Math. Anal. Appl. 184 (1994), 112–139. [4] P. Exner, S. Kondej, Bound states due to a strong 훿 interaction supported by a curved surface, J. Phys. A: Math. Gen. 36 (2003), 443–457. [5] P. Exner, K. Pankrashkin: Strong coupling asymptotics for a singular Schrödinger operator with an interaction supported by an open arc, Comm. PDE 39 (2014), 193–212. [6] P. Exner, S. Vugalter, On the existence of bound states in asymmetric leaky wires, arXiv:1505.02347 [7] P.Exner, K.Yoshitomi, Asymptotics of eigenvalues of the Schrödinger operator with a strong 훿-interaction on a loop, J. Geom. Phys. 41 (2002), 344–358. [8] T. Kato: Perturbation Theory for Linear Operators, 2nd edition, Springer, Berlin 1976. [9] V. Lotoreichik, T. Ourmières-Bonafos, On the bound states of Schrodinger operators with 훿-interactions on conical surfaces, Comm. PDE 41 (2016), 999–1028.

On the decay of almost periodic solutions of anisotropic degenerate parabolic-hyperbolic equations

Hermano Frid

Dedicated to Helge Holden on his 60th birthday

Abstract. We prove the well-posedness and decay of Besicovitch almost periodic solutions for nonlinear degenerate anisotropic hyperbolic-parabolic equations. The decay property is proven for the case where the diffusion term is given by a non-degenerate nonlinear 푑″ × 푑″ diffusion matrix and the complementary 푑′ components of flux-function form a non- degenerate flux in ℝ푑′ , with 푑′ + 푑″ = 푑. For this special case we also prove that the strong trace property at the initial time holds, which allows, in particular, to require the assumption of the initial data only in a weak sense, and gives the continuity in time of the solution 1 푑 with values in 퐿loc(ℝ ). So far, for the decay property, we need also to require that the bounded Besicovitch almost periodic initial function can be approximated in the Besicovitch 1/2 norm by almost periodic functions whose 휀-inclusion intervals 푙휀 satisfy 푙휀/| log 휀| → 0 as 휀 → 0. This includes, in particular, generalized limit periodic functions, that is, limits in the Besicovitch norm of purely periodic functions.

1. Introduction

We address the problem of the decay to the mean-value of 퐿∞ Besicovitch almost periodic solutions to nonlinear degenerate anisotropic hyperbolic-parabolic equations. Consider the Cauchy problem

푑 휕푡푢 + ∇푥 ⋅ 퐟(푢) = ∇푥 ⋅ (퐴(푢)∇푥푢), 푥 ∈ ℝ , 푡 > 0, (1.1) 푑 푢(0, 푥) = 푢0, 푥 ∈ ℝ , (1.2)

푑 where 퐟 = (푓 , … , 푓 ), 퐴(푢) = (푎 (푢)) , with 푓(푢), 푎 (푢)∶ ℝ → ℝ smooth 1 푑 푖푗 푖,푗=1 푖 푖푗 functions. 퐴(푢) is a symmetric non-negative matrix and so we may write

푑

푎푖푗(푢) = ∑ 휎푖푘(푢)휎푗푘(푢), (1.3) 푘=1 184 H. Frid

푑 with 휎 (푢)∶ ℝ → ℝ smooth functions, that is, (휎 (푢)) is the square root of 푖푗 푖푗 푖,푗=1 ∞ 푑 퐴(푢). We assume to begin with that 푢0 ∈ 퐿 (ℝ ). In this paper, we are concerned with the large-time behavior of entropy solutions of (1.1), (1.2) with initial function 푢0 satisfying ∞ 푑 푑 푢0 ∈ 퐿 (ℝ ) ∩ BAP(ℝ ). (1.4) Here, BAP(ℝ푑) denotes the space of the Besicovitch almost periodic functions (with exponent 푝 = 1), which can be defined as the completion of the space of 2휋푖휆⋅푥 √ trigonometric polynomials, i.e., finite sums ∑휆 푎휆푒 (푖 = −1 is the purely imaginary unity) under the semi-norm 1 푁 (푔) ≔ lim sup ∫ |푔(푥)| 푑푥, 1 푑 푅→∞ 푅 퐶푅 where, for 푅 > 0, 푑 퐶푅 ≔ { 푥 ∈ ℝ ∶ |푥|∞ ≔ max |푥푖| ≤ 푅/2 }. 푖=1,…,푑

We observe that the semi-norm 푁1 is indeed a norm over the trigonometric polynomials, so the referred completion through it is a well defined Banach space. 푑 Equivalently, the space BAP(ℝ ) is also the completion through 푁1 of the space of uniform (or Bohr) almost periodic functions, AP(ℝ푑), which is defined as the closure in the sup-norm of the trigonometric polynomials. We begin by stating the definition of entropy solution for (1.1), (1.2), which is in part motivated by [9]. We use the normal trace property of 퐿2-divergence measure fields (see, e.g., [6, 7]). ∞ 푑 Definition 1.1. An entropy solution for (1.1), (1.2), with 푢0 ∈ 퐿 (ℝ ), is a function 푢(푡, 푥) ∈ 퐿∞((0, ∞) × ℝ푑) such that (i) (Regularity) For any 푅 > 0, we have

푑 2 ∑ 휕푥푖훽푖푘(푢) ∈ 퐿 ((0, ∞) × 퐶푅), 푖=1 ᵆ

for 푘 = 1, … , 푑, for 훽푖푘(푢) = ∫ 휎푖푘(푣) 푑푣. (1.5)

(ii) (Chain Rule) For any function 휓 ∈ 퐶0(ℝ) with 휓(푢) ≥ 0 and any 푘 = 1, … , 푑 the following chain rule holds:

푑 푑 휓 √ 2 ∑ 휕푥푖훽푖푘(푢) = 휓(푢) ∑ 휕푥푖훽푖푘(푢) ∈ 퐿 ((0, ∞) × 퐶푅), 푖=1 푖=1 휓 ′ ′ for 푘 = 1, … , 푑, for (훽푖푘) = √휓훽푖푘, (1.6) On the decay of almost periodic solutions 185

for any 푅 > 0.

(iii) (Entropy Inequality) For any convex 퐶2 function 휂 ∶ ℝ → ℝ, and 퐪′(푢) = ′ ′ ′ ′ 휂 (푢)퐟 (푢), 푟푖푗(푢) = 휂 (푢)푎푖푗(푢), we have

푑 푑 푑 2 2 ″ 휕푡휂(푢) + ∇푥 ⋅ 퐪(푢) − ∑ 휕푥푖푥푗푟푖푗(푢) ≤ −휂 (푢) ∑ (∑ 휕푥푖훽푖푘(푢)) , (1.7) 푖푗=1 푘=1 푖=1

in the sense of distributions in (0, ∞) × ℝ푑, and

휂(푢(푡, 푥)) ⌊ {푡 = 0} = 휂(푢0(푥)), (1.8)

in the sense of the normal trace of the 퐿2-divergence measure field

푑 푑

(휂(푢), 퐪(푢) − (∑ 휕푥푗푟푖푗(푢)) ). 푗=1 푖=1

Remark 1.1. We remark that condition (iii) in the Definition 1.1 implies that for all 푘 ∈ ℝ we have

휕푡|푢(푡, 푥) − 푘| + ∇푥 ⋅ sgn(푢(푡, 푥) − 푘)(퐟(푢) − 퐟(푘)) 푑 2 − ∑ 휕푥푖푥푗 sgn(푢(푡, 푥) − 푘)(퐴푖푗(푢) − 퐴푖푗(푘)) ≤ 0, (1.9) 푖,푗=1

′ 푑 where 퐴푖푗(푢) = 푎푖푗(푢), in the sense of distributions in (0, ∞) × ℝ . Remark 1.2. We also remark that (1.8), valid for all 퐶2 convex 휂 implies, for any 푅 > 0,

lim ∫ |푢(푡, 푥) − 푢0(푥)| 푑푥 = 0, (1.10) 푡→0+ 퐶푅 as essentially follows from theorem 4.5.1 in [11] (see [15]) which establishes that (1.8) implies

lim ∫ 휂(푢(푡, 푥))휙(푥) 푑푥 = ∫ 휂(푢0(푥))휙(푥) 푑푥, 푡→0+ ℝ푑 ℝ푑

∞ 푑 for all 휙 ∈ 퐶0 (ℝ ), which by a well known convexity argument implies (1.10). 186 H. Frid

1 2 Remark 1.3. Take 휂(푢) = 2 푢 in (1.7) and as test function 휙푅(푥)휒휈(푡), with 휙푅 ∈ ∞ 푑 푑 퐶0 (ℝ ), 0 ≤ 휙푅(푥) ≤ 1, for all 푥 ∈ ℝ , 휙푅(푥) = 1, for |푥| ≤ 푅, 휙푅(푥) = 0, for 훼 |푥| ≥ 푅 + 1, and ‖퐷 휙푅‖∞ ≤ 퐶, |훼| ≤ 2, for some 퐶 > 0 independent of 푅, and 휒휈(푡) = 휃(푡 − 푡0) − 휃(푡 − 푡1), with 푡 휈푡

휃휈(푡) = ∫ 훿휈(푠) 푑푠 = ∫ 휎(푠) 푑푠, 훿휈(푠) = 휈휎(휈푠), 0 0 ∞ with 휎 ∈ 퐶0 (ℝ), supp 휎 ⊂ [0, 1], 휎 ≥ 0, ∫ℝ 휎(푠) 푑푠 = 1. Then, sending 휈 → ∞ we deduce that for some constant 퐶 > 0, independent of 푅, we have, for all 푡 > 0,

푡 푑 푑 2 푑 푑−1 ∫ ∫ ∑ (∑ 휕푥푖훽푖푘(푢)) 푑푥 푑푡 ≤ 퐶(푅 + 1) + 퐶푡(푅 + 1) . (1.11) 0 푘=1 푖=1 퐶푅 In particular, for any 푡 > 0,

푡 푑 푑 2 −푑 lim sup 푅 ∫ ∫ ∑ (∑ 휕푥푖훽푖푘(푢)) 푑푥 푑푡 ≤ 퐶. (1.12) 푅→∞ 0 푘=1 푖=1 퐶푅 For any 푔 ∈ BAP(ℝ푑), its mean value M(푔), defined by

M(푔) ≔ lim 푅−푑 ∫ 푔(푥) 푑푥, 푅→∞ 퐶푅 exists (see, e.g., [2]). The mean value M(푔) is also denoted by ⨍ℝ푑 푔 푑푥. Also, the Bohr–Fourier coefficients of 푔 ∈ BAP(ℝ푑)

−2휋푖휆⋅푥 푎휆 = M(푔푒 ), are well defined and we have that the spectrum of 푔, defined by

푔 Sp(푔) ≔ { 휆 ∈ ℝ ∶ 푎휆 ≠ 0 }, is at most countable (see, e.g., [2]). We denote by Gr(푔) the smallest additive subgroup of ℝ푑 containing Sp(푔) (cf. [22], where Gr(푔) was introduced and denoted by 푀(푔)). The first result of this paper is the following.

∞ 푑 Theorem 1.1. For any 푢0 ∈ 퐿 (ℝ ), there exists a unique weak entropy solution 푢(푡, 푥) of (1.1), (1.2). Moreover, if 푢0 satisfies (1.4), then

∞ 푑 ∞ 푑+1 푢 ∈ 퐿 ((0, ∞), BAP(ℝ )) ∩ 퐿 (ℝ+ ), (1.13) and Gr(푢(푡, ⋅ )) ⊂ Gr(푢0), for a.e. 푡 > 0. On the decay of almost periodic solutions 187

A particular case of (1.1) is the following

푑 휕푡푢 + ∇푥 ⋅ 퐟(푢) = ∇푥″(퐵(푢)∇푥″푢), 푥 ∈ ℝ , 푡 > 0, (1.14)

푑 where 퐵(푢) = (푏 (푢)) , and 1 ≤ 푑′ < 푑, so 퐵(푢) is a symmetric non- 푖푗 푖,푗=푑′+1 ″ ″ ″ ′ negative 푑 ×푑 -matrix, 푑 = 푑 −푑 , and ∇ ″ ≔ (휕 , … , 휕 ). Also, we assume 푥 푥푑′+1 푥푑 the non-degeneracy condition: For any (휏, 휅′) ∈ ℝ푑′+1, with 휏2 + 휅′2 = 1, and 휅″ ∈ ℝ푑″, with |휅″| = 1, denoting 휋푑′(퐟(푢)) = (푓1(푢), … , 푓푑′(푢)),

1 ′ ℒ { 휉 ∈ ℝ ∶ |휉| ≤ ‖푢0‖∞, 휏 + 휋푑′(퐟(휉)) ⋅ 휅 = 0 } = 0, (1.15) 1 ″푇 ″ ℒ { 휉 ∈ ℝ ∶ |휉| ≤ ‖푢0‖∞, 휅 퐵(휉)휅 = 0 } = 0. (1.16) Although (1.14) is a particular case of (1.1), under the non-degeneracy conditions (1.15) and (1.16) we may relax (1.8) in Definition 1.1 to

푢(푡, 푥) ⌊ {푡 = 0} = 푢0(푥), (1.17) in the sense of the normal trace of the 퐿2 divergence-measure field

푑 푑 ′ (푢, 퐟(푢) − (0,⏟⏟⏟ … , 0, ( ∑ 휕푥 퐵푖푗(푢)) )), 퐵푖푗(푢) = 푏푖푗(푢). 푗 푖=푑′+1 푑′ 푗=푑′+1 We call 푢(푡, 푥) ∈ 퐿∞((0, ∞) × ℝ푑) a weak entropy solution of (1.14), (1.2) if it satisfies all the corresponding conditions of Definition 1.1 except that instead of (1.8), we now impose the weaker (1.17). The second result of this paper concerns weak entropy solutions of (1.14), (1.2). Theorem 1.2. Let 푢 be weak entropy solution of (1.14), (1.2). Then,

1 푑 푢 ∈ 퐶([0, ∞), 퐿loc(ℝ )). (1.18) In particular, for any 푅 > 0,

lim ∫ |푢(푡, 푥) − 푢0(푥)| 푑푥 = 0. (1.19) 푡→0+ |푥|<푅

Moreover, if 푢0 satisfies (1.4), then

∞ 푑 ∞ 푑+1 푢 ∈ 퐿 ([0, ∞), BAP(ℝ )) ∩ 퐿 (ℝ+ ). (1.20) and Gr(푢(푡, ⋅ )) ⊂ Gr(푢0), (1.21) for a.e. 푡 > 0. 188 H. Frid

From Theorem 1.2, we deduce that weak entropy solutions of (1.14), (1.2) are indeed entropy solutions of (1.14), (1.2) in the sense of Definition 1.1, so that Theorem 1.1 applies to them. As we will see in Section 3, the proof of Theorem 1.2 amounts to show the validity of the strong trace property for the solution of (1.14), (1.2). Finally, we establish the following decay property as the third result of this paper. We remark that, in particular, the hypotheses on the initial function are clearly satisfied by the generalized limit periodic functions, that is, limits inthe Besicovitch norm induced by 푁1 of purely periodic functions.

Theorem 1.3. Assume, in addition to (1.4), that 푢0 can be approximated in the 휈 Besicovitch norm induced by 푁1 by a sequence of almost periodic functions 푢0 which, 휈 휈 1/2 for each 휀, possess 휀-inclusion intervals, 푙휀 , satisfying 푙휀 /| log 휀| → 0, as 휀 → 0. Then, the entropy solution of (1.14), (1.2) satisfies

lim M(|푢(푡, ⋅ ) − M(푢0)|) = 0. (1.22) 푡→+∞ There is a large literature related with degenerate parabolic equations, being the first important contribution by Vol’pert and Hudjaev in[27]. Uniqueness for the homogeneous Dirichlet problem, for the isotropic case, was only achieved many years later by Carrillo in [3], using an extension of Kruzhkov’s doubling of variables method [18]. The result in [3] was extended to non-homogeneous Dirichlet data by Mascia, Porretta and Terracina in [20]. An 퐿1 theory for the Cauchy problem for anisotropic degenerate parabolic equations was established by Chen and Perthame [9], based on the kinetic formulation (see [23]), and later also obtained using Kruzhkov’s approach in [1, 8] (see also, [17], [13] and the references therein). Decay of almost periodic solutions for general nonlinear systems of conservation laws of parabolic and hyperbolic types was first addressed in [14], as an extension of the ideas put forth in [4]. Only recently the problem of the decay of almost periodic solutions was retaken, specifically for scalar conservation laws, by Panov in [22], where some elegant ideas were introduced to successfully extend the result in [14] in that specific case. We first give a brief account on the way Theorem 1.1 is proven. The part of existence and uniqueness is by now well known, and for most of that we just refer to [8], which deals with the case of initial function in 퐿1(ℝ푑). Nevertheless, (1.12) is new and of great interest in the case of initial functions in 퐿∞(ℝ푑). For the invariance of the class of 퐿∞ Besicovitch almost periodic functions with exponent 푝 = 1, we use the elegant method of reduction to the periodic case introduced by Panov in [22]. Concerning Theorem 1.2, the first part, including (1.18), (1.19), and (1.20), which improves the regularity given in Theorem 1.1, is a consequence of the On the decay of almost periodic solutions 189 strong trace property enjoyed by (1.14) as is shown here. As for Theorem 1.3, namely, the decay property (1.22), it is obtained essentially using ideas in [14]. Unfortunately we cannot use the reduction to the periodic case for getting the decay of the solution, as in [22]. In particular, we cannot apply the result on the decay of periodic entropy solutions for nonlinear anisotropic degenerate parabolic-hyperbolic equations of Chen and Perthame in [10]. The reason is that we miss here the necessary non-degeneracy condition for the equation in higher space dimensions corresponding to the uplifting to the periodic context. This paper is organized as follows. After this Introduction, in Section 2, the proof of Theorem 1.1 is given, split in a number of auxiliary results, starting with the important in its own Proposition 2.1, followed by three lemmas. In Section 3, we prove Theorem 1.2, which establishes the strong trace property at the initial 1 푑 time and the continuity in time of the solution with values in 퐿loc(ℝ ). Finally, in Section 4, we prove Theorem 1.3, namely, the decay property.

2. Proof of Theorem 1.1

In this section we prove Theorem 1.1 through a number of auxiliary results and results that establish parts of the its statement. We begin with a proposition which is central in the whole strategy of reducing to the periodic case as devised in [22]. We will need the following technical lemma of [22], to which we refer for the proof. Lemma 2.1. Suppose that 푢(푥, 푦) ∈ 퐿∞(ℝ푛 × ℝ푚),

퐸 = { 푥 ∈ ℝ푛 ∶ (푥, 푦) is a Lebesgue point of 푢(푥, 푦) for a.e. 푦 ∈ ℝ푚 }.

Then 퐸 is a set of full measure and 푥 ∈ 퐸 is a common Lebesgue point of the functions 퐼(푥) = ∫ 푢(푥, 푦)휌(푦) 푑푦, for all 휌 ∈ 퐿1(ℝ푚). ℝ푚 1 ∞ 푑+1 Proposition 2.1 (mean 퐿 -contraction). Let 푢(푡, 푥), 푣(푡, 푥) ∈ 퐿 (ℝ+ ) be two ∞ 푑 entropy solutions of (1.1), (1.2), with initial data 푢0, 푣0 ∈ 퐿 (ℝ ). Then for a.e. 0 < 푡0 < 푡1 푁1(푢(푡1, ⋅ ) − 푣(푡1, ⋅ )) ≤ 푁1(푢(푡0, ⋅ ) − 푣(푡0, ⋅ )), (2.1) and also for a.e. 푡 > 0,

푁1(푢(푡, ⋅ ) − 푣(푡, ⋅ )) ≤ 푁1(푢0 − 푣0), (2.2)

Proof. We follow closely with the due adaptations the proof of proposition 1.3 in [22]. We first recall that by using the doubling of variables method of Kruzhkov 190 H. Frid

[18], as adapted by Carrillo [3] to the isotropic degenerate parabolic case and [1] to the anisotropic one, we obtain

푑 2 |푢−푣|푡 +∇⋅sgn(푢−푣)(퐟(푢)−퐟(푣)) ≤ ∑ 휕푥푖푥푗 sgn(푢−푣)(퐴푖푗(푢)−퐴푖푗(푣)) (2.3) 푖,푗=1

푑+1 in the sense of distributions in ℝ+ . As usual, we define a sequence approximating the indicator function of the interval (푡0, 푡1] , by setting for 휈 ∈ ℕ,

푡 휈푡

훿휈(푠) = 휈휎(휈푠), 휃휈(푡) = ∫ 훿휈(푠) 푑푠 = ∫ 휎(푠) 푑푠, 0 0

∞ where 휎 ∈ 퐶0 (ℝ), supp 휌 ⊂ [0, 1], 휎 ≥ 0, ∫ℝ 휎(푠) 푑푠 = 1. We see that 훿휈(푠) converges to the Dirac measure in the sense of distributions in ℝ, while 휃휈(푡) converges everywhere to the Heaviside function. For 푡1 > 푡0 > 0, if 휒휈(푡) = ∞ 휃휈(푡 − 푡0) − 휃휈(푡 − 푡1), then 휒휈 ∈ 퐶0 (ℝ+), 0 ≤ 휒휈 ≤ 1, and the sequence 휒휈(푡) converges everywhere, as 휈 → ∞, to the indicator function of the interval (푡0, 푡1]. Let ∞ 푑 us take 푔 ∈ 퐶0 (ℝ ), satisfying 0 ≤ 푔 ≤ 1, 푔(푦) ≡ 1 in the cube 퐶1, 푔(푦) ≡ 0 outside −푑 the cube 퐶푘, with 푘 > 1. We apply (1.8) to the test function 휑 = 푅 휒휈(푡)푔(푥/푅), for 푅 > 0. We then get

∞ −푑 ∫ (푅 ∫|푢(푡, 푥) − 푣(푡, 푥)| 푔(푥/푅) 푑푥)(훿휈(푡 − 푡0) − 훿휈(푡 − 푡1)) 푑푡 0 ℝ푑

−푑−1 + 푅 ∬ sgn(푢 − 푣)(퐟(푢) − 퐟(푣)) ⋅ ∇푦푔(푥/푅)휒휈(푡) 푑푥 푑푡 푑+1 ℝ+ 푑 −푑−1 − 푅 ∑ ∬ sgn(푢 − 푣)휕푥푖(퐴푖푗(푢) − 퐴푖푗(푣))휕푥푗푔(푥/푅)휒휈(푡) 푑푥 푑푡 ≥ 0. 푑+1 푖,푗=1 푅+ (2.4)

Define

퐹 = { 푡 > 0 ∶ (푡, 푥) is a Lebesgue point of |푢(푡, 푥) − 푣(푡, 푥)| for a.e. 푥 ∈ ℝ푑 }.

As a consequence of Fubini’s theorem, 퐹 is a set of full Lebesgue measure, and by Lemma 2.1 each 푡 ∈ 퐹 is a Lebesgue point of the functions

−푑 퐼푅(푡) = 푅 ∫ |푢(푡, 푥) − 푣(푡, 푥)| 푔(푥/푅) 푑푥, ℝ푑 for all 푅 > 0 and all 푔 ∈ 퐶0(ℝ). Now we assume 푡0, 푡1 ∈ 퐹 and take the limit as On the decay of almost periodic solutions 191

휈 → ∞ in (1.9) , to get

−푑−1 퐼푅(푡1) ≤ 퐼푅(푡0) + 푅 ∬ sgn(푢 − 푣)(퐟(푢) − 퐟(푣)) ⋅ ∇푦푔(푥/푅) 푑푥 푑푡

푑 (푡0,푡1)×ℝ 푑 −푑−1 − 푅 ∑ ∬ sgn(푢 − 푣)휕푥푖(퐴푖푗(푢) − 퐴푖푗(푣))휕푥푗푔(푥/푅) 푑푥 푑푡. (2.5) 푖,푗=1 푑 (푡0,푡1)×ℝ Now, we have

| | 푅−푑−1| ∬ sgn(푢 − 푣)(퐟(푢) − 퐟(푣)) ⋅ ∇ 푔(푥/푅) 푑푥 푑푡 | | 푦 | 푑 (푡0,푡1)×ℝ

−1 ≤ 푅 ‖퐟(푢) − 퐟(푣)‖∞ ∬ |∇푦푔(푦)| 푑푦 푑푡 → 0, as 푅 → ∞. (2.6)

푑 (푡0,푡1)×ℝ Also, we have

| 푑 | −푑−1| | 푅 ∑ ∬ sgn(푢 − 푣)휕푥푖(퐴푖푗(푢) − 퐴푖푗(푣))휕푥푗푔(푥/푅)휒휈(푡) 푑푥 푑푡 | 푑+1 | 푖,푗=1 푅+ 푑 푑 | | | | −푑−1| | | | ≤ 푅 ∬ ∑ (|훽푗푘(푢)| ∑ 휕푥푖훽푖푘(푢) )휕푥푗푔(푥/푅)휒휈(푡) 푑푥 푑푡 | 푑+1 | | | 푅+ 푘,푗=1 푖=1 푑 푑 | | | | −푑−1| | | | + 푅 ∬ ∑ (|훽푗푘(푣)| ∑ 휕푥푖훽푖푘(푣) )휕푥푗푔(푥/푅)휒휈(푡) 푑푥 푑푡 | 푑+1 | | | 푅+ 푘,푗=1 푖=1 푑 푑 2 푑 2 1/2 | | | | ≤ 퐶푅−1 ∑ (푅−푑 ∬ (|∑ 휕 훽 (푢)| + |∑ 휕 훽 (푣)| ) 푑푥 푑푡) | 푥푖 푖푘 | | 푥푖 푖푘 | 푘=1 푖=1 푖=1 (푡0,푡1)×퐶푘푅 1/2 2 × ( ∬ |∇푦푔(푦)| 푑푦 푑푡)

푑 (푡0,푡1)×ℝ ⟶ 0 as 푅 → ∞, (2.7) where we have used (1.12). On the other hand, we have

푑 푁1(푢(푡, ⋅ ) − 푣(푡, ⋅ )) ≤ lim sup 퐼푅(푡) ≤ 푘 푁1(푢(푡, ⋅ ) − 푣(푡, ⋅ )), 푅→∞ so taking the limit as 푅 → ∞ in (2.5), for 푡0, 푡1 ∈ 퐹, 푡0 < 푡1, we get

푑 푁1(푢(푡1, ⋅ ) − 푣(푡1, ⋅ )) ≤ 푘 푁1(푢(푡0, ⋅ ) − 푣(푡0, ⋅ )), 192 H. Frid and since 푘 > 1 is arbitrary we can make 푘 → 1+ to get the desired result. Finally, for 푡0 = 0, we use (1.10) to send 푡0 → 0+ in (2.5) and proceed exactly as we have just done.

Lemma 2.2 (Uniqueness). The problem (1.1), (1.2) has at most one entropy solution. Proof. The proof follows through standard arguments (cf., e.g., [27]). So, let 푢, 푣 ∈ ∞ 푑+1 퐿 (ℝ+ ) be two weak entropy solutions. As in Proposition 2.1, by using the doubling of variables method of Kruzhkov [18], as adapted by Carrillo [3] to the isotropic degenerate parabolic case and [1] to the anisotropic one, we obtain

∬{|푢 − 푣|휙푡 + sgn(푢 − 푣)(퐟(푢) − 퐟(푣)) ⋅ ∇휙

푑+1 ℝ+ 푑 2 + ∑ sgn(푢 − 푣)(퐴푖푗(푢) − 퐴푖푗(푣))휕푥푖푥푗휙} 푑푥 푑푡 ≥ 0, (2.8) 푖,푗=1

∞ 푑+1 −√1+푥2 for all 0 ≤ 휙 ∈ 퐶0 (ℝ+ ). We take 휙(푡, 푥) = 휌(푥)휒휈(푡), where 휌(푥) = 푒 and 휒휈 is as in the proof of Proposition 2.1. We observe that

푑 푑 2 ∑ |휕푥푖휌(푥)| + ∑ |휕푥푖푥푗휌(푥)| ≤ 퐶휌(푥), 푖=1 푖,푗=1 for some constant 퐶 > 0 depending only on 푑. Hence, making 휈 → 0, we arrive at

퐶푡˜ ∫ |푢(푡, 푥) − 푣(푡, 푥)|휌(푥) 푑푥 ≤ 푒 ∫ |푢0(푥) − 푣0(푥)|휌(푥) 푑푥, (2.9) ℝ푑 ℝ푑 which gives the desired result.

Observe that in the same way we got (2.9) from (2.8), we may get

∫(푢(푡, 푥) − 푣(푡, 푥)) 휌(푥) 푑푥 ≤ 푒퐶푡˜ ∫ (푢 (푥) − 푣 (푥)) 휌(푥) 푑푥, (2.10) + 0 0 + ℝ푑 ℝ푑 On the decay of almost periodic solutions 193 from

∬{(푢 − 푣)+휙푡 + sgn(푢 − 푣)+ (퐟(푢) − 퐟(푣)) ⋅ ∇휙

푑+1 ℝ+ 푑 2 + ∑ sgn(푢 − 푣)+ (퐴푖푗(푢) − 퐴푖푗(푣))휕푥푖푥푗휙} 푑푥 푑푡 ≥ 0, (2.11) 푖,푗=1 where (푢 − 푣)+ = max{0, 푢 − 푣} and sgn(푢 − 푣)+ = 퐻(푢 − 푣) where 퐻(푠) is the Heaviside function. Taking 푣 = 푘, with 푘 > ‖푢0‖∞, and then reversing the roles of 푢 and 푣, making 푢 = 푘 and 푣 = 푢, with 푘 < −‖푢0‖∞, we deduce that

푑 |푢(푡, 푥)| ≤ ‖푢0‖∞, for a.e. (푡, 푥) ∈ ℝ+ × ℝ . (2.12)

Lemma 2.3 (Existence). There exists an entropy solution to the problem (1.1), (1.2).

Proof. We consider (1.1), (1.2) with initial function 푢0,푅(푥) = 푢0(푥)휒퐵푅(푥), where 퐵푅 = 퐵(0, 푅) is the open ball with radius 푅 centered at the origin. By the existence theorem in [9], which holds for initial data in 퐿1(ℝ푑), we obtain an entropy solution 푢푅(푡, 푥) of (1.1), (1.2)푅. Now, using (2.9), we see that, for a.e. 푡 > 0,

∫|푢푅(푡, 푥) − 푢푅˜(푡, 푥)|휌(푥) 푑푥 ℝ푑 (2.13) 퐶푡˜ ˜ ≤ 푒 ∫ |푢0,푅(푥) − 푢0,푅˜(푥)|휌(푥) 푑푥 ⟶ 0, as 푅, 푅 → ∞. ℝ푑

1 푑 Therefore, 푢푅(푡, 푥) converges in 퐿loc((0, ∞) × ℝ ) to a function 푢(푡, 푥), which satisfies the bound in (2.12) since it holds for all 푢푅. It is now easy to deduce from the fact that the 푢푅’s satisfy all conditions of Definition 1.1 that 푢(푡, 푥) also satisfies all those conditions. We just observe that for the verification of (1.7) from the fact 1 푑 that the 푢푅’s satisfy (1.7), we use the uniform boundedness in 퐿loc(ℝ+ × ℝ ) of

푑 푑 2

∑ (∑ 휕푥푖훽푖푘(푢푅)) 푘=1 푖=1 and Fatou’s Lemma. Also, (1.8) is proved by including the initial function in (1.7), ∞ 푑+1 with 푢(푡, 푥) replaced by 푢푅(푡, 푥), tested against any function in 퐶0 (ℝ ), and taking the limit as 푅 → ∞, to conclude that (1.8) also holds.

In the next lemma, we prove that the solution operator for (1.1), (1.2) take bounded Besicovitch almost periodic functions into bounded Besicovitch almost periodic functions and that Gr(푢(푡, ⋅ )) ⊂ Gr(푢0( ⋅ )). 194 H. Frid

Lemma 2.4. Let 푢(푡, 푥) be the entropy solution of (1.1), (1.2) with 푢0 satisfying ∞ 푛 ∞ 푑+1 (1.4). Let 퐺0 = Gr(푢0). Then, 푢(푡, 푥) ∈ 퐿 ([0, ∞), BAP(ℝ )) ∩ 퐿 (ℝ+ ) and Sp(푢(푡, ⋅ )) ⊂ 퐺0, for a.e. 푡 > 0. Proof. The proof follows by the elegant method of reduction to the periodic case introduced by Panov in [22], more specifically theorems 2.1 and 2.2 in[22]. Here we limit ourselves to indicate the few adaptations that need to be made. The method begins by considering the case where the initial function 푢0 is given by a trigonometric polynomial,

2휋푖휆⋅푥 푢0(푥) = ∑ 푎휆푒 , (2.14) 휆∈Λ

푑 where Λ = Sp(푢0) ⊂ ℝ is a finite set. Since 푢0 is real we have that −Λ = Λ and 푎−휆 =휆 ̄푎 , where as usual ̄푧 is the complex conjugate of 푧 ∈ ℂ. The first observation is that we may find a basis for 퐺0, {휆1, … , 휆푚}, so that any 휆 ∈ 퐺0 can be uniquely ̄ 푚 ̄ 푚 written as 휆 = 휆(푘) = ∑푗=1 푘푗휆푗, 푘 = (푘1, … , 푘푚) ∈ ℤ , and the vectors 휆푗 are linearly independent over ℤ and so also over ℚ. Let 퐽 = { 푘̄ ∈ ℤ푚 ∶ 휆(푘)̄ ∈ Λ }.

Then 푚 2휋푖 ∑푗=1 푘푗휆푗⋅푥 푢0(푥) = ∑ 푎푘̄푒 , 푎푘̄ ≔ 푎휆(푘)̄ . (2.15) 푘∈퐽̄

We then have 푢0(푥) = 푣0(푦(푥)), where

2휋푘⋅푦̄ 푣0(푦) = ∑ 푎푘̄푒 (2.16) 푘∈퐽̄ is a periodic function, 푣0(푦 + 푒푖) = 푣0(푦), 푖 = 1, … , 푚, 푒푖 the elements of the canonical basis of ℝ푚, and

푑

푦(푥) = (푦1, … , 푦푚), 푦푗 = 휆푗 ⋅ 푥 = ∑ 휆푗푘푥푘, 휆푗 = (휆푗1, … , 휆푗푑). 푘=1 We then consider the nonlinear degenerate parabolic-hyperbolic equation

̃ 푚 푣푡 + ∇푦 ⋅ 퐟(푣) = (ℬ∇푦) ⋅ (퐴(푣)(ℬ∇푦)푣), 푣 = 푣(푡, 푦), 푡 > 0, 푦 ∈ ℝ , (2.17) ̃ ̃ ̃ with 퐟 = (푓1,…, 푓푚) and

푑 휕푦 푇 푓(푣)̃ = 휆 ⋅ 퐟(푣) = ∑ 휆 푓 (푣), 푗 = 1, … , 푚, ℬ = , 푗 푗 푗푘 푘 휕푥 푘=1 and 휕푦 푇 푚 푚 ℬ∇ = ∇ = (∑ 휆 휕 ,…, ∑ 휆 휕 ). 푦 휕푥 푦 푗1 푦푗 푗푑 푦푗 푗=1 푗=1 On the decay of almost periodic solutions 195

We consider the Cauchy problem for (2.17) with initial data

푣(0, 푦) = 푣0(푦). (2.18)

∞ 푚+1 Existence and uniqueness of the entropy solution 푣(푡, 푦) ∈ 퐿 (ℝ+ ) of (2.17), (2.18) follow from the analogs of Lemmas 2.3 and 2.2 for (2.17), (2.18), and it is easy to see that 푣(푡, 푦) is also spatially periodic, namely, 푣(푡, 푦 + 푒푖) = 푣(푡, 푦), for all 푚 푚 푦 ∈ ℝ , 푡 > 0, where 푒푗, 푗 = 1, … , 푚, is the canonical basis of ℝ . The following assertion corresponds to theorem 2.1 of [22], and its proof follows by the same lines as the proof of that result, so we just refer to [22] for the proof. Assertion #1. For a.e. 푧 ∈ ℝ푚 the function 푢(푡, 푥) = 푣(푡, 푧 + 푦(푥)) is an entropy solution of (1.1), (1.2) with initial data 푣0(푧 + 푦(푥)). The next step is another observation in [22] that it follows from Birkhoff individual ergodic theorem [12] that, for any 푤 ∈ 퐿1(Π푚), where Π푚 ≔ ℝ푚/ℤ푚, for almost all 푧 ∈ Π푚, we have

⨍ 푤(푧 + 푦(푥)) 푑푥 = ∫ 푤(푦) 푑푦. (2.19) ℝ푚 Π푚 Moreover, if 푤 ∈ 퐶(Π푚), then (2.19) holds for all 푧 ∈ Π푚, and 푤푧(푥) ≔ 푤(푧+푦(푥)) is a (Bohr) almost periodic function for each 푧 ∈ Π푚. The next main assertion corresponds to the first part of theorem 2.2 of[22], that is, it does not include the part about the decay of the entropy solution, and again its proof follows exactly as the one of the referred theorem and we refer to [22] for the proof. Also, in the present case we can no longer assert the continuity of the solution in 푡 taking values in BAP(ℝ푑), which is essentially based on the continuity of the periodic solution of the hyperbolic problem corresponding to (2.17), (2.18), which in general is not known for the degenerate parabolic-hyperbolic equation (2.17). As we will see in the next section, such continuity holds in the special case of the degenerate parabolic-hyperbolic equation (1.14), under the non-degeneracy conditions (1.15) and (1.16). We leave the claim about the decay of the weak entropy solution to be addressed in a subsequent statement by itself. Assertion #2. Let 푢(푡, 푥) be a weak entropy solution of (1.1), (1.2), and assume that the initial function 푢0(푥) is a trigonometric polynomial with 퐺0 = Gr(푢0). Then

∞ 푑 ∞ 푑+1 푢 ∈ 퐿 ([0, ∞), BAP(ℝ )) ∩ 퐿 (ℝ+ ) and Sp(푢(푡, ⋅ )) ⊂ 퐺0 for a.e. 푡 > 0. We just observe that Assertion #2 is proved (cf. [22]) by using Assertion #1 and showing, for a suitable sequence 푧푙 converging to 0 as 푙 → ∞, belonging to the set of full measure of 푧 ∈ ℝ푚 given by Assertion #1, for each fixed 푡 in a set of full 196 H. Frid

푧푙 measure in ℝ+, the convergence of the entropy solutions 푢 (푡, ⋅ ) = 푣(푡, 푧푙 + 푦(푥)) 푑 in BAP(ℝ ), as 푧푙 → 0, uniformly with respect to 푡, and using that for each 푧푙

푧 ∞ 푑 ∞ 푑+1 푢 푙 ∈ 퐿 ([0, ∞), BAP(ℝ )) ∩ 퐿 (ℝ+ )

푧푙 and Sp(푢 (푡, ⋅ )) ⊂ 퐺0 for a.e. 푡 > 0. 푑 ∞ 푑 Now, let us consider the general case where 푢0 ∈ BAP(ℝ )∩퐿 (ℝ ). Let 푢(푡, 푥) be the entropy solution of (1.1), (1.2) obtained above. Following [22], let Gr(푢0) 푑 be the minimal additive subgroup of ℝ containing Sp(푢0). We then consider a sequence 푢0푙 of trigonometrical polynomials such that 푢0푙 → 푢0 as 푙 → ∞, in 푑 BAP(ℝ ) and Sp(푢0푙) ⊂ Gr(푢0), which may be obtained from the Bochner–Fejér trigonometrical polynomials (see [2], p.105). We denote by 푢푙(푡, 푥) the weak entropy solution of (1.1), (1.2) with initial function 푢0푙(푥). By Proposition 2.1, there exists a set 퐹 ⊂ ℝ+ of full measure such that, for all 푡 ∈ 퐹 and for every 푙 ∈ ℕ, we have

푁1(푢(푡, ⋅ ) − 푢푙(푡, ⋅ )) ≤ 푁1(푢0푙 − 푢0) → 0, as 푙 → ∞. (2.20)

∞ Since 푢0푙 has finite spectrum, by Assertion #2 we see that 푢푙(푡, 푥) ∈ 퐿 ([0, ∞), 푑 BAP(ℝ )) and Sp(푢푙(푡, ⋅ )) ⊂ Gr(푢0), for all 푡 ∈ 퐹, for all 푙 ∈ ℕ. Therefore, ∞ 푑 푢 ∈ 퐿 ([0, ∞), BAP(ℝ )). Moreover, we easily see that Sp(푢(푡, ⋅ )) ⊂ Gr(푢0), for a.e. 푡 > 0.

3. Proof of Theorem 1.2

In this section we prove the first part of Theorem 1.2, namely (1.18), (1.19) and (1.20). This amounts to proving the strong trace property for the weak entropy solution of (1.14), (1.2), at any hyperplane 푡 = 푡0, for all 푡0 ≥ 0. Indeed, by the Gauss–Green Theorem (see, e.g., [6], [7]), applied to the (divergence-free) 2 퐿 -divergence-measure field (푢, 퐟(푢) − ∇푥″푏(푢)), we easily deduce that the limits ∞ 푑 lim푡→푡0± 푢(푡, 푥) exist in the weak star topology of 퐿 (ℝ ), for 푡0 > 0, and just the limit for 푡0+ when 푡0 = 0. By the same result, for 푡0 > 0, using the fact that the referred field is divergence-free, we easily deduce that the limits for 푡0+ and 푡0− must coincide. We also refer to theorem 4.5.1 of [11] whose proof establishes 1 푑 the continuity of 푢(푡, ⋅ ) from (0, ∞) into 퐿loc(ℝ ) except for a countable set of 푡 ∈ (0, ∞). As observed in [11], the continuity at 푡0 would follow if the entropy inequality included the initial time, which is not the case here, where we consider the weak initial prescription (1.17). We rewrite (1.7) for the present case. For any convex 퐶2 function 휂∶ ℝ → ℝ, On the decay of almost periodic solutions 197

′ ′ ′ ′ ′ and 퐪 (푢) = 휂 (푢)퐟(푢), 푟푖푗(푢) = 휂 (푢)푏푖푗(푢), 푖, 푗 = 푑 + 1, … , 푑, we have

푑 푑 푑 2 2 ″ 휕푡휂(푢) + ∇푥 ⋅ 퐪(푢) − ∑ 휕푥푖푥푗푟푖푗(푢) ≤ −휂 (푢) ∑ ( ∑ 휕푥푖훽푖푘(푢)) , (3.1) 푖,푗=푑′+1 푘=푑′+1 푖=푑′+1

푑 ᵆ in the sense of distributions in (0, ∞) × ℝ , where 훽푖푘(푢) = ∫ 휎푖푘(푣) 푑푣 and 푑 2 Σ(푢) = (휎푖푗(푢))푖,푗=푑′+1 satisfies 퐵(푢) = Σ(푢) . We will use the kinetic formulation for (1.1) (cf. [9]). So, we introduce the kinetic function 휒 on ℝ2:

⎧1 for 0 < 휉 < 푢, 휒(휉; 푢) = −1 for 푢 < 휉 < 0, ⎨ ⎩0 otherwise.

The following representation holds for any 푆 ∈ 퐶1(ℝ),

푆(푢) = ∫ 푆′(휉)휒(휉; 푢) 푑휉, (3.2)

ℝ which yields the following kinetic equation equivalent to (3.1):

푑 2 휕푡휒(휉; 푢)+퐚(휉)⋅∇푥휒(휉; 푢)− ∑ 푏푖푗(휉)휕푥푖푥푗휒(휉; 푢) = 휕휉(푚 +푛)(푡, 푥, 휉) (3.3) 푖,푗=푑′+1 in the sense of distributions in (0, ∞) × ℝ푑+1. In (3.3), 푚(푡, 푥, 휉), 푛(푡, 푥, 휉) are non-negative measures satisfying

∞ ∫ (푚 + 푛)(푡, 푥, 휉) 푑푥 푑푡 ≤ 휇푅,푇(휉) ∈ 퐿0 (ℝ) for all 푅, 푇 > 0, (3.4)

퐶푅,푇

∞ ∞ where 퐶푅,푇 = (0, 푇) × 퐶푅, and by 퐿0 we mean 퐿 with compact support, and

푑 푑 2

푛(푡, 푥, 휉) = 훿(휉 − 푢(푡, 푥)) ∑ ( ∑ 휕푥푖휎푖푘(푢(푡, 푥))) . (3.5) 푘=푑′+1 푖=푑′+1

1 2 Also, taking 휂(푢) = 2 푢 in (1.7), we see that

∫ ∫ (푚 + 푛)(푡, 푥, 휉) 푑푥 푑푡 푑휉 ≤ 퐶(푅, 푇), (3.6)

ℝ 퐶푅,푇 for all 푅, 푇 > 0, for some constant 퐶(푅, 푇) > 0 depending only on 푅, 푇 and ‖푢0‖∞. 198 H. Frid

Equation (3.1) implies that for any convex entropy 휂, the vector field 퐹 = 푑 (휂(푢), 퐪(푢) − (∑푑 휕 ̂푟 (푢)) ) ∈ 풟ℳ2(퐶 ), where 푖=1 푥푖 푖푗 푗=1 푅,푇

0, for 1 ≤ 푖 ≤ 푑′ or 1 ≤ 푗 ≤ 푑′, ̂푟 (푢) = { 푖푗 ′ 푟푖푗(푢), for 푑 + 1 ≤ 푖, 푗 ≤ 푑

2 and for any 푅 > 0, 푇 > 0, that is, it is an 퐿 divergence-measure field on 퐶푅,푇. By theorems 3.1 and 3.2 in [15], or essentially also from lemma 1.3.3 in [11], the 2 normal trace of the 풟ℳ -field 퐹 at the hyperplane 푡 = 푡∗ ∈ (0, 푇), from above, that is, as a part of the boundary of 퐶푅,푇 ∩ {푡 > 푡∗}, as well as from below, that is, as part of the boundary of 퐶푅,푇 ∩ {푡 < 푡∗}, is simply given by

⟨퐹 ⋅ 휈, 휙⟩푎푡=푡∗± = ∫ 휂(푢(푡∗, 푥))휙(푥) 푑푥, ℝ푑

1 푑 for a.e. 푡∗ > 0, for any 휙 ∈ 퐶푐 (ℝ ), where ⟨퐹⋅휈, ⋅ ⟩푡=푡∗+ denotes the normal trace at

{푡 = 푡∗} from above and ⟨퐹 ⋅ 휈, ⋅ ⟩푡=푡∗− the one from below. Also, from theorem 3.2 in [15] or also essentially from lemma 1.3.3 in [11], we deduce that, for any 푡0 > 0,

⟨퐹 ⋅ 휈, 휙⟩푡=푡 ± = ess lim ∫ 휂(푢(푡, 푥))휙(푥) 푑푥, (3.7) 0 푡→푡 ± 0 ℝ푑

1 푑 for any 휙 ∈ 퐶푐 (ℝ ), and for 푡0 = 0 we have, similarly,

⟨퐹 ⋅ 휈, 휙⟩푡=0+ = ess lim ∫ 휂(푢(푡, 푥))휙(푥) 푑푥. (3.8) 푡→0+ ℝ푑 Now, using (3.1) and the representation (3.2) for an arbitrary convex 휂, we deduce that, for 푓(푡, 푥, 휉) = 휒(휉; 푢(푡, 푥)), there exists the limit

lim 푓(푡, ⋅ , ⋅ ) = 푓휏( ⋅ , ⋅ ), (3.9) 푡→푡0+

∞ in the weak star topology of 퐿 (퐶푅 × (−퐿, 퐿)), for any 푅 > 0, and any 퐿 > 0 satisfying ‖푢‖ ∞ 푑+1 ≤ 퐿. Similarly, we have 퐿 (ℝ+ )

휏 lim 푓(푡, ⋅ , ⋅ ) = 푓−( ⋅ , ⋅ ), (3.10) 푡→푡0−

∞ in the weak star topology of 퐿 (퐶푅 × (−퐿, 퐿)). We observe that for 휂(푢) = 푢, for all 푡0 > 0,

ess lim ∫ 푢(푡, 푥)휙(푥) 푑푥 = ess lim ∫ 푢(푡, 푥)휙(푥) 푑푥, (3.11) 푡→푡 + 푡→푡 − 0 ℝ푑 0 ℝ푑 On the decay of almost periodic solutions 199

1 푑 for all 휙 ∈ 퐶푐 (ℝ ), as a consequence of (3.7), (3.8) and the Gauss–Green formula [6, 7] (essentially also from lemma 1.3.3 in [11]). Therefore, if the existence of strong trace of 푢(푡, 푥) at 푡 = 푡0 can be proved, both from above and below, these strong traces must coincide. Since the proof of the strong trace property from below is totally analogous to that for the strong trace from above, it will suffice to investigate the latter. Following the method in [26], in order to prove that the limits in (3.9) and 1 (3.10) can be taken as the strong convergence in 퐿 (퐶푅,푇 × (−퐿, 퐿)), it suffices to prove that 푓휏( ⋅ , ⋅ ) is a 휒-function, which is proved by using localization method introduced in [26]. For simplicity we just consider the case 푡0 = 0. 푑 ′ ″ ′ 푑′ ″ 푑″ 푑 We write for 푥 ∈ ℝ , 푥 = (푥 , 푥 ), where 푥 ∈ ℝ , 푥 ∈ ℝ . Fixing, 푥0 ∈ ℝ , we consider the sequence

푓휀(푡, 푥, 휉) ≔ 푓(휀푡, 푥0 + Λ(휀)푥, 휉),

′ 1/2 ″ where Λ(휀)푥 = (휀푥 , 휀 푥 ). So, 푓휀 satisfies

푑 ′ 1/2 ″ 2 휕푡푓휀 +퐚(휉) ⋅∇푥′푓휀 +휀 퐚(휉) ⋅∇푥″푓휀 − ∑ 푏푖푗(휉)휕푥 푥 푓휀 = 휕휉(푚휀 +푛휀), (3.12) 푖 푗 푖,푗=푑′+1 where ′ ″ ′ 퐚(휉) = (휋푑′(퐚(휉)),⏟⎵⏟⎵⏟ 0, … , 0 ), 퐚(휉) = 퐚(휉) − 퐚(휉) , 푑″ + 푑 0 0 푖 푖 and 푚휀 ∈ ℳloc((0, ∞) × ℝ × ℝ) is defined, for every 0 ≤ 푅1 < 푅2, 푅1 < 푅2, 푖 = 1, … , 푑, 퐿1 < 퐿2, by

푖 푖 (푚휀 + 푛휀) ( ∏ [푅1, 푅2] × [퐿1, 퐿2]) 0≤푖≤푑

1 0 0 푖 푖 = ″ (푚 + 푛) ([휀푅1, 휀푅2] × (푥0 + Λ(휀) ∏ [푅1, 푅2]) × [퐿1, 퐿2]) , 푑′+ 푑 휀 2 1≤푖≤푑 where Λ(휀) ∶ ℝ푑 → ℝ푑 is defined by Λ(휀)푧 ≔ (휀푧′, 휀1/2푧″). Following [26], as in [16], we have there exists a sequence 휀푛 converging to 0 푑 푑 푑 and a set ℰ ⊂ ℝ , with ℒ (ℝ ⧵ ℰ) = 0, such that for all 푥0 ∈ ℰ

lim(푚휀 + 푛휀 ) = 0, (3.13) 휀→0 푛 푛

+ 푑 in the weak topology of ℳloc((0, ∞) × ℝ × ℝ). We now observe that

휏 푓휀(0, 푥, 휉) = 푓 (푥0 + Λ(휀)푥, 휉). (3.14) 200 H. Frid

Again following [26], as in [16], we have that there exists a subsequence still ′ 푑 ′ denoted 휀푛 and a subset ℰ of ℝ such that for every 푥0 ∈ ℰ and for every 푅 > 0,

퐿 휏 휏 lim ∫ ∫ |푓 (푥0, 휉) − 푓 (푥0 + Λ(휀푛)푥, 휉)| 푑푥 푑휉 = 0. (3.15) 휀 →0 푛 −퐿 (−푅,푅)푑

Now, we claim that there exists a sequence 휀푛 which goes to 0 and a 휒-function ∞ 푑 1 푓∞ ∈ 퐿 (ℝ+ × ℝ × (−퐿, 퐿)) such that 푓휀푛 converges strongly to 푓∞ in 퐿loc(ℝ+ × ℝ푑 × (−퐿, 퐿)) and

푑 ′ 2 휕푡푓∞ + 퐚(휉) ⋅ ∇푥′푓∞ − ∑ 푏푖푗(휉)휕푥 푥 푓∞ = 0. (3.16) 푖 푗 푖,푗=푑′+1 The proof of the claim is very similar to that of proposition 3 of [26], and lemma 3.1 in [16], and relies on a particular case of the version of averaging lemma in [24] (see also [25]). Here, we need the following variation of the standard averaging lemma.

′ ″ ′ ″ Lemma 3.1. Let 푁, 푁 , 푁 be positive integers with 푁 = 푁 + 푁 , 푓푛(푦, 휉) be a 2 1 푖 푖 2 +1 bounded sequence in 퐿 (ℝ × ℝ) ∩ 퐿 (ℝ × ℝ), 퐠푛, 퐠 ∈ 퐿 (ℝ × ℝ, ℝ ) be such 푖 푖 2 +1 that 퐠푛 → 퐠 strongly in 퐿 (ℝ × ℝ, ℝ ), 푖 = 1, 2, and for 푦 ∈ ℝ we write 푦 = (푦′, 푦″), 푦′ ∈ ℝ ′, 푦″ ∈ ℝ ″. Assume

푑 ′ ″ 2 1 2 ′ ″ 훼(휉) ⋅∇푦 푓푛 +훼(휉) ⋅∇푦 푓푛 − ∑ 훽푖푗(휉)휕푦푖푦푗푓푛 = 휕휉∇푦,휉 ⋅퐠푛 +∇푦,휉 ⋅퐠푛, (3.17) 푖,푗=푑′+1 where 훼( ⋅ )′ ∈ 퐶2(ℝ; ℝ ′), 훼( ⋅ )″ ∈ 퐶2(ℝ; ℝ ″), and 훽 ∈ 퐶2(ℝ) satisfy

′ ℒ1{ 휉 ∈ ℝ ∶ 훼(휉) ⋅ 휁′ = 0 } = 0, for every 휁′ ∈ ℝ , with |휁′| = 1, (3.18) where ℒ1 is the Lebesgue measure on ℝ, and also

ℒ1{ 휉 ∈ ℝ ∶ 훽(휉) = 0 } = 0. (3.19)

∞ 휙 Then, for any 휙 ∈ 퐶푐 (ℝ), the average 푢푛(푦) = ∫ℝ 휙(휉)푓푛(푦, 휉) 푑휉 is relatively compact in 퐿2(ℝ ). The application of Lemma 3.1 to the problem at hand is made, as in [26], by ∞ 푑 multiplying (3.12) by 휙1(푡, 푥), 휙2(휉) where 휙1 ∈ 퐶0 ((1/(2푅), 2푅) × (−2푅, 2푅) ), ∞ 휙2 ∈ 퐶0 (−2퐿, 2퐿), both taking values in [0, 1], with 휙1(푡, 푥) = 1, for (푡, 푥) ∈ 푑 (1/푅, 푅) × (−푅, 푅) , 휙2(휉) = 1, for 휉 ∈ (−퐿, 퐿). We then consider the equation obtained for 휙1휙2푓휀, which is easily seen to satisfy the hypotheses of Lemma 2.1, we refer to [26] for the details. On the decay of almost periodic solutions 201

′ The final step of the proof is to prove that for every 푥0 ∈ ℰ ,

휏 푓∞(0, 푥, 휉) = 푓 (푥0, 휉), (3.20) for a.e. (푥, 휉) ∈ ℝ푑 × (−퐿, 퐿), which the result corresponding to proposition 4 of [26]. The proof is the same as the one of the referred proposition, and consists in ∞ 푑 proving that, for any 휙 ∈ 퐶0 (ℝ × (−퐿, 퐿)), the sequence

퐿 휀 ℎ휙(푡) ≔ ∫ ∫ (푓휀(푡, 푥, 휉) − 푓∞(푡, 푥, 휉))휙(푥, 휉) 푑푥 푑휉, −퐿 ℝ푑 converges to 0 in 퐵푉((0, 1)), which is done exactly as in [26]. Finally, from (3.16) and (3.20), we easily conclude that

휏 푓∞(푡, 푥, 휉) = 푓 (푥0, 휉), for almost all (푡, 푥, 휉) ∈ ℝ푑+1 × (−퐿, 퐿), which is constant with respect to (푡, 푥). 휏 Hence, since 푓∞ is a 휒-function for almost all (푡, 푥), we conclude that 푓 (푥0, ⋅ ) is a 휒-function, as was to be proved. The proof of the strong trace property at any hyperplane {푡 = 푡0}, 푡0 > 0, both from above and from below, follows exactly as just done for 푡0 = 0, from above. This establishes the strong trace property at the 1 푑 initial time and the continuity in time with values in 퐿loc(ℝ ). Finally, since we have already proved the strong assumption of the initial data, it follows that the weak entropy solution of (1.14), (1.2) is actually an entropy solution in the sense of Definition 1.1. In particular, (1.20) and (1.21) follow from Theorem 1.1.

4. Proof of Theorem 1.3

In this section we prove the decay property for the (weak) entropy solution of (1.14), (1.2). The decay property follows using ideas in [14]. We recall that the space of Stepanoff almost periodic functions (with exponent 푝 = 1) in ℝ푑, SAP(ℝ푑), is defined as the completion of the trigonometric polynomials with respect tothe norm

‖푓‖푆 ≔ sup ∫ |푓(푦)| 푑푦 = sup ∫ |푓(푦 + 푥)| 푑푦, 푥∈ℝ푟 푑 퐶1(푥) 푥∈ℝ 퐶1 where 푑 퐶푅(푥) ≔ { 푦 ∈ ℝ ∶ |푦 − 푥|∞ ≔ max |푦푖 − 푥푖| ≤ 푅/2 }. 푖=1,…,푑 202 H. Frid

Another characterization of the Stepanoff almost periodic function (S-a.p., for short) is obtained introducing the concept of 휀-period of a function 푓, that is a number 휏 satisfying ‖푓( ⋅ + 휏) − 푓( ⋅ )‖푆 ≤ 휀. (4.1)

Let 퐸푆{휀, 푓} denote the set of such numbers. If the set 퐸푆{휀, 푓} is relatively dense for all positive values of 휀, then the function 푓 is S-a.p. (see, e.g., [2]). By the set 퐸푆{휀, 푓} being relatively dense it is meant that there exists a length 푙휀, called 휀-inclusion 푑 interval, such that for any 푥 ∈ ℝ , 퐶푙휀(푥) contains an element of 퐸푆{휀, 푓}. Now, as a consequence of the fact that Gr(푢(푡, 푥)) ⊂ Gr(푢0), we have the following lemma which is of interest in its own.

Lemma 4.1. If 푢0 is a trigonometric polynomial, then the entropy solution of (1.14), (1.2), 푢(푡, 푥), is S-a.p. for all 푡 > 0, and, for any 휀 > 0, 푢(푡, 푥) possesses an 휀-inclusion ′ interval, 푙휀(푡), satisfying 푙휀(푡) = 푙휀′(휀,푡)(0), where 푙휀′(0) is an 휀 -inclusion interval of ′ −1 −퐶푡 푢0(푥), and 휀 (휀, 푡) = 휀| log 휀| 푒 , for certain 퐶 > 0.

Proof. Clearly, 푢0, being a trigonometric polynomial, is S-a.p. The fact that 푢(푡, 푥) is S-a.p. for all 푡 > 0 follows from (2.9), with 푣(푡, 푥) = 푢(푡, 푥 + 휏) and 휌(푥 − 푥0) instead of 휌(푥), from which we deduce

∫ |푢(푡, 푥 + 휏) − 푢(푡, 푥)| 푑푥

퐶1(푥0) 1 ≤ 푐(푡) ∫ |푢 (푥) − 푢 (푥 + 휏)| 휌(푥 − 푥 ) 푑푥 + 푐(푡)푂( ) 0 0 0 푅 퐶푅(푥0) 1 ≤ 푐(푅, 푡) sup ∫ |푢0(푦 + 휏) − 푢0(푦)| 푑푦 + 푐(푡)푂( ), (4.2) 푥∈ℝ푑 푅 퐶1(푥) where 푐(푡) = 푒퐶푡˜ , 퐶˜ > 0 only depending on 휌, 푐(푅, 푡) is a positive constant depending only on 푅, 푡, and 푂(1/푅) goes to zero when 푅 → ∞ uniformly with respect to 푥0. So, choosing 푅 large enough so that 푐(푡)푂(1/푅) ≤ 휀/2 and then taking any 휏 ∈ 퐸푆{휀/(2푐(푅, 푡)), 푢0}, we get that 휏 ∈ 퐸푆{휀, 푢(푡, ⋅ )}, and so 푢(푡, ⋅ ) is S-a.p. A technical computation on the terms on the right-hand side of (4.2), using 휌 to estimate 푅 as a function of 휀/(2푐(푡)), and then getting an expression for 푐(푅, 푡), ′ −1 −퐶푡 gives the estimate 푙휀(푡) = 푙휀′(휀,푡)(0), with 휀 (휀, 푡) = 휀| log 휀| 푒 , for certain 퐶 > 0, as desired.

Now we can use Lemma 4.1 to prove the decay property (1.22). Clearly, from Proposition 2.1, it suffices to consider the case where the initial function 푢0 is an 1/2 almost periodic function whose 휀-inclusion intervals 푙휀 satisfy 푙휀/| log 휀| → 0, as On the decay of almost periodic solutions 203

휀 → 0. From Lemma 4.1 we see that the 휀-inclusion interval of the solution 푢(푡, 푥) 1/2 푇 satisfies 푙휀(푡)/푡 → 0. Let us then consider the scaling sequence 푢푞 (푡, 푥) ≔ 푢(푇푡, 푇푥′, √푇푥″), and define 휉′ = 푥′/푡, 휉″ = 푥″/√푡. So, 푢푇 is a uniformly bounded 푇 sequence of weak entropy solutions of (1.1), (1.2), with initial functions 푢0 (푥) ≔ ′ ″ 푇 푢0(푇푥 , √푇푥 ). Using the Averaging Lemma 3.1, we deduce that 푢 is relatively 1 푑+1 compact in 퐿loc(ℝ+ ) and the initial functions clearly weakly converge to ̄푢0 = 푇 M(푢0). By passing to a subsequence, which we still denote by 푢 (푡, 푥), we have 푇 1 푑+1 ∞ 푑+1 that 푢 → ̄푢 as 푇 → ∞, in 퐿loc(ℝ ), for some ̄푢∈ 퐿 (ℝ+ ). We see also that ̄푢 satisfies (1.5), (1.6), (1.7) and (1.17), all of which are easy to be verified, and we observe by (1.17) that ̄푢(0, 푥) = M(푢0). Now, in view of Theorem 1.2, by 푇 1 푑+1 uniqueness, we conclude that ̄푢(푡, 푥) = M(푢0), that is 푢 → M(푢0), in 퐿loc(ℝ ). This, in particular, implies

1 | ′ ″ | ′ ″ 0 = lim ∫ ∫ 푢(푇푡, 푇푥 , √푇푥 ) − M(푢0) 푑푥 푑푥 푑푡 푇→∞ | | 0 |푥′|≤푐′, |푥″|≤푐″ 푇 1 1 ′ ″ ′ ″ = lim ∫ ′ ″ ∫ |푢(푡, 푥 , 푥 ) − M(푢0)| 푑푥 푑푥 푑푡 푇→∞ 푇 푑 +푑 /2 0 푇 |푥′|≤푐′푇, |푥″|≤푐″√푇 1 1 푇 | ′ ″√ | ′ ″ ≥ ′ ″ lim ∫ ∫ |푢(푡, 휉 푡, 휉 푡) − M(푢0)| 푑휉 푑휉 푑푡, 푑 +푑 /2 푇→∞ 푇 2 푇/2 |휉′|≤푐′, |휉″|≤푐″ which implies

푇 1 | ′ ″ | ′ ″ lim ∫ ∫ |푢(푡, 휉 푡, 휉 √푡) − M(푢0)| 푑휉 푑휉 푑푡 = 0, (4.3) 푇→∞ 푇 0 |휉′|≤푐′, |휉″|≤푐″ as is easily seen. Now, invoking Lemma 4.1, we can then make a computation similar to that in p.51 of [14] in order to get that there are constants 푐1, 푐2 > 0 depending only on the dimension, such that, given any 휀 > 0,

| ′ ″ | ′ ″ ∫ |푢(푡, 휉 푡, 휉 √푡) − M(푢0)| 푑휉 푑휉 |휉′|≤푐′,|휉″|≤푐″

≥ 푐1 M(|푢(푡, ⋅ ) − M(푢0)|) − 푐2휀. (4.4)

Therefore, by (4.3), we deduce

1 푇 lim ∫ M(|푢(푡, ⋅ ) − M(푢0)|) 푑푡 = 0. (4.5) 푇→∞ 푇 0 204 H. Frid

Now, by Proposition 2.1, we conclude

lim M(|푢(푡, ⋅ ) − M(푢0)|) 푑푡 = 0, (4.6) 푡→∞ which is the desired result.

Acknowledgements. The author gratefully acknowledges the support from CNPq, through grant proc. 303950/2009-9, and FAPERJ, through grant proc. E-26/103.019/2011.

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Factorizations and Hardy–Rellich-type inequalities

Fritz Gesztesy and Lance Littlejohn

We dedicate this note with great pleasure to Helge Holden, whose wide range of contributions to a remarkable variety of areas in mathematical physics, stochastics, partial differential equations, and integrable systems, whose exemplary involvement with students, and whose tireless efforts on behalf of the mathematical community, deserve our utmost respect and admiration. Happy Birthday, Helge, we hope our modest contribution to Hardy–Rellich-type inequalities will give some joy.

Abstract. The principal aim of this note is to illustrate how factorizations of singular, even- order partial differential operators yield an elementary approach to classical inequalities of Hardy–Rellich-type. More precisely, introducing the two-parameter 푛-dimensional homoge- −2 −2 푛 neous scalar differential expressions 푇훼,훽 ≔ −Δ+훼|푥| 푥⋅∇+훽|푥| , 훼, 훽 ∈ ℝ, 푥 ∈ ℝ ⧵{0}, + + 푛 ∈ ℕ, 푛 ≥ 2, and its formal adjoint, denoted by 푇훼,훽, we show that nonnegativity of 푇훼,훽푇훼,훽 ∞ 푛 on 퐶0 (ℝ ⧵ {0}) implies the fundamental inequality

∫ [(Δ푓)(푥)]2 푑푛푥 ≥ [(푛 − 4)훼 − 2훽] ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥 ℝ푛 ℝ푛 − 훼(훼 − 4) ∫ |푥|−4|푥 ⋅ (∇푓)(푥)|2 푑푛푥 ℝ푛 (∗) + 훽[(푛 − 4)(훼 − 2) − 훽] ∫ |푥|−4|푓(푥)|2 푑푛푥, ℝ푛 ∞ 푛 푓 ∈ 퐶0 (ℝ ⧵ {0}).

A particular choice of values for 훼 and 훽 in (∗) yields known Hardy–Rellich-type inequalities, including the classical Rellich inequality and an inequality due to Schmincke. By locality, these inequalities extend to the situation where ℝ푛 is replaced by an arbitrary open set 푛 ∞ Ω ⊆ ℝ for functions 푓 ∈ 퐶0 (Ω ⧵ {0}). Perhaps more importantly, we will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators. 208 F. Gesztesy and L. Littlejohn

1. Introduction

The celebrated (multi-dimensional) Hardy inequality,

∫ |(∇푓)(푥)|2 푑푛푥 ≥ [(푛 − 2)/2]2 ∫ |푥|−2|푓(푥)|2 푑푛푥, ℝ푛 ℝ푛 ∞ 푛 푓 ∈ 퐶0 (ℝ ⧵ {0}), 푛 ∈ ℕ, 푛 ≥ 3, (1.1) and Rellich’s inequality,

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ [푛(푛 − 4)/4]2 ∫ |푥|−4|푓(푥)|2 푑푛푥, ℝ푛 ℝ푛 ∞ 푛 푓 ∈ 퐶0 (ℝ ⧵ {0}), 푛 ∈ ℕ, 푛 ≥ 5, (1.2) the first two inequalities in an infinite sequence of higher-order Hardy-type inequalities, received enormous attention in the literature due to their ubiquity in self-adjointness and spectral theory problems associated with second and fourth- order differential operators with strongly singular coefficients, respectively (see, e.g., [2], [3], [6], [10], [15, Sect. 1.5], [16, Ch. 5], [29], [33], [34], [37], [40]–[44], [57, Ch. II], [61]). We refer to Remark 2.11 for a selection of Rellich inequality references and some pertinent monographs on Hardy’s inequality. As one of our principal results we will derive the following two-parameter family of inequalities (a special case of inequality (1.5) below): If either 훼 ≤ 0 or 훼 ≥ 4, and 훽 ∈ ℝ, then

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ [훼(푛 − 훼) − 2훽] ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥 ℝ푛 ℝ푛 + 훽[(푛 − 4)(훼 − 2) − 훽] ∫ |푥|−4|푓(푥)|2 푑푛푥, (1.3) ℝ푛 ∞ 푛 푓 ∈ 퐶0 (ℝ ⧵ {0}), 푛 ∈ ℕ, 푛 ≥ 2.

As will be shown, (1.3) contains Rellich’s inequality (1.2), and Schmincke’s one-parameter family of inequalities,

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ −푠 ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥 ℝ푛 ℝ푛 + [(푛 − 4)/4]2(4푠 + 푛2) ∫ |푥|−4|푓(푥)|2 푑푛푥, (1.4) ℝ푛 푠 ∈ [ − 2−1푛(푛 − 4), ∞), 푛 ≥ 5, Hardy–Rellich-type inequalities 209 as special cases. By locality, the inequalities (1.1)–(1.4) naturally extend to the case 푛 푛 ∞ where ℝ is replaced by an arbitrary open set Ω ⊂ ℝ for functions 푓 ∈ 퐶0 (Ω⧵{0}) (without changing the constants in these inequalities). Our approach is based on factorizing even-order differential equations. More precisely, focusing on the 4th-order case for simplicity, we introduce the two- parameter 푛-dimensional homogeneous scalar differential expressions 푇훼,훽 ≔ −Δ + 훼|푥|−2푥 ⋅ ∇ + 훽|푥|−2, 훼, 훽 ∈ ℝ, 푥 ∈ ℝ푛 ⧵ {0}, 푛 ∈ ℕ, 푛 ≥ 2, and its formal + + ∞ 푛 adjoint, denoted by 푇훼,훽. Nonnegativity of 푇훼,훽푇훼,훽 on 퐶0 (ℝ ⧵ {0}) then implies the fundamental inequality

∫ [(Δ푓)(푥)]2 푑푛푥 ≥ [(푛 − 4)훼 − 2훽] ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥 ℝ푛 ℝ푛 − 훼(훼 − 4) ∫ |푥|−4|푥 ⋅ (∇푓)(푥)|2 푑푛푥 ℝ푛 (1.5) + 훽[(푛 − 4)(훼 − 2) − 훽] ∫ |푥|−4|푓(푥푓)|2 푑푛푥, ℝ푛 ∞ 푛 푓 ∈ 퐶0 (ℝ ⧵ {0}), which in turn contains inequality (1.3) as a special case. In the meantime, our factorization approach has been applied to Hardy, Hardy– Rellich, and further refined Hardy inequalities in the considerably more general context of stratified groups and to weighted Hardy inequalities on general homogeneous groups in [59]. We conclude our note with a series of remarks putting our approach into proper context by indicating that our method is elementary and very flexible in handling a variety of generalized situations involving the inclusion of remainder terms and higher even-order differential operators.

2. Factorizations and Hardy–Rellich-type Inequalities

The principal inequality to be proven in this section is of the following form: ∞ 푛 Theorem 2.1. Let 훼, 훽 ∈ ℝ, and 푓 ∈ 퐶0 (ℝ ⧵ {0}), 푛 ∈ ℕ, 푛 ≥ 2. Then

∫ [(Δ푓)(푥)]2 푑푛푥 ≥ [(푛 − 4)훼 − 2훽] ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥 ℝ푛 ℝ푛 − 훼(훼 − 4) ∫ |푥|−4|푥 ⋅ (∇푓)(푥)|2 푑푛푥 (2.1) ℝ푛 + 훽[(푛 − 4)(훼 − 2) − 훽] ∫ |푥|−4|푓(푥)|2 푑푛푥. ℝ푛 210 F. Gesztesy and L. Littlejohn

In addition, if either 훼 ≤ 0 or 훼 ≥ 4, then

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ [훼(푛 − 훼) − 2훽] ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥 ℝ푛 ℝ푛 + 훽[(푛 − 4)(훼 − 2) − 훽] ∫ |푥|−4|푓(푥)|2 푑푛푥. (2.2) ℝ푛 Proof. Given 훼, 훽 ∈ ℝ and 푛 ∈ ℕ, 푛 ≥ 2, we introduce the two-parameter 푛-dimensional homogeneous scalar differential expressions

−2 −2 푛 푇훼,훽 ≔ −Δ + 훼|푥| 푥 ⋅ ∇ + 훽|푥| , 푥 ∈ ℝ ⧵ {0}, (2.3) + and its formal adjoint, denoted by 푇훼,훽,

+ −2 −2 푛 푇훼,훽 ≔ −Δ − 훼|푥| 푥 ⋅ ∇ + [훽 − 훼(푛 − 2)]|푥| , 푥 ∈ ℝ ⧵ {0}. (2.4) ∞ 푛 Assuming 푓 ∈ 퐶0 (ℝ ⧵ {0}) throughout this proof, employing elementary multi- + variable differential calculus, we proceed to the computation of 푇훼,훽푇훼,훽 (which, while entirely straightforward, may well produce some tears in the process),

+ 2 −2 (푇훼,훽푇훼,훽푓)(푥) = (Δ 푓)(푥) + [(푛 − 4)훼 − 2훽]|푥| (Δ푓)(푥) 푛 −4 ∑ + 훼(4 − 훼)|푥| 푥푗푥푘푓푥푗,푥푘(푥) 푗,푘=1 + [ − (푛 − 3)훼2 + 2(푛 − 2)훼 + 4훽 |푥|−4푥 ⋅ (∇푓)(푥)

+ [훽2 + 2(푛 − 4)훽 − (푛 − 4)훼훽 |푥|−4푓(푥). (2.5)

∞ 푛 Thus, choosing 푓 ∈ 퐶0 (ℝ ⧵{0}) real-valued from this point on and integrating by parts (observing the support properties of 푓, which results in vanishing surface terms) implies

2 푛 + 푛 0 ≤ ∫ [(푇훼,훽푓)(푥)] 푑 푥 = ∫ 푓(푥)(푇훼,훽푇훼,훽푓)(푥) 푑 푥 ℝ푛 ℝ푛 = ∫ [(Δ푓)(푥)]2 푑푛푥 + [(푛 − 4)훼 − 2훽] ∫ ∫ |푥|−2푓(푥)(Δ푓)(푥) 푑푛푥 ℝ푛 ℝ푛 ℝ푛 푛 ∑ ∫ −4 푛 + 훼(훼 − 4) |푥| 푓(푥)푥푗푥푘푓푥푗,푥푘(푥) 푑 푥 푗,푘=1 ℝ푛

+ [ − (푛 − 3)훼2 + 2(푛 − 2)훼 + 4훽 ∫ |푥|−4푓(푥)[푥 ⋅ (∇푓)(푥)] 푑푛푥 ℝ푛 + [훽2 + 2(푛 − 4)훽 − (푛 − 4)훼훽 ∫ |푥|−4푓(푥)2 푑푛푥. (2.6) ℝ푛 Hardy–Rellich-type inequalities 211

To simplify and exploit expression (2.6), we make two observations. First, a standard integration by parts (again observing the support properties of 푓) yields

∫ |푥|−2푓(푥)(Δ푓)(푥) 푑푛푥 = 2 ∫ |푥|−4푓(푥)[푥 ⋅ (∇푓)(푥)] 푑푛푥 푛 푛 ℝ ℝ (2.7) − ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥. ℝ푛 Similarly, one confirms that

푛 ∑ ∫ ∫ −4 푛 푥푗푥푘푓(푥)푓푥푗,푥푘(푥) = −(푛 − 3) |푥| 푓(푥)[푥 ⋅ (∇푓)(푥)] 푑 푥 푛 푛 푗,푘=1 ℝ ℝ (2.8) − ∫ |푥|−4[푥 ⋅ (∇푓)(푥)]2 푑푛푥. ℝ푛 Combining (2.6)–(2.8) then yields (2.1). Since by Cauchy’s inequality,

− ∫ |푥|−4[푥 ⋅ (∇푓)(푥)]2 푑푛푥 ≥ − ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥, (2.9) ℝ푛 ℝ푛 one concludes that as long as 훼(훼 − 4) ≥ 0, that is, as long as either 훼 ≤ 0 or 훼 ≥ 4, one can further estimate (2.1) from below and thus arrive at inequality (2.2).

As a special case of (2.2) one obtains Rellich’s classical inequality in its original form as follows:

∞ 푛 Corollary 2.2. Let 푛 ∈ ℕ, 푛 ≥ 5, and 푓 ∈ 퐶0 (ℝ ⧵ {0}). Then,

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ [푛(푛 − 4)/4]2 ∫ |푥|−4|푓(푥)|2 푑푛푥. (2.10) ℝ푛 ℝ푛 Proof. Choosing 훽 = 훼(푛 − 훼)/2 in (2.2) results in

2 푛 −4 2 푛 ∫ |(Δ푓)(푥)| 푑 푥 ≥ 퐺푛(훼) ∫ |푥| |푓(푥)| 푑 푥, (2.11) ℝ푛 ℝ푛 with 퐺푛(훼) = 훼(푛 − 훼){(푛 − 4)(훼 − 2) − [훼(푛 − 훼)/2]}/2. (2.12)

Maximizing 퐺푛(훼) with respect to 훼 (it is advantageous to introduce the new variable 푎 = 훼 − 2) yields maxima at

2 1/2 훼± = 2 ± [(푛 /2) − 2푛 + 4 , (2.13) 212 F. Gesztesy and L. Littlejohn and taking the constraints 훼 ≤ 0 or 훼 ≥ 4 into account results in 푛 ≥ 5. The fact

2 퐺푛(훼±) = [푛(푛 − 4)/4] , (2.14) then yields Rellich’s inequality (2.10).

Inequality (2.1) also implies the following result:

∞ 푛 Corollary 2.3. Let 푛 ∈ ℕ and 푓 ∈ 퐶0 (ℝ ⧵ {0}). Then

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ (푛2/4) ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥, 푛 ≥ 8, (2.15) ℝ푛 ℝ푛 and

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ 4(푛 − 4) ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥, 5 ≤ 푛 ≤ 7. (2.16) ℝ푛 ℝ푛 In addition,

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ (푛2/4) ∫ |푥|−4|푥 ⋅ (∇푓)(푥)|2 푑푛푥, 푛 ≥ 2. (2.17) ℝ푛 ℝ푛

∞ 푛 Proof. Again, we chose 푓 ∈ 퐶0 (ℝ ⧵ {0}) real-valued for simplicity throughout this proof. The choice 훽 = (푛 − 4)(훼 − 2) in (2.1) then results in

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ (푛 − 4)(4 − 훼) ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥 푛 푛 ℝ ℝ (2.18) − 훼(훼 − 4) ∫ |푥|−4|푥 ⋅ (∇푓)(푥)|2 푑푛푥. ℝ푛 If in addition 훼 < 0, then applying Cauchy’s inequality to the 2nd term on the right-hand side of (2.18) yields

2 푛 −2 2 푛 ∫ |(Δ푓)(푥)| 푑 푥 ≥ 퐻푛(훼) ∫ |푥| |(∇푓)(푥)| 푑 푥, (2.19) ℝ푛 ℝ푛 where 퐻푛(훼) = (푛 − 4 + 훼)(4 − 훼). Maximizing 퐻푛 with respect to 훼 subject to the 2 constraint 훼 < 0 yields a maximum at 훼1 = (8 − 푛)/2, with 퐻푛((8 − 푛)/2) = 푛 /4, implying inequality (2.15) for 푛 ≥ 9. On the other hand, choosing 훼 = 0 in (2.18) yields ∫ |(Δ푓)(푥)|2 푑푛푥 ≥ 4(푛 − 4) ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥. (2.20) ℝ푛 ℝ푛 Hardy–Rellich-type inequalities 213

Since 4(푛 − 4) = 푛2/4 for 푛 = 8, this proves (2.15). Actually, one can arrive at (2.15) much quicker, but since we will subsequently use (2.18), we kept the above argument in this proof: Indeed, choosing 훽 = 0 in (2.2) yields

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ 훼(푛 − 훼) ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥. (2.21) ℝ푛 ℝ푛

Maximizing 퐹푛(훼) = 훼(푛 − 훼) with respect to 훼 yields a maximum at 훼1 = 푛/2, and subjecting it to the constraint 훼 ≥ 4 proves (2.15). Choosing 훼 = 4, 훽 = 0 in (2.1) yields (2.16). For 푛 ≥ 2 and (4 − 푛) < 훼 < 4, applying Cauchy’s inequality to the 1st term on the right-hand side of (2.18) now yields

2 푛 −4 2 푛 ∫ |(Δ푓)(푥)| 푑 푥 ≥ 퐾푛(훼) ∫ |푥| |푥 ⋅ (∇푓)(푥)| 푑 푥, (2.22) ℝ푛 ℝ푛 where 퐾푛(훼) = −(훼 + 푛 − 4)(훼 − 4). Maximizing 퐾푛 subject to the constraint 2 (4 − 푛) < 훼 < 4 yields a maximum at 훼1 = (8 − 푛)/2, with 퐾푛((8 − 푛)/2) = 푛 /4, implying (2.17).

We conclude with a series of remarks that put our approach into proper context and point out natural continuations into various other directions.

Remark 2.4. (푖) The constant in inequality (2.10) is known to be optimal, see, for instance, [6, p. 222], [17], [49], [53], [62], [65].

(푖푖) A sequence of extensions of (2.15), valid for 푛 ≥ 5, and for bounded domains containing 0, was derived by Tertikas and Zographopoulos [62, Theorem 1.7]. Moreover, an extension of inequality (2.15) valid for 푛 = 4 and for bounded open domains containing 0 was proved by [1, Theorem 2.1 (b)]. An alternative inequality whose special cases also imply Rellich’s inequality (2.10) and inequality (2.15) appeared in [14]. Thus, while the constant 푛2/4 in (2.15) is known to be optimal (cf. [62] for 푛 ≥ 5), the constant 4(푛−4) in (2.16) is not, the sharp constant being known to be 푛2/4 (also for 푛 = 4, cf. [1]). ⋄

Next, we comment on a special case of inequality (2.2) originally due to Schmincke [60]:

Remark 2.5. The choice 훽 = 2−1(푛 − 4)[훼 − 2 − 4−1(푛 − 4)], and the introduction of the new variable

푠 = 푠(훼) = 훼2 − 4훼 − 2−1푛(푛 − 4), (2.23) 214 F. Gesztesy and L. Littlejohn renders the two-parameter inequality (2.2) into Schmincke’s one-parameter inequality

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ −푠 ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥 ℝ푛 ℝ푛 + [(푛 − 4)/4]2(4푠 + 푛2) ∫ |푥|−4|푓(푥)|2 푑푛푥, (2.24) ℝ푛 푠 ∈ [−2−1푛(푛 − 4), ∞), 푛 ≥ 5. Here the requirements 훼 ≤ 0, equivalently, 훼 ≥ 4, both yield the range requirement for 푠 in the form 푠 ∈ [ − 2−1푛(푛 − 4), ∞). Inequality (2.24) is precisely the content of Lemma 2 in Schmincke [60], in particular, (2.2) thus recovers Schmincke’s result. Moreover, assuming 푛 ≥ 5 (the case 푛 = 4 being trivial) permits the value 푠 = 0 and hence implies Rellich’s inequality (2.10). If 푛 ≥ 8, the value 푠 = −푛2/4 is permitted, yielding inequality (2.15). Finally, for 5 ≤ 푛 ≤ 7, 푠 ∈ [−2−1푛(푛−4), ∞) and 4푠 + 푛2 ≥ 0 permit one to choose 푠 = −푛(푛 − 4)/2, and hence to conclude

∫ |(Δ푓)(푥)|2 푑푛푥 ≥ 2−1푛(푛 − 4) ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥 ℝ푛 ℝ푛 + [(푛 − 4)/4]2(4푠 + 푛2) ∫ |푥|−4|푓(푥)|2 푑푛푥 ℝ푛 ≥ 2−1푛(푛 − 4) ∫ |푥|−2|(∇푓)(푥)|2 푑푛푥, 5 ≤ 푛 ≤ 7, (2.25) ℝ푛 but inequality (2.16) is strictly superior to (2.25). Hence the two-parameter version (2.2) yields the better result (2.16), even though the latter is not optimal either as mentioned in the previous Remark 2.4 (푖푖). ⋄ Remark 2.6. Since all differential expressions employed are local, and only integration by parts was involved in deriving (2.6) (cf., e.g., [41, Remark 4] in this context), the estimates (2.1), (2.2), (2.10), (2.15)–(2.17), (2.24), all extend to the case 푛 푛 ∞ where ℝ is replaced by an arbitrary open set Ω ⊂ ℝ for functions 푓 ∈ 퐶0 (Ω⧵{0}) (without changing the constants in these inequalities). ⋄

∞ 푛 2 푛 Remark 2.7. Since 퐶0 (ℝ ⧵ {0}) is dense in 퐻 (ℝ ) if and only if 푛 ≥ 4, equiva- 2 푛 lently, −Δ| ∞ 푛 is essentially self-adjoint in 퐿 (ℝ ) if and only if 푛 ≥ 4 (see, 퐶0 (ℝ ⧵{0}) e.g., [22, p. 412–413], see also [43], [60]), Rellich’s inequality (2.10) extends from ∞ 푛 2 푛 퐶0 (ℝ ⧵ {0}) to 퐻 (ℝ ) for 푛 ≥ 5, and inequalities (2.15), and (2.17) extend from ∞ 푛 2 푛 퐶0 (ℝ ⧵ {0}) to 퐻 (ℝ ) for 푛 ≥ 4. ⋄ Remark 2.8. This factorization approach was originally employed in the context of the classical Hardy inequality in [33] (and some of its logarithmic refinements Hardy–Rellich-type inequalities 215 in [29]). Without repeating the analogous steps in detail we just mention that given 푛 ∈ ℕ, 푛 ≥ 3, 훼 ∈ ℝ, one introduces the one-parameter family of homogeneous vector-valued differential expressions

−2 푛 푇훼 ≔ ∇ + 훼|푥| 푥, 푥 ∈ ℝ ⧵ {0}, (2.26)

+ with formal adjoint, denoted by 푇훼 ,

+ −2 푛 푇훼 = − div( ⋅ ) + 훼|푥| 푥 ⋅, 푥 ∈ ℝ ⧵ {0}, (2.27)

∞ 푛 such that (e.g., on 퐶0 (ℝ ⧵ {0})-functions),

+ −2 푇훼 푇훼 = −Δ + 훼(훼 + 2 − 푛)|푥| . (2.28)

∞ 푛 Thus, for 푓 ∈ 퐶0 (ℝ ⧵ {0}),

2 푛 + 푛 0 ≤ ∫ |푇훼푓(푥)| 푑 푥 = ∫ 푓(푥)(푇훼 푇훼푓)(푥) 푑 푥 푛 푛 ℝ ℝ (2.29) = ∫ |(∇푓)(푥)|2 푑푛푥 + 훼(훼 + 2 − 푛) ∫ |푥|−2|푓(푥)|2 푑푛푥, ℝ푛 ℝ푛 and hence

∫ |(∇푓)(푥)|2 푑푛푥 ≥ 훼[(푛 − 2) − 훼] ∫ |푥|−2|푓(푥)|2 푑푛푥. (2.30) ℝ푛 ℝ푛 Maximizing 훼[(푛 − 2) − 훼] with respect to 훼 yields the classical Hardy inequality,

2 푛 2 −2 2 푛 ∞ 푛 ∫ |(∇푓)(푥)| 푑 푥 ≥ [(푛−2)/2] ∫ |푥| |푓(푥)| 푑 푥, 푓 ∈ 퐶0 (ℝ ⧵{0}), 푛 ≥ 3. ℝ푛 ℝ푛 (2.31) Again, it is well-known that the constant in (2.31) is optimal (cf., e.g., [65]). ⋄ Actually, our factorization approach also yields a known improvement of Hardy’s inequality (see, e.g., [6, Theorem 1.2.5], specializing it to 푝 = 2, 휀 = 0). Next, we briefly sketch the corresponding argument. Remark 2.9. Given 푛 ∈ ℕ, 푛 ≥ 3, 훼 ∈ ℝ, one introduces the following modified one-parameter family of homogeneous vector-valued differential expressions

−1 −1 푛 푇˜훼 ≔ (|푥| 푥) ⋅ ∇ + 훼|푥| , 푥 ∈ ℝ ⧵ {0}, (2.32)

+ with formal adjoint, denoted by (푇˜훼) ,

+ −1 −1 푛 (푇˜훼) = −(|푥| 푥) ⋅ ∇ + (훼 − 푛 + 1)|푥| , 푥 ∈ ℝ ⧵ {0}. (2.33) 216 F. Gesztesy and L. Littlejohn

∞ 푛 Exploiting the identities (for 푓 ∈ 퐶0 (ℝ ⧵ {0}), for simplicity), 푛 −1 −1 −2 ∑ 푛 [|푥| 푥 ⋅ ∇[ |푥| 푥 ⋅ (∇푓)(푥) = |푥| 푥푗푥푘푓푥푗,푥푘(푥), 푥 ∈ ℝ ⧵ {0}, (2.34) 푗,푘=1 푥 ⋅ ∇(|푥|−1푓(푥)) = |푥|−1[푥 ⋅ (∇푓)(푥)] − |푥|−1푓(푥), 푥 ∈ ℝ푛 ⧵ {0}, (2.35) ∞ 푛 one computes (e.g., on 퐶0 (ℝ ⧵ {0})-functions), 푛 + ˜ ˜ −2 ∑ −2 (푇훼) 푇훼 = −|푥| 푥푗푥푘휕푥푗휕푥푘 − (푛 − 1)|푥| [푥 ⋅ (∇푓)(푥)] 푗,푘=1 (2.36) + 훼(훼 + 2 − 푛)|푥|−2, 푥 ∈ ℝ푛 ⧵ {0}. Thus, appropriate integration by parts yield

2 푛 + 푛 0 ≤ ∫ |(푇˜훼푓)(푥)| 푑 푥 = ∫ 푓(푥)((푇˜훼) 푇˜훼푓)(푥) 푑 푥 ℝ푛 ℝ푛 푛 ∫ −2 ∑ = − |푥| { 푥푗푥푘푓(푥)푓푥푗,푥푘(푥) + (푛 − 1)푓(푥)[푥 ⋅ (∇푓)(푥)] ℝ푛 푗,푘=1 − 훼(훼 − 푛 + 2)|푓(푥)|2} 푑푛푥

−2 2 2 ∞ 푛 = ∫ |푥| {|[푥 ⋅ (∇푓)(푥)]| + 훼(훼 − 푛 + 2)|푓(푥)| }, 푓 ∈ 퐶0 (ℝ ⧵ {0}). ℝ푛 (2.37) Here we used 푛 ∑ ∫ −2 푛 ∫ −2| |2 푛 |푥| 푥푗푥푘푓(푥)푓푥푗,푥푘(푥) 푑 푥 = − |푥| |푥 ⋅ (∇푓)(푥)| 푑 푥 푛 푛 푗,푘=1 ℝ ℝ (2.38) −2 푛 ∞ 푛 − (푛 − 1) ∫ |푥| 푓(푥)[푥 ⋅ (∇푓)(푥)] 푑 푥, 푓 ∈ 퐶0 (ℝ ⧵ {0}). ℝ푛 Thus,

2 ∫ |[|푥|−1푥 ⋅ ∇푓](푥)| 푑푛푥 ≥ 훼[(푛 − 2) − 훼] ∫ |푥|−2|푓(푥)|2 푑푛푥. (2.39) ℝ푛 ℝ푛 Maximizing 훼[(푛 − 2) − 훼] with respect to 훼 yields the improved Hardy inequality,

2 | −1 | 푛 2 −2 2 푛 ∫ |[|푥| 푥 ⋅ ∇푓 (푥)| 푑 푥 ≥ [(푛 − 2)/2] ∫ |푥| |푓(푥)| 푑 푥, ℝ푛 ℝ푛 ∞ 푛 푓 ∈ 퐶0 (ℝ ⧵ {0}), 푛 ≥ 3. (2.40) (By Cauchy’s inequality, (2.40) implies the classical Hardy inequality (2.31).) Again, it is known that the constant in (2.40) is optimal (cf., e.g., [6, Theorem 1.2.5]). ⋄ Hardy–Rellich-type inequalities 217

Remark 2.10. The case of Rellich (and Hardy) inequalities in the half-line case is completely analogous (and much more straightforward): Consider the differential expressions

푑2 훼 푑 훽 푑2 훼 푑 훼 + 훽 푇 = − + + , 푇+ = − − + , (2.41) 푑푥2 푥 푑푥 푥2 푑푥2 푥 푑푥 푥2 with 훼, 훽 ∈ ℝ, which are formal adjoints to each other. One verifies

푑4 훼 − 훼2 − 2훽 푑2 2훼2 − 2훼 + 4훽 푑 3훼훽 + 훽2 − 6훽 푇+푇 = + + + , (2.42) 푑푥4 푥2 푑푥2 푥3 푑푥 푥4 and hence upon some integrations by parts,

∞ ∞ 0 ≤ ∫ (푇푓)(푥)2 푑푥 = ∫ 푓(푥)(푇+푇푓)(푥) 푑푥 0 0 ∞ ∞ [푓′(푥)]2 = ∫ [푓″(푥)]2 푑푥 − (훼 − 훼2 − 2훽) ∫ 푑푥 푥2 0 0 ∞ 푓(푥)2 + 훽(3훼 + 훽 − 6) ∫ 푑푥, 푓 ∈ 퐶∞((0, ∞)), (2.43) 푥4 0 0 choosing 푓 real-valued (for simplicity and w.l.o.g.). Thus, one obtains

∞ ∞ |푓′(푥)|2 ∫ |푓″(푥)|2 푑푥 ≥ (훼 − 훼2 − 2훽) ∫ 푑푥 푥2 0 0 ∞ |푓(푥)|2 + 훽(6 − 훽 − 3훼) ∫ 푑푥, (2.44) 푥4 0 ∞ 푓 ∈ 퐶0 ((0, ∞)), 훼, 훽 ∈ ℝ.

Choosing 훽 = (훼 − 훼2)/2 yields the Rellich-type inequality

∞ ∞ |푓(푥)|2 ∫ |푓″(푥)|2 푑푥 ≥ [3훼 − (19/4)훼2 + 2훼3 − (1/4)훼4 ∫ 푑푥, 푥4 0 0 ∞ 푓 ∈ 퐶0 ((0, ∞)). (2.45)

Introducing 퐹(훼) = 3훼 − (19/4)훼2 + 2훼3 − (1/4)훼4 = −(1/4)(훼 − 4)(훼 − 3) (훼 − 1)훼, 훼 ∈ ℝ, one verifies that 퐹(2 + 훾) = 퐹(2 − 훾), 훾 ∈ ℝ, and factors its derivative as 퐹′(훼) = 3 − (19/2)훼 + 6훼2 − 훼3 (2.46) = −(훼 − 2)(훼 − 2 + (5/2)1/2)(훼 − 2 − (5/2)1/2). 218 F. Gesztesy and L. Littlejohn

1/2 One notes that 훼1 = 2 yields a local minimum with 퐹(2) = −1, 훼2 = 2 − (5/2) 1/2 and 훼3 = 2 + (5/2) both yield local maxima of equal value, that is, 퐹(훼2) = 9 퐹(훼3) = 16 . Thus, one obtains Rellich’s inequality for the half-line in the form

∞ 9 ∞ |푓(푥)|2 ∫ |푓″(푥)|2 푑푥 ≥ ∫ 푑푥, 푓 ∈ 퐶∞((0, ∞)). (2.47) 16 푥4 0 0 0 We refer to Birman [13, p. 46] (see also Glazman [36, p. 83–84]), who presents a sequence of higher-order Hardy-type inequalities on (0, ∞) whose second member coincides with (2.47). For a variant of (2.47) on the interval (0, 1) we refer to [16, p. 114]; the case of higher-order Hardy-type inequalities for general interval is also considered in [55]. We will reconsider this sequence of higher-order Hardy-type inequalities in [31]. In addition, choosing 훽 = 0 or 훽 = 6 − 3훼 and subsequently maximizing with respect to 훼 yields in either case

∞ 1 ∞ |푓′(푥)|2 ∫ |푓″(푥)|2 푑푥 ≥ ∫ 푑푥, 푓 ∈ 퐶∞((0, ∞)), (2.48) 4 푥2 0 0 0 however, this is just Hardy’s inequality [38], [39], with 푓 replaced by 푓′. ⋄

Remark 2.11. While we basically focused on inequalities in 퐿2(ℝ푛) (see, however, Remark 2.6), much of the recent work on Rellich and higher-order Hardy inequalities aims at 퐿푝(Ω) for open sets Ω ⊂ ℝ푛 (frequently, Ω is bounded with 0 ∈ Ω), 푝 ∈ [1, ∞), appropriate remainder terms (the latter often associated with logarithmic refinements or with boundary terms), higher-order Hardy–Rellich inequalities, and the inclusion of magnetic fields and weights. The enormous number of references on this subject, especially, in the context of Hardy-type inequalities, makes it impossible to achieve any reasonable level of completeness in such a short note as the underlying one. Hence we felt we had to restrict ourselves basically to Rellich and higher-order Hardy inequality references only and thus we refer, for instance, to [1], [2], [4], [5], [6, Ch. 6], [7], [8], [9], [11], [12], [17], [18], [19], [20], [21], [23], [24], [25], [26], [27], [28], [35], [42], [47], [48], [50], [52], [55], [56], [58], [62], [63], [64], and the extensive literature cited therein. For the case of Hardy-type inequalities we only refer to the standard monographs such as, [6], [45], [46], and [54]. In this context we emphasize once again that the factorization method is entirely independent of the choice of domain Ω. Indeed, factorizations in the context of Hardy’s inequality in balls with optimal constants and logarithmic correction terms were already studied in [29], [33], based on prior work in [40], [43], and [44], although this appears to have gone unnoticed in the recent literature Hardy–Rellich-type inequalities 219 on this subject. For instance, introducing

푒푘 푒0 = 푒1 ≔ 1, 푒푘+1 ≔ 푒 , 푘 ∈ ℕ, (2.49) one can introduce iterated logarithms of the form for 훾 > 0, 푥 ∈ ℝ푛\{0}, 푛 ∈ ℕ, 푛 ≥ 2,

(− ln(|푥|/훾)) = 1, 0 (− ln(|푥|/훾)) = (− ln(|푥|/훾)), 0 < |푥| < 훾, 1 (− ln(|푥|/훾)) = ln((− ln(|푥|/훾)) ), 0 < |푥| < 훾/푒 , 푘 ∈ ℕ, (2.50) 푘+1 푘 푘+1

and replace ∇ by 푇훼푚,푦, where

푚 푗 −1 푇 = ∇ + 2−1|푥 − 푦|−2{(푛 − 2) + ∑ ∏[(− ln(|푥 − 푦|/훾)) 훼푚,푦 푘 푗=1 푘=1 푚 −1 − 훼 ∏[(− ln(|푥 − 푦|/훾)) }(푥 − 푦), (2.51) 푚 푘 푘=1

0 < |푥| < 푟 < 훾/푒푚, 훼푚 ≥ 0, 푚 ∈ ℕ, 푛 ∈ ℕ, 푛 ≥ 2, −1 −2 푇훼0,푦 = ∇ + 2 (푛 − 2 − 훼0)|푥 − 푦| (푥 − 푦), (2.52)

0 < |푥| < 푟, 훼0 ≥ 0, 푚 = 0, 푛 ∈ ℕ, 푛 ≥ 3.

+ ∞ Then with 푇훼푚,푦 the formal adjoint of 푇훼푚,푦, one obtains for 푓 ∈ 퐶0 (퐵푛(푦; 푟)\{푦})

+ −1 −2 2 (푇훼푚,푦푇훼푚,푦푓)(푥) = (−Δ푓)(푥) − 4 |푥 − 푦| {(푛 − 2)

푚 푗 푚 −2 −2 + ∑ ∏[(− ln(|푥 − 푦|/훾)) 푓(푥) − 훼2 ∏[(− ln(|푥 − 푦|/훾)) }푓(푥), 푘 푚 푘 푗=1 푘=1 푘=1 푚 ∈ ℕ, 푛 ∈ ℕ, 푛 ≥ 2, (2.53) + −1 2 2 −2 (푇훼0,푦푇훼0,푦푓)(푥) = (−Δ푓)(푥) − 4 [(푛 − 2) − 훼0 |푥 − 푦| 푓(푥), 푚 = 0, 푛 ∈ ℕ, 푛 ≥ 3. (2.54)

푛 푛 (Here 퐵푛(푥0; 푟0) denotes the open ball in ℝ with center 푥0 ∈ ℝ and radius 220 F. Gesztesy and L. Littlejohn

푟0 > 0.) In particular, letting 푟0 ↓ 0 and 푟1 ↑ 푟 in [29, Lemma 1] implies

2 0 ≤ ∫ |(푇훼푚,푦푓)(푥)| 퐵(푦;푟)

= ∫ {|(∇푓)(푥)|2 − 4−1|푥 − 푦|−2 ⋅ 퐵(푦;푟) 푚 푗 −2 (2.55) ⋅ [(푛 − 2)2 + ∑ ∏[(− ln(|푥 − 푦|/훾)) 푘 푗=1 푘=1 푚 −2 − 훼2 ∏[(− ln(|푥 − 푦|/훾)) ]|푓(푥)|2} 푑푛푥, 푚 푘 푘=1 ∞ 0 < 푟 < 훾/푒푚, 푓 ∈ 퐶0 (퐵(푦; 푟) ⧵ {푦}), 푚 ∈ ℕ ∪ {0}, and hence (with 훼푚 = 0),

∫ |(∇푓)(푥)|2 푑푛푥 ≥ 4−1 ∫ |푥 − 푦|−2{(푛 − 2)2 퐵(푦;푟) 퐵(푦;푟) 푚 푗 −2 + ∑ ∏[(− ln(|푥 − 푦|/훾)) }|푓(푥)|2 푑푛푥, (2.56) 푘 푗=1 푘=1 ∞ 0 < 푟 < 훾/푒푚, 푓 ∈ 퐶0 (퐵(푦; 푟)\{푦}), 푚 ∈ ℕ ∪ {0}. Here 푛 ∈ ℕ, 푛 ≥ 2 if 푚 ∈ ℕ, and 푛 ∈ ℕ, 푛 ≥ 3 if 푚 = 0 in (2.55) and (2.56). (Following standard practice, a product, resp., sum over an empty index set is defined to equal 1, resp., 0.) In analogy to Remark 2.6, inequality (2.56) extends to arbitrary open bounded sets Ω ⊂ ℝ푛 as long as 훾 is chosen sufficiently large. The constants in (2.55) are best possible as it is well-known that the operators (2.53), 2 (2.54) are nonnegative if and only if 훼푚 ≥ 0, 푚 ∈ ℕ ∪ {0} (cf., e.g., [29, p. 99] or 2 [34, Theorem 2.2]). (They are unbounded from below for 훼푚 < 0, 푚 ∈ ℕ ∪ {0}, permitting temporarily a continuation to negative values of 훼2 on the right-hand sides of (2.53) and (2.54).) For the special half-line case we also refer to [34]. Higher-order logarithmic refinements of the multi-dimensional Hardy–Rellich- type inequality appeared in [1, Theorem 2.1], and a sequence of such multidimensional Hardy–Rellich-type inequalities, with additional generalizations, appeared in [62, Theorems 1.8–1.10]. ⋄ We conclude this section by mentioning that factorization also works for other singular interactions, for instance, for point dipole interactions, where |푥|−2 is replaced by |푥|−3(푑 ⋅ 푥), with 푑 ∈ ℝ푛 a constant vector. Moreover, it applies to higher-order Hardy-type inequalities where −Δ is replaced by (−Δ)ℓ, ℓ ∈ ℕ. We defer all this to future investigations [30]. Hardy–Rellich-type inequalities 221

3. An Application of Rellich’s Inequality

In our final section we sketch an application to lower semiboundedness andto form boundedness for interactions with countably many strong singularities. To keep matters short we will just aim at the particular case (−Δ)2 + 푊, where 푊 has countably many strong singularities. We start by recalling an abstract version of a result of Morgan [51] as described in [32]: Theorem 3.1. Suppose 푇, 푊 are self-adjoint operators in ℋ with dom (|푇|1/2) ⊆ 1/2 dom (|푊| ), and let 푐, 푑 ∈ (0, ∞), 푒 ∈ [0, ∞). Moreover, suppose Φ푗 ∈ ℬ(ℋ), 1/2 1/2 푗 ∈ 퐽, 퐽 ∈ ℕ an index set, leave dom (|푇| ) invariant, that is, Φ푗 dom (|푇| ) ⊆ dom (|푇|1/2), 푗 ∈ 퐽, and satisfy the following conditions (푖)–(푖푖푖): ∗ (푖) ∑푗∈퐽 Φ푗 Φ푗 ≤ 퐼ℋ. ∗ −1 1/2 (푖푖) ∑푗∈퐽 Φ푗 |푊|Φ푗 ≥ 푐 |푊| on dom (|푇| ). 1/2 2 1/2 2 2 1/2 (푖푖푖) ∑푗∈퐽 ‖|푇| Φ푗푓‖ℋ ≤ 푑‖|푇| 푓‖ℋ + 푒‖푓‖ℋ, 푓 ∈ dom (|푇| ). Then,

‖|푊|1/2Φ 푓‖2 ≤ 푎‖|푇|1/2Φ 푓‖2 + 푏‖Φ 푓‖2 , 푓 ∈ dom(|푇|1/2), 푗 ∈ 퐽, (3.1) ‖ 푗 ‖ℋ ‖ 푗 ‖ℋ 푗 ℋ implies

‖|푊|1/2푓‖2 ≤ 푎 푐 푑‖|푇|1/2푓‖2 + [푎 푐 푒 + 푏 푐]‖푓‖2 , 푓 ∈ dom(|푇|1/2). (3.2) ‖ ‖ℋ ‖ ‖ℋ ℋ Thus, the key for applications would be to have 푐 and 푑 arbitrarily close to 1 such that if 푎 < 1, also 푎푐푑 < 1. If 푊 is local and Φ푗 represents the operator of multiplication with “bump functions” 휙푗, 푗 ∈ 퐽 ⊆ ℕ, such that 휙푗, 푗 ∈ 퐽 is a family of smooth, real-valued functions defined on ℝ푛 satisfying that for each 푥 ∈ ℝ푛, there exists an open 푛 neighborhood 푈푥 ⊂ ℝ of 푥 such that there exist only finitely many indices 푘 ∈ 퐽 with supp (휙푘) ∩ 푈푥 ≠ ∅ and 휙푘|푈푥 ≠ 0, as well as

2 푛 ∑ 휙푗(푥) = 1, 푥 ∈ ℝ (3.3) 푗∈퐽

(the sum over 푗 ∈ 퐽 being finite). Then Φ푗 and 푊 commute and hence

∗ ∗ 1/2 ∑ Φ푗 Φ푗 = 퐼ℋ and ∑ Φ푗 |푊|Φ푗 = |푊| on dom (|푇| ) (3.4) 푗∈퐽 푗∈퐽 yield condition (푖) and also (푖푖) with 푐 = 1 of Theorem 3.1. Next, we will illustrate a typical situation where for all 휀 > 0, one can actually choose 푑 = 1 + 휀. 222 F. Gesztesy and L. Littlejohn

Consider 푇 = (−Δ)2, dom(푇) = 퐻4(ℝ푛) in 퐿2(ℝ푛), 푛 ≥ 5, and suppose that

dom (|푇|1/2) ⊆ dom (|푊|1/2) (3.5)

(representing a relative form boundedness condition). Assume

‖ ‖ ‖ ‖ 2 ‖ 2‖ ‖ 2‖ ∑ 휙푗(⋅) = 1, ‖ ∑ |∇휙푗(⋅)| ‖ < ∞, ‖ ∑ |(Δ휙푗)(⋅)| ‖ < ∞. 푗∈퐽 푗∈퐽 퐿∞(ℝ푛) 푗∈퐽 퐿∞(ℝ푛) (3.6) Then given 휀 > 0, the elementary estimate

2 푛 2 푛 ∑ ∫ |Δ(휙푗푓)(푥)| 푑 푥 ≤ ∫ |(Δ푓)(푥)| 푑 푥 푗∈퐽 ℝ푛 ℝ푛 ‖ ‖ + ‖ ∑ |(Δ휙 )(⋅)|2‖ ‖푓‖2 ‖ 푗 ‖ 퐿2(ℝ푛) 푗∈퐽 퐿∞(ℝ푛) ‖ ‖ ‖ ‖ 푛 + 4‖ ∑ |(Δ휙푗)(⋅)||(∇휙푗)(⋅)|‖ ∫ |(∇푓)(푥)||푓(푥)| 푑 푥 푗∈퐽 퐿∞(ℝ푛) ℝ푛 ‖ ‖ ‖ ‖ 푛 + 2‖ ∑ |(Δ휙푗)(⋅)||휙푗(⋅)|‖ ∫ |(Δ푓)(푥)||푓(푥)| 푑 푥 푗∈퐽 퐿∞(ℝ푛) ℝ푛 ‖ ‖ ‖ ‖ 푛 + 4‖ ∑ |휙푗(⋅)||(∇휙푗)(⋅)|‖ ∫ |(∇푓)(푥)||(Δ푓)(푥)| 푑 푥 푗∈퐽 퐿∞(ℝ푛) ℝ푛

2 푛 2 2 푛 ≤ (1 + 휀) ∫ |(Δ푓)(푥)| 푑 푥 + 퐶휀 ‖푓‖퐿2(ℝ푛), 푓 ∈ 퐻 (ℝ ), (3.7) ℝ푛 for some constant 퐶휀 ∈ (0, ∞), shows that

∑ ‖|푇|1/2(휙 푓)‖2 = ∑ ∫ |Δ(휙 푓)(푥)|2 푑푛푥 ‖ 푗 ‖퐿2(ℝ푛) 푗 푛 푗∈퐽 푗∈퐽 ℝ (3.8) 2 푛 2 ≤ (1 + 휀) ∫ |(Δ푓)(푥)| 푑 푥 + 퐶휀‖푓‖퐿2(ℝ푛). ℝ푛 Thus, for arbitrary 휀 > 0, also condition (푖푖푖) of Theorem 3.1 holds with 푑 = 1 + 휀. Strongly singular potentials 푊 that are covered by Theorem 3.1 are, for instance, 푛 of the following form: Let 퐽 ⊆ ℕ be an index set, and {푥푗}푗∈퐽 ⊂ ℝ , 푛 ∈ ℕ, 푛 ≥ 3, be a set of points such that

inf |푥푗 − 푥푗′| > 0. (3.9) 푗,푗′∈퐽 푗≠푗′

Let 휙 be a nonnegative smooth function which equals 1 in 퐵푛(0; 1/2) and vanishes Hardy–Rellich-type inequalities 223

2 푛 outside 퐵푛(0; 1). Let ∑푗∈퐽 휙(푥 − 푥푗) ≥ 1/2, 푥 ∈ ℝ , and set

−1/2 2 푛 휙푗(푥) = 휙(푥 − 푥푗)[ ∑ 휙(푥 − 푥푗′) ] , 푥 ∈ ℝ , 푗 ∈ 퐽, (3.10) 푗′∈퐽

2 푛 such that ∑푗∈퐽 휙푗(푥) = 1, 푥 ∈ ℝ . In addition, let 훾푗 ∈ ℝ, 푗 ∈ 퐽, 훾, 훿 ∈ (0, ∞) with 2 |훾푗| ≤ 훾 < [푛(푛 − 4)/4 , 푗 ∈ 퐽, (3.11) and consider

−4 −훿|푥−푥푗| 푛 푊0(푥) = ∑ 훾푗 |푥 − 푥푗| 푒 , 푥 ∈ ℝ ⧵ {푥푗}푗∈퐽. (3.12) 푗∈퐽 Then combining Rellich’s inequality in ℝ푛, 푛 ≥ 5 (cf. Corollary 2.2) and Theorem 3.1 (with 푐 = 1 and 푑 = 1 + 휀 for arbitrary 휀 > 0), 푊0 is form bounded with respect to 푇 = (−Δ)2 with form bound strictly less than one.

Acknowledgments. We are indebted to Mark Ashbaugh, Roger Lewis, and Michael Ruzhansky for very valuable hints to the literature on Hardy–Rellich-type inequalities and to Isaac Michael, Michael Pang, and Richard Wellman for helpful discussions.

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Katrin Grunert and Xavier Raynaud

This paper is dedicated to Helge Holden on the occasion of his sixtieth anniversary with admiration and gratefulness for all the inspiration he has been giving us in our work

Abstract. Compared with the two-component Camassa–Holm system, the modified two- component Camassa–Holm system introduces a regularized density which makes possible the existence of solutions of lower regularity, and in particular of multipeakon solutions. In this paper, we derive a new pointwise invariant for the modified two-component Camassa– Holm system. The derivation of the invariant uses directly the symmetry of the system, following the classical argument of Noether’s theorem. The existence of the multipeakon solutions can be directly inferred from this pointwise invariant. This derivation shows the strong connection between symmetries and the existence of special solutions. The observation also holds for the scalar Camassa–Holm equation and, for comparison, we have also included the corresponding derivation. Finally, we compute explicitly the solutions obtained for the peakon-antipeakon case. We observe the existence of a periodic solution which has not been reported in the literature previously. This case shows the attractive effect that the introduction of an elastic potential can have on the solutions.

1. Introduction

In [23], the authors introduce the modified two-component Camassa–Holm system (M2CH), which is given by

푚푡 + 푢푚푥 + 2푚푢푥 + 휌푥̄ 휌 = 0, (1a)

휌푡 + (푢휌)푥 = 0, (1b) where

푚 = 푢 − 푢푥푥, (1c)

휌 = 휌̄ − 휌푥푥̄ . (1d) 228 K. Grunert and X. Raynaud

This system is a generalization of the Camassa–Holm (CH) equation,

푚푡 + 푢푚푥 + 2푚푢푥 = 0, (2) with (1c) and the two-component Camassa–Holm system (2CH)

푚푡 + 푢푚푥 + 2푚푢푥 + 휌푥휌 = 0, (3) with (1b) and (1c). All these equations can be derived from a variational principle for the kinetic energy that is defined

1 퐸kin(푡) = ∫(푢2 + 푢2 )(푡, 푥) 푑푥, (4) 2 푥 ℝ and the following potential energy

1 1 퐸pot = 0, 퐸pot = ∫ 휌2(푡, 푥) 푑푥, 퐸pot = ∫(휌2̄ + 휌2̄ )(푡, 푥) 푑푥, (5) 2 2 푥 ℝ ℝ for CH, 2CH and M2CH, respectively. An advantage of M2CH is that the system of equations requires a lower regularity for the density, compared to 2CH. Indeed, given the potential energy as in (5), while the 2CH system requires that 휌 ∈ 퐿2(ℝ), the M2CH system requires that 휌̄ ∈ 퐻1(ℝ), which is equivalent to 휌 ∈ 퐻−1(ℝ), as 1 −1 the Helmholtz operator id −휕푥푥 is an isomorphism from 퐻 (ℝ) to 퐻 (ℝ). The CH equation has a rich mathematical structure which explains the very extensive literature that is available on this equation. In this work, we consider global conservative solutions which can be defined beyond the blow-up of the classical solutions. For the CH equation, the blow-up scenario is known and occurs 1 when, for some given initial data 푢0 ∈ 퐻 (ℝ), the spatial derivative 푢푥 becomes unbounded from below within finite time, while the 퐻1(ℝ)-norm of 푢, and hence also its 퐿∞-norm, remains bounded. This phenomenon, which is referred to as wave breaking, is described in [4, 5, 6, 7]. In particular, it can be predicted whether wave breaking occurs in the nearby future or not, see [13] and the references therein. In more recent works, the regularization properties of the density in the case of the 2CH system have been studied [8, 14, 17]. There, it is shown that if the density is bounded away from zero initially, a solution with smooth initial data will never experience blow-up. We find the following interpretation appropriate. The governing equations, that are obtained from the variational principle, model the velocity 푢 of an underlying flow map 휙(푡, 휉), that is, 휙푡 = 푢 ∘ 휙. The elastic energy introduced by 퐸pot prevents compression so that the flow map cannot become irregular in the sense that several particles can occupy the same place 휙(푡, 휉1) = 휙(푡, 휉2) for two particles 휉1 and 휉2. The potential energy for the M2CH Modified two-component Camassa–Holm system 229 system is weaker in the sense that, if we consider a concentration of particles at a single point, the potential energy for the 2CH system becomes infinite making this state not reachable while it is finite for the M2CH system. Indeed, formally speaking, a concentration of particles gives rise to a density 휌 equal to a Dirac delta 2 1 −|푥| 1 function which has infinite 퐿 (ℝ) norm while 휌̄ = 2 푒 ∗ 휌 remains in 퐻 (ℝ). However, compared to the 2CH system, the M2CH system has the property of having a special class of solutions. The CH equation admits a special type of soliton-like solutions that have been called multipeakons, due to the peaks that characterize them. The multipeakon solutions can be seen as a discrete version of the equation. Such solutions are dense [2], robust [10, 12, 11], and have been used to design convergent numerical schemes, which can also handle blow-up [18, 19]. It turns out that the M2CH also admits such solutions, as pointed out in [23]. In this paper, we follow the following understanding. Special solutions exist because the equation has a special structure, and structures are identified by symmetries. In this case, the symmetry ofthe system is related to the invariance with respect to relabeling of both the kinetic and potential energy. From Noether’s theorem, we know that this invariance must imply the existence of conservation laws. Since the group of diffeomorphisms has infinite dimension, we expect infinitely many invariants. As we will see,the Noether argument leads us to pointwise invariants of the form

2 (푢 − 푢푥푥)(푡, 휙(푡, 휉))휙휉(푡, 휉) (6) in the case of the CH equation, which also encodes the conservation of the left angular momentum [24]. Such invariants have been derived much earlier in [1], but here, we present a more straightforward derivation that does not require the advanced topological framework used in the fore-mentioned work. Of course, we miss some fundamental insight but simplifying the derivation, we can make it possible to adapt it directly to the case of the M2CH system later. The problem of the pointwise invariant (6) is that it is not so easy to exploit, as it mixes natural Eulerian variables (the expression of 푢푥푥 is complicated in Lagrangian variables) and Lagrangian variables (휙 it not directly available from Eulerian variables). However, it can be used to show the existence of multipeakon solutions, thus making clear the connection between the symmetries of the system and the existence of a large and non-trivial class of special solutions. In this paper, we modify the variational formulation of the M2CH system to make it suitable for the use of the Noether’s argument. We derive the pointwise invariant of the M2CH system and describe how the existence of multipeakon solutions can be inferred from it. The variational formulations are always done with respect to the flow map. Hence, Lagrangian variables are naturally introduced in this setting. The change of variables to Lagrangian variables is known to be a mean of getting rid 230 K. Grunert and X. Raynaud of non-linearity in the advection term corresponding to 푢푡 + 푢푢푥 in the equation below. We denote by ℋ = 휕 − 휕푥푥 the Helmholtz operator. After applying its inverse ℋ−1 to (1), this system of equations becomes

푢푡 + 푢푢푥 + 푃푥 = 0, (7a)

휌푡̄ + 푢휌푥̄ + 푅 + 푆푥 = 0, (7b) with

−1 2 1 2 1 2 1 2 푃 = ℋ (푢 + 2 푢푥 + 2 휌̄ − 2 휌푥̄ ), (7c) −1 푅 = ℋ (푢푥휌),̄ (7d) −1 푆 = ℋ (푢푥휌푥̄ ). (7e)

The Lagrangian variables are given by the characteristics defined as 푦푡(푡, 휉) = 푢(푡, 푦(푡, 휉)) and the Lagrangian velocity 푈(푡, 휉) = 푢(푡, 푦(푡, 휉)). In the case of the M2CH system, we need to introduce more variables to be able to handle the blow- up of the solution. Here, we follow the approach presented in [16, 25], which is very close the one introduced in [14, 21]. Once the system of equations is completely rewritten in term of purely Lagrangian variables, semi-linearities in the system enable us to obtain global solutions. Thus, the Lagrangian system defines the solutions, which are then mapped back to Eulerian variables in order to obtain some weak solutions to the original M2CH system. We show that the existence of the pointwise invariant implies the existence of multipeakon solutions but, even if this invariant can be expressed in term of purely Lagrangian variables, its form becomes then intricate. However, this fundamental invariance property is preserved by the change of Lagrangian variables so that the existence of multipeakon can also be obtained in the Lagrangian setting; see [20] for the corresponding work in the case of the CH equation. In section 5, we derive the multipeakon equations directly from the system (7), and not as in [23] where a discretization of the Hamiltionian is used. We compute explicit solutions in the case of the anti-symmetric peakon-antipeakon solution. We discover an interesting dynamic in this case, which can be decomposed into three different cases. For all cases, the peaks collide, but there are different behaviors when the peaks move away from each other after collision. In the first case, there is not enough potential energy in the system toretain the particles from completely departing from each other. In the second case, the potential energy prevents them from doing so. We can compare the situation to a classical discrete mechanical system where Hooke’s law is used to model the elastic forces. Such elastic forces act in both ways. They are repulsive when particles approach each other, over a given equilibrium state, Modified two-component Camassa–Holm system 231 and attractive when the particles move far away from each other. For M2CH, we observe that the potential energy does not yield a repulsive force that is strong enough to prevent collision, but its attractive effect can prevent the fully departure of the peaks from each other. The solution in this case is periodic, and we finally end up with a oscillatory system where the kinetic energy and the potential energy vanishes one after the other, as for a standard pendulum. This type of solution has not been observed for CH or 2CH. The last case in the description of the dynamics is the limiting case, where we do not obtain a periodic solution, but the peaks are slowed down by the attractive force until their velocity vanishes. The position of the (left) right particle tends to (minus) infinity while their velocity tend to zero, see Figure 1.

2. Conservation laws

For the M2CH system, we define the kinetic energy as

1 퐸kin(푡) = ∫(푢2 + 푢2 )(푡, 푥) 푑푥. (8) 2 푥 ℝ

1 2 The proper definition of kinetic energy from physics is 2 ∫ℝ 휌푢 푑푥. However, we are going to see that the term defined in (8) plays a role which resembles the one of the kinetic energy in standard physical systems and that is why we use this terminology. Using the same type of analogy, we refer to the quantity 퐸pot defined below as the potential energy,

1 퐸pot(푡) = ∫(휌2̄ + 휌2̄ )(푡, 푥) 푑푥. (9) 2 푥 ℝ The total energy is then given by

1 퐸tot(푡) = ∫(푢2 + 푢2 + 휌2̄ + 휌2̄ )(푡, 푥) 푑푥. (10) 2 푥 푥 ℝ The M2CH system can be derived from a variational principle for the Lagrangian

ℒ = 퐸kin − 퐸pot. (11)

We do not give the details for this computation, and refer instead to [15]. The invariance of the Lagrangian with respect to time implies through Noether’s theorem that the total energy as defined in (10) is preserved in time. More precisely, 232 K. Grunert and X. Raynaud we have the following conservation law for the energy,

(푢2 + 푢2 + 휌2̄ + 휌2̄ ) + (푢(푢2 + 푢2 + 휌2̄ + 휌2̄ )) = 푥 푥 푡 푥 푥 푥 (푢3 − 2푃푢 − 2휌ℋ̄ −1(휌푢) + 2푢휌2̄ ) . (12) 푥 Let us derive (12) from (7). One can prove that for any smooth function 푞(푡, 푥) which satisfies 푞푡 + 푢푞푥 + 푄 = 0 (13) for some given smooth 푄, one has

(푞2 + 푞2) + (푢(푞2 + 푞2)) = −푞2푢 + 푢 푞2 − 2푞푄 − 2푞 푄 . (14) 푥 푡 푥 푥 푥 푥 푥 푥 푥 We let the reader check this property. From (7b), we have

−1 −1 휌푡̄ + 푢휌푥̄ + ℋ (휌푢)푥 − 푢ℋ 휌푥 = 0.

Let us define −1 −1 푄 = ℋ (휌푢)푥 − 푢ℋ 휌푥. After some computations, we get

−1 푄푥 = ℋ (휌푢) − 푢푥휌푥̄ − 푢휌.̄

Hence 휌̄ 푄 + 휌푄̄ = (휌ℋ̄ −1(휌푢)) − 푢 휌2̄ − 2푢휌휌̄ ̄ . 푥 푥 푥 푥 푥 푥 Then, we apply (14) for 푞 = 휌̄ and get

(휌2̄ + 휌2̄ ) + (푢(휌2̄ + 휌2̄ )) = −2(휌ℋ̄ −1(휌푢)) + 푢 (휌2̄ − 휌2̄ ) + 2(휌2̄ 푢) . (15) 푥 푡 푥 푥 푥 푥 푥 푥 Now, we set 푄 = 푃푥 and we apply (14) for 푞 = 푢 and get

(푢2 + 푢2 ) + (푢(푢2 + 푢2 )) = (푢3 − 2푢푃) + 2푢 (ℋ푃 − 푢2 − 1 푢2 ). (16) 푥 푡 푥 푥 푥 푥 2 푥 We sum up (15) and (16) and obtain

(푢2 + 푢2 + 휌2̄ + 휌2̄ ) + (푢(푢2 + 푢2 + 휌2̄ + 휌2̄ )) = 푥 푥 푡 푥 푥 푥 (푢3 − 2푃푢 − 2휌ℋ̄ −1(휌푢) + 2푢휌2̄ ) + 2푢 (ℋ푃 − 푢2 − 푡 1 푢2 + 1 휌2̄ − 1 휌2̄ ). 푥 푥 2 푥 2 푥 2 The last of the two terms on the right-hand side vanishes because of (7c), so that the conservation law (12) follows. Modified two-component Camassa–Holm system 233

3. Lagrangian variables

In this section, we describe how the M2CH system (7) can be rewritten in La- grangian variables to obtain a system which is formally equivalent, but whose linear structure can be used to prove the global existence of solutions. In this section the derivation of the equivalent system is only formal. Once the system is obtained, the construction of the solution in Lagrangian variables and the mapping back to the original Eulerian variables can be done rigorously, see [16]. We introduce the characteristics defined as

푦푡(푡, 휉) = 푢(푡, 푦(푡, 휉)), (17) the Lagrangian velocity defined as

푈(푡, 휉) = 푢(푡, 푦(푡, 휉)), (18) the cumulative total energy distribution defined as

푦(푡,휉) 2 2 2 2 퐻(푡, 휉) = ∫ (푢 + 푢푥 + 휌̄ + 휌푥̄ )(푡, 푥) 푑푥, (19) −∞ the Lagrangian regularized potential energy defined as

̄푟(푡, 휉) = 휌(̄ 푡, 푦(푡, 휉)), (20) and, finally, ̄푠(푡, 휉) = 휌푥̄ (푡, 푦(푡, 휉)). (21)

We will assume in the remainder of this formal derivation that the derivative 푦휉 does not vanish. After differentiating (20) and (21), we observe that, formally,

휉̄푟 = ̄푠푦휉 (22) and 휉̄푠 = ̄푟푦휉 − 휌(푡, 푦)푦휉. (23) The inverse Helmholtz operator can be written using Green’s function as

1 [ℋ−1푞](푥) = ∫ 푒−|푥−푧|푞(푧) 푑푧. (24) 2 ℝ Hence we get from the definition (7c) of 푃 that

1 푃(푡, 푥) = ∫ 푒−|푥−푧|((푢2 + 푢2 + 휌2̄ + 휌2̄ ) + (푢2 − 2휌2̄ ))(푡, 푧) 푑푧. 4 푥 푥 푥 ℝ 234 K. Grunert and X. Raynaud

We change to Lagrangian variables and use the definition (19) of 퐻 and the identity (21) to get

1 푃(푡, 푦) = ∫ 푒−|푦(푡,휉)−푦(푡,휂)|(퐻 (푡, 휂) + (푈2(푡, 휂) − 22 ̄푠 (푡, 휂))푦 (푡, 휂)) 푑휂. (25) 4 휉 휉 ℝ The change to Lagrangian variables has the recognized advantage to get rid of the first non-linear term in(7a), which becomes

푈푡(푡, 휉) = −푃푥(푡, 푦(푡, 휉)) (26)

Introducing

1 푄(푡, 휉) = − ∫ sign(휉 − 휂)푒−|푦(푡,휉)−푦(푡,휂)| 4 ℝ 2 2 ⋅ (퐻휉(푡, 휂) + (푈 (푡, 휂) − 2 ̄푠 (푡, 휂))푦휉(푡, 휂)) 푑휂, (27) and assuming that 푦휉 remains strictly positive, we can differentiate 푃 in (25) and obtain that

푃푥(푡, 푦(푡, 휉))푦휉(푡, 휉) = 푄(푡, 휉)푦휉(푡, 휉).

We simplify the above expression by 푦휉 and thus (26) yields

푈푡 = −푄. (28)

Following the same lines we introduce the integrated variables

푅휉 = 푉푦휉 and 푆휉 = 푊푦휉. (30) Moreover, after differentiation, we get

푉휉 = −푈휉 ̄푟+ 푅푦휉 and 푊휉 = −푈휉 ̄푠+ 푆푦휉. (31) Modified two-component Camassa–Holm system 235

The conservation law (12) gives

3 −1 2 퐻푡 = (푢 − 2푃푢 − 2휌ℋ̄ (휌푢) + 2푢휌̄ ) (푡, 푦). (32)

Recalling that 휌 = 휌̄ − 휌푥푥̄ , direct computations yield that

−1 −1 2 휌ℋ̄ (휌푢) = 휌ℋ̄ (휌푢̄ − (휌푢)̄ 푥푥 + (푢푥휌)̄ 푥 + 푢푥휌푥̄ ) = 푢휌̄ + 휌푅̄ 푥 + 휌푆,̄ and (32) can be rewritten as

3 퐻푡 = 푈 − 2푃푈 − 2 ̄푟(푆 + 푉). (33)

Note that in (33), we slightly abused the notations and denoted 푃(푡, 푦) as 푃(푡, 휉). We continue to do so in the remaining. Differentiating 푃 and 푄 gives us

푃휉 = 푄푦휉, (34) 1 푄 = − 퐻 − ( 1 푈2 −2 ̄푠 − 푃)푦 . (35) 휉 2 휉 2 휉 For the Lagrangian regularized potential energy density, (7b) yields

푡̄푟 = −(푅 + 푊). (36)

Let us now gather the governing equations we have obtained in (17), (28), (33), and (36). We have seen that the governing equations (7) are formally equivalent to the system

푦푡 = 푈, (37a)

푈푡 = −푄, (37b) 3 퐻푡 = 푈 − 2푃푈 − 2 ̄푟(푆 + 푉), (37c)

푡̄푟 = −(푅 + 푊), (37d)

푡̄푠 = −(푆 + 푉), (37e) where the quantities 푃, 푄, 푅, 푆, 푉, and 푊 are defined in (25), (27), and (29), respectively. We can differentiate the first four equations in(37) and obtain

푦휉,푡 = 푈휉, (38a) 1 1 푈 = 퐻 + ( 푈2 −2 ̄푠 − 푃)푦 , (38b) 휉,푡 2 휉 2 휉 2 2 퐻휉,푡 = (3푈 − 2푃 + 2 ̄푟 )푈휉 − 2(푄푈 + ̄푟(푉 + 푊))푦휉 − 2(푅 + 푊) 휉̄푟 , (38c)

휉,푡̄푟 = ̄푠푈휉 − (푆 + 푉)푦휉. (38d) 236 K. Grunert and X. Raynaud

The system (38) reveals the semi-linear nature of the equivalent system. Indeed, the system is semi-linear with respect to the derivatives 푦휉, 푈휉, 퐻휉 and 휉̄푟 in the sense that all the other terms (included )̄푠 that enter the system are of higher regularity than these derivatives. The semi-linearity of the system is essential in the proof of the existence of solutions using Picard’s argument. The variable 퐻 is now considered as an independent variable, but when we introduced it in (19), it was clearly dependent on the other variables. Changing variables in (19) gives us

2 2 2 2 2 푦휉퐻휉 = (푈 + ̄푟 + ̄푠 )푦휉 + 푈휉 , (39) and it can be shown that the governing system (37) preserves this identity, if it holds initially. Thus, we have decoupled 퐻 from the other variables, in particular to obtain a semi-linear system; but (39) shows that the variables are not truly independent as they are constrained by the system to remain on the “manifold” defined by (39). As shown in [16], the system of ordinary differential equations (37) has global solutions in a suitable Banach space. In particular,

∞ 2 ∞ 푦 − id, 퐻 ∈ 퐿 (ℝ), 푦휉 − 1, 푈, 푈휉, ̄푟,휉 ̄푟 , ̄푠, 퐻휉 ∈ 퐿 (ℝ) ∩ 퐿 (ℝ). (40)

These global solutions in Lagrangian coordinates can then be mapped to global weak conservative solutions of the M2CH system as in [21] in the case of the CH equation. Let us be more specific. For each fixed time 푡, which we remove from the notation, we define 푢(푥) and 휌(푥) as

푢(푥) = 푈(휉), 휌(푥)̄ = ̄푟(휉) (41a) where 휉 is chosen such that 푥 = 푦(휉). Such 휉 exists as 푦 is surjective but it is not necessarily unique. The definitions (41a) are well-posed and, in addition, we can prove that 휌푥̄ (푥) = ̄푠(휉), (41b) see [16, Definition 4.4]. The energy distribution measure 휇 is defined as

휇(퐵) = ∫ 퐻휉(휉) 푑휉 for any Borel set 퐵. {푥∈푦−1(퐵)}

∞ Since ̄푠(푡, ⋅ ) ∈ 퐿 (ℝ), it follows from (41b) that 휌푥̄ remains also bounded in 퐿∞(ℝ). In particular, it means that the blow-up of the solution only occurs when 푢푥 becomes unbounded, as in the CH case, the additional variable 휌̄ of M2CH does not blow up. Before closing this section, we introduce the Lagrangian potential energy 푟 as

푟(푡, 휉) = 휌(푡, 푦(푡, 휉))푦휉(푡, 휉). (42) Modified two-component Camassa–Holm system 237

As opposed to all the other Lagrangian variables introduced until now (푦 − 휉, 푈, 퐻, ,̄푟 ),̄푠 the Lagrangian variable 푟 is not generally bounded in 퐿∞(ℝ). Formally, ̄푟 can be obtained from 푟 as 1 ̄푟(푡, 휉) = ∫ 푒−|푦(푡,휉)−푦(푡,휂)|푟(푡, 휂) 푑휂 (43) 2 ℝ and we also have the following relation between ,̄푟 ̄푠 and 푟,

푟 = −휉 ̄푠 + ̄푟푦휉, (44) from (23). From the definition of 푟, (42), and the transport equation (1b), we expect

푟푡 = 0 (45) This result can also be derived directly from the equivalent system (37) in purely Lagrangian variables. Indeed, after differentiating (44) with respect to time, we get

푟푡 = −휉푡 ̄푠 +푡 ̄푟 푦휉 + ̄푟푦휉푡. We use (37) and obtain

푟푡 = 푉휉 + 푆휉 − (푅 + 푊)푦휉 + ̄푟푈휉.

From (31) and (30), we obtain as expected that 푟푡 = 0.

4. Relabeling symmetry and local invariants

4.1. The case of the scalar Camassa–Holm equation. As we mentioned in the introduction, the CH equation can be derived from a variational principle; see [9] for a more thorough presentation. In the case of the CH equation, there is no potential energy, and the Lagrangian is given by the kinetic energy only,

1 ℒ = ∫(푢2 + 푢2 )(푡, 푥) 푑푥. CH 2 푥 ℝ The variation has to be done with respect to the particle path. We follow the notations from [9] and denote the particle path by 휙(푡, 휉), instead of 푦(푡, 휉) as in the previous section. After a change of variable, we can rewrite ℒCH as

2 1 휙푡휉 ℒ (휙) = ∫(휙2휙 + )(푡, 휉) 푑휉. (46) CH 2 푡 휉 휙 ℝ 휉 238 K. Grunert and X. Raynaud

The group of diffeomorphism on ℝ lets the Lagrangian invariant with respect to the group action of relabeling. For a given diffeomorphism 푓, the relabeling transformation of 휙(푡, 휉) with respect to 푓 is given by 휙 ∘ 푓 = 휙(푡, 푓(휉)). We can check directly that

2 2 1 (휙푡휉 ∘ 푓) 푓휉 ℒ (휙 ∘ 푓) = ∫((휙 ∘ 푓)2(휙 ∘ 푓)푓 + )(푡, 휉) 푑휉 = ℒ (휙) CH 2 푡 휉 휉 CH ℝ (휙휉 ∘ 푓)푓휉 after a change of variable. Noether’s theorem tells us that to every one-dimensional symmetry group which leaves the Lagrangian invariant, there corresponds a conservation law. For the group of diffeomorphisms, the tangent space is formally isomorphic to 퐶∞(ℝ), which is of infinite dimension, so that we expect infinitely many invariants. Let us first briefly present the Noether’s argument in afinite dimensional setting, that is, how a symmetry leads to an invariant. We consider 푞 ∈ ℝ푛 and the Lagrangian ℒ(푞, ̇푞). We assume that ℒ admits a one-dimensional symmetry group. Keeping this presentation informal, we simply assume that there exists a smooth mapping 푆∶ ℝ×ℝ푛 → ℝ푛, which represents the one-dimensional group action, such that 푆(0, ⋅ ) = id, and we denote 푞휀(푡) = 푆(휀, 푞(푡)). The invariance of the Lagrangian takes the form

ℒ(푞휀(푡),휀 ̇푞 (푡)) = ℒ(푞(푡), ̇푞(푡)). (47) The Euler–Lagrange equations for the solution are 푑 휕ℒ 휕ℒ ( ) = . (48) 푑푡 휕 ̇푞 휕푞 We differentiate (47) with respect to 휀 and obtain 휕ℒ 휕푞 휕ℒ 휕 ̇푞 휀 + 휀 = 0. 휕푞 휕휀 휕 ̇푞 휕휀 Set 휀 = 0 and use the Euler–Lagrange equation and the previous equation to get 푑 휕ℒ 휕푞 휕ℒ 휕 ̇푞 ( ) 휀 + 휀 = 0. 푑푡 휕 ̇푞 휕휀 |휀=0 휕 ̇푞 휕휀 |휀=0 Assuming the solution is smooth, we have 휕 ̇푞 푑 휕푞 = ( ) , 휕휀 |휀=0 푑푡 휕휀 |휀=0 and therefore it follows that 푑 휕ℒ 휕푞 ( 휀 ) = 0, (49) 푑푡 휕 ̇푞 휕휀 |휀=0 Modified two-component Camassa–Holm system 239 and the quantity 휕ℒ 휕푞휀 휕 ̇푞 휕휀 |휀=0 is preserved. Let us consider now the Lagrangian ℒCH. To simplify the notations we will denote the operator 휕/휕휀| as 훿 . The Lagrangian is invariant with respect to |휀=0 휀 relabeling. Formally, the tangent space at the identity of the group of smooth diffeomorphisms is the space of smooth functions 퐶∞(ℝ). For any function 푔 ∈ ∞ 퐶 (ℝ) in the tangent space, we define the diffeomorphism 푓휀(휉) = 휉 + 휀푔(휉) and consider the one-dimensional action defined as

휙휀 = 휙 ∘ 푓휀.

Assuming that the derivative of 푔 is bounded in 퐿∞(ℝ), there exists a neighborhood of zero such that, if 휀 belongs to this neighborhood, 푓휀 is a diffeomorphism. Slightly abusing the notation, we redefine ℒCH as

2 1 휓휉 ℒ (휙, 휓) = ∫(휓2휙 + )(푡, 휉) 푑휉, CH 2 휉 휙 ℝ 휉 so that ℒCH(휙, 휕푡휙) is equal to ℒCH(휙), as introduced in (46). The invariance of ℒCH with respect to relabeling implies

ℒCH(휙휀, 휕푡휙휀) = ℒCH(휙, 휕푡휙) for all 휀 ∈ ℝ close enough to zero. Following the same steps as before in the finite dimensional case, we end up with the following conservation law, corresponding to (49), 휕 훿ℒ ⟨ CH , 훿 휙⟩ = 0. (50) 휕푡 훿휓 휀 Let us give a precise meaning to each of the expressions entering (50). We have

훿ℒ 휓휉 ⟨ CH , 훿휓⟩ = ∫(휓 훿휓 휙 + 훿휓 ) 푑휉 훿휓 휉 휙 휉 ℝ 휉 so that 훿ℒ 휓휉 CH = 휓휙 − ( ) . 훿휓 휉 휙 휉 휉

Let us compute 훿휀휙휀. We have 휕 | | 훿휀휙휀 = | 휙(휉 + 휀푔(휉)) = 휙휉(휉)푔(휉). 휕휀 휀=0 240 K. Grunert and X. Raynaud

Hence, (50) can be rewritten as

휕 휙푡휉 ∫(휙 휙 − ( ) ) 휙 푔 푑휉 = 0. 휕푡 푡 휉 휙 휉 ℝ 휉 휉 Assuming that the solution is smooth and decays sufficiently fast, it follows that

휕 휙푡휉 ∫ ((휙 휙 − ( ) )휙 ) 푔 푑휉 = 0, (51) 휕푡 푡 휉 휙 휉 ℝ 휉 휉 as 푔 is independent of time. Now, we use the fact that (51) must hold for any 푔 ∈ 퐶∞, and therefore we obtain the following pointwise invariant,

휕 휙푡휉 (((휙푡휙휉 − ( ) )휙휉)(푡, 휉)) = 0, 휕푡 휙휉 휉 for all 휉 ∈ ℝ. Using the fact that 푢 ∘ 휙 = 휙푡, 푢푥 ∘ 휙 = 휙푡휉/휙휉, and 푢푥푥 ∘ 휙 = (푢푥 ∘ 휙)휉/휙휉, we can rewrite the pointwise invariant above in the form of 휕 ((푢 − 푢 )(푡, 휙(푡, 휉))휙2(푡, 휉)) = 0. (52) 휕푡 푥푥 휉 Note that, in fact, the pointwise invariant equation (52) can be used to derive the CH equation in a rather straightforward way. To see that, let us denote 푚 = ℋ푢 and expand (52). We obtain

2 (푚푡 ∘ 휙 + 푚푥 ∘ 휙휙푡)휙휉 + 2푚 ∘ 휙휙휉휙푡휉 = 0, which, after using 휙푡 = 푢 ∘ 휙 and 휙푡휉 = 푢푥 ∘ 휙휙휉, yields

2 (푚푡 + 푢푚푥 + 2푚푢푥) ∘ 휙휙휉 = 0, which, whenever 휙휉 does not vanish, is equivalent to the CH equation. Multipeakon solutions, which are a special class of solutions for the CH equation, are of the form 푛 −|푥−푞푖(푡)| 푢(푡, 푥) = ∑ 푝푖(푡)푒 (53) 푖=1 for time-dependent coefficients 푞푖(푡), which denote the position of the peaks, 푝푖(푡). Such solutions were identified in the seminal paper of Camassa and Holm[3]. Here, we want to show how the existence of this special class of solutions can be inferred directly from the pointwise invariant (52). As the pointwise invariant is a consequence of the symmetry of the problem, we can therefore establish a rather direct connection between the symmetry of the Lagrangian and the existence of Modified two-component Camassa–Holm system 241 special solutions. This approach is described in detail in [20], and we sketch it here as a preparation for the case of M2CH. After applying the Helmholtz operator ℋ to 푢 in (53), we get

(푢 − 푢푥푥)(푡, 푥) = ∑ 2푝푖(푡)훿(푥 − 푞푖(푡)), (54) 푖=1 where 훿(푥) denotes the Dirac delta distribution. For some initial data that satisfies (53) and any point 푥 ∈ ℝ away from the singularities, meaning that it does not coincide with any of the 푞푖, we have (푢 − 푢푥푥)(푥) = 0. After denoting by 푥(푡) = 푦(푡, 휉) the characteristic starting at 푥, the pointwise invariant (52) yields (푢 − 푢푥푥)(푡, 푥(푡)) = 0, as long as 푦휉(푡, 휉) ≠ 0. Hence, the structure given by (54), which defines the multipeakons, is preserved. The formulation givenby (54) cannot handle the collision of peaks as some of the coefficients 푝푖 tend to ±∞ in this case. To handle such case, we have to switch to the Lagrangian formulation. The pointwise conservation equation plays then an essential role when showing that the multipeakon structure is preserved. Between two neighboring peaks, say 푞푖(푡) and 푞푖+1(푡), we have to show that (푢−푢푥푥)(푡, 푥) = 0 for all 푥 ∈ (푞푖(푡), 푞푖+1(푡)). The peaks follow the characteristics so that, in Lagrangian coordinates, the region between the two peaks given as { (푡, 푥) ∣ 푞푖(푡) < 푥 < 푞푖+1(푡) }, which is curved in Eulerian coordinates, becomes rectangular, that is { (푡, 휉) ∣ 휉푖 < 휉 < 휉푖+1 }. Once the pointwise conservation equation is established for each of such regions, we can then deduce that the solution is indeed a multipeakon solution. The rigorous presentation of this approach is given in [20].

4.2. The case of the modified system. The Lagrangian for the M2CH system is given in (11). Let us rewrite the potential energy in terms of the Lagrangian variables we have introduced. We have 1 1 퐸pot = ∫(휌̄ − 휌̄ )휌̄ 푑푥 = ∫ 휌(푥)휌(푥)̄ 푑푥. 2 푥푥 2 ℝ ℝ We change to Lagrangian variable and obtain 1 퐸pot = ∫ ̄푟(휉)푟(휉) 푑휉. 2 ℝ We use the expression for ̄푟 derived in (43) and get 1 퐸pot = ∫ 푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂)푟(푡, 휉) 푑휂 푑휉. 4 ℝ2 The relabeling transformation for a density variable such as 푟 is defined as 푟 ↦ 푟 ∘ 푓푓휉 for any 푓 ∈ diff(ℝ). For such transformations, we can check that the 242 K. Grunert and X. Raynaud potential energy 퐸pot is invariant. In [15], when we proceed with the variation for the 2CH system, the density 휌 is treated as a function of 휙 so that the variation with respect to 휌 is not computed independently. Here, we use a different approach by decoupling the variables and introducing a Lagrangian multiplier function 휆 to enforce the mass conservation, that is 푟푡 = 0. Let 푋 = (휙, 푟, 휆), we consider the Lagrangian defined as

2 1 휙푡휉 ℒ(푋, 휕 푋) = ∫(휙2휙 + )(푡, 휉) 푑휉 푡 2 푡 휉 휙 ℝ 휉 1 − ∫ 푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂)푟(푡, 휉) 푑휂 푑휉 (55) 4 ℝ2

− ∫ 휆(푡, 휉)푟푡(푡, 휉) 푑휉. ℝ

We derive the Euler–Lagrange equation for this Lagrangian. Computations which we only sketch here give us

2 훿ℒ 1 1 휙푡,휉 = − (휙2) + ( ) 훿휙 2 푡 휉 2 휙2 휉 휉 (56a) 푟 + ∫ sign(휙(푡, 휉) − 휙(푡, 휂))푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂) 푑휂 2 ℝ with

훿ℒ 휙푡,휉 = 휙 휙 − ( ) (56b) 훿[휙 ] 푡 휉 휙 푡 휉 휉 and

훿ℒ 1 훿ℒ = − ∫ 푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂) 푑휂, = −휆, (56c) 훿푟 2 ℝ 훿[푟푡] 훿ℒ 훿ℒ = −푟푡, = 0. (56d) 훿휆 훿[휆푡]

We consider a diffeomorphism 휙 and a perturbation 훿휙, then 푟(푡, 휉) is perturbed by a corresponding 훿푟(푡, 휉). Thus varying the integral defined by

1 ℒpot = ∫ 푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂)푟(푡, 휉) 푑휂 푑휉, (57) 4 ℝ2 Modified two-component Camassa–Holm system 243 with respect to 푟 yields

훿ℒpot 1 ⟨ , 훿푟⟩ = ∫ 푒−|휙(푡,휉)−휙(푡,휂)|(훿푟(푡, 휂)푟(푡, 휉) + 푟(푡, 휂) 훿푟(푡, 휉)) 푑휂 푑휉 훿푟 4 ℝ2 1 = ∫(∫ 푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂) 푑휂) 훿푟(푡, 휉) 푑휉, 2 ℝ ℝ since we can interchange the order of integration. Varying ℒpot with respect to 휙 yields

훿ℒpot ⟨ , 훿휙⟩ 훿휙 1 = − ∫ sign(휙(푡, 휉) − 휙(푡, 휂))푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂)푟(푡, 휉) ⋅ 4 ℝ2 ⋅ (훿휙(푡, 휉) − 훿휙(푡, 휂)) 푑휂 푑휉 1 = − ∫ ∫ sign(휙(푡, 휉) − 휙(푡, 휂))푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂) 푑휂 푟(푡, 휉) 훿휙(푡, 휉) 푑휉 4 ℝ ℝ 1 + ∫ ∫ sign(휙(푡, 휉) − 휙(푡, 휂))푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂) 훿휙(푡, 휂) 푑휂 푟(푡, 휉) 푑휉 4 ℝ ℝ 1 = − ∫ ∫ sign(휙(푡, 휉) − 휙(푡, 휂))푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂) 푑휂 푟(푡, 휉) 훿휙(푡, 휉) 푑휉, 2 ℝ ℝ since we can again interchange the order of integration. The Euler–Lagrange equation 푑 훿ℒ 훿ℒ ( ) = 푑푡 훿[푋푡] 훿푋 yields 푟푡 = 0, (58) from (56d) and 1 휆 = ∫ 푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂) 푑휂, (59) 푡 2 ℝ from (56c). Using the variable ̄푟 defined as in (43), we rewrite (59) as

휆푡 = ̄푟. (60)

We can also rewrite (56a) as

휙2 ̄푟 훿ℒ 1 2 1 푡,휉 휉 = − (휙푡 )휉 + ( 2 ) − 푟 . (61) 훿휙 2 2 휙 휙휉 휉 휉 244 K. Grunert and X. Raynaud

Then (61) and (56b) yield 휙 휙2 ̄푟 푑 푡,휉 1 2 1 푡,휉 휉 (휙푡휙휉 − ( ) ) = − (휙푡 )휉 + ( ) − 푟 . (62) 푑푡 휙 휉 2 2 2 휉 휙 휉 휙휉 휉

The variable 푟 has been introduced as a primary variable, but since 푟푡 = 0, its 푟 dynamic is trivial. Setting 휌 ∘ 휙 = , 푟푡 = 0 implies that 휙휉

(휌푡 + (푢휌)푥) ∘ 휙휙휉 = 0, so that 휌 is indeed the density, if it is initially set as such. Moreover we have 1 ̄푟= ∫ 푒−|휙(푡,휉)−휙(푡,휂)|푟(푡, 휂) 푑휂 = 휌̄ ∘ 휙. 2 ℝ Hence 휉̄푟 푟 = 휌 ∘ 휙휌푥̄ ∘ 휙휙휉. 휙휉 After some computation, we can then see that (62) is equivalent to

(푚푡 + 푢푚푥 + 2푚푢푥 + 휌휌푥̄ ) ∘ 휙휙휉 = 0, that is (1a), when 휙휉 does not vanish. Let us now consider the action of relabeling on ℒ and derive pointwise invariants. The action of the group on the Lagrangian multiplier 휆 is given by (휆, 푓) ↦ 휆 ∘ 푓, for any diffeomorphism 푓. As in the scalar case of the CH equation, we consider for any 푔 ∈ 퐶∞(ℝ) the one-dimensional subgroup 푓휀(휉) = 휉 + 휀푔(휉) of diffeomorphisms. Using the notations introduced previously, we get

훿휀휙 = 휙휉푔, 훿휀푟 = 푟휉푔 + 푟푔휉 = (푟푔)휉, 훿휀휆 = 휆휉푔. (63) The pointwise conservation law (50) becomes

휕 훿ℒCH 훿ℒCH 훿ℒCH ( 훿휀휙 + 훿휀푟 + 훿휀휆) = 0 (64) 휕푡 훿[휙푡] 훿[푟푡] 훿[휆푡] in this case. Hence, using (56b), (56c), (56d) and (63), we get

휕 휙푡,휉 (∫ (휙 휙 − ( ) ) 휙 푔 푑휉 − ∫ 휆(푟푔) 푑휉) = 0. (65) 휕푡 푡 휉 휙 휉 휉 ℝ 휉 휉 ℝ Assuming that the solution is smooth and decays sufficiently fast, we move the time derivative under the integral. The first integral is the same as in the scalar case. For the second one, we get, after integration by parts, 휕 (∫ 휆(푟푔) 푑휉) = − ∫(휆 푟) 푔 푑휉, 휕푡 휉 휉 푡 ℝ ℝ Modified two-component Camassa–Holm system 245 using the fact that 푟푡 = 0. Hence (65) yields

2 ∫(푚 ∘ 휙휙휉 + 휆휉푟)푡푔 = 0, ℝ which must hold for any function 푔, so that the pointwise conservation law for the M2CH system is given by 휆 휉 2 ((푚 ∘ 휙 + 휌 ∘ 휙)휙휉) = 0, (66) 휙휉 푡 and the pointwise conserved quantity is 휆 휉 2 (푚 ∘ 휙 + 휌 ∘ 휙)휙휉. (67) 휙휉 Again, as for the case of CH, we observe that M2CH can be derived from (67) in a rather straightforward manner. Using that 푟푡 = 0 and the expression (60) for 휆푡, we get 2 (휆휉푟)푡 =휉 ̄푟 푟 = (휌휌푥̄ ) ∘ 휙 휙휉, so that (66) can be rewritten as

2 2 (푚 ∘ 휙 휙휉)푡 = −(휌휌푥̄ ) ∘ 휙 휙휉, (68) and, as before for the scalar case, we can check that (68) implies (1a). From the pointwise conservation law (66), we can deduce the existence of multipeakon solutions. These are solutions (푢(푡, 푥), 휌(푡,̄ 푥)) of the form

푛 푛 −|푥−푞푖(푡)| −|푥−푞푖(푡)| 푢(푡, 푥) = ∑ 푝푖(푡)푒 and 휌(푡,̄ 푥) = ∑ 푠푖(푡)푒 (69) 푖=1 푖=1 for time-dependent coefficients 푞푖(푡) (denoting the position of the peaks), 푝푖(푡) and 푠푖(푡). After applying the Helmholtz operator ℋ to 푢 and 휌̄ in (69) , we get

(푢 − 푢푥푥)(푡, 푥) = ∑ 2푝푖(푡)훿(푥 − 푞푖(푡)), 휌(푡, 푥) = ∑ 2푠푖(푡)훿(푥 − 푞푖(푡)), (70) 푖=1 푖=1 where 훿(푥) denotes the Dirac delta distribution. Let us consider some initial data that satisfies (70) initially. For any point 푥 ∈ ℝ away from the singularities, that is, different from any of the 푞푖, we have (푢 − 푢푥푥)(푥) = 0 and 휌(푥) = 0. Let us denote by 푥(푡) = 푦(푡, 휉) the characteristic starting at 푥. Since 푟푡 = 0, we get that 휌(푡, 푥(푡))푦휉(푡, 휉) = 0, that is, 휌(푡, 푥(푡)) = 0, as long as 푦휉(푡, 휉) ≠ 0. From the pointwise invariant (66), we infer that (푢 − 푢푥푥)(푡, 푥(푡)) = 0, as long as 246 K. Grunert and X. Raynaud

푦휉(푡, 휉) ≠ 0. Hence the structure given by (70), which defines the multipeakons, is preserved. The formulation given by (70) cannot handle the collision of peaks, as some of the coefficients 푝푖 tend to ±∞ in this case. To do so, we have to switch to the Lagrangian formulation. To show that the multipeakon structure is preserved in the Lagrangian formulation, the pointwise conservation equation plays again an essential role but, clearly as the following computations show, the derivation is significantly less tractable. Between two neighboring peaks, say 푞푖(푡) and 푞푖+1(푡), we have to show that (푢 − 푢푥푥)(푡, 푥) = 0 and 휌(푡, 푥) = 0 for all 푥 ∈ (푞푖(푡), 푞푖+1(푡)). The peaks follow the characteristics so that, in Lagrangian coordinates, the region between the two peaks given as { (푡, 푥) ∣ 푞푖(푡) < 푥 < 푞푖+1(푡) }, which is curved in Eulerian coordinates, becomes rectangular { (푡, 휉) ∣ 휉푖 < 휉 < 휉푖+1 }. Once the pointwise conservation equation is established for each of such regions, we can then deduce that the solution is indeed a multipeakon solution. The rigorous presentation of this approach is given in [20], and we only sketch here how we prove the local conservation equation in the Lagrangian setting. For each rectangular region of the form defined above, we can prove that higher regularity for the Lagrangian variables is preserved by the governing equations; see [20]. Then, we can define the following quantities

푦 2 2 휉휉 (푢 − 푢푥푥)(푡, 푦)푦휉 = 푈푦휉 − 푈휉휉 + 푈휉 푦휉 and

푟 = −휉 ̄푠 + ̄푟푦휉. (71)

Note that both quantities require higher regularity of the variables (existence of 푈휉휉, 푦휉휉, 휉̄푠 ). For simplicity, we assume that 푦휉 is different from zero. This assumption can then be removed as in [20]. The pointwise conservation equation will be established in the Lagrangian setting if we can show that the quantity 푀 defined below remains equal to zero,

푦 푑 2 휉휉 (푈푦휉 − 푈휉휉 + 푈휉) +휉 ̄푟 푟 + 휆푟푡 = 0. (72) ⏟⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⏟⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⏟푑푡 푦휉 푀

We have seen at the end of Section 3 that 푟푡 = 0 can be derived directly from the governing equations (37) in Lagrangian variables. Combining (35) and (39), we get 푈2 2 1 휉 1 2 1 2 푄휉 = −푈 푦휉 − + 푃푦휉 − ̄푟 푦휉 + ̄푠 푦휉. (73) 2 푦휉 2 2 Modified two-component Camassa–Holm system 247

Now, using the governing equations (37), we get

2 푈휉휉푈휉 푦휉휉 푦휉휉푈휉 푀 = (푈푦2) + 푄 + − 푄 − + ̄푟 푟 휉 푡 휉휉 푦 푦 휉 2 휉 ⏟⎵⏟⎵⏟ ⏟ 휉 ⏟⏟⏟휉 푦휉 퐴 퐵 퐶 with

2 퐴 = −푄푦휉 + 2푈푈휉푦휉, 2 푈휉휉푈휉 1 푈휉 푦휉휉 퐵 = −2푈푈 푦 − 푈2푦 − + + 푄푦2 + 푃푦 휉 휉 휉휉 푦 2 2 휉 휉휉 휉 푦휉 2̄푟 2̄푠 − ̄푟̄푟 푦 − 푦 + ̄푠̄푠 푦 + 푦 , 휉 휉 2 휉휉 휉 휉 2 휉휉 2 2 2 1 푈휉 푦휉휉 ̄푟 ̄푠 퐶 = −푈2푦 − + 푃푦 − 푦 + 푦 . 휉휉 2 2 휉휉 2 휉휉 2 휉휉 푦휉 Hence

푀 =휉 ̄푟 푟 − ̄푟̄푟휉푦휉 + ̄푠̄푠휉푦휉, and (72) follows from (71).

5. Double multipeakons

For the CH equation the so-called multipeakon solutions serve on the one hand as an illustrating example of how solutions may behave, and on the other hand they are dense in the set of weak conservative solutions [19, 22]. Since the M2CH system reduces in the case 휌 ≡ 0 to the CH equation, the aim of this section is to derive the time evolution of solutions until wave breaking in the case of both 푢(푡, 푥) and 휌(푡,̄ 푥) being multipeakons, which we will call from now on double multipeakons. That is we are searching for solutions of the form (69), where the positions of the peaks, 푞푖(푡), satisfy

−∞ < 푞1(푡) < ⋯ < 푞푛(푡) < ∞.

In particular, both 푢(푡, ⋅ ) and 휌(푡,̄ ⋅ ) are not differentiable at the points 푥 = 푞푖(푡) (푖 = 1, 2, … , 푛), and hence (푢(푡, 푥), 휌(푡,̄ 푥)) are going to satisfy the M2CH system in the weak sense. As a first step we have to define what it means to be a local weak solution of the M2CH system. Direct computations as in [20] for the CH equation yield that 248 K. Grunert and X. Raynaud

(1) can be rewritten as follows

푢푡 − 푢푡푥푥 + 3푢푢푥 − 2푢푥푢푥푥 − 푢푢푥푥푥 + 휌휌̄ 푥̄ − 휌푥̄ 휌푥푥̄ 3 2 1 2 1 2 1 2 1 2 = 푢푡 − 푢푡푥푥 + 2 (푢 )푥 + 2 (푢푥)푥 − 2 (푢 )푥푥푥 + 2 (휌̄ )푥 − 2 (휌푥̄ )푥 = 0. and

휌푡 + (푢휌)푥 = 휌푡̄ − 휌푡푥푥̄ + 푢휌푥̄ + 푢푥휌̄ − 푢푥휌푥푥̄ − 푢휌푥푥푥̄

= 휌푡̄ − 휌푡푥푥̄ + (푢휌)̄ 푥 − (푢휌)̄ 푥푥푥 + (푢푥휌)̄ 푥푥 + (푢푥휌푥̄ )푥 = 0. Hence we have the following definition.

1 1 1 1 Definition 1. We say that (푢, 휌)̄ ∈ 퐿loc([0, 푇], 퐻loc) × 퐿loc([0, 푇], 퐻loc) is a weak solution of the M2CH system if it satisfies

3 2 1 2 1 2 1 2 1 2 푢푡 − 푢푡푥푥 + 2 (푢 )푥 + 2 (푢푥)푥 − 2 (푢 )푥푥푥 + 2 (휌̄ )푥 − 2 (휌푥̄ )푥 = 0 (74a) 휌푡̄ − 휌푡푥푥̄ + (푢휌)̄ 푥 − (푢휌)̄ 푥푥푥 + (푢푥휌)̄ 푥푥 + (푢푥휌푥̄ )푥 = 0 (74b) in the sense of distributions. Since the local, weak multipeakon solutions are piecewise smooth solutions and following closely the computations carried out in [18] for the CH equation, we obtain after some integration by parts that all the information concerning the time ′ evolution of 푞푖(푡), 푝푖(푡), and 푠푖(푡), is contained in the coefficients of 훿푞푖 and 훿푞푖. ′ For (74a) the coefficient of 훿푞푖 must be equal to zero and is given by [푢 ] + 1 [(푢2) ] = 0, 푡 푞푖 2 푥 푞푖 where we denote by [푣]푞푖 = 푣푖(푞푖+) − 푣푖−1(푞푖−). The jumps are mainly influenced −|푥−푞푖(푡)| by the sign changes in the derivative, which come from the term 푝푖(푡)푒 at the point 푥 = 푞푖(푡). In particular, we have

′ 2 [푢푡]푞푖 = 2푝푖(푡)푞푖 (푡), [(푢 )푥]푞푖 = −4푝푖(푡)푢(푞푖(푡)), and hence ′ 2푝푖(푡)(푞푖 (푡) − 푢(푞푖(푡))) = 0.

Dividing both sides by 2푝푖(푡) yields the equation for the characteristic 푛 ′ −|푞푗(푡)−푞푖(푡)| 푞푖 (푡) = 푢(푞푖(푡)) = ∑ 푝푗(푡)푒 . 푗=1

By the same argument the coefficient of 훿푞푖 in (74a) must be equal to zero and thus [푢 ] − 1 [푢2 ] + 1 [(푢2) ] + 1 [휌2̄ ] = 0. 푡,푥 푞푖 2 푥 푞푖 2 푥푥 푞푖 2 푥 푞푖 Modified two-component Camassa–Holm system 249

Again the jumps are mainly influenced by the sign changes in the derivatives, −|푥−푞푖(푡)| −|푥−푞푖(푡)| which come from the terms 푝푖(푡)푒 and 푠푖(푡)푒 , it is therefore convenient to introduce the following abbreviations

′ 2 [푢푡,푥]푞푖 = −2푝푖 (푡), [푢푥]푞푖 = −4푝푖(푡)푓푥(푡, 푞푖(푡)), 2 2 [(푢 )푥푥]푞푖 = −8푝푖(푡)푓푥(푡, 푞푖(푡)), [휌푥̄ ]푞푖 = −4푠푖(푡)푔푥(푡, 푞푖(푡)), which implies that

′ −2푝푖 (푡) + 2푝푖(푡)푓푥(푡, 푞푖(푡)) − 4푝푖(푡)푓푥(푡, 푞푖(푡)) − 2푠푖(푡)푔푥(푡, 푞푖(푡)) = 0.

Recalling the definition of 푓(푡, 푥) and 푔(푡, 푥), we end up with

′ −|푞푖(푡)−푞푗(푡)| 푝푖 (푡) = ∑ 푝푖(푡)푝푗(푡) sign(푞푖(푡) − 푞푗(푡))푒 푗≠푖

−|푞푖(푡)−푞푗(푡)| + ∑ 푠푖(푡)푠푗(푡) sign(푞푖(푡) − 푞푗(푡))푒 . 푗≠푖

′ As far as (74b) is concerned, the coefficient of 훿푞푖 has to be equal to zero and is given by

[휌푡̄ ]푞푖 + [(푢휌)̄ 푥]푞푖 − [푢푥휌]̄ 푞푖 = 0. In particular, we have

′ [휌푡̄ ]푞푖 = 2푠푖(푡)푞푖 (푡), [푢푥휌]̄ 푞푖 = −2푝푖(푡)(푠푖(푡) + 푔(푡, 푞푖(푡))),

[(푢휌)̄ 푥]푞푖 = −2(푠푖(푡)푓(푡, 푞푖(푡)) + 푝푖(푡)푔(푡, 푞푖(푡)) + 2푠푖(푡)푝푖(푡)), and accordingly

′ 2푠푖(푡)푞푖 (푡) − 2푠푖(푡)(푓(푡, 푞푖(푡)) + 푝푖(푡)) = 0

Recalling the definition of 푓(푡, 푥) and dividing both sides by 푠푖(푡) we obtain

′ 푞푖 (푡) = 푓(푡, 푞푖(푡)) + 푝푖(푡) = 푢(푞푖(푡)). 250 K. Grunert and X. Raynaud

By the same argument the coefficient of 훿푞푖 must be equal to zero, which is equivalent to

[휌푡,푥̄ ]푞푖 + [(푢휌)̄ 푥푥]푞푖 − [(푢푥휌)̄ 푥]푞푖 − [푢푥휌푥̄ ]푞푖 = 0. Direct computations yield

′ [휌푡,푥̄ ]푞푖 = −2푠푖(푡),

[(푢휌)̄ 푥푥]푞푖 = −4푝푖(푡)푔푥(푡, 푞푖(푡)) − 4푠푖(푡)푓푥(푡, 푞푖(푡))

[푢푥휌푥̄ ]푞푖 = −2푝푖(푡)푔푥(푡, 푞푖(푡)) − 2푠푖(푡)푓푥(푡, 푞푖(푡)),

[(푢푥휌)̄ 푥]푞푖 = −2푝푖(푡)푔푥(푡, 푞푖(푡)) − 2푠푖(푡)푓푥(푡, 푞푖(푡)), and hence

′ 푠푖(푡) = 0. Thus we have the following system of ODEs

푛 ′ −|푞푖(푡)−푞푗(푡)| 푞푖 (푡) = ∑ 푝푖(푡)푒 , (75a) 푖=1 ′ −|푞푖(푡)−푞푗(푡)| 푝푖 (푡) = ∑(푝푖(푡)푝푗(푡) + 푠푖(푡)푠푗(푡)) sign(푞푖(푡) − 푞푗(푡))푒 , (75b) 푗≠푖 ′ 푠푖(푡) = 0. (75c)

6. Double peakon-antipeakon solutions

In this section, we study in detail the peakon-antipeakon solutions in the case 푛 = 2. That means both 푢(푡, 푥) and 휌(푡,̄ 푥) are the sum of one peakon and one antipeakon except when wave breaking occurs, in which case both are constantly equal to zero and part of the energy is concentrated in one point, which is represented by a 훿-distribution. To set the stage, let

−|푥−푞1(푡)| −|푥−푞2(푡)| 푢(푡, 푥) = 푝1(푡)푒 + 푝2(푡)푒 , (76a)

−|푥−푞1(푡)| −|푥−푞2(푡)| 휌(푡,̄ 푥) = 푠1(푡)푒 + 푠2(푡)푒 . (76b)

We assume that 푞1 ≤ 푞2 initially and, as the peaks travel along characteristics, this property remains true for all time. Then the corresponding time independent total energy, which we denote 퐸 is given by

2 2 2 2 푞1(푡)−푞2(푡) 퐸 = 푝1(푡) + 푝2(푡) + 푠1(푡) + 푠2(푡) + 2(푝1푝2(푡) + 푠1푠2(푡))푒 . (77) Modified two-component Camassa–Holm system 251

Introducing the variables 푞 = 푞1 − 푞2, 푄 = 푞1 + 푞2, 푝 = 푝1 − 푝2, 푃 = 푝1 + 푝2, 푠 = 푠1 − 푠2, and 푆 = 푠1 + 푠2,(76) and (77) rewrite as 1 1 −|푥− (푞+푄)(푡)| −|푥− (푄−푞)(푡)| 1 2 1 2 푢(푡, 푥) = 2 (푝 + 푃)(푡)푒 + 2 (푃 − 푝)(푡)푒 , (78a) 1 1 −|푥− (푞+푄)(푡)| −|푥− (푄−푞)(푡)| 1 2 1 2 휌(푡,̄ 푥) = 2 (푠 + 푆)(푡)푒 + 2 (푆 − 푠)(푡)푒 , (78b) 1 2 2 2 2 1 2 2 2 2 푞(푡) 퐸 = 2 (푝 + 푃 + 푠 + 푆 )(푡) + 2 (푃 − 푝 + 푆 − 푠 )(푡)푒 . (78c) According to (75), the functions 푞, 푄, 푝, 푃, 푠, and 푆 satisfy the following system of ordinary differential equations

푞(푡) 푞(푡) 푞푡(푡) = 푝(푡)(1 − 푒 ), 푄푡(푡) = 푃(푡)(1 + 푒 ), (79a) 1 2 1 푝푡(푡) = 2 푝(푡) + 2 퐶, 푃푡(푡) = 0, (79b) 푠푡(푡) = 0, 푆푡(푡) = 0, (79c) where 퐶 = (푃2(푡) + 푆2(푡) + 푠2(푡) − 2퐸). We observe that if 푄(푡) = 푃(푡) = 푆(푡) = 0 holds for some 푡, then it holds for all 푡. This means, since 푄(푡) = 푞1(푡) + 푞2(푡), 푃(푡) = 푝1(푡) + 푝2(푡), and 푆(푡) = 푠1(푡) + 푠2(푡), that there exist two peakon solutions (푢(푡, 푥), 휌(푡,̄ 푥)) such that

−|푥−푞1(푡)| −|푥+푞1(푡)| 푢(푡, 푥) = 푝1(푡)(푒 − 푒 ) (80a)

−|푥−푞1(푡)| −|푥+푞1(푡)| 휌(푡,̄ 푥) = 푠1(푡)(푒 − 푒 ). (80b) Such solutions are called peakon-antipeakon solutions, since both 푢(푡, ⋅ ) and 휌(푡,̄ ⋅ ) are antisymmetric for all 푡 ∈ ℝ. In the remaining of this section, we compute these solutions explicitly. Wave breaking occurs when two peakons ∗ ∗ ∗ occupy the same position, that is 푞(푡 ) = 푞1(푡 ) − 푞2(푡 ) = 0. In this case, we have ∗ ∗ 푢푥(푡, 푥) → ∓∞ as 푡 → 푡 ∓, which implies 푝(푡) → ±∞ as 푡 → 푡 ∓. As mentioned earlier, 휌̄ and its derivative 휌푥̄ remain bounded. We now turn to the computation of 푝(푡), 푞(푡), and 푢(푡, 푞1(푡)), the value of 푢(푡, 푥) at the left peak, depending on the value of 푠2/2 compared with the total energy 퐸. We observe that the governing equations are invariant with respect to the transformation

푡 ↦ 훼푡, 푢 ↦ 훼푢, 휌̄ ↦ 훼휌.̄

Therefore, we do not restrict ourselves by considering only a single value of 퐸. For 1 simplicity, we choose 퐸 = 2 so that (78c) yields (푝(푡)2 + 푠(푡)2)(1 − 푒푞(푡)) = 1. (81)

Moreover, the equation is also invariant by the transformation 휌̄ ↦ −휌̄ so that, without loss of generality, we assume 푠 ≥ 0. Let us denote by 푢† and 휌†̄ the values 252 K. Grunert and X. Raynaud of 푢 and 휌̄ at the peaks, that is

1 푞(푡) 푢†(푡) = 푢(푡, 푞1(푡)) = −푢(푡, 푞2(푡)) = 2 푝(푡)(1 − 푒 ) and 1 푞(푡) 휌†̄ (푡) = 휌(푡,̄ 푞1(푡)) = −휌(푡,̄ 푞2(푡)) = 2 푠(푡)(1 − 푒 ).

From these expressions, we can express 푠 and 푝 as function of 푢† and 휌†̄ and plug the results in (81). We obtain

2 2 푞 4푢† + 4휌†̄ = (1 − 푒 ).

We use again the definition of 휌†̄ and get

1 2 1 2 푢2 + (휌̄ − ) = ( ) . (82) † † 4푠 4푠

Since 푠 is constant, the trajectories of (휌†̄ , 푢†) lie on circles as depicted in Figure 1. Let us know consider the following three cases, depending on the value of 푠, which cover all the possible types of dynamics for the system, and compute explicitly the solution for each case with initial data 푞(0) = 푞0 and 푝(0) = 푝0.

6.1. Case 0 ≤ 푠 < 1. In this case, we have 1 푝 = (푝 − √−퐶)(푝 + √−퐶) 푡 2 We integrate this expression and obtain

1 + 퐴푒√−퐶푡 푝(푡) = √−퐶 , (83a) 1 − 퐴푒√−퐶푡 (1 − 퐴푒√−퐶푡)2 푞(푡) = − ln(1 + 푒−√−퐶푡 (푒−푞0 − 1)) (83b) (1 − 퐴)2 where 퐴 = (푝0 − √−퐶)/(푝0 + √−퐶) and

√−퐶푡 √−퐶푡 −푞0 1 −√−퐶푡 (1 − 퐴푒 )(1 + 퐴푒 )(푒 − 1) 푢†(푡) = √−퐶푒 . 2 (1 − 퐴)2 + 푒−√−퐶푡(1 − 퐴푒√−퐶푡)2(푒−푞0 − 1)

2 2 2 푞0 By definition, we have 퐶 = 푠 − 1 and (푝0 + 푠 )(1 − 푒 ) = 1. Hence

2 2 2 푞0 푝0 = −퐶 + (푝0 + 푠0)푒 ≥ −퐶 Modified two-component Camassa–Holm system 253 and 퐴 > 0 so that wave breaking occurs at time 푡∗ = ln(1/퐴)/√−퐶. We shift time so that the collision takes place at 푡 = 0. To do so, we let 푝0 tend to infinity and 푞0 2 2 푞0 to zero, while preserving (81), that is, (푝0 + 푠 )(1 − 푒 ) = 1. Let us denote

2 푝∞ = √1 − 푠 = √−퐶.

The solution (83) is equivalent, up to shift in time, to

1 + 푒푝∞푡 푝(푡) = 푝∞ , 1 − 푒푝∞푡 cosh(푝∞푡) − 1 푞(푡) = − ln (1 + 2 ) . 2푝∞ Moreover, we have

sinh(푝 푡) 푢 (푡) = −푝 ∞ . † ∞ 2 4푝∞ + 2(cosh(푝∞푡) − 1)

When 푡 tends to ±∞, we get the following limits

푝∞ lim 푢†(푡) = ∓ . 푡±∞ 2

Let us write 푢†(±∞) = lim푡±∞ 푢†(푡), and use the same notation for 푝(±∞) and 2 2 휌†̄ (±∞). Taking the same limit in (81), we get 푝(±∞) + 푠 = 1, which implies

2 2 1 푢†(±∞) + 휌†̄ (±∞) = 4 . (84)

This circle is plotted in Figure 1 and represents the limiting values for (휌†̄ , 푢†).

6.2. Case 푠 > 1. The solution reads for all 푡 ∈ ℝ

√ 1 √ 푝(푡) = 퐶 tan ( 2 퐶푡 + 퐷) , 2 1 √ 푞(푡) = − ln (1 + 퐵 cos ( 2 퐶푡 + 퐷)) and we have 1 퐵 sin(√퐶푡 + 2퐷) 푢†(푡) = √퐶 4 2 1 √ 1 + 퐵 cos ( 2 퐶푡 + 퐷) where −푞0 푝0 푒 − 1 퐷 = arctan ( ) and 퐵 = 2 . (85) √퐶 cos(퐷) 254 K. Grunert and X. Raynaud

We shift time as before and set the collision time to zero. The solution is then given by

푝(푡) = −√퐶 cot (√퐶푡/2) , 1 푞(푡) = − ln(1 + sin2(√퐶푡/2)), 퐶 and we have 1 √ 1 퐶 sin( 퐶푡) 푢†(푡) = − √퐶 , 4 1 2 √ 1 + 퐶 sin ( 퐶푡/2)

Especially the last double peakon-antipeakon solution comes as a surprise, since such peakon-antipeakon solutions do not exist for the CH equation. In the case of the CH equation, i.e., 푠 = 푆 = 0 for all 푡 ∈ ℝ, the constant 퐶 reduces to

2 푞0 퐶 = −2 = −푝0(1 − 푒 ), which is less than 0 under the assumption that 푞0 ≠ 0 and 푢(0, 푥) ≢ 0. For the M2CH system, on the other hand,

2 푞0 2 푞0 퐶 = −푝0(1 − 푒 ) + 푠 푒 . and choosing 푠 big in contrast to 푝0, one ends up in the case 퐶 ≥ 0. Thus the last case is intrinsic for the M2CH system.

6.3. Case 푠 = 1. Direct calculations in that case yield 2푝 푝(푡) = 0 , 2 − 푡푝0 2 2 − 푡푝0 푞(푡) = − ln (1 + ( ) (푒−푞0 − 1)) , 2 and 2−푡푝0 −푞0 푝0 4 (푒 − 1) 푢†(푡) = , 2−푡푝 2 0 −푞0 1 + ( 2 ) (푒 − 1) As in the previous case, we set the collision time to zero and obtain 1 푝(푡) = − , 푡/2 푞(푡) = − ln (1 + (푡/2)2) , Modified two-component Camassa–Holm system 255 and 푡 푢 (푡) = − . † 푡2 + 4

In this limiting case 푢†(∞) = 0.

Figure 1. Plot of the trajectory of (휌†̄ , 푢†) for different values of 푠. The outer half-circle represents the limiting values of the solution when 푡 → ±∞ when 푠 ≤ 1, see (84). The circles in the middle represent the periodic solution for 푠 ≥ 1.

Acknowledgments. K.G. gratefully acknowledges the hospitality of the Mittag- Leffler Institute, creating a great working environment for research during the fall 2016. Research supported by the grant Waves and Nonlinear Phenomena (WaNP) (FRIPRO Toppforsk project 250070) from the Research Council of Norway. 256 K. Grunert and X. Raynaud

Figure 2. Case 0 ≤ 푠 < 1. Plot of the solution 푢 (blue) and 휌̄ (red) at different times. Modified two-component Camassa–Holm system 257

Figure 3. Case 푠 > 1. The solution is periodic with period 2휋 . The first plot (top, left) shows √퐶 the solution right after a collision. 258 K. Grunert and X. Raynaud

Figure 4. Case 푠 = 1. Limiting case. The solution decays to zero. Modified two-component Camassa–Holm system 259

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Graziano Guerra and Wen Shen

Dedicated to Helge Holden on the occasion of his 60th birthday

Abstract. We consider the solutions of Riemann problems for polymer flooding models. In a suitable Lagrangian coordinate the systems take a triangular form, where the equation for thermodynamics is decoupled from the hydrodynamics, leading to the study of scalar conservation laws with discontinuous flux functions. We prove three equivalent admissibility conditions for shocks for scalar conservation laws with discontinuous flux. Furthermore, we show that a variation of minimum path of [10] proposed in [18] is the vanishing viscosity limit of a partially viscous model with viscosity only in the hydro-dynamics.

1. Introduction

Consider the model for polymer flooding in two phase flow for secondary oil recovery [6, 18] 푠푡 + 푔(푠, 푐)푥 = 0, { (1.1) (푐푠) + (푐푔(푠, 푐)) = 0. 푡 푥 Here 푠 is the saturation of the water phase, and 푐 is the fraction of the polymer dissolved in the water phase. The function 푔(푠, 푐) denotes the fractional flow, which is the classical S-shaped Buckley–Leverett flux function [7]. A related model, which takes into consideration the adsorption effect of the porous media, takes the form

푠푡 + 푔(푠, 푐)푥 = 0, { (1.2) (푚(푐) + 푐푠) + (푐푔(푠, 푐)) = 0, 푡 푥 where the term 푚(푐) models the adsorption of polymer in the rock. Physical property of the porous media usually prompts the assumptions

푚′(푐) ≥ 0 and 푚″(푐) ≤ 0.

The system in (1.1) can be viewed as a special case of (1.2) with 푚(푐) = constant. 262 G. Guerra and W. Shen

The two systems (1.1) and (1.2) share some common features. When 푐 is constant, the second equation is equivalent to the first equation. Thus, one family of integral curves are all straight lines where 푐 is constant. We call this the 푠-family. The other family, referred to as the 푐-family, will have waves that connect different straight lines of 푠-integral curves. For each given 푐, there exists at least one 푠 value where the two eigenvalues and two eigenvectors both coincide. Thus, there exist at least one curve 푠 = 풮(푐) in the domain where the system is parabolic degenerate. Along these degenerate curves, non-linear resonance occurs, and the total variation of the unknown 푠(푡, 푥) could blow up in finite time [20]. Another very interesting feature shared by these two systems is that they partially decouple using a suitably defined Lagrangian coordinate, see [17, 21]. We define the Lagrangian coordinate (휓, 휙) as

휙푥 = −푠, 휙푡 = 푔, 휓 = 푥. (1.3)

We see that 휙 is the potential function of the first equation in (1.1). Using (휓, 휙) as the independent variables, the system (1.1) becomes 휕 1 휕 푠 ⎧ ( ) − ( ) = 0, ⎪휕휓 푔(푠, 푐) 휕휙 푔(푠, 푐) (1.4) ⎨ 휕푐 ⎪ = 0. ⎩ 휕휓 For the system (1.2), the same coordinate change gives 휕 1 휕 푠 ⎧ ( ) − ( ) = 0, ⎪휕휓 푔(푠, 푐) 휕휙 푔(푠, 푐) (1.5) ⎨ 휕푐 휕 ⎪ + 푚(푐) = 0. ⎩ 휕휓 휕휙 The equivalence between weak solutions of the systems in the two coordinates is proved in the seminal paper of Wagner [21]. We observe that the systems (1.5) and (1.4) are in triangular form, where the second equation is decoupled from the first one. Referring to physical principles, the first equation in (1.5) (and (1.4)) gov- erns the hydrodynamics, while the second equation describes the thermodynamics. This decoupled feature indicates that the thermodynamic process is independent of the hydrodynamics. The solutions of (1.5) and (1.4) could be obtained by first solving the second equation for 푐, then plugging the solution of 푐 into the first equation and solving for 푠. The solution for 푐, either being constant in “time” 휓 for (1.4) or as the solution of a scalar conservation law in (1.5), might well contain discontinuities. This motivates the study of a scalar conservation law with discontinuous flux. Models of polymer flooding 263

Therefore, we consider the solution of the Riemann problem for scalar conservation law with discontinuous flux in a general setting. We consider

푢 + 푓(푎(푥), 푢) = 0, (1.6) 푡 푥 where 푎− (푥 < 0), 푎(푥) = { (1.7) 푎+ (푥 > 0), associated with the initial Riemann data

푢퐿 (푥 < 0), 푢(0, 푥) = { (1.8) 푢푅 (푥 > 0).

We observe that the solutions to (1.6)–(1.8) can be obtained as limits of two combined approximations: (i) One may approximate the jump function 푎( ⋅ ) by a sequence of smooth functions. For example, let ̂푎(푥) be a smooth function such that lim푥→±∞ ̂푎(푥) = ± ′ 푎 . Given a decreasing sequence 휀푛 → 0, one can define 푥 푎푛(푥) ≐ ̂푎 ( ′ ) . 휀푛 One may also take any other convergent sequence.

(ii) One can add a viscosity term 휀푛푢푥푥 on the right hand side of (1.6). From modeling considerations, it is natural to consider the solutions which are (푛) (푛) obtained as limits of approximations: 푢 = lim푛→∞ 푢 , where 푢 denotes a solution to the viscous conservation law with smooth flux

푢 + 푓(푎 (푥), 푢) = 휀 푢 . (1.9) 푡 푛 푥 푛 푥푥 In this setting, three main issues arise: (1) For a given initial Riemann data, is the limit solution 푢 = lim 푢(푛) uniquely ′ defined, or does it depend on the relative rate at which the sequences 휀푛, 휀푛 approach zero? (2) If in general the solution depends on the ratio, what is the sufficient assumptions one can make such that all these limits are the same? (3) How can one determine the traces 푢− = 푢(푡, 0−) and 푢+ = 푢(푡, 0+) in this limit solution? 264 G. Guerra and W. Shen

In a parallel paper [19], we show through several detailed counterexamples that the answer to the first question is the latter. The same paper also addresses the answer for the second question, showing that a suitable monotonicity condition on the flux 푐 ↦ 푔(푠, 푐), among others, will ensure the uniqueness of the double limit. The third question is addressed in this paper, and we consider the case ′ 휀푛 = 0. Here we present three equivalent admissibility conditions, and a detailed construction of the solution of a Riemann problem, and the proof that it is the vanishing viscosity limit. Scalar conservation laws with discontinuous flux functions have been the subject of much research activity in the past few decades, and numerous results are available. We refer to a survey paper [1] and the references therein, and apologize that a comprehensive list of reference is without the scope of this paper. In this paper we seek vanishing viscosity solutions of Riemann problem, leading to certain entropy conditions. In the literature, various forms of entropy conditions have been proposed and studied. For some related works, see e.g. [2, 3, 4, 8, 12, 13, 14, 15]. In connection with the adsorption model (1.2) in the Lagrangian coordinate (1.5), we may consider the triangle system in a general setting

푢푡 + 푓(푎, 푢)푥 = 0, { (1.10) 푎푡 + 푚(푎)푥 = 0, associated with initial Riemann data

푎− (푥 < 0), 푢퐿 (푥 < 0), 푎(0, 푥) = { 푢(0, 푥) = { (1.11) 푎+ (푥 > 0), 푢푅 (푥 > 0).

For the triangle system (1.10), let 푎 = 푎(푡, 푥) be a solution to the second equation of conservation law, consisting of a single entropy-admissible shock with left and right states 푎− and 푎+, respectively. By performing a linear transformation of the 푡-푥 variables, we can assume that the shock speed is zero, so that 푎(푡, 푥) = 푎(푥) as in (1.7) for all times 푡 ≥ 0. Inserting this solution in the first equation one obtains (1.6). Thus, the results for (1.6) can be applied immediately to (1.10).

The rest of the paper is organized as follows. In section 2 we study the scalar conservation law with discontinuous flux, and prove partially three equivalent admissibility conditions. These results are used to construct a Riemann solver, stated in Theorem 3.1 in section 3, which is proved to be the vanishing viscosity limit. Finally, in section 4 we go back to the polymer flooding models, and show that the Riemann solver proposed in [19] is the vanishing viscosity limit where viscosity is added only to the hydrodynamics. Models of polymer flooding 265

2. Equivalent admissible conditions for the trace

We seek a weak solution of the Riemann problem (1.6)–(1.8) in the sense that

∞ 0 ∞ ∞ − + ∫ ∫ [푢휙푡 + 푓(푎 , 푢)휙푥] 푑푥 푑푡 + ∫ ∫ [푢휙푡 + 푓(푎 , 푢)휙푥] 푑푥 푑푡 = 0 0 −∞ 0 0 for all smooth test function 휙(푡, 푥) with compact support. Here the mapping 푢 ↦ 푓 is continuous. For notational convenience, we denote the functions

푓−(푢) ≐ 푓 (푎−, 푢) , 푓+(푢) ≐ 푓 (푎+, 푢) .

It is well-known that solutions to such Riemann problems are self-similar, consisting of left-going waves, a stationary discontinuity at 푥 = 0, and some right-going waves. We denote the solution by 푢(푡, 푥) = 푈(푥/푡), and let

푢− ≐ 푈(0−), 푢+ ≐ 푈(0+) (2.1) denote the left and right state of the stationary jump. The key step in the construction of the solution lies in the selection of the values 푢−, 푢+. Once they are selected, one can solve two Riemann problems for two scalar conservation laws and obtain the left-going and right-going waves. Therefore, the entropy weak solution of the Riemann problem

퐿 − 푢 if 푥 < 0, 푢푡 + 푓 (푢)푥 = 0, 푢(0, 푥) = { (2.2) 푢− if 푥 > 0, must generate only waves with speed ≤ 0. We denote by 푊 −(푢퐿) the set of suitable 푢− values. At the same time, the entropy weak solution of the Riemann problem

+ + 푢 if 푥 < 0, 푢푡 + 푓 (푢)푥 = 0, 푢(0, 푥) = { (2.3) 푢푅 if 푥 > 0, must contain only waves of speed ≥ 0. We denote by 푊 +(푢푅) the set of suitable 푢+ values. The Rankine–Hugoniot jump condition for the stationary jump at 푥 = 0 gives

푓−(푢−) = 푓+(푢+). (2.4)

We conclude that the possible candidates for 푢−, 푢+ must satisfy

푢− ∈ 푊 −(푢퐿), 푢+ ∈ 푊 +(푢푅), 푓−(푢−) = 푓+(푢+). (2.5) 266 G. Guerra and W. Shen

In general, conditions in (2.5) yield multiple (or even infinitely many) choices for the trace 푢−, 푢+. Additional entropy conditions are needed to single out a unique solution. We now seek conditions on traces 푢−, 푢+ such that the piecewise constant functions 푢−, 푥 < 0, 푢(푡, 푥) = ̂푢(푥) ≐ { (2.6) 푢+, 푥 > 0, can be obtained as the limit of a sequence of viscous approximations 푢휀 of

푢 + 푓(푎(푥), 푢) = 휀푢 , 푢(0, 푥) = ̂푢(푥), (2.7) 푡 푥 푥푥 with 푎(푥) in (1.7), as the viscosity coefficient 휀 → 0+. Motivated by [9, 10], we introduce the monotone functions

max { 푓+(푤) ∶ 푤 ∈ [푢+, 푢] }, if 푢 ≥ 푢+, 퐺♯ (푢; 푢+) ≐ { (2.8) min { 푓+(푤) ∶ 푤 ∈ [푢, 푢+] }, if 푢 ≤ 푢+, and min { 푓−(푤) ∶ 푤 ∈ [푢−, 푢] }, if 푢 ≥ 푢−, 퐺♭(푢; 푢−) ≐ { (2.9) max { 푓−(푤) ∶ 푤 ∈ [푢, 푢−] }, if 푢 ≤ 푢−.

Here, 푢 ↦ 퐺♯ (푢; 푢+) is non-decreasing, and 푢 ↦ 퐺♭(푢; 푢−) is non-increasing. See Figure 1 for an illustration. To ensure the solvability of the Riemann problem with Riemann data (푢퐿, 푢푅), we assume that the range of the two functions 퐺♭( ⋅ ; 푢퐿) and 퐺♯ ( ⋅ ; 푢푅) have non- empty intersection. To be precise, we assume that, for the given data 푢퐿, 푢푅, there exists some ̃푢∗ such that

퐺♭ ( ̃푢∗; 푢퐿) = 퐺♯ ( ̃푢∗; 푢푅) . (2.10)

Note that, although the point ̃푢∗ in (2.10) might not be unique, the common value of the two fluxes 퐺♭ and 퐺♯ is always uniquely determined, thanks to the monotonicity properties of the functions 퐺♭ ( ⋅ ; 푢−) and 퐺♯ ( ⋅ ; 푢+).

Next Theorem states three equivalent admissible conditions for the jump at 푥 = 0.

Theorem 2.1 (Equivalent admissibility conditions). Given (푢−, 푢+), let ̂푢 be the jump function in (2.6), and let 푎(푥) be the jump function in (1.7). The following three conditions are equivalent. Models of polymer flooding 267

푓− 푓−

퐺♭ 퐺♭ 푢q − 푢−q

푓+ 푓+

푢q + 푢+q

퐺♯ 퐺♯

Figure 1. Illustrations of the functions 푢 ↦ 퐺♯(푢; 푢+) and 푢 ↦ 퐺♭(푢; 푢−) for several cases of 푢+ and 푢−.

(I) There exists a family of monotone viscous solutions 푢휀(푡, 푥) of (2.7) such that

휀 lim ‖푢 (푡, ⋅ ) − ̂푢( ⋅ )‖ 1 = 0, (2.11) 휀→0+ 퐋 uniformly on every bounded time interval [0, 푇]. (II) The Rankine–Hugoniot condition (2.4) holds, i.e., 푓−(푢−) = 푓+(푢+) ≐ 푓̄ (2.12) together with the following generalized Oleinik-type conditions [16]: (i) If 푢− < 푢+, then there exists an intermediate state 푢∗ ∈ [푢−, 푢+] such that 푓−(푢) ≥ 푓̄ for 푢 ∈ [푢−, 푢∗] , { (2.13) 푓+(푢) ≥ 푓̄ for 푢 ∈ [푢∗, 푢+] . (ii) If 푢− > 푢+, then there exists an intermediate state 푢∗ ∈ [푢+, 푢−] such that 푓+(푢) ≤ 푓̄ for 푢 ∈ [푢+, 푢∗] , { (2.14) 푓−(푢) ≤ 푓̄ for 푢 ∈ [푢∗, 푢−] . (III) There exists a state ̃푢∗, between 푢− and 푢+, such that

푓̄ = 푓−(푢−) = 퐺♭ ( ̃푢∗; 푢−) = 퐺♯ ( ̃푢∗; 푢+) = 푓+ (푢+) . (2.15) 268 G. Guerra and W. Shen

Remark 1. Condition (II) is useful to check whether a path between 푢−, 푢+ is admissible, by only using the information of 푓−, 푓+ on the interval between 푢− and 푢+. The behavior of the flux functions outside the interval between 푢− and 푢+ is not important for the admissible condition. See Figure 2 for examples of admissible and non-admissible paths.

푓+ + 푓− 푓− 푓 푓−

푓̄ 푓̄ 푓̄ 푓+

푢− 푢+ = 푢∗ 푢− 푢∗ 푢+ 푢− 푢+

Figure 2. Left and center: two cases where the jump 푢−, 푢+ is admissible. Right: the jump 푢−, 푢+ is not admissible.

Proof. We will first prove the implication (II) ⟹ (I) and the equivalence (II) ⟺ (III). We leave the last implication (I) ⟹ (III) after the proof of Theorem 3.1. The proof will take several steps. 1. We first prove the implication (II) ⟹ (I). To fix the ideas, assume that 푢− < 푢+, while the other case being entirely similar. We have three cases: (a) We can choose 푢∗ such that 푢− < 푢∗ < 푢+. In this case we must have both (푓+)′ (푢+) ≤ 0 and (푓−)′ (푢−) ≥ 0. See for example Figure 2 center plot. (b) We can only choose 푢∗ = 푢−. In this case we must have (푓+)′ (푢+) ≤ 0. (c) We can only choose 푢∗ = 푢+. In this case we must have (푓−)′(푢−) ≥ 0. See for example Figure 2 left plot. In this step we only deal with the easier case where all the inequality signs are strict in (2.13)–(2.14). To be precise, we make the following additional assumptions. Case (a): If 푢− < 푢∗ < 푢+, then we further assume that 푓−(푢) > 푓̄ for 푢 ∈ ]푢−, 푢∗], (푓−)′(푢−) > 0, { and { (2.16) 푓+(푢) > 푓̄ for 푢 ∈ [푢∗, 푢+[, (푓+)′(푢+) < 0. Case (b): If 푢∗ = 푢− only, then we also assume 푓+(푢) > 푓̄ for 푢 ∈ [푢−, 푢+[, and (푓+)′(푢+) < 0. (2.17) Models of polymer flooding 269

Case (c): If 푢∗ = 푢+ only, then we assume in addition

푓−(푢) > 푓̄ for 푢 ∈ ]푢−, 푢+], and (푓−)′(푢−) > 0. (2.18)

Under these stricter assumptions, we now show that there exists a family of traveling wave solutions such that condition (I) holds. Let 휀 = 1, and let 푈1(푥) be a stationary traveling wave profile of (2.7) with unit viscosity, with the boundary conditions

1 − 1 + 1 1 lim 푈 (푥) = 푢 , lim 푈 (푥) = 푢 , lim 푈푥 (푥) = lim 푈푥 (푥) = 0. 푥→−∞ 푥→+∞ 푥→−∞ 푥→+∞

If it exists, the viscous traveling wave 푈1(푥) must satisfy the ODE:

푓(푎(푥), 푈1) = 푈1 . 푥 푥푥 Integrating it once in 푥, and using the boundary condition at 푥 = −∞, we get

1 1 − − 1 ̄ 푈푥 = 푓(푎(푥), 푈 ) − 푓(푎 , 푢 ) = 푓(푎(푥), 푈 ) − 푓. (2.19)

Similarly, using the boundary condition at 푥 = +∞, we get

1 1 + + 1 ̄ 푈푥 = 푓(푎(푥), 푈 ) − 푓(푎 , 푢 ) = 푓(푎(푥), 푈 ) − 푓. (2.20)

Combining (2.19)–(2.20), we consider the following initial value problem

− 1 ̄ 1 푓 (푈 (푥)) − 푓, if 푥 < 0, 1 ∗ 푈푥 (푥) = { , 푈 (0) = 푢 . (2.21) 푓+(푈1(푥)) − 푓,̄ if 푥 > 0,

1 Under the assumptions in (2.16)–(2.18), we have 푈푥 ≥ 0, so the ODE (2.21) has a unique monotone solution which satisfies

lim 푈1(푥) = 푢−, lim 푈1(푥) = 푢+. 푥→−∞ 푥→+∞ The solution is strictly increasing for case (a), strictly increasing on 푥 > 0 for case (b), and strictly increasing on 푥 < 0 for case (c). Furthermore, we have

0 +∞ ‖푈1 − ̂푢‖ = ∫ |푈1(푥) − 푢−| 푑푥 + ∫ |푈1(푥) − 푢+| 푑푥 < ∞. (2.22) ‖ ‖퐋1(ℝ) | | | | −∞ 0 Here, the differences |푈(푥) − 푢±| are integrable thanks to the stricter assumptions (2.16)–(2.18) which ensure that the limits as 푥 → ±∞ in (2.22) are approached at least at an exponential rate. 270 G. Guerra and W. Shen

We observe that the functions 푢(푡, 푥) = 푈1(푥) provide a traveling wave solution to the viscous system (2.7) with 휀 = 1. In turn, for every 휀 > 0, the rescaled function

푢휀(푡, 푥) ≐ 푈1 (푥/휀) (2.23) gives a traveling wave solution to (2.7). The variable rescaling implies

‖푢휀(푡, ⋅ ) − ̂푢( ⋅ )‖ = 휀 ‖푈1 − ̂푢‖ , 퐋1(ℝ) ‖ ‖퐋1(ℝ)

휀 thus the norm ‖푢 (푡, ⋅ ) − ̂푢( ⋅ )‖퐋1(ℝ) approaches 0 as 휀 → 0. This proves the implication (II) ⟹ (I), under the stricter assumptions (2.16)–(2.18). 2. However, if the stricter assumptions (2.16)–(2.18) are removed, viscous traveling wave profiles might not exist or converge to the shockin 퐋1. For a counterexample, consider

푓−(푢) = 푢2 if 푥 ≤ 0, 푢푡 + 푓(푥, 푢)푥 = 휀푢푥푥, with 푓(푥, 푢) = { 푓+(푢) = 푢2 − 1 if 푥 > 0.

The function (1 − 푥/휀)−1 if 푥 ≤ 0, 푈휀(푡, 푥) = { 1 if 푥 > 0, is a stationary traveling wave which converges pointwise to the stationary shock

0 if 푥 < 0, 푈(푥) = { 1 if 푥 ≥ 0,

휀 but ‖푈 (푡, ⋅ ) − 푈( ⋅ )‖퐋1 = ∞ for every 휀 > 0. 3. By slightly modifying the construction in step 1, we now show that (2.11) remains valid even without the stricter assumptions (2.16)–(2.18). We discuss the three cases separately. Case (a), with 푢− < 푢∗ < 푢+. For any 훿 > 0, consider the modified flux function

+ − 푓훿(푎, 푢) ≐ 푓(푎, 푢) + 훿(푢 − 푢)(푢 − 푢 ), (2.24) so

− − + − 푓훿 (푢) ≐ 푓 (푢) + 훿(푢 − 푢)(푢 − 푢 ), + + + − 푓훿 (푢) ≐ 푓 (푢) + 훿(푢 − 푢)(푢 − 푢 ). Models of polymer flooding 271

− − + + ̄ − + Note that 푓훿 (푢 ) = 푓훿 (푢 ) = 푓. Moreover, the functions 푓훿 (푢) and 푓훿 (푢) satisfy the stricter inequalities in (2.16). Hence the ODE 푓−(푈(푥)) − 푓,̄ if 푥 < 0, 푈 (푥) = { 훿 푈(0) = 푢∗ (2.25) 푥 + ̄ 푓훿 (푈(푥)) − 푓, if 푥 > 0, has a unique solution, denoted by 푈훿( ⋅ ), which is strictly increasing and satisfies − + lim 푈훿(푥) = 푢 , lim 푈훿(푥) = 푢 , (2.26) 푥→−∞ 푥→+∞ 0 +∞ − + ‖푈훿 − ̂푢‖퐋1(ℝ) = ∫ |푈훿(푥) − 푢 | 푑휉 + ∫ |푈훿(푥) − 푢 | 푑푥 < ∞. (2.27) −∞ 0 We now have that, for every 훿, 휀 > 0, the function 휀,훿 푢 (푡, 푥) ≐ 푈훿(푥/휀) (2.28) provides a traveling profile solution to the Cauchy problem 푢 + (푓 (푎, 푢)) = 휀푢 , 푢(0, 푥) = 푈 (푥/휀). (2.29) 푡 훿 푥 푥푥 훿 Next, we observe that for every 휀 > 0 the evolution equation

푢푡 + 푓(푎, 푢)푥 = 휀푢푥푥 (2.30) 1 휀 휀 generates a contractive semigroup w.r.t. the 퐋 distance. Denote by 푡 ↦ 푢 (푡) = 풮푡 ̄푢 the solution to (2.30) with initial data 푢(0) = ̄푢. If 푡 ↦ 푤(푡) is any approximate solution, with the same initial data 푤(0) = ̄푢, then for every 휏 > 0 we have the error estimate 휏 휀 1 휀 ‖푤(휏) − 푢 (휏)‖ 1 ≤ ∫ ( lim ‖푤(푡 + ℎ) − 풮ℎ푤(푡)‖) 푑푡. (2.31) 퐋 (ℝ) ℎ→0+ ℎ 0 Regarding 푤(푡, 푥) ≐ 푢휀,훿(푡, 푥) as an approximation of (2.30), defining 푢휀(푡, 푥) as the solution to (2.30) with initial data 푢휀(0, 푥) = 푢휀,훿(0, 푥), the formula (2.31) leads to the following error estimate

∫|푢휀(휏, 푥) − 푢휀,훿(휏, 푥)| 푑푥

휏 | | ≤ ∫ ∫|[푓 (푎, 푢휀,훿(푡, 푥)) − 푓 (푎, 푢휀,훿(푡, 푥)) | 푑푥 푑푡 | 훿 푥| 0 휏 | | = ∫ ∫ 훿|{(푢+ − 푢휀,훿(푡, 푥))(푢휀,훿(푡, 푥) − 푢−)} | 푑푥 푑푡 | 푥| 0 휏 + − | 휀,훿 | ≤ 2훿 (푢 − 푢 ) ∫ ∫|푢푥 (푡, 푥)| 푑푥 푑푡 0 = 2훿 (푢+ − 푢−) ⋅ 휏 (푢+ − 푢−) . (2.32) 272 G. Guerra and W. Shen

Combining (2.32) with (2.27), we have, for every 휏 > 0,

∫|푢휀(휏, 푥) − ̂푢(푥)| 푑푥

≤ ∫|푢휀(휏, 푥) − 푢휀,훿(휏, 푥)| 푑푥 + ∫|푢휀,훿(휏, 푥) − ̂푢(푥)| 푑푥

+ − 2 ≤ 2휏훿 (푢 − 푢 ) + 휀 ‖푈훿 − ̂푢‖퐋1(ℝ) . (2.33) Finally, we choose 훿 = 훿(휀) such that

lim 훿(휀) = 0, lim 휀 ‖푈훿(휀) − ̂푢‖ = 0. (2.34) 휀→0 휀→0 퐋1(ℝ)

휀 휀 This yields a family of solutions 푢 (푡, ⋅ ) = 풮푡푈훿(휀) of (2.7), for which (2.11) holds. Case (b), where we must choose 푢∗ = 푢−. The approach here is very similar to that of Case (a). For any 훿 > 0, we define the modified flux functions 푎 − 푎− 푓 (푎, 푢) ≐ 푓(푎, 푢) + 훿 (푢+ − 푢) , 훿 푎+ − 푎− such that − − + + + 푓훿 (푢) ≐ 푓 (푢), 푓훿 (푢) ≐ 푓 (푢) + 훿 (푢 − 푢) .

Let 푈훿 be the solution to the ODE

푈′(푥) = 푈(푥) ⋅ [푓+(푈(푥)) − 푓̄ , (푥 > 0), { 훿 푈(푥) = 푢−, (푥 ≤ 0).

With this modified flux, the stricter assumptions in (2.17) hold, and we also have for every 휏 > 0

∫|푢휀(휏, 푥) − 푢휀,훿(휏, 푥)|푑푥

휏 +∞ | | ≤ 훿 ∫ ∫ |[(푢+ − 푢휀,훿(푡, 푦)) | 푑푥 푑푡 ≤ 훿 휏 (푢+ − 푢−) . | 푥| 0 0 This leads to the estimate

휀 + − ∫|푢 (휏, 푥) − ̂푢(푥)|푑푥 ≤ 훿 휏 (푢 − 푢 ) + 휀 ‖푈훿 − ̂푢‖퐋1(ℝ) .

The rest follows. Case (c), where we must choose 푢∗ = 푢+, is completely similar to Case (b). Models of polymer flooding 273

4. The equivalence (III) ⟺ (II) is straightforward. Indeed, assume 푢− < 푢+, then

퐺♭(푢∗; 푢−) = 푓−(푢−) iff 푓−(푤) ≥ 푓−(푢−) ∀푤 ∈ [푢−, 푢∗], 퐺♯(푢∗; 푢+) = 푓+(푢+) iff 푓+(푤) ≥ 푓+(푢+) ∀푤 ∈ [푢∗, 푢+].

Hence (III) ⟺ (II). A completely similar argument shows the equivalency for the case 푢− > 푢+. This proves most of Theorem 2.1, leaving only the implication (I) ⇒ (III), which will be established after proving Theorem 3.1.

Remark 2. Condition (III) can be used to construct the unique solution for the Riemann problem. After having constructed the functions 퐺♯, 퐺♭, one can take the unique minimum path to determine 푢−, 푢+. See Figure 3 for an example. See also [10].

퐺♭(푢; 푢퐿)

퐺♯(푢; 푢푅)

푢퐿 푢푅 푢− 푢+ 푢

Figure 3. A Riemann Solver for (1.6)–(1.8). For 푡 > 0, the traces 푢− = 푢(푡, 0−) and 푢+ = 푢(푡, 0+) are those for which the horizontal distance between the two curves is minimized.

3. The Riemann Solver by Vanishing Viscosity

The partial result (III) ⟺ (II) ⇒ (I) in Theorem 2.1 motivates a Riemann solver for (1.6)–(1.7) with Riemann data (1.8), as commented in Remark 2. This Riemann solver, described in the next Theorem, generates solutions which are the vanishing viscosity limit as 휀 → 0 of

푢퐿, if 푥 < 0, 푢푡 + 푓(푎(푥), 푢) = 휀푢푥푥, 푢(0, 푥) = { (3.1) 푥 푢푅, if 푥 > 0,

Theorem 3.1 (Vanishing viscosity solution to the Riemann problem). Given a left and right states (푢퐿, 푢푅), let 퐺♯(푢; 푢푅), 퐺♭(푢; 푢퐿) be defined as in (2.8)–(2.9), and let 274 G. Guerra and W. Shen

푓̄be the unique value such that 푓̄ = 퐺♭ (푢∗; 푢퐿) = 퐺♯ (푢∗; 푢푅) (3.2) for some 푢∗. We define the trace 푢−, 푢+ of 푢 along 푥 = 0 as follows:

푢− ≐ argmin { |푢 − 푢퐿| ∶ 푓−(푢) = 푓̄}, (3.3) 푢+ ≐ argmin { |푢 − 푢푅| ∶ 푓+(푢) = 푓̄}. (3.4) We call the path (푢−, 푢+) the minimum path connecting fluxes 푓−, 푓+ with data (푢퐿, 푢푅). Then the vanishing viscosity solution 푢(푡, 푥) of (1.6)–(1.8) is obtained by piecing together the solutions to

퐿 − 푢 , if 푥 < 0, 푢푡 + 푓 (푢)푥 = 0, 푢(0, 푥) = { (3.5) 푢−, if 푥 > 0, for 푥 < 0, and the solution to

+ + 푢 , if 푥 < 0, 푢푡 + 푓 (푢)푥 = 0, 푢(0, 푥) = { (3.6) 푢푅, if 푥 > 0, for 푥 > 0. In particular, for every 푡 > 0 we have lim 푢(푡, 푥) = 푢−, lim 푢(푡, 푥) = 푢+, (3.7) 푥 → 0− 푥 → 0+ and 휀 lim ‖푢 (푡, ⋅ ) − 푢(푡, ⋅ )‖ 1 = 0 (3.8) 휀→0 퐋 (ℝ) uniformly on every bounded time interval [0, 푇], where 푢휀 is a solution to the viscous equation (3.1). Proof. The proof takes several steps. 1. By the definitions of 푢− and 푢+ in (3.3)–(3.4), the following hold. (i) The entropy-admissible solution 푣(푡, 푥) ≐ 푈−(푥/푡) to the Riemann problem (3.5) contains only waves of speed smaller than 0. Indeed lim 푈−(푥) = 푢−, and 푈−(푥) = 푢− ∀푥 ≥ 0. (3.9) 푥 → 0−

(ii) The entropy-admissible solution 푤(푡, 푥) ≐ 푈+(푥/푡) to the Riemann problem (3.6) contains only waves of speed larger than 0. Indeed lim 푈+(푥) = 푢+, and 푈+(푥) = 푢+ ∀푥 ≤ 0. (3.10) 푥 → 0+ Models of polymer flooding 275

(iii) The left and right states of 푢 where 푎(푥) has a jump, denoted as 푢− and 푢+, satisfy the condition (III) in Theorem 2.1. Therefore there exist three families of viscous approximations 푣휀, 푤휀, 푧휀, satisfying

휀 − 휀 휀 푣푡 + 푓 (푣 )푥 = 휀푣푥푥, 휀 + 휀 휀 푤푡 + 푓 (푤 )푥 = 휀푤푥푥, (3.11) 푧휀 + 푓(푎(푥), 푧휀) = 휀푧휀 , 푡 푥 푥푥 where 푎(푥) is given in (1.7). Moreover, as 휀 → 0 one has

휀 휀 ‖푣 (푡, ⋅ ) − 푣(푡, ⋅ )‖퐋1(ℝ) → 0, ‖푤 (푡, ⋅ ) − 푤(푡, ⋅ )‖퐋1(ℝ) → 0, (3.12) and 0 +∞ ∫ |푧휀(푡, 푥) − 푢−| 푑푥 + ∫ |푧휀(푡, 푥) − 푢+| 푑푥 → 0 (3.13) −∞ 0 uniformly as 푡 ranges on bounded intervals. The functions 푣휀 and 푤휀 can be uniquely determined by imposing the initial data

푢퐿, (푥 < 0), 푢+, (푥 < 0), 푣휀(0, 푥) = { 푤휀(0, 푥) = { (3.14) 푢−, (푥 > 0), 푢푅, (푥 > 0), while 푧휀 is obtained by the construction in Step 3 of the proof for Theorem 2.1. We note that all function 푣휀, 푤휀 and 푧휀 are monotone, either non-increasing or non-decreasing w.r.t. the variable 푥. Thanks to this fact, we conclude that the 퐋1 convergence in (3.12) implies pointwise convergence, at every point (푡, 푥) where 푣, 푤 are continuous. 2. For any given 푇, 훿 > 0, define the domains

1/4 1/2 푟푙Ω훿 ≐ { (푡, 푥) ∶ 푡 ∈ [훿 , 푇 + 1], |푥| ≤ 훿 } , (3.15) ′ 1/4 1/2 Ω훿 ≐ { (푡, 푥) ∶ 푡 ∈ [훿 , 푇 + 1], |푥| ≤ 2훿 }. We recall that, by (3.9)–(3.10), the functions 푣 and 푤 are continuous at all points (푡, 0) with 푡 > 0. For any 훿 > 0, we can thus find 휀 = 휀(훿) > 0 small enough such that

sup { |푧휀(푡, 푥) − 푢−| ∶ 푡 ∈ [0, 푇] , 푥 ≤ −훿 } ≤ 훿 , (3.18) sup { |푧휀(푡, 푥) − 푢+| ∶ 푡 ∈ [0, 푇] , 푥 ≥ 훿 } ≤ 훿 . (3.19)

Without loss of generality we can assume that the map 훿 ↦ 휀(훿) is continuous, strictly increasing, and satisfies 휀(훿) ∈ 0, 훿2 . Its inverse 휀 ↦ 훿(휀) is thus well defined and satisfies

훿(휀) ≥ √휀 , lim 훿(휀) = 0 . (3.20) 휀→0

3. To construct a family of vanishing viscosity solutions to the Riemann problem (3.1), we need to patch together the three solutions 푣, 푤, 푧 on different domains. To remove the discontinuity of 푣휀, 푤휀 at the point (0, 0), we perform a time shift and define ̃푣휀(푡, 푥) = 푣휀(푡 + 훿(휀), 푥), { (3.21) ˜푤휀(푡, 푥) = 푤휀(푡 + 훿(휀), 푥). Next, let 휑 ∶ ℝ ↦ [0, 1] be a smooth, non-decreasing function such that

0 if 푦 ≤ 0, 휑(푦) = { (3.22) 1 if 푦 ≥ 1.

For any 휀 > 0, set 훿 = 훿(휀) and define the interpolated function

√훿 + 푥 √훿 − 푥 ̃푢휀,훿(푡, 푥) ≐ [휑( ) + 휑( ) − 1]푧휀(푡, 푥) 훿 훿 √훿 + 푥 √훿 − 푥 + [1 − 휑( )] ̃푣휀(푡, 푥) + [1 − 휑( )]˜푤휀(푡, 푥) . (3.23) 훿 훿

For 훿 < 1, the interpolated function satisfies

휀 √ ⎧ ̃푣 (푡, 푥), if 푥 ≤ − 훿, ̃푢휀,훿(휀)(푡, 푥) = 푧휀(푡, 푥), if − √훿 + 훿 ≤ 푥 ≤ √훿 − 훿, ⎨ 휀 ⎩ ˜푤 (푡, 푥), if 푥 ≥ √훿.

In the region −√훿 ≤ 푥 ≤ −√훿 + 훿, it interpolates between ̃푣휀 and 푧휀, while in the region √훿 − 훿 ≤ 푥 ≤ √훿, it interpolates between 푧휀 and ˜푤휀. We call 푢 = 푢(푡, 푥) the solution to the Riemann problem (1.6)–(1.8) obtained by the Riemann solver described in this Theorem, i.e., by piecing together the two Models of polymer flooding 277 solutions at (3.5)–(3.6). Then, for any choice of 훿 = 훿(휀) such that lim휀→0+ 훿(휀) = 0, we have with 푢휀,훿(휀) = ̃푢휀,훿(휀)

lim ‖푢휀,훿(휀)(푡, ⋅ ) − 푢(푡, ⋅ )‖ = 0 (3.24) 휀→0+ 퐋1(ℝ) uniformly for 푡 in bounded set.

4. Since each 푢휀,훿(휀) is only an approximate solution to the viscous Cauchy problem

푢 + 푓(푎(푥), 푢) = 휀푢 , with 푢(0, 푥) = 푢휀,훿(휀)(0, 푥), (3.25) 푡 푥 푥푥 it remains to prove that the exact solution 푢휀 of (3.25) satisfies

lim ‖푢휀(푡, ⋅ ) − 푢휀,훿(휀)(푡, ⋅ )‖ = 0 (3.26) 휀→0+ 퐋1(ℝ) uniformly for 푡 in bounded sets. Indeed, observe that, for any 휏 > 0 the error estimate (2.31) yields

푥 ∈ [−√훿, −√훿 + 훿] ∪ [√훿 − 훿, √훿] .

Fix 푡 > 0, we consider the second interval

퐼훿 ≐ [√훿 − 훿, √훿] where 푢휀,훿(휀) interpolates between 푧휀 and ˜푤휀. We define

√훿 − 푥 휑훿(푥) ≐ 휑( ). 훿

Recalling (3.23), for 푥 ∈ 퐼훿 we have

푢휀,훿(휀)(푡, 푥) = 휑훿(푥)푧휀(푡, 푥) + (1 − 휑훿(푥)) ˜푤휀(푡, 푥) . 278 G. Guerra and W. Shen

We compute

푢휀,훿(휀) + 푓(푎+, 푢휀,훿(휀)) − 휀푢휀,훿(휀) 푡 푥 푥푥 훿 휀 훿 휀 = [휑 푧푡 + (1 − 휑 )˜푤푡 + (푓+)′(푢휀,훿(휀)) ⋅ [(휑훿푧휀) + ((1 − 휑훿)˜푤휀) 푥 푥 − 휀[(휑훿푧휀) + ((1 − 휑훿)˜푤휀) 푥푥 푥푥

≐ 퐴1 + 퐴2 + 퐴3, (3.28) where

훿 휀 + ′ 휀,훿(휀) 훿 휀 훿 휀 퐴1 = 휑 푧푡 + (푓 ) (푢 ) ⋅ 휑 푧푥 − 휀휑 푧푥푥 ,

훿 휀 + ′ 휀,훿(휀) 훿 휀 훿 휀 퐴2 = (1 − 휑 )˜푤푡 + (푓 ) (푢 ) ⋅ (1 − 휑 )˜푤푥 − 휀(1 − 휑 )˜푤푥푥 ,

+ ′ 휀,훿(휀) 훿 휀 휀 훿 휀 휀 훿 휀 휀 퐴3 = (푓 ) (푢 )휑푥(푧 − ˜푤 ) − 휀휑푥푥(푧 − ˜푤 ) − 2휀휑푥(푧푥 − ˜푤푥)

We have the estimates

훿 + ′ 휀,훿(휀) + ′ 휀 휀 |퐴1| = 휑 |(푓 ) (푢 ) − (푓 ) (푧 )||푧푥| 훿 훿 + ″ 휀 휀 휀 ≤ 휑 (1 − 휑 ) ⋅ ‖(푓 ) ‖ ∞ ⋅ ‖푧 − ˜푤 ‖ ∞ ⋅ |푧 | 퐋 퐋 (퐼훿) 푥 + ″ 휀 휀 휀 ≤ ‖(푓 ) ‖ ∞ ⋅ ‖푧 − ˜푤 ‖ ∞ ⋅ |푧 |, (3.29) 퐋 퐋 (퐼훿) 푥 and

훿 + ′ 휀,훿(휀) + ′ 휀 휀 |퐴2| = (1 − 휑 )|(푓 ) (푢 ) − (푓 ) (˜푤 )| ⋅ |˜푤푥| 훿 훿 + ″ 휀 휀 휀 ≤ 휑 (1 − 휑 ) ⋅ ‖(푓 ) ‖ ∞ ⋅ ‖푧 − ˜푤 ‖ ∞ ⋅ |˜푤 | 퐋 퐋 (퐼훿) 푥 + ″ 휀 휀 휀 ≤ ‖(푓 ) ‖ ∞ ⋅ ‖푧 − ˜푤 ‖ ∞ ⋅ |˜푤 |. (3.30) 퐋 퐋 (퐼훿) 푥

The last term can be estimated as

1 ′ + ′ 휀 휀 ‖퐴 ‖ ≤ ‖휑 ‖ ∞ ⋅ ‖(푓 ) ‖ ∞ ⋅ ‖푧 − ˜푤 ‖ ∞ 3 훿 퐋 퐋 퐋 (퐼훿) 휀 ″ 휀 휀 2휀 ′ 휀 휀 + ‖휑 ‖ ∞ ⋅ ‖푧 − ˜푤 ‖ ∞ + ‖휑 ‖ ∞{|푧 | + |˜푤 |}. (3.31) 훿2 퐋 퐋 (퐼훿) 훿 퐋 푥 푥 Models of polymer flooding 279

Combining the estimates (3.29)–(3.31) we obtain

휀,훿(휀) + 휀,훿(휀) 휀,훿(휀) ∫ |푢푡 + 푓(푎 , 푢 )푥 − 휀푢푥푥 | 푑푥 퐼훿 + ″ 휀 휀 휀 휀 ≤ ‖(푓 ) ‖ ∞ ⋅ ‖푧 − ˜푤 ‖ ∞ ‖푧 ‖ 1 + ‖˜푤 ‖ 1 퐋 퐋 (퐼훿)( 푥 퐋 (퐼훿) 푥 퐋 (퐼훿)) ′ + ′ 휀 ″ 휀 휀 + (‖휑 ‖ ∞ ⋅ ‖(푓 )‖ ∞ + ‖휑 ‖ ∞) ⋅ ‖푧 − ˜푤 ‖ ∞ 퐋 퐋 훿 퐋 퐋 (퐼훿) 2휀 ′ 휀 휀 + ‖휑 ‖ ∞ ⋅ (‖푧 ‖ 1 + ‖˜푤 ‖ 1 ) . (3.32) 훿 퐋 푥 퐋 (퐼훿) 푥 퐋 (퐼훿) Since ˜푤휀 and 푧휀 are all monotone functions of 푥, for every fixed time 푡 ≥ 0 their total variation is computed simply by

휀 푅 + 휀 + − ‖˜푤푥‖퐋1(ℝ) = |푢 − 푢 |, ‖푧푥‖퐋1(ℝ) = |푢 − 푢 |. Thanks to the estimates (3.16)–(3.19) we have

휀 휀 ‖푧 (푡, ⋅ ) − ˜푤 (푡, ⋅ )‖ ∞ → 0 퐋 (퐼훿(휀)) as 휀 → 0, uniformly for 푡 ∈ [0, 푇]. Moreover, (3.20) implies 휀/훿(휀) → 0 as 휀 → 0. We thus conclude that the right hand side of (3.32) approaches zero as 휀 → 0, uniformly for 푡 ∈ [0, 푇]. Of course, an entirely similar estimate is valid for the integral over the interval

[−√훿, −√훿 + 훿] .

From (3.27) we thus conclude

휀 휀 휀,훿(휀) lim ‖푢 (휏) − 푢(휏)‖ 1 ≤ lim ‖푢 (휏) − ̃푢 (휏)‖ 휀→0 퐋 (ℝ) 휀→0 퐋1(ℝ) + lim ‖ ̃푢휀,훿(휀)(휏) − 푢(휏)‖ = 0, 휀→0 퐋1(ℝ) proving that the solution 푢 of the Riemann problem (1.6)–(1.8) described in this Theorem is indeed a limit of vanishing viscosity approximations of the viscous model (3.1). This completes the proof of Theorem 3.1.

We now go back to Theorem 2.1, and complete the last part of the proof.

Proof (of Theorem 2.1). We now prove the implication (I) ⟹ (III). Assuming condition (I), i.e., there exists a family of monotone viscous solution 푢휀(푡, 푥) of (2.7) such that (2.11) holds. A standard argument shows that the Rankine–Hugoniot condition (2.12) must hold. To prove (2.13)–(2.14), we argue with contradiction. Suppose that (III) fails. Then, by Theorem 3.1, we can construct a family of viscous 280 G. Guerra and W. Shen solution ̃푢휀(푡, 푥) that converges in 퐋1 to the solution ̃푢(푡, 푥) of the Riemann solver stated in Theorem 3.1. The solution ̃푢(푡, 푥) consists of a stationary jump at 푥 = 0, and at least one left-going or right-going wave. Clearly, ‖ ̂푢( ⋅ ) − ̃푢(푡, ⋅ )‖퐋1 is not 0 for 푡 > 0. Since the equation (2.7) generates a contractive semigroup, we must have for 0 < 푡 < 푇

휀 휀 lim ‖푢 (푡, ⋅ ) − ̃푢 (푡, ⋅ )‖ 1 = ‖ ̂푢( ⋅ ) − ̃푢(푡, ⋅ )‖ 1 = 0, 휀→0+ 퐋 퐋 reaching a contradiction. This completes the proof for Theorem 2.1.

4. Riemann problem for polymer flooding models

We now go back to the polymer flooding models, and consider (1.2) while treating (1.1) as a special case with 푚(푐) = constant. We consider (1.2) with the Riemann data 푠퐿, (푥 < 0) 푐−, (푥 < 0) 푠(0, 푥) = { , 푐(0, 푥) = { , 푐− < 푐+. (4.1) 푠푅, (푥 > 0) 푐+, (푥 > 0)

Consider the Lagrangian coordinate (휓, 휙), defined in (1.3), and the associated polymer flooding system (1.5). In this coordinate, since 푚″ < 0, the solution for 푐 contains a single admissible jump, traveling with speed 푚(푐−) − 푚(푐+) 휎 = . 푐− − 푐+ We consider the model where viscosity is added only for the hydrodynamics 휕 1 휕 푠 휕2 1 ⎧ ( ) − ( ) = 휀 ( ) , ⎪휕휓 푔(푠, 푐) 휕휙 푔(푠, 푐) 휕휙2 푔(푠, 푐) (4.2) ⎨ 휕푐 휕 ⎪ + 푚(푐) = 0. ⎩ 휕휓 휕휙 With a simple coordinate shift 휙̃ ≐ 휙 − 휎휓, the 푐-jump will be stationary at 휙̃ = 0, and the first equation in (4.2) becomes 휕 1 휕 푠 + 휎 휕2 1 ( ) − ( ) = 휀 ( ) . (4.3) 휕휓 푔(푠, 푐) 휕휙̃ 푔(푠, 푐) 휕휙2̃ 푔(푠, 푐) We now assume 푠퐿 > 0, 푠푅 > 0. For notational convenience, we denote the conserved quantity and the flux in (4.3) as 1 푠 + 휎 푊(푠, 푐) = , 퐻∶ 푊 ↦ − . 푔(푠, 푐) 푔(푠, 푐) Models of polymer flooding 281

In particular, the fluxes 퐻 at the left and right of the 푐-jump are denoted as 푠 + 휎 푠 + 휎 퐻− ∶ 푊(푠, 푐−) ↦ − , 퐻+ ∶ 푊(푠, 푐+) ↦ − . 푔(푠, 푐−) 푔(푠, 푐+) Since 푠 ↦ 푔(푠, 푐) is strictly increasing for any given 푐, the mappings 퐻−, 퐻+ are well-defined. We can now rewrite (4.3) as − ̃ ̃ ̃ 퐻 (푊), if 휙 < 0, 푊휓 + 퐻(푊, 휙) ̃ = 휀푊 ̃ ̃, 퐻(푊, 휙) = { (4.4) 휙 휙휙 퐻+(푊), if 휙̃ > 0, which is associated with the Riemann data 푊 퐿 ≐ 푊(푠퐿, 푐−), if 휙̃ < 0, 푊(0, 휙)̃ = { 푊 푅 ≐ 푊(푠푅, 푐+), if 휙̃ > 0. By Theorem 3.1, as 휀 → 0, the vanishing viscosity solution of (4.4) can be obtained by patching up solutions of two Riemann problems, where the traces 푊 − = 푊(휓, 0−), 푊 + = 푊(휓, 0+) are determined as the minimum path connecting the fluxes 퐻−, 퐻+ with data 푊 −, 푊 +. We go back to the original system (1.2) in Eulerian coordinate. A Riemann solver was proposed in [18] for the case 푚(푐) = constant. One can easily extend this solver for the more general case with 푚″ > 0, which we present in our final Theorem. Furthermore, this Riemann solver is equivalent to the one in Lagrangian coordinate described above. Theorem 4.1. Consider the model (1.2) with Riemann data (4.1) and 푠퐿 > 0, 푠푅 > 0. Let (푠(푡, 푥), 푐(푡, 푥)) be the solution of the following Riemann solver. Define the flux functions 푔(푠, 푐) 푔−(푠) 푔+(푠) ℎ(푠, 푐) ≐ , ℎ−(푠) ≐ , ℎ+(푠) ≐ . 푠 + 휎 푠 + 휎 푠 + 휎 Let (푠−, 푠+) be the minimum path connecting the fluxes ℎ−, ℎ+ with data (푠퐿, 푠푅) as defined in Theorem 3.1, and let 푔+(푠+) 푔−(푠−) 휎 = = . 푐 푠+ + 휎 푠− + 휎

Then 푐(푡, 푥) contains a single jump traveling with speed 휎푐, and 푠(푡, 푥) is obtained by piecing together the solutions to

퐿 − 푠 , if 푥 < 0, 푠푡 + 푔 (푠)푥 = 0, 푠(0, 푥) = { 푠−, if 푥 > 0, 282 G. Guerra and W. Shen for 푥 < 휎푐푡, and the solution to

+ + 푠 , if 푥 < 0, 푠푡 + 푔 (푠)푥 = 0, 푠(0, 푥) = { 푠푅, if 푥 > 0, for 푥 > 휎푐푡. Then this Riemann solver solution is the vanishing viscosity limit of (4.2), as 휀 → 0+, where the convergence is in 퐋1-norm in 푥, and uniformly for 푡 in bounded sets. Proof. It suffices to show that condition (II) in Theorem 2.1 is equivalent in these two Riemann solvers. By the Rankine–Hugoniot condition, we have 1 퐻−(푊 −) = 퐻+(푊 +) ≐ 퐻,̄ ℎ−(푠−) = ℎ+(푠+) ≐ ℎ,̄ 퐻̄ = − . ℎ̄ Consider the case 푠− < 푠+. The above condition gives 푠− + 휎 푠+ + 휎 1 1 = ⇒ > ⇒ 푊 − > 푊 +. 푔−(푠−) 푔+(푠+) 푔−(푠−) 푔+(푠+) Then, for the system in the Eulerian coordinate, (2.13) in condition (II) holds, i.e., there exists an 푠∗ between 푠− and 푠+ such that ℎ+(푠) ≥ ℎ̄ for 푠 ∈ [푠−, 푠∗] , { (4.5) ℎ−(푠) ≥ ℎ̄ for 푠 ∈ [푠∗, 푠+] . For the system in the Lagrangian coordinate, (2.14) in condition (II) holds, i.e., there exists a 푊 ∗ between 푊 + and 푊 − such that 퐻+(푊) ≤ 퐻 for 푊 ∈ [푊 +, 푊 ∗] , { (4.6) 퐻−(푊) ≤ 퐻 for 푊 ∈ [푊 ∗, 푊 −] . Observe that 푠 ↦ 푊 is strictly decreasing for any fixed 푐. Thus, it suffices to show that the derivatives 휕ℎ 휕퐻 ( ̄푠, 푐), (푊, 푐), where 푊 = 푊( ̄푠, 푐) (4.7) 휕푠 휕푊 have opposite signs. Indeed, by the definition of 퐻, ℎ, for a given 푐, both mappings 푊 ↦ 퐻 and 푠 ↦ ℎ have a unique maximum. Let ̃푠 be the point where 푠 ↦ ℎ(푠, 푐) reaches the maximum value. Then the maximum value of the mapping 푊 ↦ 퐻(푊, 푐) is reached at 푊˜ = 푊( ̃푠, 푐). Furthermore, since 푠 ↦ 푊(푠, 푐) is strictly decreasing, the derivatives in (4.7) have opposite signs. See Figure 4 for an illustration. This completes the proof for Theorem 4.1. Models of polymer flooding 283

퐻 61 푊˜ 푊 - 6ℎ

(1 + 휎)−1 −1 − 휎

- ̃푠 1 푠

Figure 4. Plots of the mapping 푠 ↦ ℎ(푠, 푐) and 푊 ↦ 퐻(푊, 푐). The colors show the correspondence of the two graphs where 푊 = 푊(푠, 푐) for the same 푠 values.

Remark 3. The Riemann solver in Theorem 4.1, adapted to the polymer flooding model by setting 푚(푐) = constant, is used in [18] in a front tracking approximation, which generates unique entropy solutions for (1.1), even with the effect of gravitation. However, for the gravitation model of (1.1), the flux 푔 is changed into

풢(푠, 푐) = 푔(푠, 푐)(1 − 퐾푔휆(푠, 푐)), where the term 퐾푔휆(푠, 푐) represents the effect of the gravitation force, see [6, 18]. Here, 풢(푠, 푐) could be 0 for some (푠, 푐) where 푠 > 0, 푐 > 0. At that point, the Lagrangian coordinate (휓, 휙) in (1.3) is no longer valid. However, using another Lagrangian coordinate (휏, 휉) defined as (see [21])

휉푥 = 푠, 휉푡 = −풢, 휏 = 푡 the system (1.1) becomes

1 풢(푠, 푐) ( ) − ( ) = 0, 푠 휏 푠 휉

푐휏 = 0.

Theorem 4.1 can be applied here with very little modifications.

References

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[19] W. Shen, On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding. NoDEA Nonlinear Differential Equations Appl. 24 (2017), no. 4, Art. 37, 25pp. [20] B. Temple. Stability and decay in systems of conservation laws. In Proc. Nonlinear Hyperbolic Problems, St. Etienne, France, C. Carasso, P. A. Raviart, D. Serre, eds., Springer-Verlag, Berlin, New York, 1986. [21] D. Wagner. Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions. J. Differential Equations, 68 (1987), pp. 118–136.

Efficient computation of all speed flows using an entropy stable shock-capturing space-time discontinuous Galerkin method

Andreas Hiltebrand and Siddhartha Mishra

Dedicated to Helge Holden on his 60th birthday

Abstract. We present a shock-capturing space-time discontinuous Galerkin method to approximate all speed flows modeled by systems of conservation laws with multiple time scales. The method provides a very general and computationally efficient framework for approximating such systems on account of its ability to incorporate large time steps. Numer- ical examples ranging from computing the incompressible limit (robustness with respect to Mach number) of the Euler equations to accelerating convergence to steady state are presented for illustrating the method.

1. Introduction

1.1. The models. Systems of conservation laws are nonlinear systems of partial differential equations that arise in a wide variety of problems in physics and engineering. Examples include the shallow water equations of oceanography, the Euler equations of compressible fluid dynamics, the magnetohydrodynamics (MHD) equations of plasma physics, and the equations of nonlinear elasticity [5]. A generic form for a multi-dimensional system of conservation laws is

푑 ∑ 푘 퐔푡 + 퐅 (퐔)푥푘 = 0, (푥, 푡) ∈ Ω × ℝ+. (1.1) 푘=1

Here, Ω ⊂ ℝ푑 (푑 = 1, 2, 3) is a bounded spatial domain and 퐔∶ Ω → ℝ푚 is the vector of unknowns. 퐅푘 is the (smooth) flux vector in the 푘-th direction. The conservation law (1.1) is equipped with suitable initial and boundary conditions. The system (1.1) is termed hyperbolic if the flux Jacobian (along each normal direction) has real eigenvalues [5]. Hyperbolic systems are characterized by the fact that solutions of (1.1) can be expressed in terms of waves that travel at finite speeds. Furthermore, it is well known that solutions of (1.1) develop discontinuities such 288 A. Hiltebrand and S. Mishra as shock waves in finite time, even when the initial data are smooth. Hence, the solutions of (1.1) are sought as integrable functions that satisfy (1.1) in the sense of distributions [5]. These weak solutions are not necessarily unique. Admissibility criteria in the form of entropy conditions need to be imposed in order to select a unique weak solution [5]. In fact, recent numerical work [14] indicates that an even weaker notion of solutions, that of entropy measure valued solutions [9], is an appropriate framework of solutions for (1.1).

1.2. Numerical methods. A large variety of numerical methods, such as finite volume, conservative finite difference, discontinuous Galerkin finite element and spectral viscosity methods, have been developed to efficiently approximate systems of conservation laws (1.1). Finite volume methods are often the preferred discretization framework [32]. Such methods rely on evolving the cell average of the solution of (1.1) in terms of numerical fluxes that are based on the exact (approximate) solutions of Riemann problems at each interface. Higher-order spatial accuracy is obtained by employing non-oscillatory piecewise polynomial reconstruction procedures such as TVD [32], ENO [21], and WENO [35]. An alternative high-order spatial discretization is provided by the discontinuous Galerkin method [4]. Although the above mentioned methods are highly successful and widely used, rigorous stability and convergence results, particularly for multi-dimensional systems of conservation laws, are lacking. Recently developed schemes as the TeCNO schemes [13] combine arbitrary high-order of accuracy with entropy stability (the only known notion of nonlinear stability for systems of conservation laws). Fur- thermore, these schemes can be shown to converge to entropy measure valued solutions of (1.1)[14].

Time stepping. Hyperbolic conservation laws are characterized by finite speeds of propagation. Hence, and in contrast with parabolic problems, explicit time stepping methods are often employed for time integration of the high-resolution finite volume and DG schemes. A particularly attractive choice is the strong stability preserving (SSP) Runge–Kutta methods [16]. Alternatives include the ADER time stepping procedure [38]. The time step in explicit methods is related to the spatial mesh size in terms of fastest wave speed of the hyperbolic system (1.1)[32].

1.3. Multiple time scales. The 푚 × 푚 hyperbolic system (1.1) possesses 푚 different waves traveling at different speeds. In many problems of interest, these wave speeds can differ by several orders of magnitude, resulting in the presence of multiple time scales in the system. A prototypical example for such systems All speed flows 289 is the incompressible limit of the Euler equations [32]. It is well known that solutions to the Euler equations contain acoustic waves and matter waves (density fluctuations, contact discontinuities, and velocity shear waves). The Mach number characterizes the difference in wave speed between these two families of waves. The incompressible limit is determined letting the Mach number go to zero. In this regime, the small density fluctuations ensure that the acoustic waves are orders of magnitude faster than the matter waves. However, it is only the matter waves that carry significant information about the incompressible limit. The numerical resolution of the incompressible limit by standard explicit (such as the SSP-RK) time stepping methods is notoriously difficult ([32] and references therein) as the time step in such methods is dictated by the very fast moving acoustic waves. The resulting time step can be very small and the overall computational cost prohibitively expensive, particularly as the resolution of these time scales (corresponding to the fast acoustic waves) is of little interest. Another prototypical example for a system with multiple time scales is given by Radiation hydrodynamics and Radiation MHD ([15]). Again, the fastest time scale in the system is dictated by the radiation waves that travel at the speed of light. However, these waves carry little information and the main objective of any simulation is to approximate the time scales corresponding to the sonic and magneto-sonic waves, that are 4–5 orders of magnitude slower than the light waves. Multiphase flows also contain such examples of systems with multiple time scales, [10] and references therein. As a final example of systems with multiple time scales, we can also include systems of conservation laws where the steady state is the main interest of the computation, for instance in aerodynamic calculations [22]. One of the standard approaches for computing steady states is to start the computation with some initial conditions and drive the system to converge to steady state. The time evolution to steady state, if computed by an explicit method, is again enormously expensive as the time step can be very small. Furthermore, the computation of the transient is of little significance as we are interested in only the steady state. The above examples illustrate that when a system of conservation laws contains multiple time scales that differ in orders of magnitude, one is mainly interested in resolving the slow time scales. Explicit methods, that are designed to resolve the fastest time scale, are very expensive computationally when they are employed to approximate such all speed flows and an alternative framework needs to be designed.

1.4. A brief survey of existing methods to compute all speed flows. A large number of methods have been developed to deal with multiple time scales in conservation laws. An overarching feature of these methods is that almost all of them 290 A. Hiltebrand and S. Mishra are designed with a specific application in mind. In particular, efficient computation of the incompressible limit of the compressible Euler equations has received a lot of attention beginning with the implicit continuous fluid Eulerian (ICE) technique of Harlow and Amsdan [19, 20]. These methods use incompressible techniques such as staggered meshes to simulate compressible flows in the low mach number limit. Other popular methods include the splitting methods of Bijl and Wesseling [3], the multiple pressure variable methods of Munz et al [33, 34], the asymptotic preserving (AP) methods of Degond, Jin, Liu, and co-workers [8, 6, 7, 18] and references therein, semi-implicit methods of Klein and co-workers [30, 31], and the conservative pressure methods of Huel and Wesseling [39]. Similarly, a large number of semi-implicit numerical methods have been developed to deal with radiation hydrodynamics and radiation magnetohydrodynamics (see [15] for a literature survey of these methods). A literature survey for methods to compute all speed multiphase flows is given in[10]. Finally, accelerating convergence to steady state in aerodynamic simulations has been considered in the pioneering works of Jameson [27]. The wide variety of methods described above employ some form of implicit and semi-implicit time stepping in order to factor out the fast waves of the system and resolve the slow waves of interest. The methodologies are mostly of an ad hoc nature and work well for particular applications. It would be fair to say that there is scope to develop a broad-based and fairly general numerical method that is able to approximate the system of conservation laws (1.1) in a stable and efficient manner while being robust to the presence of multiple time scales in the system. We aim to describe such a numerical method in this paper.

1.5. Aims and scope of the current paper. The main aim of the current paper is to present a robust numerical method for approximating (1.1) that can efficiently compute all speed flows ranging from the incompressible limit of the Euler equations to convergence to steady state in aerodynamical calculations. Our method is a space-time discontinuous Galerkin (DG) method that was described in a recent paper [23]. In turn, this method was based on earlier works such as [29, 28, 2]. The method is based on the discretization of the space-time computational domain into finite elements and the subsequent approximation of a suitable variational formulation of the system of conservation laws (1.1). Entropy variables are the degrees of freedom of the variational formulation. Suitable numerical fluxes (those designed by Tadmor in [37], see also [13]) are employed to ensure nonlinear entropy stability. Further stabilization operators such as streamline diffusion as well as shock-capturing operators ensure sufficient intra-element stabilization and (essentially) oscillation free shock-capturing. The method was presented in [23] and was shown to be entropy stable as well as convergent to an entropy measure All speed flows 291 valued solution of the conservation law (1.1). The design of efficient preconditioners for the method was the subject of another recent paper [24] and space-time adaptivity aspects of the method are presented in [25]. As shown in [23], the space-time DG method is unconditionally stable i.e., entropy stability holds without any restriction on the time step. In particular, the time step is not bound by the fastest wave speed of the system (1.1). Given this observation, the shock-capturing space-time DG method can be readily adapted for the computation of all speed flows (multiple time scales) by setting the time step such that only the time scales corresponding to the slow waves of interest need to be resolved. Hence, the fastest waves that impede computational efficiency can be factored out automatically in this method. We present this approach in the current paper and illustrate its success with a large number of numerical experiments The rest of the paper is organized as follows: in section 2, we present the shock- capturing space-time DG method. The general methodology for computing all speed flows is presented in section 3 and is illustrated for a model linear hyperbolic system in section 4. In section 5, we present numerical examples for all speed flows modeled by the Euler equations of gas dynamics.

2. The shock-capturing space-time DG method

Following the recent paper [23], we assume that the system of conservation laws (1.1) is equipped with a convex entropy function 푆∶ ℝ푚 → ℝ and the corresponding entropy variables are 퐕 = 푆퐔. As 푆 is convex, the mapping 퐕 = 퐕(퐔) is invertible [5], and the conservation law (1.1) can be expressed in terms of the vector of entropy variables 퐕 as

푑 ∑ 푘 퐔(퐕)푡 + 퐅 (퐕)푥푘 = 0, (푥, 푡) ∈ Ω × ℝ+, (2.1) 푘=1

Here, we have used the change of variable 퐔 = 퐔(퐕) and retained the notation 퐅푘(퐕) = 퐅푘(퐔(퐕)) for all 푘, for notational convenience. This is the form of the conservation law that we are going to discretize using a space-time DG method.

2.1. The mesh. At the 푛-th time level 푡푛, we denote the time step as Δ푡푛 = 푡푛+1 −푡푛 and the update time interval as 퐼푛 = [푡푛, 푡푛+1). For simplicity, we assume that the spatial domain Ω ⊂ ℝ푑 is polyhedral and divide it into a triangulation 풯, i.e., a set of open convex polyhedra 퐾 ⊂ ℝ푑 with plane faces. Furthermore, we assume mesh regularity [28] and quasiuniformity. For a generic element (cell) 퐾, 292 A. Hiltebrand and S. Mishra we denote

Δ푥퐾 = diam(퐾), ′ ′ ′ 풩(퐾) = { 퐾 ∈ 풯 ∶ 퐾 ≠ 퐾 ∧ meas푑−1(퐾 ∩ 퐾 ) > 0 }.

The mesh width of the triangulation is Δ푥(풯) = max퐾 Δ푥퐾. A generic space-time element is the prism: 퐾 × 퐼푛. We also assume that there exists an (arbitrarily large) constant 퐶 > 0 such that

(1/퐶)Δ푥 ≤ Δ푡푛 ≤ 퐶Δ푥, (2.2) for all time levels 푛.

2.2. Variational formulation. On a given triangulation 풯 with mesh width Δ푥, we seek entropy variables

Δ푥 푚 퐕 ∈ 풱푝 = (ℙ푝(Ω × [0, 푇])) 1 푚 = { 퐖 ∈ (퐿 (Ω × [0, 푇])) ∶ 퐖|퐾×퐼푛 is a polynomial (2.3) of degree 푝 in each component }

Δ푥 such that the following quasilinear variational form is satisfied for each 퐖 ∈ 풱푝:

Δ푥 Δ푥 Δ푥 Δ푥 Δ푥 Δ푥 Δ푥 Δ푥 ℬ(퐕 , 퐖 ) ≔ ℬ퐷퐺(퐕 , 퐖 ) + ℬ푆퐷(퐕 , 퐖 ) + ℬ푆퐶(퐕 , 퐖 ) = 0. (2.4)

We elaborate on each of the three quasilinear forms (nonlinear in the first argument and linear in the second) in the following.

2.3. The DG quasilinear form. The form ℬ퐷퐺 is given by,

Δ푥 Δ푥 ℬ퐷퐺(퐕 , 퐖 ) 푑 ∑ ∫ ∫ Δ푥 Δ푥 ∑ 푘 Δ푥 Δ푥 = − (⟨퐔(퐕 ), 퐖푡 ⟩ + ⟨퐅 (퐕 ), 퐖푥푘 ⟩) 푑푥 푑푡 푛,퐾 퐼푛 퐾 푘=1

Δ푥 Δ푥 Δ푥 + ∑ ∫ ⟨핌(퐕푛+1,−, 퐕푛+1,+), 퐖푛+1,−⟩ 푑푥 푛,퐾 퐾

Δ푥 Δ푥 Δ푥 − ∑ ∫ ⟨핌(퐕푛,−, 퐕푛,+), 퐖푛,+⟩ 푑푥 푛,퐾 퐾 푑 푘,∗ Δ푥 Δ푥 Δ푥 푘 + ∑ ∑ ∫ ∫ ( ∑ ⟨픽 (퐕퐾,−, 퐕퐾,+), 퐖퐾,−⟩ 휈퐾퐾′) 푑휎(푥) 푑푡 − ′ 푛 푛,퐾 퐾 ∈풩(퐾) 퐼 휕퐾퐾′ 푘=1 All speed flows 293

1 Δ푥 Δ푥 Δ푥 − ∑ ∑ ∫ ∫ ⟨퐖퐾,−, 퐃(퐕퐾,+ − 퐕퐾,−)⟩ 푑휎(푥) 푑푡. (2.5) 2 ′ 푛 푛,퐾 퐾 ∈풩(퐾) 퐼 휕퐾퐾′ Here we have employed the notation 푛 퐖푛,±(푥) = 퐖(푥, 푡±), ′ 휕퐾퐾′ = 퐾 ∩ 퐾 , ′ 휈퐾퐾′ = unit normal for edge 퐾퐾 pointing outwards from element 퐾,

퐖퐾,±(푥, 푡) = lim 퐖(푥 ± ℎ휈, 푡), ∀푥 ∈ 휕퐾퐾′, ℎ→0 Δ푥 Δ푥 퐃 = 퐃(퐕퐾,−, 퐕퐾,+; 휈퐾퐾′) for all 퐖 ∈ 풱푝. Moreover, we denote the intercell temporal numerical flux as 핌. The spatial flux across the cell interfaces has two components, an entropy conservative component 픽 that is consistent with the flux in the direction normal to the interface and a stabilizing numerical diffusion denoted by 퐃. The temporal and spatial fluxes are specified below. We remark that the boundary condition is ignored in the above variational form by considering compactly supported (in the spatial domain) solutions and test functions.

Numerical fluxes. Both the temporal and spatial numerical fluxes need to be specified in order to complete the DG quasilinear form. In order to obtain causality (marching) after each time step, we choose the temporal numerical flux to be the upwind flux: 핌(푎, 푏) = 퐔(푎). (2.6) This ensures that we can use the values at the previous time step in order to compute an update at the time level 푡푛. A different choice of temporal numerical fluxes will imply that all the degrees of freedom (for all times) are coupledand force us to solve a very large non-linear algebraic system of equations. The spatial numerical flux consists of the following two components,

Entropy conservative flux. The entropy conservative flux (in the 푘-th direction) is any flux [36] that satisfies the relation ⟨푏 − 푎, 픽푘,∗(푎, 푏)⟩ = Ψ푘(푏) − Ψ푘(푎). (2.7) Here, Ψ푘 = ⟨퐕, 퐅푘⟩ − 푄푘 is the entropy potential. The existence of such fluxes (for any generic conservation law with an entropy framework) was shown by Tadmor in [36]. More recently, explicit expressions of entropy conservative fluxes for specific systems of interest like the shallow water equations [12] and Euler equations [26] have been obtained. 294 A. Hiltebrand and S. Mishra

Numerical diffusion operators. Following [37, 12, 13], we choose the numerical diffusion operator as

⊤ 퐃(푎, 푏; 휈) = 퐑휈퐏(Λ휈( ⋅ ); 푎, 푏)퐑휈 . (2.8)

Here, Λ휈, 퐑휈 are the eigenvalue and eigenvector matrices of the Jacobian 휕퐔(⟨퐅, 휈⟩) in the normal direction 휈. 퐑휈 is evaluated at an averaged state, e.g. (푎 + 푏)/2, and ⊤ scaled such that 퐑휈퐑휈 = 퐔퐕. 퐏 is a non-negative matrix function. Examples of 퐏 include 퐏(Λ휈( ⋅ ); 푎, 푏) = |Λ휈((푎 + 푏)/2)|, which leads to a Roe type scheme, and 퐏(Λ휈( ⋅ ); 푎, 푏) = max {휆max(푎; 휈), 휆max(푏; 휈)}퐈퐃, which leads to a Rusanov type scheme [13], where 휆max(퐔; 휈) is the maximal wave speed in direction of 휈, i.e. 휆max(퐔; 휈) is the spectral radius of Λ휈(퐔).

2.4. Streamline diffusion operator. There is no numerical diffusion in the interior of the space-time element 퐾 × 퐼푛. In order to suppress the resulting unphysical oscillations near shocks, we choose the following streamline diffusion operator,

Δ푥 Δ푥 ℬ푆퐷(퐕 , 퐖 ) 푑 (2.9) ∑ ∫ ∫ Δ푥 Δ푥 ∑ 푘 Δ푥 Δ푥 푆퐷 = ⟨(퐔퐕(퐕 )퐖푡 + 퐅퐕(퐕 )퐖푥푘 ), 퐃 Res⟩ 푑푥 푑푡 푛,퐾 퐼푛 퐾 푘=1 with intra-element residual: 푑 Δ푥 ∑ 푘 Δ푥 Res = 퐔(퐕 )푡 + 퐅 (퐕 )푥푘, (2.10) 푘=1 and the scaling matrix is chosen as

푆퐷 푆퐷 푛 −1 Δ푥 퐃 = 퐶 Δ푡 퐔퐕 (퐕 ), (2.11) for some positive constant 퐶푆퐷. Note that the intra-element residual is well defined as we are taking first-derivatives of a polynomial function.

2.5. Shock capturing operator. The streamline diffusion operator adds numerical diffusion in the direction of the streamlines. However, we need further numerical diffusion in order to reduce possible oscillations at shocks. We usethe following shock-capturing operator:

Δ푥 Δ푥 푆퐶 Δ푥 ˜ Δ푥 ℬ푆퐶(퐕 , 퐖 ) = ∑ ∫ ∫ 퐷푛,퐾(⟨퐖푡 , 퐔퐕(퐕푛,퐾)퐕푡 ⟩ 푛,퐾 퐼푛 퐾 (2.12a) 푑 Δ푥 2 + ∑ 퐾 ⟨퐖Δ푥, 퐔 (퐕˜ )퐕Δ푥⟩) 푑푥 푑푡, 푛 2 푥푘 퐕 푛,퐾 푥푘 푘=1 (Δ푡 ) All speed flows 295 with ˜ 1 Δ푥 퐕푛,퐾 = 푛 ∫ ∫ 퐕 (푥, 푡) 푑푥 푑푡. meas(퐼 × 퐾) 퐼푛 퐾 being the cell average and the scaling factor, Δ푡푛퐶푆퐶 Res 퐷푆퐶 = 푛,퐾 , (2.12b) 푛,퐾 √ √ Δ푥 ˜ Δ푥 √ ⎛⟨퐕푡 , 퐔퐕(퐕푛,퐾)퐕푡 ⟩ ⎞ √ √∫ ∫ ⎜ 푑 2 ⎟ 푑푥 푑푡 + 휖 √ Δ푥퐾 퐼푛 퐾 ⎜ + ∑ ⟨퐕Δ푥, 퐔 (퐕˜ )퐕Δ푥⟩⎟ (Δ푡푛)2 푥푘 퐕 푛,퐾 푥푘 √ ⎝ 푘=1 ⎠

1 푛 −1/2 휃 with 휖 = |퐾| 2 (Δ푡 ) (Δ푥/diam(Ω)) and 휃 ≥ 1/2 (chosen as 1), and

−1 Δ푥 Res = ∫ ∫ ⟨Res, 퐔퐕 (퐕 ) Res⟩ 푑푥 푑푡. (2.12c) 푛,퐾 √ 퐼푛 퐾 Here, 퐶푆퐶 is a positive constant.

2.6. Entropy stability and convergence. The entire design of the shock-capturing space-time DG method (2.4) is motivated by the need to prove entropy stability for nonlinear conservation laws. To this end, we proved the following theorem in the recent paper [23]: Theorem 2.1. Consider the system of conservation laws (1.1) with strictly convex entropy function 푆 and entropy flux functions 푄푘 (1 ≤ 푘 ≤ 푑). For simplicity, assume that the exact and approximate solutions have compact support inside the spatial domain Ω. Let the final time be denoted by 푡 . Then the streamline diffusion shock- capturing discontinuous Galerkin scheme (2.4) approximating (1.1) has the following properties: (i) The scheme (2.4) is conservative, i.e., the approximate solutions 퐔Δ푥 = 퐔(퐕Δ푥) satisfy

Δ푥 Δ푥 0 ∫ 퐔 (푥, 푡− ) 푑푥 = ∫ 퐔 (푥, 푡−) 푑푥. (2.13) Ω Ω (ii) The scheme (2.4) is entropy stable, i.e., the approximate solutions satisfy

∗ 0 Δ푥 Δ푥 0 ∫ 푆(퐔 (푡−)) 푑푥 ≤ ∫ 푆(퐔 (푥, 푡− )) 푑푥 ≤ ∫ 푆(퐔 (푥, 푡−)) 푑푥, (2.14) Ω Ω Ω with 퐔∗ being the domain average:

∗ 0 1 0 퐔 (푡−) = ∫ 퐔(퐕(푥, 푡−)) 푑푥. meas(Ω) Ω 296 A. Hiltebrand and S. Mishra

Hence, the space-time DG method is nonlinearly stable for any system of conservation laws that is equipped with a uniformly convex entropy function. The uniform convexity of the entropy readily implies that the approximate solution 퐔Δ푥 is bounded in 퐿2. Furthermore, under the additional assumption that the approximate solutions are bounded (uniformly) in 퐿∞, we proved that the approximate solutions converge to an entropy measure valued solution of the system of conservation laws (1.1) when Δ푥 → 0. Entropy measure valued solutions are a weaker but possibly more relevant solution concept for systems of conservation laws than entropy solutions [14]. Moreover, one can show that the approximate solutions converge to the weak solution of a linear symmetrizable system as well as to a weak solution for scalar conservation laws, see [25].

3. Methodology for computing all speed flows

We remark that the entropy stability result (Theorem 2.1) as well as the convergence results do not require any restriction on the time step Δ푡, apart from the very mild mesh regularity requirement (2.2). Note that the constant 퐶 in (2.2) can have any finite value. Thus the method (2.4) is unconditionally stable with respect to time step size. Nevertheless, it is customary to relate the time step and the (spatial) mesh size (for the purpose of accuracy of the approximation) through a CFL type condition (see [23]), Δ푥 Δ푡푛 = 퐶CFL min 퐾 , (3.1) Δ푥 푛 퐾∈풯,푥∈퐾 휆max(퐔 (푥, 푡 )) in one space dimension and |퐾|/Δ푥 Δ푡푛 = 퐶CFL min 퐾 , (3.2) Δ푥 푛 퐾∈풯,푥∈퐾 휆max(퐔 (푥, 푡 )) in two space dimensions. Here 휆max(퐔) = max휈 휆max(퐔; 휈) is the maximal wave speed (eigenvalue of the flux Jacobian) in all directions. Note that the CFL number 퐶CFL can be taken arbitrarily large and the stability result still holds. However, accuracy may suffer from a large CFL number asthe temporal error is 풪(Δ푡푠), with 푠 being related to the order of the method (degree of the underlying polynomials) and a large 퐶CFL results in a large time step Δ푡 and consequently, a possibly large error. However, and as mentioned in the introduction, there is a large class of problems with multiple time scales where the waves of interest travel with a speed slow bounded by 휆max and slow 휆max ≪ 휆max. (3.3) All speed flows 297

In other words, the fastest wave speed is considerably larger than the slow wave speed in the system. On the other hand, the interest of the computation is to compute the slow waves. For instance, the matter (shear) waves are much more relevant than the acoustic waves in the incompressible limit of the Euler equations [7]. Given this context, we take advantage of (3.3) and change the time step sizes (3.1) and (3.2) and set

Δ푥 Δ푡푛 = 퐶CFL min 퐾 , (3.4) red slow Δ푥 푛 퐾∈풯,푥∈퐾 휆max (퐔 (푥, 푡 )) in one space dimension and

|퐾| Δ푥 Δ푡푛 = 퐶CFL min 퐾 , (3.5) red slow Δ푥 푛 퐾∈풯,푥∈퐾 휆max (퐔 (푥, 푡 )) in two space dimensions. From (3.1) (resp. (3.2)) and (3.4) (resp. (3.5)), we obtain that

CFL CFL 휆max 퐶 ≈ 퐶red slow . (3.6) 휆max

CFL We typically choose 퐶red = 풪(1) (for the sake of high accuracy). Hence, from (3.6), we see that the effective CFL number 퐶CFL can be very large on account of (3.3). The resulting method has the following features,

• Unconditional stability of the space-time DG method implies that the effective CFL number 퐶CFL can be arbitrarily high and the method will still be stable.

• The fact that our interest is in resolving the slow waves of the system implies that the numerical error due to time discretization will still be low as the time step is based on the slow wave speed.

• The computational cost will be low, as the slow wave speed is considerably smaller than the fast wave speed. This results in large time steps, and significantly reduces the number of time steps that are required to reachthe desired final time.

Hence the shock-capturing DG method (2.4) with the time step decided by (3.4), (3.5) is well poised to resolve all speed flows efficiently. The method is readily modified to approximate convergence to steady state also. 298 A. Hiltebrand and S. Mishra

3.1. Brief description of implementation. The implementation of the shock- capturing space-time DG method is described in detail in the recent paper [24], see also [25]. As the test and trial spaces for (2.4) involve piecewise polynomials, we choose a suitable basis for this space as the span of scaled and shifted monomials, see [24]. Then, the space-time DG formulation is recast into a large system of nonlinear algebraic equations for the degrees of freedom (entropy variables). Given the upwind temporal flux (2.6), one can perform time marching i.e., the degrees of freedom for a given time slab can be solved once the degrees of freedom for the previous time slab have been computed. Nevertheless, a large nonlinear algebraic system needs to be solved at every time step. We employ a damped Newton method to solve this nonlinear system (see [24]). Given the structure of the Newton method, a large, sparse and non-symmetric linear system needs to be inverted at every step of the Newton iterate. This linear system is solved using an iterative procedure such as GMRES. Such iterative schemes rely on the availability of efficient preconditioners to ensure convergence within a reasonable number of iterations. We have designed and analyzed efficient block Jacobi and block Gauss–Seidel preconditioners for this purpose. These preconditioners are also described in [24].

4. A toy model: Linear symmetric system involving two wave speeds

Next, we will investigate whether our general methodology, as presented in the last section, is able to efficiently approximate all speed flows modeled by systems of conservation laws such as (1.1). To this end, we consider a simple one-dimensional linear symmetric 2 × 2 system of the form:

퐔푡 + 퐅(퐔)푥 = 0, (4.1) with 푢 1 (푎 + 푏)푢 + (푏 − 푎)푢 퐔 = ( 1) , 퐅(퐔) = ( 1 2) . (4.2) 푢2 2 (푏 − 푎)푢1 + (푎 + 푏)푢2 Here, 푎 and 푏 are assumed to be positive constants with 푎 ≤ 푏. Clearly, the above system (4.1) is hyperbolic and has two wave speeds given by 푎 and 푏. Based slow on our assumptions, 푎 ≤ 푏, we denote 휆max = 푏 and 휆max = 푎. Furthermore, 1 the energy 2 ⟨퐔, 퐔⟩ serves as the canonical entropy for linear symmetric systems. Hence the entropy variables 퐕 = 퐔 coincide with the conservative variables. We will apply the shock-capturing space-time DG method (2.4) to approximate the linear system (4.1). The only parameter that needs to be specified is the spatial All speed flows 299 numerical flux. It is well known (see [13]) that the entropy conservative flux (2.7) for linear systems is the arithmetic average of the two interfacial states. We will consider both the Rusanov diffusion operator with the wave speed being 휆 = 푎 as well as the Roe type diffusion operator in the numerical diffusion operator (2.8). Note that the Roe type diffusion operator, together with the arithmetic average as the entropy conservative flux, implies that the spatial numerical flux is the upwind flux for this linear system. We consider (4.1) in the domain [−1, 1] with initial Riemann data,

1, 푥 < 0 푢1(푥, 0) = { 0, 푥 ≥ 0 (4.3)

푢2(푥, 0) = 0.

In the first numerical experiment, we would like to investigate whether the shock-capturing space-time DG method (2.4) can approximate a flow with vastly different wave speeds (time scales). To this end, weset 푎 = 1 and 푏 = 100 in (4.1). Thus, there is a slow wave traveling at speed 1 and a fast wave traveling at speed 100. Our interest is in computing the slow moving wave accurately as the fast wave quickly exits the domain. We set Dirichlet boundary conditions on the left boundary and Neumann type transparent boundary conditions on the right boundary. The exact solution can be readily computed by a characteristic decomposition of the system (4.1). We plot this exact solution at time 푇 = 0.5 in figure 1. As seen in the figure, the fast wave, traveling at a speed 100, has clearly exited the computational domain and the slow wave has moved the initial discontinuity to 0.5. To simulate this example, we use the shock-capturing space-time DG method slow (2.4) with the time step being determined by the condition (3.4) with 휆max = 푎. We CFL CFL set 퐶red = 0.5 and from (3.6), we see that the effective CFL number is 퐶 = 50. The results obtained with a piecewise quadratic (푝 = 2) elements and a Roe type numerical diffusion operator in (2.4), at different mesh resolutions, are plotted in figure 1. The figure clearly shows that the space-time DG method (1.1), even at a high CFL number of 50, is able to resolve both components of the solution very sharply. As expected, there are very small and localized oscillations near the discontinuities but the overall quality of approximation is really good, even for such large time steps as dictated by an effective CFL number of 50.

4.1. Approximation of fast waves. The aim of the above computation was to approximate the slow waves accurately as the fast waves have already exited the system. In order to explore the approximation of fast waves, we enlarge the computational domain to (−1, 10) and compute only to the final time 푇 = 0.05. 300 A. Hiltebrand and S. Mishra

1.1 0.6 N =20 c N =80 1 0.5 c N =320 c exact 0.9 0.4

0.8 0.3

0.7 0.2

0.6 N =20 0.1 c N =80 c 0.5 N =320 0 c exact 0.4 −0.1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

(a) ᵆ1 (b) ᵆ2

Figure 1. Approximate solution of (4.1) at time 푇 = 0.5 with Neumann boundary conditions, for the two speed advection problem using polynomial degree 푝 = 2, Roe type numerical diffusion operator and a fast wave speed of 푏 = 100.

In this case, the fast wave does not leave the domain. The results with same parameters as the previous example are shown in figure 2. We clearly see that the slow wave is sharply resolved, even at coarse mesh resolutions. However, the fast wave is diffused (smeared) by the approximation. This is not unexpected asthe whole design of the method was to resolve slow waves accurately at the expense of smearing (inaccurate resolution) of fast waves.

1.2 0.6 N =220 N =220 c c N =880 N =880 1 c 0.5 c N =3520 N =3520 c c exact exact 0.8 0.4

0.6 0.3

0.4 0.2

0.2 0.1

0 0

−0.2 −0.1 0 2 4 6 8 10 0 2 4 6 8 10

(a) ᵆ1 (b) ᵆ2

Figure 2. Approximate solution of (4.1) at time 푇 = 0.05 on a larger domain using 푝 = 2, Roe type numerical diffusion operator, and 푏 = 100. All speed flows 301

4.2. Computational efficiency. The results, shown in figure 1 clearly establish that the method is able to approximate the waves of interest (slow waves) accurately, even at a high effective CFL number of 50. It is natural to examine if this high accuracy, even for large time steps, comes at a computational cost. Given the nature of the shock-capturing space-time DG method (2.4), we need to solve a large nonlinear system of equations at every time step even though the underlying PDE (4.1) is linear. The Newton method converged to the desired tolerance within three iterations for this problem. Thus, the main contributor to the computational cost per time step was the GMRES solve per Newton iterate. As described in detail in [24], and mentioned in the last section, we will employ block Jacobi and block Gauss–Seidel preconditioners for GMRES. The average number of iterations for the preconditioned GMRES to converge is presented in figure 3. We compare three different cases, that of 푏 = 1, 10, and 100, i.e., when there is no separation, intermediate separation, and a large separation in wave speeds (time scales) for the system (4.1). The results obtained with a Roe type numerical diffusion operator are presented in figure 3, and show that

• The number of iterations (wrt both preconditioners) is approximately independent of the mesh resolution.

• There is very little difference between the number of iterations between piecewise linear (푝 = 1) and piecewise quadratic (푝 = 2) elements.

• The block Gauss–Seidel preconditioner converges in one iteration, irrespec- tive of mesh size, polynomial degree, or effective CFL number (size of the time step). This is on account of the fact that the Gauss–Seidel preconditioner amounts to a direct solve for an upwind flux [24].

• There is a very modest (at most three fold) increase in the number of iterations for the block Jacobi preconditioner, even when the size of time step (effective CFL number) changes by two orders of magnitude.

Given the above observations, we conclude that there is very little increase in computational cost per time step, even when the time step size increases by two orders of magnitude. Thus, for large time step sizes, the method is computationally efficient. As a final result in this section, we compare the computational cost oftheRoe type and Rusanov type numerical diffusion operators, i.e. the average number of block Jacobi preconditioned GMRES iterations per Newton iterate vis a vis the number of cells in figure 4. The results show that, in contrast to the rather low (around 10) iterations for the Roe type numerical diffusion, the number of iterations registers a very sharp increase with respect to the wave speed ratio 푏/푎 302 A. Hiltebrand and S. Mishra

20 20 BlockJac, b=1 BlockJac, b=1 18 BlockGS, b=1 18 BlockGS, b=1 BlockJac, b=10 BlockJac, b=10 16 BlockGS, b=10 16 BlockGS, b=10 BlockJac, b=100 BlockJac, b=100 14 BlockGS, b=100 14 BlockGS, b=100

12 12

10 10

8 8

6 6

4 4

2 2

0 0

Average number of Krylov iterations per Newton iteration 1 2 3 4 Average number of Krylov iterations per Newton iteration 1 2 3 10 10 10 10 10 10 10 Number of cells Number of cells (a) 푝 = 1 (b) 푝 = 2

Figure 3. Number of Krylov iterations for (4.1) using a Roe type of diffusion with varying fast wave speed 푏 and with different preconditioners. for the Rusanov type diffusion operator. Furthermore, the piecewise quadratic version of the method with the Rusanov type diffusion operator may not even converge. Thus, this is a good example to illustrate that the Roe method, based on a characteristic decomposition of the system, is vastly superior to an approximate characteristic decomposition such as the Rusanov type numerical diffusion operator.

200 200 Roe, b=1 Roe, b=1 180 Roe, b=10 180 Roe, b=10 Roe, b=100 Roe, b=100 160 Rusanov, b=1 160 Rusanov, b=1 Rusanov, b=10 Rusanov, b=10 140 Rusanov, b=100 140 Rusanov, b=100

120 120

100 100

80 80

60 60

40 40

20 20

0 0

Figure 4. Number of Krylov iterations for the linear system (4.1) using the block Jacobi preconditioner with varying fast wave speed 푏 and diffusive fluxes. All speed flows 303

5. Euler equations

For the final set of numerical experiments, we consider the compressible Euler equations of gas dynamics. In two space dimensions, the Euler equations are of the form

1 2 퐔푡 + 퐅 (퐔)푥 + 퐅 (퐔)푦 = 0, 퐔 = (휌, 휌푢, 휌푣, 휌퐸), (5.1) 퐅1(퐔) = (휌푢, 휌푢2 + 푝, 휌푢푣, 휌푢퐻) , 퐅2(퐔) = (휌푣, 휌푢푣, 휌푣2 + 푝, 휌푣퐻) .

Here, 휌 is the density, 푢 and 푣 are the 푥 and 푦 components of the velocity, respectively, and 휌퐸 is the total energy. Furthermore, auxiliary quantities are the pressure 푝, sound speed 푐, and the enthalpy 퐻 given by

푝 푐2 1 푝 = (훾 − 1)(휌퐸 − 1 휌(푢2 + 푣2)), 푐 = 훾 , 퐻 = + (푢2 + 푣2) 2 √ 휌 훾 − 1 2 and 훾 is the adiabatic exponent, which is set to 1.4 (diatomic gas) in all experiments. The Euler equations are equipped with the standard thermodynamic specific entropy 푠 = log 푝 − 훾 log 휌. The resulting entropy function is given by, −휌푠 푆 = . 훾 − 1

The corresponding entropy flux functions and entropy variables can be readily calculated and are described in [13]. We will approximate the Euler equations with the streamline diffusion shock-capturing DG method (2.4). The entropy conservative flux that we use is the one designed in[26] and the numerical diffusion operators of the Roe and Rusanov type are described in [13].

5.1. One-dimensional pulse propagation. As a first numerical example, we consider the problem proposed by Klein in [30] as a model problem for computing the incompressible limit of the Euler equations. As in [30], we consider the one- dimensional version of the Euler equations (5.1) by setting 푣 = 0 and considering only the 푥 direction. The aim is to simulate a high amplitude, short wave length layering of the density that is set in motion by a periodic train of right-running (fast) acoustic waves. 푀 = 1/51 represents the nondimensionalisationing parameter (reference Mach number), and we consider a rescaled version of the initial data of [30] (problem 304 A. Hiltebrand and S. Mishra

II):

1 휌(푥, 0) = 휌0̄ + Φ(푥)휌0̃ sin(40휋푥/퐿) + 푀휌0̃ 2 (1 + cos(휋푥/퐿)), 1 2 푝(푥, 0) = ( 0̄푝 + 푀0 ̃푝 2 (1 + cos(휋푥/퐿)))/푀 , (5.2) 1 푢(푥, 0) = ̃푢0 2 (1 + cos(휋푥/퐿)), where 푥 is in the domain [−퐿, 퐿] with 퐿 = 1/푀 and

휌0̄ = 1, 휌0̃ = 1/2,0 ̄푝 = 1,0 ̃푝 = 2훾, ̃푢0 = 2√훾. (5.3)

The function

1 (1 − cos(5휋푥/퐿)), 0 ≤ 푥 ≤ 2 퐿, Φ(푥) = { 2 5 (5.4) 0, otherwise, is used to smoothly restrict the large amplitude, short wave length density variation to the region [0, 2퐿/5]. The initial density is visualized in figure 5 (a). Periodic boundary conditions are used, and the equations are solved from 푡 = 0 up to 푇 = 5.071. In this time span, the long wave length acoustic pulse crosses the domain about two and a half times. As we are interested in computing the slowly moving density fluctuations, we use a discrete version of

slow 푛 휆max = max |푢(푥, 푡 )| (5.5) 푥∈[−퐿,퐿] at every time step 푡푛 to set the time step (CFL number) as in (3.5). The resulting effective CFL number computed with (3.6) and the computed solution averages to approximately 18. The approximate density, computed with the piecewise quadratic shock-capturing DG method (2.4) with a Roe type numerical diffusion operator and an effective CFL number of approximately 18, is shown in figure 5 (b). The exact solution is a (slow) rightward propagation of the initial short wave length density fluctuations. The computed solutions (particularly at moderate to fine resolutions) are a very good approximation of the exact solution even if the time step, being based on the slow wave speed, is quite large and does not resolve the fast moving acoustic waves. Thus, the method is quite effective at computing the density pulse. Given the large time steps, the key contributor to the computational cost is the cost per time step. Again, the dominant factor in the cost per time step is the number of GMRES iterations per Newton step. We present the average number of GMRES iterations per Newton step in figure 6. We observe that All speed flows 305

1.5 1.5 1.5 N =200 N =200 c c 1.4 1.4 N =400 1.4 N =400 c c N =800 N =800 1.3 1.3 c 1.3 c N =1600 N =1600 c c 1.2 1.2 1.2

1.1 1.1 1.1

1 1 1

0.9 0.9 0.9

0.8 0.8 0.8

0.7 0.7 0.7

0.6 0.6 0.6

0.5 0.5 0.5 −60 −40 −20 0 20 40 60 −60 −40 −20 0 20 40 60 10 15 20 25 (a) 푡 = 0 (b) 푡 = 5.071 (c) 푡 = 5.071, closeup

Figure 5. Approximate density for the one-dimensional pulse propagation in the Euler equations, computed initially and 푇 = 5.071, with the shock-capturing DG method with piecewise quadratic elements and a Roe type numerical diffusion operator at an effective CFL number of 18.

• Both the block Jacobi and block Gauss Seidel preconditioners, developed in the recent paper [24], are robust in the sense that the number of iterations is constant (or even decreasing) with respect to increasing mesh resolutions or increasing polynomial degree.

• Even if the time step is quite large (effective CFL number of 18), the number of GMRES iterations with the block Jacobi preconditioner and a Roe type diffusion operator is about 20.

• The number of iterations is minuscule, about 2, for the symmetric version of the block Gauss Seidel preconditioner and the Roe type diffusion operator, making this combination a very effective computational framework.

• The Rusanov diffusion operator does not function as well. Either the number of iterations are quite high (about 120 for piecewise linear elements) or the iteration does not converge (for the piecewise quadratic elements).

Summarizing, we observe that the shock-capturing space-time DG method (2.4), together with a Roe type diffusion operator and a Block Gauss–Seidel (or Block Jacobi) preconditioner, is very effective at computing the slow moving density fluctuations, even for a large time step that does not resolve the time scales corresponding to the fast moving acoustic waves.

5.2. Flow past a cylinder: computing the incompressible limit. Similar to [11], we consider an Euler flow (solution of the Euler equations (5.1)) past a two-dimensional cylinder with a low free stream Mach number Ma∞. The cylinder is centered at the origin and has diameter 1. The computational domain is bounded 306 A. Hiltebrand and S. Mishra

120 120 BlockJac Roe BlockJac Roe BlockGS Roe BlockGS Roe 100 BlockJac, Rusanov 100

80 80

60 60

40 40

20 20

0 0

Average number of Krylov iterations per Newton iteration 2 3 4 Average number of Krylov iterations per Newton iteration 2 3 4 10 10 10 10 10 10 Number of cells Number of cells (a) 푝 = 1 (b) 푝 = 2

Figure 6. Average number of GMRES iterations for the pulse propagation problem for one-dimensional Euler equations, comparing different diffusive fluxes and preconditioners. by an (artificial) circle of radius 10 around the origin. The following free stream variables are imposed at the outer boundary: 푝 푝 = 101325, 휌 = ∞ , 푢 = Ma 푐 , 푣 = 0, (5.6) ∞ ∞ 287.05 ⋅ 288.15 ∞ ∞ ∞ ∞ with 푐∞ = √훾푝∞/휌∞. Slip boundary conditions are imposed on the boundary of the cylinder. It is essential (particularly for computations with second and higher order polynomials) that the boundary of the cylinder is resolved accurately. To this end, we use a polynomial mapping of degree 푝 (for 푝 ≥ 1) to generate curved boundary elements. The equations are solved from 푡 = 0 up to 푇 = 0.02/Ma∞. We use a Roe type diffusive flux. As we will be interested in computing the incompressible limit, we determine the time step using the slow wave speed (5.5) and the formula (3.5). In particular, the acoustic waves (sound speed 푐∞) is ignored while determining the time step. We will compute for three different free stream mach numbers, Ma∞ = 0.1, Ma∞ = 0.01, and Ma∞ = 0.001. Thus, three different regimes of moderate Mach number, low Mach number, and very low Mach number are covered in this calculation. As is well known, Ma∞ → 0 is (formally) the incompressible limit of the Euler equations, corresponding to the steady state solution of the underlying boundary value problem. Thus, we will also compute a potential flow (solution of the Laplace equation) around the cylinder to represent the incompressible limit. CFL We have used a reduced CFL number of 퐶red = 10 to determine the time step from (3.5). From (3.6) and the computation, we report that the effective CFL All speed flows 307 numbers are 65, 590, and 6000 for the free stream Mach numbers of 0.1, 0.01, and 0.001, respectively. The results with piecewise quadratic elements are presented in figure 7. In this figure, we have depicted the pressure coefficient 푝 − 푝 푐 = ∞ 푝 1 2 2 휌∞ ‖퐮∞‖ for the three different free stream Mach numbers. The results from figure 7 clearly show that the flow is accurately resolved, even for the lowest Mach number of 0.001 and the incompressible limit is approximated very well when the Mach number → 0. Thus, the method clearly resolves the incompressible limit without any ad hoc fixes. In particular, we study also the effect of the numerical diffusion. In[17], it is shown that the pressure fluctuations (max 푝−min 푝)/max 푝 for certain numerical schemes wrongly scale with the Mach number while they scale with the square of the Mach number for the continuous problem. These schemes fail to compute the low Mach number limit correctly. We plot the pressure fluctuations for the flow around the cylinder for different Mach numbers in figure 8. This shows very clearly the correct scaling of the pressure fluctuations with the square of the Mach number, and provides anaddi- tional argument that our method is able to approximate the incompressible limit accurately. Given the very large time steps that are associated with a very high effective CFL number of 6000, the total number of time steps is rather moderate. The computational cost per time step is determined by the number of preconditioned GMRES iterations per Newton step. The average number of multi block Gauss–Sei- del preconditioned GMRES iterations are 18, 94, and 598 for Mach numbers of 0.1, 0.01, and 0.001, respectively. Thus, we see that the number of iterations increases by at most an order of magnitude, even if the Mach number (and consequently the time step size (effective CFL number)) decreases (increases) by two orders of magnitude. Thus, the computational cost is still very moderate for very low Mach number flows. In the above experiment, we have demonstrated the ability of the shock-capturing space-time DG scheme to compute low to very low mach number flows very efficiently. The scheme is of course well suited to compute moderate to highMach number flows. As an example, we consider a flow past a cylinder with amoderate free stream Mach number of 0.75. The results (푐푝), obtained with a piecewise quadratic version of the space-time DG method (2.4) and 13282 elements, are shown in figure 9. The results show that the resulting transsonic flow is very well resolved with accurate and (essentially) non-oscillatory approximations of 308 A. Hiltebrand and S. Mishra

(a) Ma∞ = 0.1 (b) Ma∞ = 0.01

Figure 7. Pressure coefficient 푐푝 of a flow around a cylinder at different Mach numbers, using polynomial degree 푝 = 2 and 푁푐 = 13282 spatial elements.

Figure 8. Pressure fluctuations of a flow around a cylinder at different Mach numbers, using polynomial degree 푝 = 2 and 푁푐 = 13282 spatial elements. All speed flows 309 shocks. The effective CFL number in this computation is 1.6 and at most 4 block Gauss–Seidel preconditioned GMRES iterations are needed per time step. Thus, the results show that the space-time DG method is very effective and robust in computing flows with underlying Mach numbers that differ by several ordersof magnitude.

(a) Complete domain (b) Closeup around the cylinder

Figure 9. Pressure coefficient of a flow around a cylinder with Ma∞ = 0.75 using polynomial degree 푝 = 2 and 푁푐 = 13282 spatial elements.

5.3. Flows past aerofoils: computing convergence to steady state. As mentioned in the introduction, the aim of most aerodynamic computations is to accurately compute the steady state flow (cruise conditions). Although one can compute the steady state directly by solving the steady (time-independent) version of the Euler equations, it is fairly common to compute the steady state by starting with an initial condition and driving the system to steady state [27]. The transient flow need not be accurately resolved in such a computation. Giventhis context, convergence to steady state constitutes another set of problems where large time steps are necessary in order to accelerate convergence to steady state. The shock-capturing space-time DG method is well suited for this purpose. In order to demonstrate the ability of the space-time DG method (2.4) to accelerate convergence to steady state, we consider an Euler flow around a NACA 0012 aerofoil [1]. The aerofoil is placed along the 푥 axis, ranging from 푥 = 0 (head) to 푥 = 1 (tail). Slip boundary conditions are used on the aerofoil. An artificial outer boundary is placed on a circle around (2, 0) with radius 4, where the following free-stream values are prescribed: Mach number Ma∞ = 0.75, pressure 310 A. Hiltebrand and S. Mishra

∘ 푝∞ = 8.5419, density 휌∞ = 11.4452, and an angle of attack of 4 . We will compute 1 2 and display the pressure coefficient 푐푝 = (푝 − 푝∞)/( 2 휌∞ ‖퐮∞‖ ), where 퐮∞ is the free-stream flow velocity. At 푡 = 0, the flow is initialized by free-stream values. The equations are then solved up to 푡 = 3.5; the time by which the steady state is approximately reached. An unstructured mesh (consisting of triangles) is generated around the aerofoil. This mesh is finer near the head of the aerofoil than near the tail. Asafurther modification, we replace the shock-capturing operator with a pressure scaled variant suggested in [23], i.e., (2.12b) is replaced by

푝 푛 푆퐶 퐷 Δ푡 퐶 Res푛,퐾 퐷푆퐶 = 푛,퐾 (5.7) 푛,퐾 √ √ Δ푥 ˜ Δ푥 √ ⎛⟨퐕푡 , 퐔퐕(퐕푛,퐾)퐕푡 ⟩ ⎞ √ √ ⎜ 푑 ⎟ √∫ ∫ Δ푥2 푑푥 푑푡 + 휖 퐼푛 퐾 ⎜ + ∑ ⟨퐕Δ푥, 퐔 (퐕˜ )퐕Δ푥⟩⎟ (Δ푡푛)2 푥푘 퐕 푛,퐾 푥푘 √ ⎝ 푘=1 ⎠ with

√ 푑 1 1 √ 2 푛 ∫ ∫ ∑ 푝푥푘푥푘 푑푥 푑푡 Δ푡 |퐾| 푛 퐷푝 = Δ푥2 퐼 퐾 √푘=1 . (5.8) 푛,퐾 퐾 1 1 ∫ ∫ 푝 푑푥 푑푡 Δ푡푛 |퐾| 퐼푛 퐾 We will determine the time step using the condition (3.2) and consider four different time steps corresponding to effective CFL numbers of 0.5, 5, 50, and 500 respectively. The resulting pressure coefficient 푐푝, computed with piecewise linear elements is shown in figure 10. The results clearly show that increasing the time step (even by three orders of magnitude) did not result in a significant deterioration of the accuracy with which the steady state is resolved. In fact, the results obtained with a very large time step, corresponding to an effective CFL number of 500 are quite accurate and resolve the transsonic shocks as well as the smooth features of the solution rather well. Furthermore, the average number of Krylov iterations increases only moderately given the three orders of magnitude increase of the CFL number: it is 3, 6, 17, and 53 for CFL numbers of 0.5, 5, 50, and 500, respectively. As a next numerical experiment, we show four different flows with underlying free stream Mach numbers of 0.5, 0.75, 1.3, and 3.0 and a very large time step, corresponding to an effective CFL number of 500 in figure 11. The results show that even for a large time step, the method resolves various regimes of flow i.e., subsonic, transsonic, and supersonic flow, very accurately, at least at the steady All speed flows 311

(a) CFL = 0.5 (b) CFL = 5

Figure 10. Pressure coefficient 푐푝 of a Ma∞ = 0.75 Euler flow over a NACA 0012 aerofoil using 푝 = 1 and 푁푐 = 16704 at 푡 = 3.5. 312 A. Hiltebrand and S. Mishra state. Thus, the space time DG method is well suited to approximate problems requiring rapid convergence to and high resolution of the steady state.

(a) Ma∞ = 0.5 (b) Ma∞ = 0.75

Figure 11. Pressure coefficient 푐푝 of Euler flows over a NACA 0012 aerofoil using 푝 = 1 and 푁푐 = 16704 at 푡 = 3.5.

Finally, we repeat the experiment presented in [17]. This corresponds to a low Mach number flow over the NACA 0012 aerofoil at 0 degree angle of attack. The Mach number Ma∞ is varied over 0.1, 0.01, and 0.001. The approximate solutions at 푡 = 0.35/Ma∞ using piecewise quadratic polynomials and a reduced CFL number CFL 3 4 of 퐶red = 10 (resulting in effective CFL numbers of about 175, 1.7⋅10 , and 1.7⋅10 ) are shown in figure 12. The pressure coefficients are almost indistinguishable, and approximate the incompressible limit very accurately. In particular, the pressure fluctuations (max 푝 − min 푝) / max 푝 scale with the square of the Mach number as shown in figure 13. This is in accordance with the continuous solutions, and further confirms that the scheme is also able to compute the low Mach number All speed flows 313 limit accurately.

(a) Ma∞ = 0.1 (b) Ma∞ = 0.01

Figure 12. Pressure coefficient 푐푝 of Euler flows over a NACA 0012 aerofoil using 푝 = 2 and 푁푐 = 16704 at 푡 = 0.35/Ma∞.

6. Conclusion

Many problems of interest modeled by systems of conservation laws (1.1) include phenomena that occur at multiple time scales. Often, the main interest in a simulation of such problems is to resolve the slow waves accurately, without requiring an accurate approximation of fast waves. Examples include the incompressible (zero Mach number) limit of the compressible Euler equations (5.1), where the interest is in resolving the matter waves rather than the fast acoustic waves. Radiation hydrodynamics and MHD also fall into this class of problems, as it is not necessary 314 A. Hiltebrand and S. Mishra

Figure 13. Pressure fluctuations of a flow around a NACA 0012 aerofoil at different Mach numbers, using polynomial degree 푝 = 2 and 푁푐 = 16704 spatial elements. to accurately resolve the radiative waves (traveling at the speed of light), which are several orders of magnitude faster than the sonic and magneto-sonic waves of interest. Furthermore, those problems where the steady state of the flow is the object of interest also fall into this category when time stepping methods are used to drive the system into steady state. The corresponding transient need not be accurately approximated. Standard explicit time stepping methods fail at the efficient approximation of such all speed flows as the time step is bound by the fastest time scale, leading to a prohibitively expensive method. Although many implicit and semi-implicit methods are proposed to deal with this problem, they are either tailored to a particular problem of interest or require ad hoc arguments. We present an alternative framework to compute all speed flows in this paper. Our approach is based on an entropy stable shock-capturing streamline diffusion space-time discontinuous Galerkin (DG) method proposed in a recent paper [23]. As the space-time DG method (2.4) is unconditionally stable, one can choose suitable time steps. In particular, we choose the time step to resolve the time scales of interest by (3.4), (3.5). Consequently, the method automatically resolves the slow waves at the expense of smearing fast waves. The method is very general, and requires no ad hoc fixes or alterations in the code. The only change isinthe CFL parameter that determines the time step. We illustrate the method on four different sets of problems,

• A toy linear symmetric system (4.1) with two time scales.

• A one-dimensional (slow) propagation of density fluctuations in the Euler equations proposed by Klein in [30].

• A two-dimensional Euler flow past a cylinder with the specific objective of All speed flows 315

computing the incompressible (zero Mach number) limit. • A two-dimensional Euler flow past a NACA aerofoil with the aim to computing the steady state accurately. Based on the numerical results presented here, we conclude that the shock-capturing space-time DG method (2.4), particularly with a Roe type numerical diffusion operator and a block Gauss–Seidel type preconditioned GMRES solver, is able to • compute the (interesting time scales) of the flow very accurately, • is robust to several orders of magnitude variation in the Mach number, in particular it can compute the incompressible limit, • allows for very large time steps (with effective CFL numbers running from 500 to 6000), thus resulting in significant decrease in the overall computational cost. Hence, at one stroke, we present a method that can approximate all speed flows as well as accelerate convergence to steady state, all with reasonable computational cost. It is instructive to compare the computational cost of the space-time DG method presented here with a standard explicit finite volume scheme to approximate systems of conservation laws. Clearly, the cost of a single time-step with the space- time DG method is (considerably) more than a explicit finite volume method. This is primarily due to the fact that a (large) non-linear system of algebraic equations is solved at every time step with the DG method, in contrast to the explicit scheme. Even with efficient (damped) Newton iterations and suitable pre-conditioners for the resulting linear systems in each Newton step, the computational cost per time step with the space-time DG scheme is higher than the finite volume method. However, the real computational advantage of the space-time DG scheme is revealed when problems with multiple time scales, such as all-speed flows or computation of steady states, are dealt with. Here, depending on the separation of time scales, the number of time-steps with the explicit method can be several orders of magnitude more than an implicit scheme. Thus, even if each time step is an order of magnitude more expensive with the space-time DG method, it will still be several orders of magnitude faster, in terms of computational time, than a competing explicit method. Hence we believe that the shock-capturing DG method is particularly suitable for problems that contain multiple time scales in the system. The current paper was focused on the two-dimensional Euler equations of hydrodynamics. Extensions to three dimensional flows, radiation hydrodynamics, and magnetohydrodynamics as well as combustion will be the object of future work. 316 A. Hiltebrand and S. Mishra

Acknowledgments. This work was partially supported by ERC STG NN. 306279 SPARCCLE.

References

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Markus Holzleitner, Aleksey Kostenko, and Gerald Teschl

To Helge Holden, inspiring colleague and friend, on the occasion of his 60th birthday

Abstract. We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger 1 operators, where the angular momentum takes the critical value 푙 = − 2 . We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.

1. Introduction

The stationary one-dimensional radial Schrödinger equation 푑2 푙(푙 + 1) i휓(푡,̇ 푥) = 퐻휓(푡, 푥), 퐻 ≔ − + + 푞(푥), (푡, 푥) ∈ ℝ × ℝ , (1.1) 푑푥2 푥2 + is a well-studied object in quantum mechanics. Starting from the Schrödinger equation with a spherically symmetric potential in three dimensions, one obtains (1.1) with 푙 a nonnegative integer. However, other dimensions will lead to different values for 푙 (see e.g. [34, Sect. 17.F]). In particular, the half-integer values arise in two dimensions and hence are equally important. Moreover, the integer case is 1 1 typically not more difficult than the case 푙 > − 2 , but the borderline case 푙 = − 2 usually imposes additional technical problems. For example in [19] we investigated the dispersive properties of the associated radial Schrödinger equation, but were 1 not able to cover the case 푙 = − 2 . This was also partly due to the fact that several 1 results we relied upon were only available for the case 푙 > − 2 . The present paper aims at filling this gap by investigating 푑2 1 i휓(푡,̇ 푥) = 퐻휓(푡, 푥), 퐻 ≔ − − + 푞(푥), (푡, 푥) ∈ ℝ × ℝ , (1.2) 푑푥2 4푥2 + Research supported by the Austrian Science Fund (FWF) under Grants No. P26060 and W1245. 320 M. Holzleitner, A. Kostenko, and G. Teschl with real locally integrable potential 푞. We will use 휏 to describe the formal Sturm– 2 Liouville differential expression, and 퐻 the self-adjoint operator acting in 퐿 (ℝ+) and given by 휏 together with the Friedrichs boundary condition at 푥 = 0:

lim 푊(√푥, 푓(푥)) = 0. (1.3) 푥→0 More specifically, our goal is to provide dispersive decay estimates for these equations. To this end we recall that under the assumption

∞ ∫ 푥(1 + |log(푥)|) |푞(푥)| 푑푥 < ∞ 0 the operator 퐻 has a purely absolutely continuous spectrum on [0, ∞) plus a finite number of eigenvalues in (−∞, 0) (see, e.g., [25, Theorem 5.1] and [29, Sect. 9.7]). Then our main result reads as follows: Theorem 1.1. Assume that 1 ∞ ∫ |푞(푥)| 푑푥 < ∞ and ∫ 푥 log2(1 + 푥) |푞(푥)| 푑푥 < ∞, (1.4) 0 1 and suppose there is no resonance at 0 (see Definition 2.17). Then the following decay holds −i푡퐻 −1/2 ‖e 푃푐(퐻)‖ 1 ∞ = 풪(|푡| ), 푡 → ∞. (1.5) 퐿 (ℝ+)→퐿 (ℝ+) 2 Here 푃푐(퐻) is the orthogonal projection in 퐿 (ℝ+) onto the continuous spectrum of 퐻. Such dispersive estimates for Schrödinger equations have a long tradition, and here we refer to a brief selection of articles [4, 5, 8, 10, 11, 14, 19, 20, 24, 32, 33], where further references can be found. We will show this result by establishing a corresponding low energy result, Theorem 3.2 (see also Theorem 3.1), and a corresponding high energy result, Theorem 3.3. Our proof is based on the approach proposed in [19]; however, the main technical difficulty is the analysis of the low and high energy behavior of the corresponding Jost function. Let us also mention that the potential 푞 ≡ 0 does not satisfy the conditions of Theorem 1.1, that is, there is a resonance at 0 in this case. However, it is known that the dispersive decay (1.5) holds true if 푞 ≡ 0 [17], and hence Theorem 1.1 states that the corresponding estimate remains true under additive non-resonant perturbations. For related results on scattering theory for such operators we refer to [2, 3]. Finally, let us briefly describe the content of the paper. Section 2 is of preliminary character, where we collect and derive some necessary estimates for solutions, the Green’s function and the high and low energy behavior of the Jost function Dispersion estimates 321

(2.29). However, we would like to emphasize that the behavior of the Jost function near the bottom of the essential spectrum is still not understood satisfactorily, and for this very reason the resonant case had to be excluded from our main theorem. The proof of Theorem 1.1 is given in Section 3. In order to make the exposition self-contained, we gathered the appropriate version of the van der Corput lemma and necessary facts on the Wiener algebra in Appendix A. Appendix B contains relevant facts about Bessel and Hankel functions.

2. Properties of solutions

In this section we will collect some properties of the solutions of the underlying differential equation required for our main results.

2.1. The regular solution. Suppose that 1 1 푞 ∈ 퐿loc(ℝ+) and ∫ 푥(1 − log(푥)) |푞(푥)| 푑푥 < ∞. (2.1) 0 Then the ordinary differential equation 푑2 1 휏푓 = 푧푓, 휏 ≔ − − + 푞(푥), 푑푥2 4푥2 has a system of solutions 휙(푧, 푥) and 휃(푧, 푥) which are real entire with respect to 푧 and such that 휋푥 2푥 휙(푧, 푥) = 휙(푧,̃ 푥), 휃(푧, 푥) = − log(푥)휃(푧,̃ 푥), (2.2) √ 2 √ 휋 where 휙(푧,̃ ⋅ ) ∈ 푊 1,1[0, 1], 휃(푧,̃ ⋅ ) ∈ 퐶[0, 1], and 휙(푧,̃ 0) = 휃(푧,̃ 0) = 1. Moreover, we can choose 휃(푧, 푥) such that lim푥→0 푊(√푥 log(푥), 휃(푧, 푥)) = 0 for all 푧 ∈ ℂ. Here 푊(푢, 푣) = 푢(푥)푣′(푥) − 푢′(푥)푣(푥) is the usual Wronski determinant. For a detailed construction of these solutions we refer to, e.g., [17]. We start with two lemmas containing estimates for the Green’s function of the unperturbed equation

퐺 1 (푧, 푥, 푦) = 휙 1 (푧, 푥)휃 1 (푧, 푦) − 휙 1 (푧, 푦)휃 1 (푧, 푥) − 2 − 2 − 2 − 2 − 2 and the regular solution 휙(푧, 푥) (see, e.g., [15, Lemmas 2.2, A.1, and A.2]). Here 휋푥 휙− 1 (푧, 푥) = 퐽0(√푧푥), 2 √ 2 (2.3) 휋푥 1 휃− 1 (푧, 푥) = ( log(푧)퐽0(√푧푥) − 푌0(√푧푥)) , 2 √ 2 휋 322 M. Holzleitner, A. Kostenko, and G. Teschl where 퐽0 and 푌0 are the usual Bessel and Neumann functions (see Appendix B). All branch cuts are chosen along the negative real axis unless explicitly stated otherwise. The first two results are essentially from [15, Appendix A]. However, since the focus there was on a finite interval, some small adaptions are necessary tocover the present case of a half-line. Lemma 2.1 ([15]). The following estimates hold:

1 2 | 2 | 푥 |Im 푘|푥 |휙− 1 (푘 , 푥)| ≤ 퐶 ( ) e , (2.4) 2 1 + |푘|푥 1 2 | 2 | 푥 | 1 + |푘|푥 | |Im 푘|푥 |휃− 1 (푘 , 푥)| ≤ 퐶 ( ) (1 + |log ( )|) e , (2.5) 2 1 + |푘|푥 | 푥 | for all 푥 > 0, and

1 1 2 2 | 2 | 푥 푦 푥 |Im 푘|(푥−푦) |퐺− 1 (푘 , 푥, 푦)| ≤ 퐶 ( ) ( ) (1 + log ( )) e (2.6) 2 1 + |푘|푥 1 + |푘|푦 푦 for all 0 < 푦 ≤ 푥 < ∞. Proof. The first two estimates are clear from the asymptotic behavior of the Bessel function 퐽0 and the Neumann function 푌0 (see (B.1), (B.2) and (B.4), (B.5)). To consider the third one, first of all we have

2 휋 퐺− 1 (푘 , 푥, 푦) = − √푥푦[퐽0(푘푥)푌0(푘푦) − 퐽0(푘푦)푌0(푘푥)] 2 2 (2.7) i휋 (1) (2) (1) (2) = − √푥푦[퐻 (푘푥)퐻 (푘푦) − 퐻 (푘푦)퐻 (푘푥)]. 4 0 0 0 0 We divide the proof of (2.6) in three steps. Step (i): |푘푦| ≤ |푘푥| ≤ 1. Using the first equality in (2.7) and employing (B.1) and (B.2), we get

| 2 | |푘|푥 푥 |퐺− 1 (푘 , 푥, 푦)| ≤ 퐶√푥푦 (1 + log ( )) = 퐶√푥푦 (1 + log ( )) , 2 |푘|푦 푦 which immediately implies (2.6). Step (ii): |푘푦| ≤ 1 ≤ |푘푥|. Using the asymptotics (B.1)–(B.5) from Appendix B, we get

| 2 | 1 |Im 푘|(푥−푦) |퐺− 1 (푘 , 푥, 푦)| ≤ 퐶√푥푦 e (1 − log(|푘|푦)) . 2 √|푘|푥 Dispersion estimates 323

We arrive at (2.6) by noting that

0 < − log(|푘|푦) ≤ log(푥/푦) since |푘|푦 ≤ 1 ≤ |푘|푥. Step (iii): 1 ≤ |푘푦| ≤ |푘푥|. For the remaining case it suffices to use the second equality in (2.7) and (B.6)–(B.7) to arrive at

| 2 | 1 |Im 푘|(푥−푦) 퐶 |Im 푘|(푥−푦) |퐺− 1 (푘 , 푥, 푦)| ≤ 퐶√푥푦 e = e , 2 √|푘|푥|푘|푦 |푘| which implies the claim.

Lemma 2.2 ([15]). Assume (2.1). Then 휙(푧, 푥) satisfies the integral equation

푥 휙(푧, 푥) = 휙 1 (푧, 푥) + ∫ 퐺 1 (푧, 푥, 푦)휙(푧, 푦)푞(푦) 푑푦. (2.8) − 2 − 2 0 Moreover, 휙( ⋅ , 푥) is entire for every 푥 > 0 and satisfies the estimate

1/2 | 2 2 | 푥 |Im 푘|푥 |휙(푘 , 푥) − 휙− 1 (푘 , 푥)| ≤ 퐶 ( ) e 2 1 + |푘|푥 푥 푦 푥 × ∫ (1 + log ( )) |푞(푦)| 푑푦 (2.9) 1 + |푘|푦 푦 0 for all 푥 > 0 and 푘 ∈ ℂ.

Proof. The proof is based on the successive iteration procedure. As in the proof of Lemma 2.2 in [15], set

∞ 푥 2 2 2 휙 = ∑ 휙푛, 휙0 = 휙 1 , 휙푛(푘 , 푥) ≔ ∫ 퐺 1 (푘 , 푥, 푦)휙푛−1(푘 , 푦)푞(푦) 푑푦 − 2 − 2 푛=0 0 for all 푛 ∈ ℕ. The series is absolutely convergent since

퐶푛+1 푥 1/2 |휙 (푘2, 푥)| ≤ ( ) e|Im 푘|푥 푛 푛! 1 + |푘|푥 푛 (2.10) 푥 푦 푥 × (∫ (1 + log ( )) |푞(푦)| 푑푦) , 푛 ∈ ℕ. 1 + |푘|푦 푦 0 This is all we need to finish the proof of this lemma.

We also need the estimates for derivatives. 324 M. Holzleitner, A. Kostenko, and G. Teschl

Lemma 2.3. The following estimates hold

3/2 2 푥 |Im 푘|푥 |휕푘휙− 1 (푘 , 푥)| ≤ 퐶|푘|푥 ( ) e (2.11) 2 1 + |푘|푥 for all 푥 > 0, and

3/2 1/2 | 2 | 푥 푦 |휕푘퐺− 1 (푘 , 푥, 푦)| ≤ 퐶|푘|푥 ( ) ( ) 2 1 + |푘|푥 1 + |푘|푦 (2.12) 푥 × (1 + log ( )) e|Im 푘|(푥−푦), 푦 for all 0 < 푦 ≤ 푥 < ∞. Proof. The first inequality follows from the identity (see[23,(10.6.3)])

2 휋푥 휕푘휙− 1 (푘 , 푥) = −푥 퐽1(푘푥) 2 √ 2 along with the asymptotic behavior of the Bessel function 퐽1 (cf. [19, Lemma 2.1]). To prove (2.12), we first calculate

2 휋 휕푘퐺− 1 (푘 , 푥, 푦) = √푥푦[푥퐽1(푘푥)푌0(푘푦) − 푦퐽1(푘푦)푌0(푘푥) 2 2

− 푥퐽0(푘푦)푌1(푘푥) + 푦퐽0(푘푥)푌1(푘푦)] (2.13) i휋 (1) (2) (1) (2) = √푥푦 [푥퐻 (푘푥)퐻 (푘푦) − 푦퐻 (푘푦)퐻 (푘푥) 4 1 0 1 0 (1) (2) (1) (2) +푥퐻0 (푘푦)퐻1 (푘푥) − 푦퐻0 (푘푥)퐻1 (푘푦)] , where we have used formulas (2.7) and the identities for derivatives of Bessel and Hankel functions (cf. Appendix B). Step (i): |푘푦| ≤ |푘푥| ≤ 1. Employing the series expansions (B.1)–(B.2) we get from the first equality in (2.13)

2 휕푘퐺 1 (푘 , 푥, 푦) − 2 휋 푘푥 2 log(푘푦) 푘푦 2 log(푘푥) 1 2 log(푘푥) 푘푥 = √푥푦[푥 − 푦 − 푥( + ) 2 4 휋 4 휋 2휋푘푥 휋 4 1 2 log(푘푦) 푘푦 + 푦( + )](1 + 풪(1)) 2휋푘푦 휋 4 휋 = √푥푦(푘푥2 + 푘푦2)( log(푘푦) − log(푘푥))(1 + 풪(1)) 2 휋 = √푥푦푘푥2 log(푦/푥)(1 + 풪(1)). 2 Dispersion estimates 325

This immediately implies the desired claim. Step (ii): |푘푦| ≤ 1 ≤ |푘푥|. Again we employ the asymptotics (B.1)–(B.5) from Appendix B to get:

2 휕푘퐺 1 (푘 , 푥, 푦) − 2 휋√푥푦 2푥 3휋 2 log(푘푦) 2 휋 = [ cos (푘푥 − ) − 푦푘푦 cos (푘푥 − ) 2 √휋푘 4 휋 √휋푘푥 4 2푥 3휋 2 휋 1 − cos (푘푥 − ) + 푦 cos (푘푥 − ) ](1 + 풪(1)) √휋푘 4 √휋푘푥 4 2휋푘푦 휋√푥푦 2푥 3휋 2 = [ cos (푘푥 − )( log(푘푦) − 1) 2 √휋푘 4 휋 2 휋 1 + cos (푘푥 − )( − 푦푘푦)](1 + 풪(1)). √휋푘푥 4 2휋푘 This gives the desired estimate, where we have to use 1/|푘| ≤ 푥 to estimate the second summand and the logarithmic expression appropriately (cf. step (ii) of 2.1). Step (iii): 1 ≤ |푘푦| ≤ |푘푥|. To deal with the remaining case we shall use the second equality in (2.13) and the asymptotic expansions of Hankel functions (B.6)–(B.7): i휋 푥푦 2 √ 2 i푘(푥−푦)−i휋/2 2 i푘(푦−푥)−i휋/2 휕푘퐺− 1 (푘 , 푥, 푦) = [푥 e − 푦 e 2 4 휋푘√푥푦 휋푘√푥푦 2 2 + 푥 ei푘(푦−푥)+i휋/2 − 푦 ei푘(푥−푦)+i휋/2](1 + 풪(1)) 휋푘√푥푦 휋푘√푥푦 푥 + 푦 = sin(푘(푥 − 푦))(1 + 풪(1)). 2i푘 This again immediately implies (2.12).

2 Lemma 2.4. Assume (2.1). Then 휕푘휙(푘 , 푥) is a solution to the integral equation

2 2 휕푘휙(푘 , 푥) = 휕푘휙 1 (푘 , 푥) − 2 푥 2 2 2 2 + ∫ [휕푘퐺 1 (푘 , 푥, 푦)휙(푘 , 푦) + 퐺 1 (푘 , 푥, 푦)휕푘휙(푘 , 푦) 푞(푦) 푑푦 (2.14) − 2 − 2 0 and satisfies the estimate 3/2 | 2 2 | 푥 |Im 푘|푥 |휕푘휙(푘 , 푥) − 휕푘휙− 1 (푘 , 푥)| ≤ 퐶|푘|푥 ( ) e (2.15) 2 1 + |푘|푥 푥 푦 푥 × ∫ (1 + log ( )) |푞(푦)| 푑푦. 1 + |푘|푦 푦 0 326 M. Holzleitner, A. Kostenko, and G. Teschl

2 Proof. Let us show that 휕푘휙(푘 , 푥) given by

∞ 2 휕푘휙 = ∑ 훽푛, 훽0(푘, 푥) = 휕푘휙 1 (푘 , 푥), (2.16) − 2 푛=0 푥 2 2 훽푛(푘, 푥) = ∫ 휕푘퐺 1 (푘 , 푥, 푦) 휙푛−1(푘 , 푦)푞(푦) 푑푦 − 2 0 푥 (2.17) 2 + ∫ 퐺 1 (푘 , 푥, 푦)훽푛−1(푘, 푦)푞(푦) 푑푦, 푛 ∈ ℕ, − 2 0 satisfies (2.14). Here 휙푛 is defined in Lemma 2.2. Using (2.10) and (2.11), we can bound the first summand in (2.17) as follows

퐶푛+1 푥 3/2 |1st term| ≤ |푘|푥 ( ) e|Im 푘|푥 (푛 − 1)! 1 + |푘|푥 푛−1 푥 푥 푦|푞(푦)| 푦 푦 푡|푞(푡)| ∫ (1 + log ( )) (∫ (1 + log ( )) 푑푡) 푑푦 푦 1 + |푘|푦 푡 1 + |푘|푡 0 0 3 푥 푛 퐶푛+1 푥 2 푥 푦|푞(푦)| ≤ |푘|푥 ( ) e|Im 푘|푥 (∫ (1 + log ( )) 푑푦) . 푛! 1 + |푘|푥 푦 1 + |푘|푦 0 Next, using induction, one can show that the second summand admits a similar bound and hence we finally get

3 푥 푛 퐶푛+1 푥 2 푥 푦|푞(푦)| |훽 (푘, 푥)| ≤ |푘|푥 ( ) e|Im 푘|푥 (∫ (1 + log ( )) 푑푦) . 푛 푛! 1 + |푘|푥 푦 1 + |푘|푦 0 This immediately implies the convergence of (2.16) and, moreover, the estimate

∞ 2 2 |휕푘휙(푘 , 푥) − 휕푘휙 1 (푘 , 푥)| ≤ ∑ |훽푛(푘, 푥)|, − 2 푛=1 from which (2.15) follows under the assumption (2.1).

Furthermore, by [9, 7, 30] (see also [12]), the regular solution 휙 admits a representation by means of transformation operators preserving the behavior of solutions at 푥 = 0 (see also [6, Chap. III] for further details and historical remarks).

1 Lemma 2.5. Suppose 푞 ∈ 퐿loc([0, ∞)). Then

푥 휙(푧, 푥) = 휙 1 (푧, 푥) + ∫ 퐵(푥, 푦)휙 1 (푧, 푦) 푑푦 = (퐼 + 퐵)휙 1 (푧, 푥), (2.18) − 2 − 2 − 2 0 Dispersion estimates 327

2 where the so-called Gelfand–Levitan kernel 퐵∶ ℝ+ → ℝ satisfies the estimate

1 푥 + 푦 푥 |퐵(푥, 푦)| ≤ 휎 ( ) e휍1(푥), 휎 (푥) = ∫ 푠푗|푞(푠)| 푑푠, (2.19) 2 0 2 푗 0 for all 0 < 푦 < 푥 and 푗 ∈ {0, 1}. In particular, this lemma immediately implies the following useful result. Corollary 2.6. Suppose 푞 ∈ 퐿1((0, 1)). Then 퐵 is a bounded operator on 퐿∞((0, 1)). Proof. If 푓 ∈ 퐿∞((0, 1)), then using the estimate (2.19) we get

푥 푥 | | | | |(퐵푓)(푥)| = |∫ 퐵(푥, 푦)푓(푦) 푑푦| ≤ ‖푓‖∞ ∫ |퐵(푥, 푦)| 푑푦 0 0 1 푥 푥 + 푦 1 ≤ ‖푓‖ e휍1(1) ∫ 휎 ( ) 푑푦 ≤ ‖푓‖ e휍1(1)휎 (1), 2 ∞ 0 2 2 ∞ 0 0 which proves the claim.

Remark 2.7. Note that 퐵 is a bounded operator on 퐿2((0, 푎)) for all 푎 > 0. However, the estimate (2.19) allows to show that its norm behaves like 풪(푎) as 푎 → ∞ and 2 hence 퐵 might not be bounded on 퐿 (ℝ+).

2.2. The Jost solution and the Jost function. In this subsection, we assume that the potential 푞 belongs to the Marchenko class, i.e., in addition to (2.1), 푞 also satisfies ∞ ∫ 푥 log(1 + 푥) |푞(푥)| 푑푥 < ∞. (2.20) 1 Recall that under these assumptions on 푞 the spectrum of 퐻 is purely absolutely continuous on (0, ∞) with an at most finite number of eigenvalues 휆푛 ∈ (−∞, 0). A solution 푓(푘, ⋅ ) to 휏푦 = 푘2푦 with 푘 ≠ 0 satisfying the following asymptotic normalization

푓(푘, 푥) = ei푘푥(1 + 표(1)), 푓′(푘, 푥) = i푘ei푘푥(1 + 표(1)) (2.21) as 푥 → ∞, is called the Jost solution. In the case 푞 ≡ 0, we have (cf. (B.6))

i휋/4 휋푥푘 (1) 푓− 1 (푘, 푥) = e 퐻0 (푘푥), (2.22) 2 √ 2

(1) which is analytic in ℂ+ and continuous in ℂ+⧵{0}. Here 퐻휈 is the Hankel function of the first kind (see Appendix B). Using the estimates for Hankel functions we 328 M. Holzleitner, A. Kostenko, and G. Teschl obtain

|푘|푥 1/2 |푘|푥 | | −|Im 푘|푥 −|Im 푘|푥 (2.23) |푓− 1 (푘, 푥)| ≤ 퐶 ( ) e (1 − log ( )) ≤ 퐶e 2 1 + |푘|푥 1 + |푘|푥 for all 푥 > 0. Notice that for the second inequality in (2.23) we have to use the fact that the function 푥 ↦ √푥/(푥 + 1) log (푥/(푥 + 1)) is bounded on ℝ+. Lemma 2.8. Assume (2.20). Then the Jost solution satisfies the integral equation

∞ 2 푓(푘, 푥) = 푓 1 (푘, 푥) − ∫ 퐺 1 (푘 , 푥, 푦)푓(푘, 푦)푞(푦) 푑푦. (2.24) − 2 − 2 푥 For all 푥 > 0, 푓( ⋅ , 푥) is analytic in the upper half plane, and can be continuously extended to the real axis away from 푘 = 0, and

1/2 푥 −|Im 푘| 푥 |푓(푘, 푥) − 푓− 1 (푘, 푥)| ≤ 퐶 ( ) e (2.25) 2 1 + |푘|푥 ∞ 푦 1/2 푦 × ∫ ( ) (1 + log ( )) |푞(푦)| 푑푦. | | 푥 푥 1 + 푘 푦 Proof. The proof is based on the successive iteration procedure. Set

∞ ∞ 2 푓 = ∑ 푓푛, 푓0 = 푓 1 , 푓푛(푘, 푥) = − ∫ 퐺 1 (푘 , 푥, 푦)푓푛−1(푘, 푦)푞(푦) 푑푦 − 2 − 2 푛=0 푥 for all 푛 ∈ ℕ. The series is absolutely convergent since

퐶푛+1 푥 1/2 |푓 (푘, 푥)| ≤ ( ) e−|Im 푘|푥 푛 푛! 1 + |푘|푥 푛 ∞ 푦 1/2 푦 × (∫ ( ) (1 + log ( )) |푞(푦)| 푑푦) | | 푥 푥 1 + 푘 푦 holds for all 푛 ∈ ℕ. The latter also proves (2.25).

Furthermore, by [9, 7, 26, 27] (see also [12]), the Jost solution 푓 admits a representation by means of transformation operators preserving the behavior of solutions at infinity.

Lemma 2.9 ([26, 27]). Assume (2.20) and let 푘 ≠ 0. Then

∞ 푓(푘, 푥) = 푓 1 (푘, 푥) + ∫ 퐾(푥, 푦)푓 1 (푘, 푦) 푑푦 = (퐼 + 퐾)푓 1 (푘, 푥), (2.26) − 2 − 2 − 2 푥 Dispersion estimates 329 where the so-called Marchenko kernel 퐾∶ ℝ2 → ℝ satisfies the estimate

∞ 푥+푦 푐0 푥 + 푦 푐 휍̃ (푥)−휍̃ ( ) 푗 |퐾(푥, 푦)| ≤ ̃휎 ( ) e 0 1 1 2 , ̃휎 (푥) = ∫ 푠 |푞(푠)| 푑푠, (2.27) 2 0 2 푗 푥 for all 푥 < 푦 < ∞. Here 푐0 is a positive constant given by

∞ 2 1/2 1/2, 1/2 1/2 ((1/2)푛) 푛 푐0 ≔ sup (1 − 푠) 2퐹1( ; 푠) = sup (1 − 푠) ∑ 2 푠 . 푠∈(0,1) 1 푠∈(0,1) 푛=0 (푛!)

Notice that 푐0 is finite in view of[23,(15.4.21)]. Moreover, this lemma immediately implies the following useful result.

Corollary 2.10. If (2.20) holds, then 퐾 is a bounded operator on 퐿∞((1, ∞)).

Proof. If 푓 ∈ 퐿∞((1, ∞)), then using the estimate (2.27) we get

∞ ∞ | | | | |(퐾푓)(푥)| = |∫ 퐾(푥, 푦)푓(푦) 푑푦| ≤ ‖푓‖∞ ∫ |퐾(푥, 푦)| 푑푦 푥 푥 ∞ 푐0 1 + 푦 ≤ ‖푓‖ e푐0휍̃1(푥) ∫ ̃휎 ( ) 푑푦 2 ∞ 0 2 1 ∞ 푐0휍̃1(1) 푐0휍̃1(1) ≤ 푐0‖푓‖∞e ∫ 0̃휎 (푠) 푑푠 = 푐0‖푓‖∞( 1̃휎 (1) −0 ̃휎 (1))e , 1 which proves the claim.

By Lemma 2.8, the Jost solution is analytic in the upper half plane, and can be continuously extended to the real axis away from 푘 = 0. We can extend it to the lower half plane by setting 푓(푘, 푥) = 푓(−푘, 푥) = 푓(푘∗, 푥)∗ for Im(푘) < 0 (here and below we denote the complex conjugate of 푧 by 푧∗). For 푘 ∈ ℝ ⧵ {0} we obtain two solutions 푓(푘, 푥) and 푓(−푘, 푥) = 푓(푘, 푥)∗ of the same equation whose Wronskian is given by (cf. (2.21)) 푊(푓(−푘, ⋅ ), 푓(푘, ⋅ )) = 2i푘. (2.28) The Jost function is defined as

푓(푘) ≔ 푊(푓(푘, ⋅ ), 휙(푘2, ⋅ )) (2.29) and we also set 푔(푘) ≔ 푊(푓(푘, ⋅ ), 휃(푘2, ⋅ )) such that 푓(푘, 푥) = 푓(푘)휃(푘2, 푥) − 푔(푘)휙(푘2, 푥). (2.30) 330 M. Holzleitner, A. Kostenko, and G. Teschl

In particular, the function given by 푔(푘) 푚(푘2) ≔ − , 푘 ∈ ℂ , 푓(푘) + is called the Weyl 푚-function (we refer to [16, 18] for further details). Note that both 푓(푘) and 푔(푘) are analytic in the upper half plane, and 푓(푘) has simple zeros at i휅푛 = √휆푛 ∈ ℂ+. Since 푓(푘, 푥)∗ = 푓(−푘, 푥) for 푘 ∈ ℝ ⧵ {0}, we obtain 푓(푘)∗ = 푓(−푘) and 푔(푘)∗ = 푔(−푘). Moreover, (2.28) shows 푓(−푘) 푓(푘) 휙(푘2, 푥) = 푓(푘, 푥) − 푓(−푘, 푥), 푘 ∈ ℝ ⧵ {0}, (2.31) 2i푘 2i푘 and by (2.30) we get

2i Im(푓(푘)푔(푘)∗) = 푓(푘)푔(푘)∗ − 푓(푘)∗푔(푘) = 푊(푓(−푘, ⋅ ), 푓(푘, ⋅ )) = 2i푘.

Moreover,

Im (푓(푘)∗푔(푘)) 푘 Im 푚(푘2) = − = , 푘 ∈ ℝ ⧵ {0}. (2.32) |푓(푘)|2 |푓(푘)|2 Note that

2 −i 휋 푓 1 (푘) = 푊(푓 1 (푘, ⋅ ), 휙 1 (푘 , ⋅ )) = √푘e 4 , 0 ≤ arg(푘) < 휋. − 2 − 2 − 2 Thus, by [18, Theorem 2.1] (see also Eq. (5.15) in [18] or [13]), on the real line we have |푓(푘)| = √|푘|(1 + 표(1)), 푘 → ∞. (2.33)

2.3. High and low energy behavior of the Jost function. Consider the following function 푓(푘) 퐹(푘) = = ei휋/4 푘−1/2푓(푘) = ei휋/4 푘−1/2푊(푓(푘, .), 휙(푘2, .)), Im 푘 ≥ 0. 푓 1 (푘) − 2 (2.34) Let us summarize the basic properties of 퐹.

Lemma 2.11. The function 퐹 defined by (2.34) is analytic in ℂ+ and continuous in ∗ ℂ+ ⧵ {0}. Moreover, 퐹(푘) = 퐹(−푘) ≠ 0 for all 푘 ∈ ℝ ⧵ {0} and

|퐹(푘)| = 1 + 표(1) (2.35) as 푘 ∈ ℝ tends to ∞. Dispersion estimates 331

Proof. The first claim follows from the corresponding properties of the Jostfunc- tion. Next, (2.31) implies that 푓(푘) ≠ 0 for all 푘 ∈ ℝ ⧵ {0}. Finally, (2.35) follows from (2.33).

The analysis of the behavior of 퐹 near zero is much more delicate. We start with the following integral representation.

Lemma 2.12 ([18]). Assume (2.1) and (2.20). Then the function 퐹 admits the integral representation

∞ i휋/4 −1/2 2 퐹(푘) = 1+e 푘 ∫ 푓 1 (푘, 푥)휙(푘 , 푥)푞(푥) 푑푥 (2.36) − 2 0 ∞ i휋/4 −1/2 2 = 1 + e 푘 ∫ 푓(푘, 푥)휙 1 (푘 , 푥)푞(푥) 푑푥 − 2 0 for all 푘 ∈ ℂ+ ⧵ {0}. Proof. To prove the integral representations (2.36), we need to replace 휙 and 푓 in (2.34) by (2.8) and (2.24), respectively, use the asymptotic estimates for 휙, 푓 and 퐺 1 , and then take the limits 푥 → +∞ and 푥 → 0. − 2 Corollary 2.13. Assume in addition that 푞 satisfies

∞ ∫ 푥 log2(1 + 푥) |푞(푥)| 푑푥 < ∞. (2.37) 1 Then for 푘 > 0 the integral representation (2.36) can be rewritten as follows

∞ 2 2 퐹(푘) = 1+ ∫ 휃 1 (푘 , 푥)휙(푘 , 푥)푞(푥) 푑푥 − 2 0 ∞ (2.38) 1 2 2 2 + (i − log(푘 )) ∫ 휙− 1 (푘 , 푥)휙(푘 , 푥)푞(푥) 푑푥. 휋 2 0 Proof. Indeed, the integrals converge for all 푘 ∈ ℝ ⧵ {0} due to (2.4), (2.5) and (2.9). Then (2.38) follows from the first formula in (2.36) since (cf. (2.3) and (2.22))

1 휋 1 2 2 2 i 4 − 2 휃− 1 (푘 , 푥) − log(−푘 )휙− 1 (푘 , 푥) = e 푘 푓− 1 (푘, 푥). 2 휋 2 2 Notice also that it suffices to consider only positive 푘 > 0 since 퐹(−푘) = 퐹(푘)∗ by Lemma 2.12.

Before proceeding further, we need the following simple facts. 332 M. Holzleitner, A. Kostenko, and G. Teschl

Lemma 2.14. Suppose that 푞 satisfies (2.1) and (2.37). Then

∞ 휋 ∫ 휙− 1 (0, 푠)휙(0, 푠)푞(푠) 푑푠 = lim 푊(√푥, 휙(0, 푥)), (2.39) 2 √ 2 푥→∞ 0 ∞ 2 ∫ 휃− 1 (0, 푠)휙(0, 푠)푞(푠) 푑푠 = −1 − lim 푊(√푥 log(푥), 휙(0, 푥)). (2.40) 2 √휋 푥→∞ 0 Proof. First observe that the integrals on the left-hand side are finite, since

휋푥 2푥 휙− 1 (0, 푥) = , 휃− 1 (0, 푥) = − log(푥), 2 √ 2 2 √ 휋 and 푞 satisfies (2.1) and (2.37). Now notice that 푥 푥 ″ 1 ∫ 휙− 1 (0, 푠)휙(0, 푠)푞(푠) 푑푠 = ∫ 휙− 1 (0, 푠)(휙 (0, 푠) + 휙(0, 푠)) 푑푠 2 2 4푠2 0 0 ″ 2 since 휏휙 = 0. Integrating by parts and noting that 휙 1 (0, 푥) solves 푦 +1/4푥 푦 = 0, − 2 we get 푥 휋 ∫ 휙− 1 (0, 푠)휙(0, 푠)푞(푠) 푑푠 = 푊(√푥, 휙(0, 푥)) 2 √ 2 0 since 푊(√푥, 휙(0, 푥)) → 0 as 푥 → 0. Passing to the limit as 푥 → ∞, we arrive at (2.39). The proof of (2.40) is analogous. Lemma 2.15. Assume the conditions of Lemma 2.14. Then the equation 1 휏푦 = −푦″ − 푦 + 푞(푥)푦 = 0 4푥2 has two linearly independent solution 푦1 and 푦2 such that

′ 1 푦1(푥) = √푥(1 + 표(1)), 푦1(푥) = (1 + 표(1)) (2.41) 2√푥 and

′ log(√푥) 푦2(푥) = √푥 log(푥)(1 + 표(1)), 푦2(푥) = (1 + 표(1)) (2.42) √푥 as 푥 → ∞. Proof. The proof is based on successive iteration. Namely, each solution to 휏푦 = 0 solves the integral equation ∞ 푓(푥) = 푎√푥 + 푏√푥 log(푥) − ∫ √푥푠 log(푥/푠)푓(푠)푞(푠) 푑푠. 푥 Dispersion estimates 333

Since the argument is fairly standard we only provide some details for 푦2(푥); the calculations for 푦1(푥) are similar. For simplicity we set 푥 > e, which is no restriction since we only need estimates for large 푥 anyway. As in, e.g., Lemma 2.2 we set ∞

푦2(푥) = ∑ 휙푛, 휙0(푥) ≔ √푥 log(푥), 푛=0 ∞

휙푛(푥) ≔ − ∫ √푥푠 log(푥/푠)휙푛−1(푠)푞(푠) 푑푠. 푥 Since log(푠/푥) ≤ log(푥) log(푠) for all e ≤ 푥 ≤ 푠 < ∞, we immediately get ∞ ∞ 2 |휙1(푥)| ≤ ∫ √푥푠 log(푠/푥)√푠 log(푠) |푞(푠)| 푑푠 ≤ √푥 log(푥) ∫ 푠 log (푠) |푞(푠)| 푑푠, 푥 푥 and then inductively we obtain that

∞ ′ 1 + log(√푥) 푦2(푥) = ∑ 훽푛, 훽0(푥) ≔ , 푛=0 √푥 ∞ 푠 훽 (푥) ≔ − ∫ (1 + log(√푥/푠)) 훽 (푠)푞(푠) 푑푠. 푛 √푥 푛−1 푥 Iteration then gives

푛 퐶푛+1 1 + log(√푥) ∞ |훽 (푥)| ≤ (∫ 푠 log2(푠) |푞(푠)| 푑푠) 푛 푛! √푥 푥 for all 푛 ∈ ℕ and 푥 ≥ e, since

1 + log(푥/푠) ≤ (1 + log(푥))(1 + log(푠)) ≤ 2 log(푠)(1 + log(푥)) 334 M. Holzleitner, A. Kostenko, and G. Teschl for all e ≤ 푥 ≤ 푠 < ∞. Thus we end up with the estimate | | ∞ ′ 1 + log(√푥) 1 + log(√푥) 2 |푦2(푥) − | ≤ 퐶 ∫ 푠 log (푠) |푞(푠)| 푑푠, 푥 ≥ e, (2.44) | √푥 | √푥 푥 which completes the proof. Now we are in position to characterize the behavior of 퐹 near 0. Lemma 2.16. Suppose that 푘 > 0 and 푞 satisfies (2.1) and (2.37). Then 1 퐹(푘) = 퐹 (푘) + (i − log(푘2))퐹 (푘), 푘 ≠ 0, (2.45) 1 휋 2 where 퐹1 and 퐹2 are continuous real-valued functions on ℝ. Moreover, 휋 퐹2(0) = lim 푊(√푥, 휙(0, 푥)) = 0 (2.46) √ 2 푥→∞ precisely when 휙(0, 푥) = 풪(√푥) as 푥 → ∞. In the latter case

2 2 퐹(푘) = 퐹1(0) + 풪(푘 log(−푘 )), 푘 → 0, (2.47) with 2 퐹1(0) = − lim 푊(√푥 log(푥), 휙(0, 푥)) ≠ 0. (2.48) √휋 푥→∞ Proof. The first claim follows from the integral representation (2.38) since the corresponding integrals are continuous in 푘 by the dominated convergence theorem. 2 2 Moreover, 휙(푘 , 푥) and 휃(푘 , 푥) are real if 푘 ∈ ℝ, and hence so are 퐹1 and 퐹2. By Lemma 2.15, 휙(0, 푥) = 푎푦1(푥) + 푏푦2(푥), where the asymptotic behavior of 푦1 and 푦2 is given by (2.41) and (2.42), respectively. Combining Lemma 2.14 with the representation (2.38), we conclude that 퐹2(0) = 푏√휋/2 ≠ 0 in (2.45) precisely when 푏 ≠ 0, and hence the second claim follows. Assume now that 퐹2(0) = 0, which is equivalent to the equality 휙(0, 푥) = 푎푦1(푥) with 푎 = √휋/2퐹1(0) ≠ 0. Noting that both 휙 1 ( ⋅ , 푥) and 휙( ⋅ , 푥) are analytic for − 2 each 푥 > 0 and applying the dominated convergence theorem once again, we conclude that ∞ 2 2 2 ∫ 휙 1 (푘 , 푥)휙(푘 , 푥)푞(푥) 푑푥 = 풪(푘 ), 푘 → 0. − 2 0 This immediately proves (2.47).

Definition 2.17. We shall say that there is a resonance at 0 if 휙(0, 푥) = 풪(√푥) as 푥 → ∞. Dispersion estimates 335

Let us mention that there is a resonance at 0 if 푞 ≡ 0 since in this case 휙(0, 푥) =

휙 1 (0, 푥) = √휋푥/2. − 2 We finish this section with the following estimate. Lemma 2.18. Assume that 푞 satisfies (2.1) and (2.20). Then 퐹 is differentiable for all 푘 ≠ 0, and 퐶 |퐹′(푘)| ≤ , 푘 ≠ 0. |푘| Proof. Setting

푓− 1 (푘, 푥) ̃ 2 i휋/4 −1/2 푓− 1 (푘, 푥) ≔ = e 푘 푓− 1 (푘, 푥), 2 푓 1 (푘) 2 − 2 we find that its derivative is given by (cf.[23,(10.6.3)])

̃ 휋푥 (1) 휕푘푓− 1 (푘, 푥) = −i푥 퐻1 (푘푥). 2 √ 2 Similar to (2.23), we obtain the estimate √푥(1 + |푘|푥) | ̃ | −|Im 푘|푥 |휕푘푓− 1 (푘, 푥)| ≤ 퐶 e , (2.49) 2 |푘| which holds for all 푥 > 0. Using (2.36), we get ∞ ′ ̃ 2 ̃ 2 퐹 (푘) = ∫ (휕푘푓 1 (푘, 푥)휙(푘 , 푥) + 푓 1 (푘, 푥)휕푘휙(푘 , 푥)) 푞(푥) 푑푥. − 2 − 2 0 The integral converges absolutely for all 푘 ≠ 0. Indeed, we have 푥 1 + log ( ) ≤ (1 + | log(푥)|)(1 + | log(푦)|), 0 < 푦 ≤ 푥. (2.50) 푦 By (2.15), (2.23) and also (2.50), we obtain | ∞ | ̃ 2 |∫ 푓 1 (푘, 푥)휕푘휙(푘 , 푥)푞(푥) 푑푥| − 2 | 0 | ∞ 푥 3/2 ≤ 퐶 ∫ √|푘|푥 ( ) (1 + | log(푥) |)|푞(푥)| 푑푥 1 + |푘|푥 0 퐶 ∞ ≤ ∫ 푥(1 + | log(푥)|) |푞(푥)| 푑푥. |푘| 0 Using (2.9) and (2.49) (again in combination with (2.50)), we get the following estimates for the first summand: | ∞ | ∞ ̃ 2 퐶 |∫ 휕푘푓− 1 (푘, 푥)휙(푘 , 푥)푞(푥) 푑푥| ≤ ∫ 푥(1 + | log(푥)|) |푞(푥)| 푑푥. 2 |푘| | 0 | 0 Now the claim follows. 336 M. Holzleitner, A. Kostenko, and G. Teschl

3. Dispersive decay

In this section we prove the dispersive decay estimate (1.5) for the Schrödinger equation (1.2). In order to do this, we divide the analysis into low and high energy regimes. In the analysis of both regimes we make use of variants of the van der Corput lemma (see Appendix A), combined with a Born series approach for the high energy regime suggested in [10] and adapted to our setting in [19].

3.1. The low energy part. For the low energy regime, it is convenient to use −i푡퐻 the following well-known representation of the integral kernel of e 푃푐(퐻), ∞ 2 2 [e−i푡퐻푃 (퐻) (푥, 푦) = ∫ e−i푡푘 휙(푘2, 푥)휙(푘2, 푦) Im 푚(푘2)푘 푑푘 푐 휋 −∞ ∞ 2 2 2 2 2 휙(푘 , 푥)휙(푘 , 푦)푘 = ∫ e−i푡푘 푑푘 (3.1) 휋 2 −∞ |푓(푘)| ∞ 2 2 휙(푘,̃ 푥)휙(푘,̃ 푦) = ∫ e−i푡푘 푑푘, 휋 2 −∞ |퐹(푘)| where the integral is to be understood as an improper integral. In fact, adding an additional energy cut-off (which is all we will need below), the formula is immediate from the spectral transformation [16, §3], and the general case can then be established by taking limits (see [19] for further details). In the last equality we have used

1 2 휙(푘,̃ 푥) ≔ |푘| 2 휙(푘 , 푥), 푘 ∈ ℝ. (3.2)

Note that

1 푥 |푘|푥 2 푥 푦|푞(푦)| |휙(푘,̃ 푥)| ≤ 퐶 ( ) e| Im 푘|푥 (1 + ∫ (1 + log ( )) 푑푦) , (3.3) 1 + |푘|푥 푦 1 + |푘|푦 0 − 1 푥 |푘|푥 2 푥 푦|푞(푦)| |휕 휙(푘,̃ 푥)| ≤ 퐶푥 ( ) e| Im 푘|푥 (1 + ∫ (1 + log ( )) 푑푦) , 푘 1 + |푘|푥 푦 1 + |푘|푦 0 (3.4) which follow from (2.4), (2.9) and the equality

1 − 1 2 1 2 휕 휙(푘,̃ 푥) = sgn(푘)|푘| 2 휙(푘 , 푥) + |푘| 2 휕 휙(푘 , 푥) 푘 2 푘 together with (2.11), (2.15). We begin with the following estimate. Dispersion estimates 337

∞ Theorem 3.1. Assume (2.1) and (2.37). Let 휒 ∈ 퐶푐 (ℝ) with supp(휒) ⊂ (−푘0, 푘0). Then | −i푡퐻 | − 1 |[e 휒(퐻)푃푐(퐻) (푥, 푦)| ≤ 퐶√푥푦 |푡| 2 (3.5) for all 푥, 푦 ≤ 1. Proof. We want to apply the van der Corput Lemma A.1 to the integral ∞ 2 2 휙(푘,̃ 푥)휙(푘,̃ 푦) 퐼(푡, 푥, 푦) ≔ [e−i푡퐻휒(퐻)푃 (퐻) (푥, 푦) = ∫ e−i푡푘 휒(푘2) 푑푘. 푐 휋 2 −∞ |퐹(푘)| Denote 휙(푘,̃ 푥)휙(푘,̃ 푦) 퐴(푘) = 휒(푘2)퐴 (푘), 퐴 (푘) = . 0 0 |퐹(푘)|2 Note that

′ ′ ′ ‖퐴‖∞ ≤ ‖휒‖∞‖퐴0‖∞, ‖퐴 ‖1 ≤ ‖휒 ‖1‖퐴0‖∞ + ‖휒‖1‖퐴0‖∞. By Lemma 2.11, 퐹(푘) ≠ 0 for all 푘 ∈ ℝ ⧵ {0}. Moreover, combining (2.35) with Lemma 2.16, we conclude that ‖1/퐹‖∞ < ∞. Using (3.3) and noting that log(푥/푦) ≤ log(1/푦) for all 0 < 푦 ≤ 푥 ≤ 1, we get |푘|푥 1/2 |휙(푘,̃ 푥)| ≤ 퐶 ( ) e| Im 푘|푥, 푥 ∈ (0, 1]. (3.6) 1 + |푘|푥 Therefore, 2 sup |퐴0(푘)| ≤ 퐶 ‖1/퐹‖∞|푘0|√푥푦, (3.7) 푘∈[−푘0,푘0] which holds for all 푥, 푦 ∈ (0, 1] with some uniform constant 퐶 > 0. Next, we get ̃ ̃ ̃ ̃ ′ ′ 휕푘휙(푘, 푥)휙(푘, 푦) + 휙(푘, 푥)휕푘휙(푘, 푦) 퐹 (푘) 퐴0(푘) = − 퐴0(푘) Re . |퐹(푘)|2 퐹(푘) To consider the second term, we infer from (3.6), Lemma 2.16, and Lemma 2.18 that | 퐹′(푘)| |휙(푘,̃ 푥)휙(푘,̃ 푦)||퐹′(푘)| |퐴 (푘) Re | ≤ | | ≤ 퐶√푥푦. | 0 퐹(푘) | |퐹(푘)|2 | 퐹(푘) | The estimate for the first term follows from (3.6) and (3.4) since ̃ ̃ ̃ ̃ |휕푘휙(푘, 푥)휙(푘, 푦) + 휙(푘, 푥)휕푘휙(푘, 푦)| |푘|푥 1/2 |푘|푦 1/2 1 + |푘|푥 1 + |푘|푦 ≤ 퐶 ( ) ( ) ( + ) 1 + |푘|푥 1 + |푘|푦 |푘| |푘| 1 + |푘|푥 + 1 + |푘|푦 ≤ 퐶√푥푦 ≤ 2퐶(1 + |푘|)√푥푦, 푥, 푦 ∈ (0, 1]. √(1 + |푘|푥)(1 + |푘|푦) 338 M. Holzleitner, A. Kostenko, and G. Teschl

The claim now follows by applying the classical van der Corput Lemma (see [28, page 334]), or by noting that 퐴 ∈ 풲0(ℝ) in view of Lemma A.2, and then it remains to apply Lemma A.1.

Theorem 3.2. Assume

1 ∞ ∫ |푞(푥)| 푑푥 < ∞ and ∫ 푥 log2(1 + 푥) |푞(푥)| 푑푥 < ∞. (3.8) 0 1

∞ Let also 휒 ∈ 퐶푐 (ℝ) with supp(휒) ⊂ (−푘0, 푘0). If 휙(0, 푥)/√푥 is unbounded near ∞, then | −i푡퐻 | − 1 |[e 휒(퐻)푃푐(퐻) (푥, 푦)| ≤ 퐶|푡| 2 , (3.9) whenever max(푥, 푦) ≥ 1.

Proof. Assume that 0 < 푥 ≤ 1 ≤ 푦. We proceed as in the previous proof, but use Lemma 2.5 and Lemma 2.9 to write ̃ ̃ (퐼 + 퐵푥)휙− 1 (푘, 푥) ⋅ (퐼 + 퐾푦)휙− 1 (푘, 푦) 퐴(푘) = 휒(푘2) 2 2 , 푘 ≠ 0. |퐹(푘)|2

Indeed, for all 푘 ∈ ℝ ⧵ {0}, 휙(푘2, ⋅ ) admits the representation (2.31). Therefore, ̃ ̃ by Lemma 2.9, 휙(푘, 푦) = (퐼 + 퐾푦)휙 1 (푘, 푦) for all 푘 ∈ ℝ ⧵ {0}. − 2 By symmetry, 퐴(푘) = 퐴(−푘), and hence our integral reads

∞ 4 2 퐼(푡, 푥, 푦) = ∫ e−i푡푘 퐴(푘) 푑푘. 휋 0 Let us show that the individual parts of 퐴(푘) coincide with a function which is the Fourier transform of a finite measure. Clearly, we can redefine 퐴(푘) for 푘 < 0. To ̃ 2 this end note that 휙 1 (푘 , 푥) = 퐽(|푘|푥), where 퐽(푟) = √푟퐽0(푟). Note that 퐽(푟) ∼ √푟 − 2 √ 휋 −1 as 푟 → 0 and 퐽(푟) = 2/휋 cos(푟 − 4 ) + 푂(푟 ) as 푟 → +∞ (see (B.4)). Moreover, ′ ′ √ 휋 −1 퐽 (푟) ∼ 1/2√푟 as 푟 → 0 and 퐽 (푟) = 2/휋 cos(푟 + 4 ) + 푂(푟 ) as 푟 → +∞ (see (B.8)). Moreover, we can define 퐽(푟) for 푟 < 0 such that it is locally in 퐻1 and √ 휋 ̃ 2 퐽(푟) = 2/휋 cos(푟 − 4 ) for 푟 < −1. By construction we then have 퐽 ∈ 퐿 (ℝ) and 푝 퐽̃ ∈ 퐿 (ℝ) for all 푝 ∈ (1, 2). By Lemma A.2, 퐽̃ ∈ 풲0 and hence 퐽̃ is the Fourier 휋 transform of an integrable function. Moreover, cos(푟 − 4 ) is the Fourier transform of the sum of two Dirac delta measures and so 퐽 is the Fourier transform of a finite measure. By scaling, the total variation of the measures corresponding to 퐽(푘푥) is independent of 푥. 2 −2 Let us show that 휒(푘 )|퐹(푘)| belongs to the Wiener algebra 풲0(ℝ). As in Lemma A.3, we define the functions 푓0 and 푓1. Since 휙(0, 푥)/√푥 is unbounded Dispersion estimates 339 near ∞, by Lemma 2.16 we conclude that 퐹(푘) = log(푘2)(푐 + 표(1)) as 푘 → 0 with some 푐 ≠ 0. Hence Lemma 2.18 yields

| 푑 1 | | 1 퐹′(푘) | |퐹′(푘)| 퐶 | | = |− 2 Re ( )| ≤ 2 ≤ |푑푘 |퐹(푘)|2 | | |퐹(푘)|2 퐹(푘)∗ | |퐹(푘)|3 |푘|| log(푘)|3 for 푘 near zero, which implies that 1 푓1(푘) ≤ 퐶 , 푘 ∈ (0, 1). 푘 log3(2/푘)

Therefore, we get

1 1 푑푘 1/2 푑푘 퐶 ∫ log (2/푘)푓 (푘) 푑푘 ≤ 퐶 ∫ = 퐶 ∫ = < ∞. 1 2 2 log 2 0 0 푘 log (2/푘) 0 푘 log (푘) Noting that the second condition in (A.3) is satisfied since 휒 has compact support 2 −2 and hence so are 푓0 and 푓1. Therefore Lemma A.3 implies that 휒(푘 )|퐹(푘)| belongs to the Wiener algebra 풲0(ℝ). Lemma A.1 then shows

∞ 휙̃ 1 (푘, 푥)휙̃ 1 (푘, 푦) 퐶 4 2 − − |퐼(푡,̃ 푥, 푦)| ≤ , 퐼(푡,̃ 푥, 푦) ≔ ∫ e−i푡푘 휒(푘2) 2 2 푑푘. 휋 2 √푡 0 |퐹(푘)|

But by Fubini we have 퐼(푡, 푥, 푦) = (1 + 퐵푥)(1 + 퐾푦)퐼(푡,̃ 푥, 푦), and the claim follows since both 퐵∶ 퐿∞((0, 1)) → 퐿∞((0, 1)) and 퐾∶ 퐿∞((1, ∞)) → 퐿∞((1, ∞)) are bounded in view of Corollary 2.6 and Corollary 2.10, respectively. By symmetry, we immediately obtain the same estimate if 0 < 푦 ≤ 1 ≤ 푥. The case min(푥, 푦) ≥ 1 can be proved analogously; we only need to write

̃ ̃ (퐼 + 퐾푥)휙− 1 (푘, 푥) ⋅ (퐼 + 퐾푦)휙− 1 (푘, 푦) 퐴(푘) = 휒(푘2) 2 2 , 푘 ≠ 0. |퐹(푘)|2

3.2. The high energy part. For the analysis of the high energy regime we use the following – also well-known – alternative representation:

1 ∞ e−i푡퐻푃 (퐻) = ∫ e−i푡휔[ℛ (휔 + i0) − ℛ (휔 − i0) 푑휔 푐 2휋i 퐻 퐻 0 ∞ 1 2 = ∫ e−i푡푘 ℛ (푘2 + i0) 푘 푑푘, (3.10) 휋i 퐻 −∞ 340 M. Holzleitner, A. Kostenko, and G. Teschl

−1 where ℛ퐻(휔) = (퐻 − 휔) is the resolvent of the Schrödinger operator 퐻 and the limit is understood in the strong sense (see, e.g., [29]). We recall that for 푘 ∈ ℝ ⧵ {0} the Green’s function is given by

푓(±푘, 푦) [ℛ (푘2 ± i0) (푥, 푦) = [ℛ (푘2 ± i0) (푦, 푥) = 휙(푘2, 푥) , 푥 ≤ 푦. (3.11) 퐻 퐻 푓(±푘)

∞ Fix 푘0 > 0 and let 휒∶ ℝ → [0, ∞) be a 퐶 function such that

0, |푘| < 2푘 , 휒(푘2) = { 0 (3.12) 1, |푘| > 3푘0.

The purpose of this section is to prove the following estimate.

1 Theorem 3.3. Suppose 푞 ∈ 퐿 (ℝ+) satisfies (2.20). Then

−i푡퐻 − 1 |[e 휒(퐻)푃푐(퐻)](푥, 푦)| ≤ 퐶|푡| 2 .

Our starting point is the fact that the resolvent ℛ퐻 of 퐻 can be expanded into the Born series ∞ 2 2 2 푛 ℛ퐻(푘 ± i0) = ∑ ℛ 1 (푘 ± i0)(−푞 ℛ 1 (푘 ± i0)) , (3.13) − 2 − 2 푛=0 where ℛ 1 stands for the resolvent of the unperturbed radial Schrödinger operator. − 2 To this end we begin by collecting some facts about ℛ 1 . Its kernel is given by − 2

2 1 ℛ− 1 (푘 ± i0, 푥, 푦) = 푟− 1 (±푘, 푥, 푦), 2 푘 2 where

(1) 푟 1 (푘; 푥, 푦) = 푟 1 (푘; 푦, 푥) = 푘√푥푦 퐽0(푘푥)퐻0 (푘푦), 푥 ≤ 푦. − 2 − 2

Lemma 3.4. The function 푟 1 (푘, 푥, 푦) can be written as − 2

i푘푝 −i푘푝 ∗ 푟 1 (푘, 푥, 푦) = 휒(−∞,0](푘) ∫ e 푑휌푥,푦(푝) + 휒[0,∞)(푘) ∫ e 푑휌푥,푦(푝) − 2 ℝ ℝ with a measure whose total variation satisfies

‖휌푥,푦‖ ≤ 퐶.

Here 휌∗ is the complex conjugated measure. Dispersion estimates 341

Proof. Let 푥 ≤ 푦 and 푘 ≥ 0. Write

푟 1 (푘, 푥, 푦) = 퐽(푘푥)퐻(푘푦), − 2 where (1) 퐽(푟) = √푟 퐽0(푟), 퐻(푟) = √푟 퐻0 (푟). We continue 퐽(푟), 퐻(푟) to the region 푟 < 0 such that they are continuously differentiable and satisfy

2 휋 2 i(푟− 휋 ) 퐽(푟) = cos(푟 − ), 퐻(푟) = e 4 , √휋 4 √휋 for 푟 < −1. It’s enough to show that

2 휋 2 i(푟− 휋 ) 퐽(푟)̃ = 퐽(푟) − cos(푟 − ) and 퐻(푟)̃ = 퐻(푟) − e 4 √휋 4 √휋 are elements of the Wiener Algebra 풲0(ℝ). In fact, they are continuously differentiable, and hence it suffices to look at their asymptotic behavior. To do this,we need the results about Bessel and Hankel functions, collected in Appendix B. For 푟 < −1 both 퐽(푟)̃ and 퐻(푟)̃ are zero. 퐽̃is integrable near 0, and for 푟 > 1 it behaves like 푂(푟−1) and 푂(푟−1) for the derivative. So 퐽̃is contained in 퐻1(ℝ) and therefore in 풲0 by Lemma A.2. As for 퐻̃, near 0 it behaves like √푟 log 푟, and hence its derivative belongs to 퐿푝 for all 푝 ∈ (1, 2) near zero. Since 퐻(푟)̃ and its derivative also −1 behave like 푂(푟 ) for 푟 > 1, Lemma A.2 applies, and thus we also have 퐻̃ ∈ 풲0. As a consequence, both 퐽 and 퐻 are Fourier transforms of finite measures. By scaling the total variation of the measures corresponding to 퐽(푘푥) and 퐻(푘푦) are independent of 푥 and 푦, respectively. This finishes the proof.

Now we are in position to finish the proof of the main result.

Proof of Theorem 3.3. As a consequence of Lemma 3.4 we note

2 퐶 |ℛ− 1 (푘 ± i0, 푥, 푦)| ≤ , 2 |푘|

2 1 and hence the operator 푞 ℛ 1 (푘 ± i0) is bounded on 퐿 with − 2

2 퐶 ‖ 1 ‖ ‖ ‖ ‖푞 ℛ− (푘 ± i0)‖ 1 ≤ ‖푞‖ 1. 2 퐿 |푘| 퐿 342 M. Holzleitner, A. Kostenko, and G. Teschl

Thus we get

2 2 푛 |⟨ℛ 1 (푘 ± i0)(−푞 ℛ 1 (푘 ± i0)) 푓, 푔⟩| | − 2 − 2 | 2 푛 2 = |⟨(−푞 ℛ 1 (푘 ± i0)) 푓, ℛ 1 (푘 ∓ i0)푔⟩| | − 2 − 2 | ‖ 2 푛 ‖ ‖ 2 ‖ ≤ ‖(−푞 ℛ− 1 (푘 ± i0)) 푓‖ ‖ℛ− 1 (푘 ∓ i0)푔‖ 2 퐿1 2 퐿∞ 푛+1 푛 퐶 ‖푞‖ 1 ≤ 퐿 ‖푓‖ ‖푔‖ . |푘|푛+1 퐿1 퐿1 This estimate holds for all 퐿1 functions 푓 and 푔, and hence the series (3.13) weakly 1 converges whenever |푘| > 푘0 = 퐶(푙)‖푞‖퐿1. Namely, for all 퐿 functions 푓 and 푔 we have ∞ 2 2 2 푛 ⟨ℛ퐻(푘 ± i0)푓, 푔⟩ = ∑ ⟨ℛ 1 (푘 ± i0)(−푞 ℛ 1 (푘 ± i0)) 푓, 푔⟩. (3.14) − 2 − 2 푛=0 Using the estimates (2.9), (2.25), (2.34), and (2.35) for the Green’s function (3.11), one can see that 2 ∞ ℛ퐻(푘 ± i0) 푔 ∈ 퐿 whenever 푔 ∈ 퐿1 and |푘| > 0. Therefore, we get

| 2 2 푛 | ⟨ℛ퐻(푘 ± i0)(−푞 ℛ 1 (푘 ± i0)) 푓, 푔⟩ | − 2 | | 2 푛 2 | = ⟨(−푞 ℛ 1 (푘 ± i0)) 푓, ℛ퐻(푘 ∓ i0)푔⟩ | − 2 | 2 푛 2 ‖ 1 ‖ ‖ ‖ ≤ ‖(−푞 ℛ− (푘 ± i0)) 푓‖ ‖ℛ퐻(푘 ∓ i0)푔‖ ∞ 2 퐿1 퐿 푛 퐶‖푞‖ 1 ≤ ( 퐿 ) ‖ℛ (푘2 ∓ i0)푔‖ , 푘 퐻 퐿∞

2 2 푛 which means that ℛ퐻(푘 ± i0)(−푞 ℛ 1 (푘 ± i0)) weakly tends to 0 whenever − 2 |푘| > 푘0. Let us consider again a function 휒 as in (3.12) with 푘0 = 퐶‖푞‖1. From (3.10) i푡퐻 i푡퐻 we get, since e 휒(퐻)푃푐 = e 휒(퐻), ∞ 1 2 ⟨e−i푡퐻휒(퐻)푓, 푔⟩ = ∫ e−i푡푘 휒(푘2)푘⟨ℛ (푘2 + i0)푓, 푔⟩ 푑푘. 휋i 퐻 −∞ Using (3.14) and noting that we can exchange summation and integration, we get

⟨e−i푡퐻휒(퐻)푓, 푔⟩ ∞ ∞ 1 −i푡푘2 2 2 2 푛 = ∑ ∫ e 휒(푘 )푘⟨ℛ− 1 (푘 + i0)(−푞 ℛ− 1 (푘 + i0)) 푓, 푔⟩ 푑푘. 휋i 2 2 푛=0 −∞ Dispersion estimates 343

2 2 푛 The kernel of the operator ℛ 1 (푘 + i0)(−푞 ℛ 1 (푘 + i0)) is given by − 2 − 2

푛 푛−1 1 푛+1 ∫ 푟− 1 (푘; 푥, 푦1) ∏ 푞(푦푖) ∏ 푟− 1 (푘; 푦푖, 푦푖+1)푟− 1 (푘; 푦푛, 푦) 푑푦1 ⋯ 푑푦푛. 푘 푛 2 2 2 ℝ+ 푖=1 푖=1 Applying Fubini’s theorem, we can integrate in 푘 first, and hence we need to obtain a uniform estimate of the oscillatory integral

−푛 푛 −i푡푘2 2 푘 퐼푛(푡; 푢0, … , 푢푛+1) = ∫ e 휒(푘 ) ( ) ∏ 푟− 1 (푘; 푢푖, 푢푖+1) 푑푘 2푘 2 ℝ 0 푖=0 since, recalling that 푘0 = 퐶(푙)‖푞‖퐿1, one obtains ∞ 1 1 | −i푡퐻 | | |‖ ‖ ‖ ‖ |⟨e 휒(퐻)푓, 푔⟩| ≤ ∑ 푛 sup |퐼푛(푡; 푢0, … , 푢푛+1)| 푓 퐿1 푔 퐿1. 휋 (2퐶) 푛+1 푛=0 {ᵆ푖}푖=0

2 −푛 Consider the function 푓푛(푘) = 휒(푘 )(푘/2푘0) . Clearly, 푓0 is the Fourier transform 1 of a measure 휈0 satisfying ‖휈0‖ ≤ 퐶1. For 푛 ≥ 1, 푓푛 belongs to 퐻 (ℝ) with ‖푓푛‖퐻1 ≤ −1/2 휋 퐶1(1 + 푛). Hence by Lemma A.1 and Lemma 3.4 we obtain

2퐶푣퐶1 푛+1 |퐼푛(푡; 푢0, … , 푢푛+1)| ≤ (1 + 푛)퐶 √푡 implying ∞ 2퐶 퐶 퐶 1 + 푛 |⟨e−i푡퐻휒(퐻)푓, 푔⟩| ≤ 푣 1 ‖푓‖ ‖푔‖ ∑ . | | 퐿1 퐿1 2푛 √푡 푛=0 This proves Theorem 3.3.

A. The van der Corput Lemma

We will need the the following variant of the van der Corput lemma (see, e.g., [19, Lemma A.2] and [28, page 334]). Lemma A.1. Let (푎, 푏) ⊆ ℝ and consider the oscillatory integral

푏 퐼(푡) = ∫ ei푡푘2퐴(푘) 푑푘. 푎 If 퐴 ∈ 풲(ℝ), i.e., 퐴 is the Fourier transform of a signed measure

퐴(푘) = ∫ ei푘푝 푑훼(푝), ℝ 344 M. Holzleitner, A. Kostenko, and G. Teschl then the above integral exists as an improper integral, and satisfies 1 − 2 |퐼(푡)| ≤ 퐶2|푡| ‖퐴‖풲, |푡| > 0. 8/3 where ‖퐴‖풲 ≔ ‖훼‖ = |훼|(ℝ) denotes the total variation of 훼, and 퐶2 ≤ 2 is a universal constant.

Note that if 퐴1, 퐴2 ∈ 풲(ℝ), then (cf. p. 208 in [1])

1 i푘푝 (퐴1퐴2)(푘) = 2 ∫ e 푑(훼1 ∗ 훼2)(푝) (2휋) ℝ is associated with the convolution

훼1 ∗ 훼2(Ω) = ∬ ퟙΩ(푥 + 푦) 푑훼1(푥) 푑훼2(푦), where ퟙΩ is the indicator function of a set Ω. Note that

‖훼1 ∗ 훼2‖ ≤ ‖훼1‖‖훼2‖.

Let 풲0(ℝ) be the Wiener algebra of functions 퐶(ℝ) which are Fourier transforms of 퐿1 functions,

i푘푥 1 풲0(ℝ) = { 푓 ∈ 퐶(ℝ) ∶ 푓(푘) = ∫ e 푔(푥) 푑푥, 푔 ∈ 퐿 (ℝ) }. ℝ

Clearly, 풲0(ℝ) ⊂ 풲(ℝ). Moreover, by the Riemann–Lebesgue lemma, 푓 ∈ 퐶0(ℝ), that is, 푓(푘) → 0 as 푘 → ∞ if 푓 ∈ 풲0(ℝ). A comprehensive survey of necessary and sufficient conditions for 푓 ∈ 퐶(ℝ) to be in the Wiener algebras 풲0(ℝ) and 풲(ℝ) can be found in [21], [22]. We need the following statement, which extends the well-known Beurling condition (see [11, Lemma B.3]). Lemma A.2. If 푓 ∈ 퐿2(ℝ) is locally absolutely continuous and 푓′ ∈ 퐿푝(ℝ) with 푝 ∈ (1, 2], then 푓 is in the Wiener algebra 풲0(ℝ) and ′ ‖푓‖풲 ≤ 퐶푝(‖푓‖퐿2(ℝ) + ‖푓 ‖퐿푝(ℝ)), (A.1) where 퐶푝 > 0 is a positive constant, which depends only on 푝. We also need the following result from [22].

Lemma A.3. Let 푓 ∈ 퐶0(ℝ) be locally absolutely continuous on ℝ ⧵ {0}. Set ′ 푓0(푥) ≔ sup |푓(푦)|, 푓1(푥) ≔ ess sup |푓 (푦)|, (A.2) |푦|≥|푥| |푦|≥|푥| for all 푥 ≠ 0. If 1 ∞ ∞ 1/2

∫ log (2/푥)푓1(푥) 푑푥 < ∞, ∫ (∫ 푓0(푦)푓1(푦) 푑푦) 푑푥 < ∞, (A.3) 0 1 푥 then 푓 ∈ 풲0(ℝ). Dispersion estimates 345

B. Bessel functions

Here we collect basic formulas and information on Bessel and Hankel functions (see, e.g., [23, 31]). First of all assume 푚 ∈ ℕ0. We start with the definitions: ∞ 푧 푚 (−푧2/4)푛 퐽 (푧) = ( ) ∑ , (B.1) 푚 2 푛=0 푛!(푛 + 푚 + 1)! 푚−1 (−푧/2)−푚 (푚 − 푛 − 1)!(푧2/4)푛 2 푧 푌 (푧) = − ∑ + log( )퐽 (푧) 푚 휋 푛! 휋 2 푚 푛=0 ∞ (푧/2)푚 (−푧2/4)푛 + ∑ (휓(푛 + 1) + 휓(푛 + 푚 + 1)) , (B.2) 휋 푛=0 푛!(푛 + 푚 + 1)! (1) (2) 퐻푚 (푧) = 퐽푚(푧) + i푌푚(푧), 퐻푚 (푧) = 퐽푚(푧) − i푌푚(푧). (B.3) Here 휓 is the digamma function [23,(5.2.2)]. The asymptotic behavior as |푧| → ∞ is given by

2 퐽 (푧) = (cos(푧 − 휋푚/2 − 휋/4) + e| Im 푧|풪(|푧|−1)) , | arg 푧| < 휋, (B.4) 푚 √휋푧 2 푌 (푧) = (sin(푧 − 휋푚/2 − 휋/4) + e| Im 푧|풪(|푧|−1)) , |arg 푧| < 휋, (B.5) 푚 √휋푧

(1) 2 푖(푧− 2푚+1 휋) −1 퐻 (푧) = e 4 (1 + 풪(|푧| )) , −휋 < arg 푧 < 2휋, (B.6) 푚 √휋푧

(2) 2 −푖(푧− 2푚+1 휋) −1 퐻 (푧) = e 4 (1 + 풪(|푧| )) , −2휋 < arg 푧 < 휋. (B.7) 푚 √휋푧 Using [23,(10.6.2)], one can show that the derivative of the remainder satisfies

′ 휋푧 ( 퐽 (푧) − cos(푧 − 휋/4)) = e| Im 푧|풪(|푧|−1), (B.8) √ 2 0

(1) (2) as |푧| → ∞. The same is true for 푌푚, 퐻푚 and 퐻푚 .

Acknowledgments. We thank Vladislav Kravchenko and Sergii Torba for providing us with the paper [26]. We are also grateful to Iryna Egorova for the copy of A. S. Sohin’s PhD thesis.

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Sixty years of moments for random matrices

Werner Kirsch and Thomas Kriecherbauer

Dedicated to Helge Holden on the occasion of his 60th birthday

Abstract. This is an elementary review, aimed at non-specialists, of results that have been obtained for the limiting distribution of eigenvalues and for the operator norms of real symmetric random matrices via the method of moments. This method goes back to a remarkable argument of Eugene Wigner some sixty years ago which works best for independent matrix entries, as far as symmetry permits, that are all centered and have the same variance. We then discuss variations of this classical result for ensembles for which the variance may depend on the distance of the matrix entry to the diagonal, including in particular the case of band random matrices, and/or for which the required independence of the matrix entries is replaced by some weaker condition. This includes results on ensembles with entries from Curie–Weiss random variables or from sequences of exchangeable random variables that have been obtained quite recently.

1. Introduction

Approximately at the time when Helge Holden was born the physicist Eugene Wigner presented a result in [45] that may be considered to be the starting signal for an extremely fruitful line of investigations creating the now ample realm of random matrices. The reader may consult the handbook [3] to obtain an impression of the richness of the field. Its ongoing briskness is well documented by over700 publications listed in MathSciNet after the printing of [3] in 2011. In view of later developments that often use heavy machinery to provide very detailed knowledge about specific spectral statistics, Wigner’s observation im- presses by its simplicity and fine combinatorics. For certain matrix ensembles, which in various generalizations are nowadays called Wigner ensembles, he was able to determine the limiting density of eigenvalues by the moment method. More precisely, he computed the expectations of all moments of the empirical eigenvalue distribution measures in the limit of matrix dimensions tending to infinity. Fur- thermore, he observed that these limits agree with the moments of the semicircle 350 W. Kirsch and T. Kriecherbauer distribution thus proving the semicircle law that bears his name (see Sections 2 and 3 for definitions of the phrases in italics). It is quite remarkable that the moment method continues to provide new insights into the distribution of random eigenvalues. With this article we take the reader on a tour that starts with Wigner’s discovery and ends with the description of recent results, some yet unpublished. Along the way we try to explain a few developments in more detail while briefly pointing at others. The first application of the moment method to the analysis of random eigenvalues appears almost accidental. In an effort to understand “the wave functions of quantum mechanical systems which are assumed to be so complicated that statistical considerations can be applied to them”, Wigner introduces in [45] three types of ensembles of “real symmetric matrices of high dimensionality”. Although he considers his results not satisfactory from a physical point of view, he expresses the hope that “the calculation which follows may have some independent interest”. Moreover, the reader learns that one of the three models considered just serves “as an intermediate step”. And it is only this auxiliary ensemble that we would now call a Wigner ensemble. Wigner names it the “random sign symmetric matrix” by which he understands (2푁 + 1) × (2푁 + 1) matrices for which the diagonal elements are zero and “non diagonal elements 푣푖푘 = 푣푘푖 = ±푣 have all the same absolute value but random signs”. In the short note [46] that appeared a few years later, Wigner remarks that the arguments of [45] show the semicircle law for a much larger class of real symmetric ensembles. He observes that, except for technical assumptions, two features of the model were essential for his proof: Firstly, stochastic independence of the matrix entries (as far as the symmetry permits) and, secondly, that all (or at least most) matrix entries are centered and have the same variance. In Section 3 we present Wigner’s proof with enough detail to make the significance of these two assumptions apparent. The remaining sections are then devoted to the discussion of results where at least one of these essential assumptions are weakened. In Section 4 independence and centeredness of the matrix entries are kept. However, we allow the variances to vary as a function of the distance to the diagonal. The most prominent examples in this class are band matrices and we discuss them in detail. A first step to loosen the assumption of independence is presented in Section 5. Its central result provides conditions on the number and location of matrix entries that may be dependent without affecting the validity of Wigner’s reasoning. We call such dependence structures sparse. Sparse dependence structures appear for example in certain types of block random matrices that are used in modelling Sixty years of moments for random matrices 351 disordered systems in mesoscopic physics (see e. g. [1]) . In the last three sections 6–8 we report on results for ensembles with a dependence structure that is not sparse. This is largely uncharted territory. However, in recent years a number of special cases were analyzed using the method of moments. They show interesting phenomena that should be explored further. We divide the models into three groups. In the first group the correlations decay to 0 as the distance of the matrix entries becomes large in some prescribed metric. Then we look at those ensembles for which the entries are drawn from Curie–Weiss random variables. Here the correlations have no spatial decay, but decay for supercritical temperatures as the matrix dimension becomes large. Finally, we pick the matrix entries from an infinite sequence of exchangeable random variables. Here the correlations between matrix entries depend neither on their locations nor on the size of the matrix. We close this introduction by stating what is not contained in this survey. One of the striking features of random matrix theory is the observation that local statistics of the eigenvalues obey universal laws that, somewhat surprisingly, have also arisen in certain combinatorial problems, in some models from statistical mechanics and even in the distribution of the non-trivial zeros of zeta-functions. By local statistics we mean statistics after local but deterministic rescaling so that the spacings between neighboring eigenvalues are of order 1. Examples are the statistics of spacings or the distribution of extremal eigenvalues. Such results were first obtained for Gaussian ensembles, i.e., Wigner ensembles with normally distributed entries. In this special case it is possible to derive an explicit formula for the joint distribution of eigenvalues that can then be analyzed using the method of orthogonal polynomials. In the Gaussian case this requires detailed asymptotic formulas for Hermite polynomials of large degree that had already been derived in the beginning of the twentieth century. The first step to prove universality beyond Gaussian ensembles was then taken about twenty years ago within the class of ensembles that are invariant under change of orthonormal bases. For such ensembles the eigenvectors are distributed according to Haar measure, the joint distribution of eigenvalues is still explicit, and the method of orthogonal polynomials works, albeit they generally do not belong to the well studied families of classical orthogonal polynomials (see for example [11, 12, 37] and references therein). It is only seven years ago that universality results for local statistics became available for Wigner matrices (see e. g. [14, 15, 43, 44, 22] and references therein). Since all of these results do not use the moment method, we will not discuss them in this paper. There is one notable exception to what has just been said. The distribution of extremal eigenvalues (and consequently of the operator norm) can and has 352 W. Kirsch and T. Kriecherbauer been investigated for Wigner ensembles on the local level, using the method of moments [40], see also [39] and references therein. However, this requires quite substantial extensions of the ideas that we explain and goes way beyond the scope of this paper. We therefore only state weaker results that might be considered as laws of large numbers for the operator norm and that can be proved with much less effort. Nevertheless, we do not discuss their proofs either and refer the reader to Section 2.3 of the textbook [42]. Finally, we mention that the moment method can also be applied to complex Hermitian matrices and to sample covariance matrices (also known as Wishart ensembles), but in the present article we always restrict ourselves to the case of real symmetric matrices to keep the presentation as elementary as possible.

2. Setup

We begin by setting the scene and fixing some notation.

Definition 2.1. A (real symmetric) matrix ensemble is a family 푋 (푖, 푗), for 푖, 푗 = 1, … , 푁 and 푁 ∈ ℕ, of real valued random variables on a probability space (Ω, ℱ, ℙ) such that 푋 (푖, 푗) = 푋 (푗, 푖). We then denote by 푋 the corresponding 푁 × 푁 matrix, i.e.,

푋 (1, 1) 푋 (1, 2) ⋯ 푋 (1, 푁) ⎛ ⎞ 푋 (2, 1) 푋 (2, 2) ⋯ 푋 (2, 푁) 푋 = ⎜ ⎟ (1) ⎜ ⋮ ⋮ ⋱ ⋮ ⎟ ⎝푋 (푁, 1) 푋 (푁, 2) ⋯ 푋 (푁, 푁)⎠ Since we deal exclusively with real symmetric matrices by ‘matrix ensemble’ we always mean a real symmetric one.

Definition 2.2. A (real symmetric) matrix ensemble is called independent if for each 푁 ∈ ℕ the random variables 푋 (푖, 푗), 1 ≤ 푖 ≤ 푗 ≤ 푁 are independent. It is called identically distributed, if all 푋 (푖, 푗) have the same distribution. An independent and identically distributed matrix ensemble 푋 is called a 2 Wigner ensemble if 피(푋 (푖, 푗)) = 0 and 피(푋 (푖, 푗) ) = 1.

By a slight abuse of language we use the phrase ‘푋 is a Wigner matrix’ to indicate that the family 푋 of random (symmetric) matrices form a Wigner ensemble. Some authors allow for Wigner ensembles a probability distribution for the diagonal elements which differs from the distribution for the non-diagonal entries. Sixty years of moments for random matrices 353

Definition 2.3. The 푘th moment of a random variable 푋 is the expectation 피(푋푘). We say that all moments of 푋 exist, if 피(|푋|푘) < ∞ for all 푘 ∈ ℕ.

Unless stated otherwise, we always assume that all random variables occurring in this text have all moments existing.

For any symmetric 푁 × 푁-matrix 푀 we denote the eigenvalues of 푀 by 휆푗(푀). We order these eigenvalues such that

휆1(푀) ≤ 휆2(푀) ≤ ⋯ ≤ 휆 (푀) where degenerate eigenvalues are repeated according to their multiplicity.

The empirical eigenvalue distribution measure 휈 of 푀 is defined by 1 휈 (퐴) = |{ 푗 | 휆 (푀) ∈ 퐴 }| 푁 | 푗 |

1 = ∑ 훿 (퐴) 푁 휆푗(푀) 푗=1 where |퐵| denotes the number of points in 퐵; 푁 – as above – is the dimension of the matrix 푀; 퐴 is a Borel-subset of ℝ; and 훿푎 is the Dirac measure in 푎, i.e.,

1 if 푎 ∈ 퐴 훿푎(퐴) = { (2) 0 otherwise .

It turns out that for a Wigner matrix 푋 the empirical eigenvalue distribution measure 휈 of 푋 has no chance to converge as 푁 → ∞ as the following back-of- the-envelope calculations show. We have

1 1 ∫ 휆2 푑휈 (휆) = ∑ 휆 (푋 )2 = tr 푋2 . (3) 푁 ℓ 푁 ℓ=1

If the 푁 × 푁-matrix 푋 has entries ±1 (random or not), then (3) shows

2 ∫ 휆 푑휈 (휆) = 푁 , (4)

2 and if the 푋 are random matrices with 피(푋 (푖, 푗) ) = 1, we get

2 피(∫ 휆 푑휈 (휆)) = 푁 . (5)

This shows that (at least the second moment of) the empirical eigenvalue distribution measure of 푋 is divergent. 354 W. Kirsch and T. Kriecherbauer

Moreover, the same calculation suggests that the empirical eigenvalue distribution measure of the normalized matrices

−1/2 푀 = 푁 푋 might converge, as for 푀 1 피 ∫ 휆2 푑휈 = tr 푀2 = 1 . 푁 As we shall see below, this is indeed the case not only for Wigner ensembles, but for a huge class of random matrices.

A similar reasoning applies to the operator norm of a matrix ensemble 푋 :

‖푋 ‖ = max {|휆1(푋 )|, |휆 (푋 )|} . (6)

Since for any real symmetric 푁 × 푁-matrix 푀:

1 1 tr 푀2 = ∑ 휆 (푀)2 ≤ ‖푀‖2 ≤ tr 푀2 (7) 푁 푁 ℓ ℓ=1 a matrix 푀 with ±1-entries satisfies

√푁 ≤ ‖푀‖ ≤ 푁

2 and similarly for 피(푋 (푖, 푗) ) = 1

2 1 √푁 ≤ 피(‖푋 ‖ ) 2 ≤ 푁 .

−1/2 Again, one is led to look at the norm of 푀 = 푁 푋 . Indeed, for Wigner ensembles the norm of 푀 will stay bounded as 푁 → ∞. In fact, it will converge to 2.

However, this fact is more subtle than the convergence of 휈 , and so is its proof (cf. Theorem 3.13 that was proved by Füredi and Komlós in [20], see also [5], and [42] for a textbook presentation). To illustrate this, let us look at a particular example within the class considered in (4), namely the 푁 × 푁-matrices

ℰ (푖, 푗) = 1 for all 1 ≤ 푖, 푗 ≤ 푁 . (8)

The matrix ℰ can be written as

ℰ = 푁 ⋅ 푃푒

−1/2 where 푃푒 is the orthogonal projection onto the vector 푒, with 푒(푖) = 푁 for 푖 = 1, … , 푁. Sixty years of moments for random matrices 355

Consequently ℰ is of rank 1, and

푁 for 푗 = 푁 휆푗(ℰ ) = { (9) 0 otherwise . Thus we obtain ‖ ℰ ‖ ‖ ‖ = √푁 → ∞ , ‖√푁‖ but the eigenvalue distribution function 휈 of ℰ /√푁 is given by 1 1 휈 = (푁 − 1)훿 + 훿 ⟹ 훿 푁 0 푁 √ 0 where ⇒ means weak convergence (see Definition 3.1).

3. Wigner’s Semicircle Law

In this section we present and discuss the classical semicircle law for Wigner ensembles. −1/2 So, let 푋 be a Wigner ensemble (see Definition 2.2), set 푀 = 푁 푋 , and denote the empirical eigenvalue distribution measure of 푀 by 휎 , thus

1 | | 1 | 휎 (퐴) = |{ 푗 | 휆푗( 푋 ) ∈ 퐴 }| . 푁 | | √푁 |

The semicircle law, in its original form due to Wigner ([45], [46]), states that 휎 converges to the semicircle distribution 휎 given through its Lebesgue density

1 √4 − 푥2 for |푥| ≤ 2, 휎(푥) = { 2휋 (10) 0 otherwise.

휎 describes a semicircle of radius 2 around the origin, hence the name.

So far, we have avoided to explain in which sense 휎 converges. This is what we do now. Let us first look at the convergence of measures on ℝ.

Definition 3.1. Suppose 휇 and 휇 are probability measures on ℝ (equipped with the Borel 휎-algebra 퐵(ℝ)). We say that 휇 converges weakly to 휇, in symbols 휇 ⇒ 휇, if

∫ 푓(푥) 푑휇 (푥) → ∫ 푓(푥) 푑휇(푥) for all 푓 ∈ 퐶푏(ℝ), the space of bounded continuous functions. 356 W. Kirsch and T. Kriecherbauer

If the matrix 푋 is random and

1 푋 휎 = ∑ 훿휆 ( ) 푁 푗 √푁

−1/2 is the empirical eigenvalue distribution measure of 푁 푋 , then the measure 휎 itself is random. Consequently, we have to define not only in which sense the measures converge (namely weakly), but also how this convergence is meant with respect to randomness, i.e., to the ‘parameter’ 휔 ∈ Ω. There are various ways to do this.

휔 휔 Definition 3.2. Let (Ω, ℱ, ℙ) be a probability space and let 휇 and 휇 be random probability measures on (ℝ, 퐵(ℝ)).

휔 휔 1) We say that 휇 converges to 휇 weakly in expectation, if for every 푓 ∈ 퐶푏(ℝ)

휔 휔 피(∫ 푓(푥) 푑휇 (푥)) → 피( ∫ 푓(푥) 푑휇 (푥)) (11)

as 푁 → ∞.

휔 휔 2) We say that 휇 converges to 휇 weakly in probability, if for every 푓 ∈ 퐶푏(ℝ) and any 휖 > 0

| | | 휔 휔 | ℙ(|∫ 푓(푥) 푑휇 (푥) − ∫ 푓(푥) 푑휇 (푥)| > 휖) → 0

as 푁 → ∞.

휔 휔 3) We say that 휇 converges to 휇 weakly ℙ-almost surely if there is a set Ω0 ⊂ Ω 휔 휔 with ℙ(Ω0) = 1 such that 휇 ⇒ 휇 for all 휔 ∈ Ω0.

Theorem 3.3 (Semicircle Law). Suppose that 푋 is a Wigner ensemble for which 푘 피(|푋 (푖, 푗)| ) < ∞ for all 푘 ∈ ℕ, and let 휎 denote the empirical eigenvalue distri- −1/2 bution measure of 푀 = 푁 푋 . Then 휎 converges to the semicircle distribution 휎 weakly ℙ-almost surely.

Remarks 3.4.

1. Wigner [45, 46] proved this theorem for weak convergence in expectation.

2. Grenander [23] showed under the same conditions that the convergence holds weakly in probability. Sixty years of moments for random matrices 357

3. Arnold [4] proved that the convergence is weakly ℙ-almost surely. He also relaxed the moment condition to

6 피 (푋 (푖, 푗) ) < ∞

for ℙ-almost sure weak convergence and to

4 피 (푋 (푖, 푗) ) < ∞

for weak convergence in probability. 4. According to Definition 2.2, the entries in a Wigner ensemble are indepen- 푘 dent and identically distributed. Hence a condition of the form 피(|푋 (푖, 푗)| ) < ∞, as it appears for example in the previous remark, actually implies

푘 sup 피(|푋 (푖, 푗)| ) < ∞ . ,푖,푗

Besides the moment method we discuss in this article, there is another important technique to prove the semicircle law. This is the Stieltjes transform method originating in [27], [35] and [36], see also [37] and references given there. Both methods are discussed in [2] and in [42]. The moment method is based on the observation that the following result is true.

Proposition 3.5. If 휇 and 휇 are probability measures on ℝ such that all moments of 휇 exist and ∫|푥|푘 푑휇(푥) ≤ 퐴퐶푘푘! (12) for all 푘 and some constants 퐴, 퐶, then

푘 푘 ∫ 푥 푑휇 (푥) → ∫ 푥 푑휇(푥) for all 푘 ∈ ℕ implies that 휇 ⇒ 휇 . For a proof see for example [8], [32], or [28]. Since the semicircle distribution 휎 has compact support, it obviously satisfies (12). The moments of 휎 are given by:

퐶 if 푘 is even, ∫ 푥푘 푑휎(푥) = { 푘/2 (13) 0 if 푘 is odd. 358 W. Kirsch and T. Kriecherbauer where

1 2ℓ 퐶 = ( ) (14) ℓ ℓ + 1 ℓ are the Catalan numbers. (For a concise introduction to Catalan numbers see e. g. [33] or [41].)

The moments of 휎 can be expressed through traces of the matrices 푋

1 푋 푘 1 푋 푘 피(∫ 푥푘 푑휎 (푥)) = 피(∑ 휆 ( ) ) = 피(tr ( ) ) 푁 푗 푁 푗=1 √푁 √푁

1 = ∑ 피(푋 (푖 , 푖 ) ⋅ 푋 (푖 , 푖 ) ⋅ ⋯ ⋅ 푋 (푖 , 푖 )) . (15) 푁1+푘/2 1 2 2 3 푘 1 푖1,…,푖푘=1

The sum in (15) contains 푁푘 terms. So, at a first glance, the normalizing factor 푁1+푘/2 seems too small to compensate the growth of the sum. Fortunately, many of the summands are zero, as we shall see later.

For the purpose of bookkeeping it is useful to think of 푖1, 푖2, …, 푖푘 in terms of a graph.

Definition 3.6. The multigraph 풢 with vertex set

풱 ≔ {푖1, 푖2, … , 푖푘} (16) and ℓ (undirected) edges between 푖 and 푗 if {푖, 푗} occurs ℓ times in the sequence

{푖1, 푖2}, {푖2, 푖3}, … , {푖푘, 푖1} (17) is called the multigraph associated with (푖1, 푖2, … , 푖푘).

Remark 3.7. The sequence (푖1, 푖2, … , 푖푘) defines a multigraph since there may be several edges between the vertices 푖휈. Definition 3.8. If 풢 is a multigraph we define the associated (simple) graph 풢˜ in the following way. The set of vertices of 풢˜ is the same as the vertex set of 풢 and 풢˜ has a single edge between 푖 and 푗 whenever 풢 has at least one edge between 푖 and 푗.

Remark 3.9. The sequence (17) describes not only a multigraph but in addition a closed path through the multigraph which uses each edge exactly once. Such paths are called Eulerian circuits. They occur for example in the famous problem of the ‘Seven Bridges of Königsberg’ (see e. g. [7]). The existence of an Eulerian circuit implies in particular that the multigraph is connected. Sixty years of moments for random matrices 359

Now, we order the sum in (15) according to the number |풱| = |{푖1, … , 푖푘}| of different indices (vertices) occurring in the sequence 푖1, 푖2, … , 푖푘.

∑ 피(푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ ⋯ ⋅ 푋 (푖푘, 푖1)) 푖1,…,푖푘=1 푘

= ∑ ∑ 피(푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ ⋯ ⋅ 푋 (푖푘, 푖1)) (18) 푟=1 |{푖1,…,푖푘}|=푟

푟 The number of index tuples (푖1, … , 푖푘) with |{푖1, … , 푖푘}| = 푟 is of order 풪(푁 ) and can be bounded above by 푟푘 푁푟. In fact, to choose the 푟 different numbers in {1, … , 푁} we have less than 푁푟 possibilities. Then, to choose which one to put at a given position, we have at most 푟 choices for each of the 푘 positions. Therefore, the sum

∑ 피(푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ ⋯ ⋅ 푋 (푖푘, 푖1)) (19) |{푖1,…,푖푘}|=푟

푟 is of order 풪(푁 ) as well. Thus the terms with 푟 = |{푖1, … , 푖푘}| < 1 + 푘/2 in (15) can be neglected compared to prefactor 푁−(1+푘/2). Consequently 1 ∑ 피(푋 (푖 , 푖 ) ⋅ ⋯ ⋅ 푋 (푖 , 푖 )) ⟶ 0 (20) 푁1+푘/2 1 2 푘 1 |{푖1,…,푖푘}|<1+푘/2

To handle those terms with |{푖1, … , 푖푘}| > 1 + 푘/2 we need the following two observations. For comparison with results in Section 7 we formulate the first one as a lemma.

Lemma 3.10. Whenever an edge {푖, 푗} occurs only once in (17) then

피(푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ ⋯ ⋅ 푋 (푖푘, 푖1)) = 0 . (21)

This follows from independence and the assumption 피(푋 (푖, 푗)) = 0. The second observation is:

Proposition 3.11. If |{푖1, … , 푖푘}| > 1 + 푘/2 there is an edge {푖, 푗} which occurs only once in {푖1, 푖2}, {푖2, 푖3}, … , {푖푘, 푖1}.

Proof. Set 푟 = |{푖1, … , 푖푘}| and denote the distinct elements of |{푖1, … , 푖푘}| by 푗1, …, 푗푟. To connect the vertices 푗1, …, 푗푟 we need at least 푟 − 1 edges. To double each of these connections we need 2푟 − 2 edges. So, if we have 푘 edges we need that 푘 ≥ 2푟 − 2 to double each connection. Hence, if 푟 > 1 + 푘/2, at least one edge occurs only once. 360 W. Kirsch and T. Kriecherbauer

Remark 3.12. A similar reasoning as in the proof above shows: If a graph 풢 with 푘 edges and 푘 + 1 vertices is connected then 풢 is a tree, i.e., 풢 contains no loops. Indeed, if 풢 contained a loop we could remove an edge without destroying the connectedness of the graph. But the new graph would have 푘 − 1 edges and 푘 + 1 vertices, so it cannot be connected.

From Proposition 3.11 and (21) we learn that

∑ 피(푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ ⋯ ⋅ 푋 (푖푘, 푖1)) = 0 . (22) |{푖1,…,푖푘}|>1+푘/2

To summarize, what we proved so far is

1 ∑ 피(푋 (푖 , 푖 ) ⋅ 푋 (푖 , 푖 ) ⋅ ⋯ ⋅ 푋 (푖 , 푖 )) 푁1+푘/2 1 2 2 3 푘 1 푖1,…,푖푘=1 1 ≈ ∑ 피(푋 (푖 , 푖 ) ⋅ 푋 (푖 , 푖 ) ⋅ ⋯ ⋅ 푋 (푖 , 푖 )) (23) 푁1+푘/2 1 2 2 3 푘 1 |{푖1,…,푖푘}|=1+푘/2 all {푖,푗} occur exactly twice Let us set () | ℐ푘 = { (푖1, … , 푖푘) ∈ {1, … , 푁} | |{푖1, … , 푖푘}| = 1 + 푘/2 and all {푖, 푗} occur exactly twice. } (24)

() For odd 푘 the set ℐ푘 is empty, so the sum (23) is obviously zero. 2 Due to independence and the assumptions 피(푋 (푖, 푗)) = 0 and 피(푋 (푖, 푗) ) = 1, we have

피(푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ ⋯ ⋅ 푋 (푖푘, 푖1)) = 1 whenever all {푖, 푗} occur exactly twice. Consequently, 1 Right side of (23) = |ℐ() | . (25) 푁1+푘/2 푘

() For even 푘, let us consider the multigraph 풢 associated with (푖1, … , 푖푘) ∈ ℐ푘 . Since 풢 has 1 + 푘/2 vertices and 푘 double vertices, the corresponding simple graph 풢˜ is a connected graph with 1 + 푘/2 vertices and 푘/2 edges. Thus, this 풢˜ is a tree by Remark 3.12. Moreover the path (푖1, … , 푖푘, 푖1) defines an ordering on 풢˜. The number of ordered trees [33, 41] with ℓ edges (and hence ℓ + 1 vertices) is known to be the Catalan number 퐶ℓ (see (14)). Sixty years of moments for random matrices 361

Given an (abstract) ordered tree with ℓ = 1 + 푘/2 vertices we find all corresponding paths (푖1, 푖2, … , 푖푘, 푖1) with 푖푗 ∈ {1, … , 푁} by assigning 1 + 푘/2 (different) numbers (indices) from {1, … , 푁} to the vertices of the tree. There are 푁!/(푁 − (1 + 푘/2))! ≈ 푁1+푘/2 ways to do this. Thus

1 피(∫ 푥푘 푑휎 (푥)) ≈ |ℐ() | (26) 푁1+푘/2 푘 퐶 for 푘 even , → { 푘/2 (27) 0 for 푘 odd , and these are the moments of the semicircle distribution 휎 (see (13)). In view of Proposition 3.5, this proves that 휎 converges to 휎 weakly in expectation (cf. Defi- nition 3.2). For more details on the semicircle law and its proof see [2], [42] or [28]. −1/2 From Theorem 3.3 and (10) we conclude that lim inf ‖푁 푋 ‖ ≥ 2 almost surely, since for symmetric 푁 × 푁-matrices 퐴 the matrix norm ‖퐴‖, as an operator on the Euclidean space ℝ , satisfies ‖퐴‖ = max {|휆1(퐴)|, |휆 (퐴)|}. −1/2 However, Theorem 3.3 does not imply that lim inf ‖푁 푋 ‖ ≤ 2! Wigner’s result does imply that the majority of the eigenvalues will be less than 2 + 휀 finally, however some (in fact even 표(푁)) eigenvalues could be bigger and might even go to ∞. In Sections 4, 7, and 8 we encounter ensembles for which exactly this happens. −1/2 However, for Wigner ensembles it is correct that the norm of 푁 푋 goes to 2. This can be shown by a more sophisticated variant of the moment method.

푘 Theorem 3.13. Suppose 푋 is a Wigner ensemble with 피(|푋 (푖, 푗)| ) < ∞ for all 푘 ∈ ℕ and let

∗ | 1 | | 1 | ‖ 1 ‖ 휆 = max{|휆1( 푋 )|, |휆 ( 푋 )|} = ‖ 푋 ‖ | √푁 | | √푁 | ‖√푁 ‖

−1/2 be the operator norm of 푀 = 푁 푋 , then

∗ 휆 → 2 as 푁 → ∞ ℙ-almost surely .

This theorem was proved by Füredi and Komlós in [20], see also [5]. 푡ℎ To prove the semicircle law we considered the 푘 moment 푚푘 of 휎 for fixed 푘 as 푁 goes to infinity. For the norm estimate we need bounds on 푚푘 for 푘 = 푘 for a sequence 푘 which is growing with 푁. See [42, Section 2.3] for a pedagogical explanation. 362 W. Kirsch and T. Kriecherbauer

4. Random Band Matrices

In a first variation of Wigner’s semicircle law we abandon the assumption of identical distribution of the 푋 (푖, 푗), by assuming that entries away from a band around the diagonal are zero, while the other entries are still iid, apart from the symmetry 푋 (푖, 푗) = 푋 (푗, 푖). More precisely, let 푋˜ (푖, 푗) be a Wigner ensemble and set

푋˜ (푖, 푗) for |푖 − 푗| ≤ 푏 , 푋 (푖, 푗) = { (28) 0 otherwise . where 푏 is a sequence of integers with 푏 → ∞ and 2푏 + 1 ≤ 푁. We call such matrices banded Wigner matrices with band width 훽 = 2푏 + 1. There is a ‘Semicircle Law’ for banded Wigner matrices due to Bogachev, Molchanov and Pastur [6].

Theorem 4.1. Suppose 푋 is a banded Wigner matrix with band width 훽 = 2푏 + 1 ≤ 푁, and assume that all moments of 푋 (푖, 푗) exist. Set 푀 = (1/√훽 )푋 , and denote by 휎 the empirical eigenvalue distribution measure of 푀 .

1) If 훽 → ∞ but 훽 /푁 → 0 then the 휎 converges to the semicircle distribution weakly in probability.

2) If 훽 ≈ 푐푁 for some 푐 > 0 then 휎 converges weakly in probability to a measure ̃휎 which is not the semicircle distribution.

It turns out that the moment method used to prove Wigner’s result can also be applied to banded random matrices. Let us look at the products

푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ ⋯ ⋅ 푋 (푖푘, 푖1) which occur in evaluating traces as in (15).

We have 푁 possibilities to choose 푖1. In principle, for 푖2 we have again 푁 possibilities. However, unlike to the Wigner case, at most 훽 of these possibilities are not identically zero. This observation makes it plausible that

푘/2 ∑ 피(푋 (푖1, 푖2) ⋅ ⋯ ⋅ 푋 (푖푘, 푖1)) ≈ 푁 훽 푖1,…,푖푘 since – again – only those terms with each {푖, 푗} occurring exactly twice count in the limit. Note that our assumption 훽 → ∞ is needed here. Without this assumption, pairs {푖, 푗} occurring more than twice are not negligible. Sixty years of moments for random matrices 363

Unfortunately, the above argument is not quite correct. It is true that most columns (and rows) contain 훽 entries 푋 (푖, 푗) which are not identically equal to zero. However, this is wrong for the rows with row number 푗 when

푗 ≤ 푏 or 푗 > 푁 − 푏 , i.e., in the ‘corners’ of the matrix.

Thus for any 1 ≤ ℓ < 푘 for which the vertex 푖ℓ+1 is new in the path, i.e., 푖ℓ+1 ∉ {푖1, … , 푖ℓ} we have at least 훽 − ℓ choices for 푖푙+1 only if

푏 < 푖ℓ ≤ 푁 − 푏 .

If 푏 /푁 → 0 (as in case 1 of the theorem) the number of exceptions (i.e., 푖ℓ ≤ 푏 or 푖푙 > 푁 − 푏 ) is negligible and the semicircle law is again valid. However, if 푏 grows proportional to 푁 the ‘exceptional’ terms are not exceptional any more but rather contribute in the limit 푁 → ∞. For details of the proof see [6] or [10].

The above argument suggests that for 푏 ≈ 푐푁 the limit distribution might be again the semicircle distribution if we ‘fill the corners’ of the matrix appropriately. This can be achieved by the following modification of (28). Definition 4.2. Set (for 푖 ∈ ℕ)

| 푖 | = min {| 푖 |, |푁 − 푖|} (29) and let 푋˜ be a Wigner ensemble. Then we call the matrix

푋˜ (푖, 푗) for |푖 − 푗| ≤ 푏 푋 (푖, 푗) = { (30) 0 otherwise . a periodic band matrix.

Here, |푖 − 푗| measures the distance of 푖 and 푗 on ℤ/푁ℤ. The choice of | ⋅ | guarantees that each column (and each row) contains exactly 훽 = 2푏 + 1 non zero (i.e., not identically zero) entries. As we anticipated, we have

Theorem 4.3. If 푋 is a periodic band random matrix with band width 훽 ≤ 푁 and 훽 → ∞, then the empirical eigenvalue distribution measure 휎 of (1/√훽 )푋 converges weakly in probability to the semicircle distribution 휎.

A proof of this result due to Bogachev, Molchanov and Pastur can be found in [6] or in [10]. 364 W. Kirsch and T. Kriecherbauer

Catalano [10] has generalized the above result to matrices of the form

|푖 − 푗| 푋 (푖, 푗) = 훼( )푋˜ (푖, 푗) (31) 푁 where 푋˜ is a Wigner matrix and 훼∶ [0, 1] → ℝ a Riemann integrable function. This class of matrices contains both random band matrices with 푏 ≈ 푐푁 and periodic random band matrices, take either 훼(푥) = 휒[0,푐](푥) or 훼(푥) = 휒[0,푐]∪[1−푐,1](푥), where 1 if 푥 ∈ 퐴, 휒퐴(푥) = { 0 otherwise.

Theorem 4.4. Let 푋 be a matrix ensemble as in (31), set

1 1 Φ ≔ ∫ ∫ 훼2(|푥 − 푦|) 푑푥 푑푦 0 0 and let 휎 be the empirical eigenvalue distribution measure for (1/√Φ푁)푋 . Then 휎 converges weakly in probability to a limit measure 휏. The limit 휏 is the semicircle law if and only if

|훼(푥)| = |훼(1 − 푥)| (32) for almost all 푥 ∈ ℝ.

Note that in the case of band matrices with bandwidth proportional to 푁, condition (32) is fulfilled for the periodic case, but not for the non periodic case(28). As for the Wigner case the question arises whether the norm of band matrices is bounded in the limit 푁 → ∞. In fact we have:

Theorem 4.5. Let 푋 be a banded Wigner ensemble (as in (28)) with bandwidth 훽 ≤ 푁 and assume that all moments of 푋 (푖, 푗) exist. If there are positive constants 훾 훾 and 퐶 such that 훽 ≥ 퐶 푁 for all 푁, then

−1/2 lim sup ‖훽 푋 ‖ ≤ 2 (33) →∞ ℙ-almost surely.

A proof of Theorem 4.5 is contained in the forthcoming paper [30]. This theorem applies to periodic band matrices as well.

Bogachev, Molchanov and Pastur [6] show that the norm of (1/√훽 ) 푋 can go to infinity if 훽 grows only on a logarithmic scale with 푁. Sixty years of moments for random matrices 365

We mention that there are various other results about matrices with independent, but not identically distributed random variables. Already the papers [35] and [36] consider matrix entries with constant variances but not necessarily identical distribution. The identical distribution of the entries is replaced by a (far weaker) condition of Lindeberg type. In the paper [21] even the condition of constant variances is relaxed. Moreover, these authors replace independence by a martingale condition.

5. Sparse Dependencies

Now we turn to attempts to weaken the assumption of independence between the 푋 (푖, 푗) of a matrix ensemble. We start with what we call ‘sparse dependencies’. This means that, while we don’t care how some of the 푋 (푖, 푗) depend on each other, we restrict the number of dependencies in a way specified below. We follow Schenker and Schulz-Baldes [38] in this section. 2 We assume that for each 푁 there is an equivalence relation ∼ on ℕ with ℕ = {1, 2, … , 푁} and we suppose that the random variables 푋 (푖, 푗) and 푋 (푘, ℓ) for 1 ≤ 푗, 푘 ≤ ℓ are independent unless (푖, 푗) and (푘, ℓ) belong to the same equivalence class with respect to ∼ .

Definition 5.1. We call the equivalence relations ∼ sparse if the following conditions are fulfilled:

3 2 1) max |{ (푗, 푘, ℓ) ∈ ℕ | (푖, 푗) ∼ (푘, ℓ) }| = 표(푁 ), 푖∈ℕ푁

3 2 2) |{ (푖, 푗, ℓ) ∈ ℕ | (푖, 푗) ∼ (푗, ℓ) and ℓ ≠ 푖 }| = 표(푁 ),

3) max |{ ℓ ∈ ℕ | (푖, 푗) ∼ (푘, ℓ) }| ≤ 퐵 for an 푁-independent constant 퐵. 푖,푗,푘∈ℕ푁

Definition 5.2. A symmetric random matrix ensemble 푋 (푖, 푗) with

2 피(푋 (푖, 푗)) = 0, 피(푋 (푖, 푗) ) = 1, 푘 and sup 피(푋 (푖, 푗) ) < ∞ for all 푘 ∈ ℕ ,푖,푗 is called a generalized Wigner ensemble with sparse dependence structure if there are sparse equivalence relations ∼ such that 푋 (푖, 푗) and 푋 (푘, ℓ) are independent if (푖, 푗) ≁ (푘, ℓ).

Examples 5.3. If 퐴 and 퐵 are Wigner matrices, then the 2푁 × 2푁-matrices 366 W. Kirsch and T. Kriecherbauer

퐴 퐵 푋 = ( ) 퐵 −퐴 and

′ 퐴 퐵 푋 = ( ) 퐵 퐴 are generalized Wigner ensembles with sparse dependence structure. Many more example classes can be found in [26].

Theorem 5.4. If 푋 is a generalized Wigner ensemble with sparse dependence −1/2 structure and 휎 is the empirical eigenvalue distribution measure of 푀 = 푁 푋 , then 휎 converges to the semicircle distribution weakly in probability.

This theorem is due to Schenker and Schulz-Baldes [38] who proved weak convergence in expectation, for convergence in probability see [10]. Catalano [10] combines sparse dependence structures with generalized band structures as in (31).

6. Decaying Correlations

In this section we discuss some matrix ensembles for which the random variables 푋 (푖, 푗) have decaying correlations. We begin by what we call ‘diagonal’ ensembles. ′ ′ By this we mean that the random variables 푋 (푖, 푗) and 푋 (푖 , 푗 ) are independent if the index pairs (푖, 푗) and (푖′, 푗′) belong to different diagonals, i. e. if 푖 − 푗 ≠ 푖′ − 푗′ (for 푖 ≤ 푗 and 푖′ ≤ 푗′).

(ℓ) Definition 6.1. Suppose 푌푛 is a sequence of random variables and 푌푛 are independent copies of 푌푛 for ℓ ∈ ℕ, then the matrix ensemble

(|푖−푗|) 푋 (푖, 푗) = 푌푖 for 1 ≤ 푖 ≤ 푗 ≤ 푁 (34) is called the matrix ensemble with independent diagonals generated by 푌푛.

Of course, if the random variables 푌푛 themselves are independent then we obtain an independent matrix ensemble. If, on the other hand, 푌푛 = 푌1, we get a matrix with constant entries along each diagonal, which vary randomly from diagonal to diagonal. Such a matrix is thus a random Toeplitz matrix. Random Toeplitz matrices were considered by Bryc, Dembo and Jiang in [9]. They prove: Sixty years of moments for random matrices 367

Theorem 6.2. Suppose that 푋 (푖, 푗) is the random Toeplitz matrix ensemble asso- 2 퐾 ciated with 푌푛 = 푌 with 피(푌) = 0, 피(푌 ) = 1 and 피(푌 ) < ∞ for all 퐾, then the −1/2 empirical eigenvalue distribution measures 휎 of 푁 푋 converge weakly almost surely to a nonrandom measure 훾 which is independent of the distribution of 푌 and has unbounded support. In particular, 훾 is not the semicircle distribution.

Friesen and Löwe [18] consider matrix ensembles with independent diagonals generated by a sequence 푌푛 of weakly correlated random variables. In their case, the limit distribution is the semicircle law again.

Theorem 6.3. Let 푌푛 be a stationary sequence of random variables with 피(푌1) = 0, 2 퐾 피(푌1 ) = 1, and 피(푌1 ) < ∞ for all 퐾. Assume ∞

∑ |피(푌1푌1+ℓ)| < ∞ . (35) ℓ=1

Let 푋 be the matrix ensemble with independent diagonals generated by 푌푛. Then the −1/2 empirical eigenvalue distribution measures 휎 of 푁 푋 converge to the semicircle distribution ℙ-almost surely.

The next step away from independence is to start with a sequence {푍푛}푛∈ℕ of random variables and to distribute them in some prescribed way on the matrix entries 푋 (푖, 푗). It turns out (see [34]) that the validity of the semicircle law depends on the way we fill the matrix with the random number 푍푛. One main example of a filling is the ‘diagonal’ one, resulting in:

푍 푍 푍 ⋯ ⋯ ⋯ 푍 ⎛ 1 +1 2 (+1)/2 ⎞ ⎜ 푍+1 푍2 푍+2 ⋯ ⋯ ⋯ ⋯ ⎟ ⎜ 푍 푍 푍 푍 ⋯ ⋯ ⋯ ⎟ 푋 = 2 +2 3 +3 (36) ⎜ ⋯ 푍 푍 푍 ⋯ ⋯ ⋯ ⎟ ⎜ 2+1 +3 4 ⎟ ⎜ ⋮ ⋮ ⋮ ⎟ ⎝푍(+1)/2 ⋯ ⋯ ⋯ ⋯ 푍2−1 푍 ⎠

Löwe and Schubert define abstractly: Definition 6.4. A filling is a sequence of bijective mappings

2 휑 ∶ {1, 2, … , 푁(푁 + 1)/2} ⟶ { (푖, 푗) ∈ {1, 2, … , 푁} | 푖 ≤ 푗 } (37)

If 푍푛 is a stochastic process and {휑 } is a filling we say that

푋 (푖, 푗) = 푍 −1 for 1 ≤ 푖 ≤ 푗 ≤ 푁 (38) 휑푁 (푖,푗) is the matrix ensemble corresponding to {푍푛} with filling {휑 } 368 W. Kirsch and T. Kriecherbauer

Another example of a filling, besides the ‘diagonal’ one, is the (symmetric) ‘row by row’ filling:

푍 푍 푍 ⋯ ⋯ ⋯ 푍 ⎛ 1 2 3 ⎞ ⎜푍2 푍+1 푍+2 푍+3 ⋯ ⋯ 푍2−1 ⎟ ⎜푍 푍 푍 푍 ⋯ ⋯ 푍 ⎟ 푋 = 3 +2 2 2+1 3−3 (39) ⎜ ⋯ 푍 푍 푍 ⋯ ⋯ ⋯ ⎟ ⎜ +3 2+1 3−2 ⎟ ⎜ ⋮ ⋮ ⋮ ⎟ ⎝푍 푍2−1 ⋯ ⋯ ⋯ ⋯ 푍(+1)/2 ⎠

Among other results, Löwe and Schubert prove:

Theorem 6.5. Suppose 푍푛 is an ergodic Markov chain with finite state space 푆 ⊂ ℝ started in its stationary measure, and assume

피(푍푛1푍푛2 ⋯ 푍푛푘) = 0 , (40) 2 피(푍푛 ) = 1 (41) for any 푛 and any 푛1, … , 푛푘 with 푘 odd. If 푋 is the matrix ensemble corresponding to {푍푛} with diagonal filling, then the −1/2 empirical eigenvalue distribution measures 휎 of 푁 푋 converge to the semicircle distribution ℙ-almost surely.

The assumptions we made in Theorem 6.5 both on 푍푛 and on the filling are only an example of the abstract assumptions given in [34]. These authors also show:

Theorem 6.6. There is an ergodic Markov chain {푍푛} with finite state space 푆 ⊂ ℝ started in its stationary measure satisfying (40) and (41) such that for the matrix ensemble 푋 corresponding to {푍푛} with row by row filling the empirical eigenvalue −1/2 distribution measures 휎 of 푁 푋 do not converge to the semicircle distribution.

Consequently, the convergence behavior of 휎 depends not only on the process {푍푛} but also on the way we fill the matrices with this process. For details werefer to [34].

7. Curie–Weiss Ensembles

In Section 6 we discussed matrix ensembles 푋 (푖, 푗) which are generated through stochastic processes with decaying correlations. Thus, for fixed 푁, the correlations 피 (푋 (푖, 푗)푋 (푘, ℓ)) become small for (푖, 푗) and (푘, ℓ) far apart, in some appropriate sense. Sixty years of moments for random matrices 369

In the present section we investigate matrix ensembles 푋 (푖, 푗) for which 피(푋 (푖, 푗)) = 0 and the correlations 피 (푋 (푖, 푗)푋 (푘, ℓ)) do not depend on 푖, 푗, 푘, ℓ for most (or at least many) choices of 푖, 푗, 푘, and ℓ, but the correlations depend on 푁 instead. More precisely, we will have that for given (푖, 푗)

피(푋 (푖, 푗)푋 (푘, ℓ)) ∼ 퐶 ≥ 0

2 for (푘, ℓ) ∈ 퐵 with |퐵 | ∼ 푁 or even |퐵 | ∼ 푁 , and, as a rule, 퐶 → 0. However, in Theorem 7.13 we will encounter an example for which 퐶 does not decay. The main example we discuss comes from statistical physics, more precisely from the Curie–Weiss model.

푀 Definition 7.1. Curie–Weiss random variables 휉1, … , 휉푀 take values in {−1, 1} with probability

푀 2 푀 −1 훽/(2푀)(∑푖=1 푥푖) ℙ훽 (휉1 = 푥1, … , 휉푀 = 푥푀) = 푍 푒 (42)

푀 where 푍 = 푍훽,푀 is a normalization constant (to make ℙ훽 a probability measure) and 훽 ≥ 0 is a parameter which is interpreted in physics as ‘inverse temperature’, 훽 = 1/푇.

If 훽 = 0 (i.e., 푇 = ∞) the random variables 휉푖 are independent, while for 훽 > 0 there is a positive correlation between the 휉푖, so the 휉푖 tend to have the same value +1 or −1. This tendency is growing as 훽 → ∞. The Curie–Weiss model is used in physics as an easy model to describe magnetism. The 휉푖 represent small magnets (‘spins’) which can be directed upwards (‘휉푖 = 1’) or downwards (‘휉푖 = −1’). At low temperature (high 훽) such systems tend to be aligned, i.e., a majority of the spins have the same direction (either upwards or downwards). For high temperature they behave almost like independent spins. These different types of behavior are described in the following theorem.

푀 Theorem 7.2. Suppose 휉1, … , 휉푀 are ℙ훽 -distributed Curie–Weiss random variables. 1 푀 Then the mean 푀 ∑푖=1 휉푖 converges in distribution, namely

푀 1 풟 훿 if 훽 ≤ 1 , ∑ 휉 ⟹ { 0 (43) 푀 푖 1 푖=1 2 (훿−푚(훽) + 훿푚(훽)) if 훽 > 1 . where 푚 = 푚(훽) is the (unique) strictly positive solution of

tanh(훽푚) = 푚 (44) 370 W. Kirsch and T. Kriecherbauer

풟 Above we used ⟹ to indicate convergence in distribution: Random variables 휁푖 converge in distribution to a measure 휇 if the distributions of 휁푖 converge weakly to 휇. Also, 훿푥 denotes the Dirac measure (see (2)). For a proof of the above theorem see e. g. [13] or [28].

Theorem 7.2 makes the intuition from physics precise: The 휉푖 satisfy a law of large numbers, like independent random variables do, if 훽 ≤ 1, in the sense 1 푀 that the distribution of 푚푀 = 푀 ∑푖=1 휉푖 converges weakly to zero, while 푚푀, the 1 ‘mean magnetization’, equals ±푚(훽) ≠ 0 in the limit, with probability 2 each, for 훽 > 1. In physics jargon, there is a phase transition for the Curie–Weiss model at 훽 = 1, the ‘critical inverse temperature’. We now discuss two matrix ensembles connected with Curie–Weiss random variables. The first one, which we call the diagonal Curie–Weiss ensemble, was introduced in [19]. It has independent ‘diagonals’, and the matrix entries within the same diagonal are Curie–Weiss distributed. Thus, it is closely related to the diagonal filling as defined in(36).

Definition 7.3. Let the random variables 휉1, 휉2, … , 휉 be ℙ훽 -distributed Curie– Weiss random variables, and take 푁 independent copies of the 휉푖, which we call

1 1 1 2 2 2 휉1 , 휉2 , … , 휉 , 휉1 , 휉2 , … , 휉 , … , 휉1 , 휉2 , … , 휉 .

Then we call the random matrix

ℓ 푋 (푖, 푖 + ℓ) ≔ 휉푖 for ℓ = 0, … , 푁 − 1 and 푖 = 1, … , 푁 − ℓ (45)

푋 (푖, 푗) ≔ 푋 (푗, 푖) for 푖 > 푗 (46)

the diagonal Curie–Weiss ensemble (with diagonal distribution ℙ훽 ).

For the diagonal Curie–Weiss ensemble, Friesen and Löwe [19] prove the following result.

Theorem 7.4. Suppose 푋 is a diagonal Curie–Weiss ensemble with diagonal dis- −1/2 tribution ℙ훽 . Then the empirical eigenvalue distribution measure 휎 of 푁 푋 converges weakly almost surely to a measure 휎훽. 휎훽 is the semicircle law 휎 if and only if 훽 ≤ 1.

Remarks 7.5.

1. The theorem shows that there is a phase transition for the eigenvalue distribution of the diagonal Curie–Weiss ensemble at 훽 = 1. Sixty years of moments for random matrices 371

2. The proof in [19] uses the moment method. It allows the authors to give an expression for the moments of 휎훽 in terms of 푚(훽) (see (43)). For large 훽 the empirical eigenvalue distribution measure of the diagonal Curie– Weiss ensemble approaches the eigenvalue distribution measure of random Toep- litz matrices we discussed in Theorem 6.2 (see Bryc, Dembo and Jiang [9]). The second Curie–Weiss-type matrix ensemble, which we call the ‘full Curie– Weiss ensemble’, is defined as follows. 2 Definition 7.6. Take 푁 Curie–Weiss random variables 푋˜ (푖, 푗) with distribution 2 ℙ훽 and set

푋˜ (푖, 푗) for 푖 ≤ 푗 , 푋 (푖, 푗) = { (47) 푋˜ (푗, 푖) otherwise .

We call the random matrix 푋 defined above the full Curie–Weiss ensemble. To our knowledge this ensemble was first considered in [25], where the following result was proved.

Theorem 7.7. Let 푋 be the full Curie–Weiss matrix ensemble with inverse temper- −1/2 ature 훽 ≤ 1. Then the empirical eigenvalue distribution measure 휎 of 푁 푋 converges weakly in probability to the semicircle distribution 휎. The proof is based on the moment method we discussed in section 3. In [25] the authors prove this result just using assumptions on correlations of the 푋 (푖, 푗) which are in particular satisfied by the full Curie–Weiss model if 훽 ≤ 1. Here, we only discuss this special case and refer to [25] for the more general case. The main difficulty in this proof is the fact that for the Curie–Weiss ensemble it is not true that

2 피훽 (푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ ⋯ ⋅ 푋 (푖푘, 푖1)) (48) is zero if an edge {푖, 푗} occurs only once in (48) (cf. (21) for the independent case). In other words, we need an appropriate substitute for Lemma 3.10. So, we need a way to handle expectations as in (48) when there are edges (index pairs, see Definition 3.6) which occur only once. Let us call such index pairs ‘single edges’. Correlation estimates as we need them can be obtained from a special way of 푀 푀 writing expectations 피훽 with respect to the measure ℙ훽 . (1) Definition 7.8. For 푡 ∈ [−1, 1] we denote by 푃푡 the probability measure on {−1, 1} given by

(1) 1 (1) 1 푃푡 (1) = 2 (1 + 푡) and 푃푡 (−1) = 2 (1 − 푡) . 372 W. Kirsch and T. Kriecherbauer

(푀) (1) 푀 푃푡 denotes the 푀-fold product of 푃푡 on {−1, 1} . If 푀 is clear from the con- (푀) (푀) text, we write 푃푡 instead of 푃푡 . By 퐸푡 resp. 퐸푡 we denote the corresponding expectation. Proposition 7.9. For any function 휙 on {−1, 1}푀 we have 1 푒−푀퐹훽(푡)/2 피푀(휙(푋 , … , 푋 )) = ∫ 퐸 (휙(푋 , … , 푋 )) 푑푡 (49) 훽 1 푀 푡 1 푀 1 − 푡2 −1 where 1 1 1 + 푡 2 퐹 (푡) = ( ln ) + ln(1 − 푡2). 훽 훽 2 1 − 푡 This proposition can be proved using the so called Hubbard–Stratonovich transformation. For a proof see [25] or [28]. The way to write expectations with respect to 푀 ℙ훽 as a combination of independent measures is typical for exchangeable random variables and is known as de Finetti representation [16]. We will discuss this issue in detail in Section 8 and in particular in [29, 30, 31]. The advantage of the representation (49) comes from the observation that under the probability measure 푃푡 the random variables 푋1,…, 푋푀 are independent, and the fact that the integral is in a form which is immediately accessible to the Laplace method for the asymptotic evaluation of integrals. The Laplace method and Proposition 7.9 yield the required correlation estimates. 푀 Proposition 7.10. Suppose 푋1, … , 푋푀 are ℙ훽 -distributed Curie–Weiss random variables. If ℓ is even, then as 푀 → ∞ 1. if 훽 < 1 훽 ℓ/2 1 피(푀)(푋 ⋅ 푋 ⋅ ⋯ ⋅ 푋 ) ≈ (푙 − 1)!! ( ) 훽 1 2 ℓ 1 − 훽 푀ℓ/2

2. if 훽 = 1 there is a constant 푐ℓ such that 1 피(푀)(푋 ⋅ 푋 ⋅ ⋯ ⋅ 푋 ) ≈ 푐 훽 1 2 ℓ ℓ 푀ℓ/4 3. if 훽 > 1 (푀) ℓ 피훽 (푋1 ⋅ 푋2 ⋅ ⋯ ⋅ 푋ℓ) ≈ 푚(훽) where 푡 = 푚(훽), as in (44), is the strictly positive solution of tanh 훽푡 = 푡.

(푀) If ℓ is odd, then 피훽 (푋1 ⋅ 푋2 ⋅ … ⋅ 푋ℓ) = 0 for all 훽. We remind the reader that for an odd number 푘 we set 푘!! = 푘 ⋅ (푘 − 2) ⋯ 3 ⋅ 1. For proof of Proposition 7.10 see again [25] or [28]. From Proposition 7.10 we get immediately the following Corollary, which substitutes Lemma 3.10. Sixty years of moments for random matrices 373

Corollary 7.11. Let 푋 be the full Curie–Weiss matrix ensemble with inverse temperature 훽, and let the graph corresponding to the sequence 푖1, 푖2, … , 푖푘 contain ℓ single edges.

1. If 훽 < 1 then

| | −ℓ | 피(푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ … ⋅ 푋 (푖푘, 푖1)) | ≤ 퐶 푁 . (50)

2. If 훽 = 1 then

| | −ℓ/2 | 피(푋 (푖1, 푖2) ⋅ 푋 (푖2, 푖3) ⋅ … ⋅ 푋 (푖푘, 푖1)) | ≤ 퐶 푁 . (51)

In the next step we have to prove a quantitative version of Proposition 3.11.

Proposition 7.12. If |{푖1, … , 푖푘}| ≥ 1 + 푘/2 + 푠 for some 푠 > 0, then there are at least 2푠 + 2 single edges in {푖1, 푖2}, {푖2, 푖3}, … , {푖푘, 푖1}. Proof. The proof is a refinement of the proof of Proposition 3.11. Suppose 풢 is a multigraph with 푟 vertices and 푘 edges. Then, as we saw already, 푘 ≥ 푟 − 1 if 풢 is connected. So, there are at most 푘 − 푟 + 1 edges left for ‘double’ connections. This means that there are at least ℓ = 푟 − 1 − (푘 − 푟 + 1) single edges, and

ℓ = 푟 − 1 − (푘 − 푟 + 1) = 2푟 − 푘 − 2 ≥ (푘 + 2 + 2푠) − 푘 − 2 = 2푠 (52) by assumption on 푟. So, by the above simple argument we are off the assertion by two only.

Now, we take into account that the sequence (푖1, … , 푖푘, 푖1) defines a closed path through the graph. Since |{푖1, … , 푖푘}| > 1 + 푘/2 there is at least one single edge. If we remove one of the single edges from the graph, this new graph 풢′ is still connected. 풢′ has 푟 vertices and 푘 − 1 edges. We redo the above argument with the graph 풢′ and get for the minimal number ℓ′ of single edges in 풢′ equation (52) with 푘 replaced by 푘 − 1, and thus obtain

ℓ′ = 푟 − 1 − (푘 − 1 − 푟 + 1) = 2푟 − 푘 − 1 ≥ 푘 + 2 + 2푠 − 푘 − 1 = 2푠 + 1. (53)

Since we have removed a single edge from 풢, the graph 풢 has at least 2푠 + 2 single edges. 374 W. Kirsch and T. Kriecherbauer

Corollary 7.11 and Proposition 7.12 together allow us to do the moment argument as in Section 2.2. We turn to the case 훽 > 1 for the full Curie–Weiss model. Part 3 of Proposition 7.10 shows that there are strong correlations in this case, so one is tempted to believe that there is no semicircle law for 훽 > 1. −1−푘 2푘 In fact, it is easy to see that for 훽 > 1 the expectations of 푁 tr (푋 ) cannot converge for 푘 ≥ 2 as 푁 → ∞. For example, for 푘 = 2 we have 1 피 ( tr (푋 4)) 푁3 1 = ∑ 피 (푋 (푖 , 푖 )푋 (푖 , 푖 )푋 (푖 , 푖 )푋 (푖 , 푖 )) + 풪(1) 푁3 1 2 2 3 3 4 4 1 푖1,푖2,푖3,푖4 all different 푁(푁 − 1)(푁 − 2)(푁 − 3) ≈ 푚(훽)4 → ∞, (54) 푁3 so the moment method will not work here. A closer analysis of the problem shows that the divergence of the moments −1/2 of traces is due to a single eigenvalue of 푁 푋 which goes to infinity. All the other eigenvalues behave ‘nicely’. Informally speaking, for 훽 > 1 the matrices 푋 fluctuate around the matrices ±푚(훽) ℰ (see (8)) with probability 1/2 each. As we saw in Section 2 these matrices have rank one. So, one may hope that they do not change the empirical eigenvalue distribution measure in the limit.

Analyzing the fluctuations around ±푚(훽) ℰ one can apply the moment meth- 2 od to 푋 ∓ 푚(훽) ℰ . The variance of the matrix entries is 푣(훽) = 1 − 푚(훽) , so this has a chance to converge to the semicircle distribution, but scaled due to the variance 푣(훽) < 1. In fact we have:

Theorem 7.13. Let 푋 be the full Curie–Weiss matrix ensemble with arbitrary inverse temperature 훽 ≥ 0. Then the empirical eigenvalue distribution measure 휎 −1/2 of 푁 푋 converges weakly in probability to the rescaled semicircle distribution 휎푣(훽), given by:

√4푣(훽) − 푥2/(2휋푣(훽)) for |푥| ≤ 2√푣(훽), 휎푣(훽)(푥) = { (55) 0 otherwise. Here, 푣(훽) = 1 − 푚(훽)2 with 푚(훽) = 0 for 훽 ≤ 1 and 푚 = 푚(훽) is the unique positive solution of tanh(훽푚) = 푚 for 훽 > 1 (cf.(44)). A detailed proof will be contained in [29]. −1/2 Already in (54) we saw that the norm of 푁 푋 does not converge for the full Curie–Weiss ensemble if 훽 > 1. This is made precise in the following theorem. Sixty years of moments for random matrices 375

Theorem 7.14. Suppose 푋 is a full Curie–Weiss ensemble. 1. If 훽 < 1 then ‖ 1 ‖ ‖ 푋 ‖ → 2 as 푁 → ∞ ‖√푁 ‖ ℙ-almost surely. 2. If 훽 = 1 then 1 ‖ 푋 ‖ → 0 as 푁 → ∞ ‖푁훾 ‖ for every 훾 > 1/2 ℙ-almost surely. 3. If 훽 > 1 then 1 ‖ 푋 ‖ → 푚(훽) as 푁 → ∞ ‖푁 ‖ ℙ-almost surely.

Theorem 7.14.3 was proved in [25], 1 and 2 can be found in [31].

8. Ensembles with Exchangeable Entries

The results presented in the previous section for Curie–Weiss ensembles with subcritical temperatures (훽 > 1) suggest that models with correlations that do not decay sufficiently fast as 푁 tends to infinity (e.g. in the sense of Corollary 7.11) may display a wealth of spectral phenomena depending on the specific features of the model. This is largely uncharted territory. One step into this world is to consider matrix ensembles with entries chosen from a sequence of exchangeable random variables.

A sequence (휉푖)푖∈ℕ of real valued random variables with underlying probability space (Ω, ℱ, ℙ) is called exchangeable, if for all integers 푁 ∈ ℕ, all permutations 휋 on {1, … , 푁}, and 퐹 ∈ ℬ(ℝ ) it is true that

ℙ((휉1, … , 휉 ) ∈ 퐹) = ℙ((휉휋(1), … , 휉휋() ) ∈ 퐹) . Generalizing a result of de Finetti [16, 17] for random variables that only take on two values, Hewitt and Savage [24, Theorem 7.4] showed in a very general setting that such probability measures ℙ may be represented as averages of i.i.d. sequences with respect to some probability measure 휇. In our context we impose the additional condition that all moments of the random variables 휉푖 exist (cf. Definition 2.3). This leads us to the following general definition of ensembles of real symmetric matrices with exchangeable entries. 376 W. Kirsch and T. Kriecherbauer

Definition 8.1. Let 휇 denote a probability measure on some measurable space (0) (푇, 풯) and let Λ∶ 푇 → ℳ1 (ℝ) be a measurable map that assigns every element 휏 of 푇 to a Borel probability measure Λ휏 on ℝ for which all moments exist (we call (0) ℳ1 (ℝ) the set of all such probability measures on ℝ). Define

∞ ℙ 휇,Λ ≔ ∫ 푃 푑휇(휏) , with 푃 ≔ Λ , (56) 휏 휏 ⨂ 휏 푇 푖=1 as the 휇-average of i.i.d. sequences of real random variables with distributions Λ휏. The corresponding matrix ensemble with exchangeable entries consists of matrices 푋 with entries 푋 (푖, 푗) for 1 ≤ 푖 ≤ 푗 ≤ 푁, given by the first 푁(푁 + 1)/2 members of the sequence (휉푖)푖 of exchangeable random variables that is distributed according 휇,Λ to ℙ of (56). The remaining entries 푋 (푖, 푗), where 1 ≤ 푗 < 푖 ≤ 푁, are then fixed by symmetry 푋 (푖, 푗) = 푋 (푗, 푖). Observe that due to the exchangeability of (휉푖)푖 it is of no relevance in which order the upper triangular part of 푋 is filled by 휉1, …, 휉(+1)/2 . Moreover, one could have chosen any 푁(푁 + 1)/2 distinct members of (휉푖)푖 to fill the entries of 푋 without changing the ensemble.

It is instructive to consider the special case of ensembles that allow only for matrix entries 푋 (푖, 푗) ∈ {1, −1}. We refer to it as the spin case. Observe that the probability measures with support contained in {1, −1} are all represented by the 1 family Λ휏 = 2 [(1 + 휏)훿1 + (1 − 휏)훿−1], 휏 ∈ 푇 ≔ [−1, 1]. Hence all ensembles of the spin case are given by (56) with the just mentioned choices for 푇 and Λ휏. They are parameterized by the probability measures 휇 on [−1, 1]. Recall that Λ휏 already (1) appeared in Definition 7.8 as the building block 푃푡 for Curie–Weiss ensembles. What is different from Section 7 is that there the averaging measure 휇 depends on the matrix size 푁 and is of a special form. Let us return to the general ensembles with exchangeable entries of Defini- tion 8.1. The key for analyzing both the empirical eigenvalue distribution measure and the operator norm is that for every 휏 ∈ 푇 the measure 푃휏 generates i.i.d. entries for 푋 . For the latter ensembles 푃휏 the following observations that can already be found in [20] are useful: Subtracting the mean of the entries yields a Wigner ensemble (multiplied by the standard deviation of Λ휏) for which Theorem 3.3 is applicable. Considering first the empirical eigenvalue distribution measure, we note that the mean is some multiple of the matrix ℰ defined in (8). Since ℰ has rank 1, the subtraction of the mean will not have an influence on the limiting 휇,Λ spectral measure. As ℙ is the 휇-average over all measures 푃휏, it is plausible that the limit of the empirical eigenvalue distribution measures is an average of scaled semicircles w.r.t. the measure 휇, where the scaling factors are given by the Sixty years of moments for random matrices 377 standard deviation of Λ휏. Accordingly, we define

휎휇 ≔ ∫ 휎푣(휏) 푑휇(휏) , (57) 푇 where 푣(휏) denotes the variance of Λ휏 and 휎푣 is the semicircle distribution with support [−2√푣, 2√푣 ] (cf. Definition (55)). We prove in [30]

휇,Λ Theorem 8.2. Denote by ℙ , 휎휇 the measures introduced in Definition 8.1 and in (57). Then the empirical eigenvalue distribution measures 휎 of 푋 /√푁 converge 휇,Λ weakly in expectation to 휎휇 w.r.t. the measure ℙ .

Moreover, it is shown in [30] that 휎휇 is a semicircle if and only if the function 휏 ↦ 푣(휏) is constant 휇-almost surely.

For the operator norm the situation is quite different. Since ‖ℰ ‖ = 푁, the operator norm of 푋 w.r.t. the measure 푃휏 is determined to leading order by the mean of 푋 , if the mean does not vanish. Therefore the operator norm scales with 푁, except for the special case that the matrix entries are ℙ 휇,Λ-almost surely centered. We prove in addition in [30] that the 푁-scaling of the norm is due to a single outlier of the spectrum by showing that the second largest eigenvalue (in modulus) possesses a √푁-scaling that is consistent with the law for the limiting spectral measure. In [30] we also generalize the just mentioned results to band matrices. Here an additional difficulty arises, because the mean of 푋 is no longer a multiple of ℰ and will have large rank. Nevertheless it is shown that all results obtained for full matrices can be saved, except for the result on the second largest eigenvalue (in modulus).

Acknowledgment. The authors would like to thank the referee for an exceptionally careful reading of the manuscript and for a number of valuable sugges- tions.

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Bound states of Schrödinger type operators with Heisenberg sub-Laplacian

Ari Laptev and Andrei Velicu

Dedicated to Helge Holden on the occasion of his 60th birthday

Abstract. Using the technique from [8] we find a new constant in a Cwikel–Lieb–Rozen- blum type inequality that estimate the number of negative eigenvalues of a Schrödinger operator involving the Heisenberg sub-Laplacian with a potential that is proportional to the characteristic function of a measurable set.

1. Introduction

Consider the Schrödinger operator

퐻 = Δ − 푉, where 푛 휕2 Δ = − ∑ 2 푖=1 휕푥푖 푛 is the usual Laplacian on ℝ and 푉 is a decaying potential. Let 휆1 ≤ 휆2 ≤ ⋯ < 0 be its negative eigenvalues, and let 푁(푉) be the number of such eigenvalues (counted with multiplicity). The celebrated Lieb–Thirring inequalities [15] give estimates on the sum of powers of the negative eigenvalues of 퐻, namely

훾 푛/2+훾 푆훾(푉) ≔ ∑ |휆푖| ≤ 퐿푛,훾 ∫ 푉+(푥) d푥, 푖 ℝ푛 with some constant 퐿푛,훾. Here and elsewhere below we use the notation 푓+(푥) = 1 max {푓(푥), 0}. This inequality holds for 훾 > 0 if 푛 ≥ 2 or 훾 ≥ 2 if 푛 = 1. In the case 훾 = 0, 푛 ≥ 3, this inequality gives bounds on 푁(푉). It was proved independently by Rozenblum, Cwikel and Lieb, and is known as the Cwikel–Lieb–Rozenblum inequality (see [3], [12] and [16]). The best constant in this inequality is due to Lieb. 382 A. Laptev and A. Velicu

In [8], the Cwikel–Lieb–Rozenblum inequality was proved using elementary methods in the case in which the potential is proportional to the indicator function of a finite measure set, giving sharper estimates on the constant. In this short article we consider a similar result in the case of a Schrödinger type operator which involves the Heisenberg sub-Laplacian. An important element of this proof is the property which connects the Heisenberg sub-Laplacian to a Schrödinger operator with constant magnetic field, given by the relation

−1 2 ℱΔℍℱ 푓(푥, 푦, 푡) = (푖∇(푥,푦) + 푡퐴(푥, 푦)) , 1 where 퐴(푥, 푦) = 2 (−푦, 푥), 1 ℱ푓(푥, 푦, 휉) = ∫ 푓(푥, 푦, 푡)푒−푖푡⋅휉 d푡 √2휋 is the usual Fourier transform with respect to the last coordinate, and Δℍ is the Heisenberg sub-Laplacian as defined below. In particular, this connection has also been exploited in [5] in deriving an inequality between Neumann and Dirichlet eigenvalues of the sub-Laplacian. It was also used in the paper [7], where the authors obtained Li–Yau type inequalities for the spectrum of the Dirichlet boundary problem for such operators. We also refer to a related paper [4], where the authors studied the Dirichlet eigenvalues for Laplacians with constant magnetic fields. The Schrödinger type operators with Heisenberg sub-Laplacians are operators that generate a Markov semigroup. CLR inequalities for operators 퐵 − 푉, where 퐵 > 0 is an operator that generates a Markov semigroup, were first considered in [10] and in [17], where such inequalities were extended to generators of positively dominated semigroups. This result was also given in [6]. In particular, the inequality for the number 푁(푉) of the negative eigenvalues of the operator −Δℍ − 푉, with 푉 ≥ 0, follows from [6, Theorem 2.1 (see also page 8)]

2 푁(푉) ≤ 퐿0 ∫ 푉 (푤) 푑푤, (1) ℝ3 where ∞ −1 퐾 −1 푎 −휆 −1 퐿0 = min 푎 푒 (1 − 푎 ∫ 푒 (휆 + 푎) 푑휆) . 2 푎>0 0 −2 Here the constant 퐾 appears in the estimate ‖exp(−푡Δℍ)‖1→∞ < 퐾푡 , and equals 퐾 = 1/16. The latter follows immediately from the explicit value of the kernel exp(−푡Δℍ) on the diagonal. For the respective Lieb–Thirring inequalities the authors [6] have obtained

훾 Tr (Δ − 푉) ≤ 퐿 ∫ 푉훾+2(푤) 푑푤, (2) ℍ − 훾 Bound States of operators with Heisenberg sub-Laplacian 383 where 훾훾+1 Γ(휃 + 3)Γ(훾 − 휃) 퐿훾 = 퐿0 inf . Γ(훾 + 3) 휃<훾 휃휃(훾 − 휃)훾−휃 When proving (2), the authors have used a combination of the Aizenman–Lieb type of argument together with some interpolation. For example, the constant 퐿1 = 0.08…. This is almost two times better than if one uses the classical Lieb–Thirring method from [15] directly. Note that sharp constants in the inequalities (1) and (2) for any 훾 ≥ 0 are unknown. The main result of this paper is Theorem 1, where we prove that if Ω ⊂ ℝ3 is a measurable set of finite measure and 푉 = 푐휒Ω, where 휒 is the characteristic function of the set Ω, then 푐2|Ω| 푁(푐휒 ) ≤ . Ω 6

Note that the constant in (1) equals 퐿0 = 0.18856… > 0.166… = 1/6. In the remainder of this introduction we provide a brief background to the Heisenberg group, while in Section 2 we follow [8] and prove a special case of the Cwikel–Lieb–Rozenblum inequality. The Heisenberg group ℍ is the group with underlying set ℝ3 and group operation defined by

′ ′ ′ ′ ′ ′ 1 ′ ′ (푥, 푦, 푡) ∘ (푥 , 푦 , 푡 ) = (푥 + 푥 , 푦 + 푦 , 푡 + 푡 − 2 (푥푦 − 푥 푦)). We will write the elements of ℍ in general as 푤 = (푥, 푦, 푡) and denote 푤 = (푥, 푦). The following vector fields generate the whole Lie algebra:

휕 1 휕 휕 1 휕 푋 = + 푦 , 푌 = − 푥 . 휕푥 2 휕푡 휕푦 2 휕푡 We define the Heisenberg sub-Laplacian to be the operator

2 2 Δℍ ≔ −푋 − 푌 .

This is a self-adjoint operator whose quadratic form is 퐻1(ℝ3). In what follows we will be interested in Schrödinger operators of the form

퐻 = Δℍ − 푉, where 푉 ≥ 0. Keeping the same notation as above, let 휆1 ≤ 휆2 ≤ ⋯ < 0 be the negative eigenvalues of 퐻 and denote by 푁(푉) the number of such eigenvalues. We also recall the following facts about the spectral decomposition of the sub-Laplacian, see for example [18]. 384 A. Laptev and A. Velicu

∞ 1 ′ 1 ′ 2 Δℍ푓(푤) = 2 ∫ ∑ (2푘 + 1)|휆| ∫ 푓(푤 )퐿푘 ( 2 |휆| ⋅ |푤 − 푤 | ) ⋅ (2휋) ℝ 푘=0 ℝ3 − 1 |휆|⋅|푤−푤′|2 푖 (푥푦′−푦푥′)+푖휆(푡′−푡) ′ ⋅ 푒 4 푒 2 |휆| d푤 d휆, where 퐿0, 퐿1,… are the Laguerre polynomials. For a Borel measurable function 휑 ∶ ℝ → ℝ it is natural to define

∞ 1 ′ 1 ′ 2 휑(Δℍ)푓(푤) = 2 ∫ ∑ 휑((2푘+1)|휆|) ∫ 푓(푤 )퐿푘( 2 |휆| ⋅ |푤 − 푤 | ) ⋅ (2휋) ℝ 푘=0 ℝ3 − 1 |휆|⋅|푤−푤′|2 푖 (푥푦′−푦푥′)+푖휆(푡′−푡) ′ ⋅ 푒 4 푒 2 |휆| d푤 d휆.

Let

∞ ′ 1 1 ′ 2 퐾휑(푤, 푤 ) = 2 ∫ ∑ 휑((2푘 + 1)|휆|)퐿푘( 2 |휆| ⋅ |푤 − 푤 | ) ⋅ (2휋) ℝ 푘=0 − 1 |휆|⋅|푤−푤′|2 푖 (푥푦′−푦푥′)+푖휆(푡′−푡) ⋅ 푒 4 푒 2 |휆| d휆.

Then ′ ′ ′ 휑(Δℍ)푓(푤) = ∫ 푓(푤 )퐾휑(푤, 푤 ) d푤 , ℝ3 and we have

Tr(휑(Δℍ)) = ∫ 퐾휑(푤, 푤) d푤. ℝ3

2. The Cwikel–Lieb–Rozenblum inequality

Here we consider potentials that are given by the characteristic function of a measurable set. Let Ω ⊂ ℝ3 be a measurable set of finite measure, and consider 푉 = 푐휒Ω for a positive constant 푐. The main result is the following.

Theorem 1. Recall that 푁(푐휒Ω) denotes the number of negative eigenvalues of

퐻 = Δℍ − 푐휒Ω.

Then we have 푐2|Ω| 푁(푐휒 ) ≤ . Ω 6 Bound States of operators with Heisenberg sub-Laplacian 385

In order to prove this result we use the Birman–Schwinger principle and the Berezin–Lieb trace inequality, whose statement adapted for our case is given below (see [2] and [9] for the full generality of the Berezin–Lieb trace inequality). Define, for 휉 > 0, the operator

−1 푇휉 ≔ 푐휒Ω(Δℍ + 휉) 휒Ω, and denote by 푛(휉) the number of eigenvalues of 푇휉 that are greater or equal to one.

Proposition 2 (Berezin–Lieb trace inequality). Let 휑∶ ℝ → [0, ∞) be a convex function with 휑(0) = 0. Then, for any 휉 > 0, we have the inequality

−1 −1 Tr 휑(푇휉) = Tr 휑(푐휒Ω(Δℍ + 휉) 휒Ω) ≤ Tr(휒Ω휑(푐(Δℍ + 휉) )휒Ω).

Proof of Theorem 1. For any 0 < 푎 < 1, consider the convex function 휑푎 ∶ ℝ → ℝ+ defined by 푡 − 푎, if 푡 > 푎 휑푎(푡) = (푡 − 푎)+ = { 0, otherwise. Fix 휉 > 0. We can bound the number of eigenvalues larger or equal to 1 of the operator 푇휉 in terms of Tr 휑푎(푇휉), i.e., we have 1 푛(휉) ≤ Tr 휑 (푇 ). 1 − 푎 푎 휉 Applying the Berezin–Lieb trace inequality, we have furthermore 1 푛(휉) ≤ Tr(휒 휑(푐(Δ + 휉)−1)휒 ). (3) 1 − 푎 Ω ℍ Ω

−1 But the operator 휒Ω휑푎(푐(Δℍ + 휉) )휒Ω has integral kernel

퐾(푤, 푤′) = 휒 (푤)휒 (푤′)퐾 (푤, 푤′), Ω Ω 휙휉

−1 for a function 휙휉(푡) = 휑푎(푐(푡 + 휉) ). Therefore, we can compute explicitly

Tr (휒 휑 (푐(Δ − 휉)−1)휒 ) = ∫ 퐾 (푤, 푤) d푤 Ω 푎 ℍ Ω 휙휉 Ω |Ω| ∞ = 2 ∫ |휆| ∑ 휙휉((2푘 + 1)|휆|) d휆, (2휋) ℝ 푘=0 386 A. Laptev and A. Velicu where we used the fact that 퐿푘(0) = 1 for all 푘. By the dominated convergence theorem, as 휉 → 0, this converges to

|Ω| ∞ 2 ∫ |휆| ∑ 휙0((2푘 + 1)|휆|) d휆 (2휋) ℝ 푘=0 ∞ |Ω| 푐 = 2 ∫ |휆| ∑ ( − 푎) d휆 (2휋) ℝ 푘=0 (2푘 + 1)|휆| + 푐 ∞ 0 |Ω| (2푘+1)푎 푐 푐 = ∑ (∫ ( − 푎휆) d휆 + ∫ ( + 푎휆) d휆) 2 2푘 + 1 푐 2푘 + 1 (2휋) 푘=0 0 − (2푘+1)푎 ∞ 푐2|Ω| 1 푐2|Ω| = ∑ = . 4푎휋2 2 24푎 푘=0 (2푘 + 1)

We note that we have lim 푛(휉) = 푁(푐휒Ω), so, passing to the limit 휉 → 0 in (3), we 휉→0 have proved that 푐2|Ω| 푁(푐휒 ) ≤ . Ω 24푎(1 − 푎) The maximum of the function 1/(푎(1 − 푎)) on the interval (0, 1) is attained at 1 푎 = 2 , thus obtaining 푐2|Ω| 푁(푐휒 ) ≤ . Ω 6

Acknowledgements. AL is grateful to R. Frank for useful discussions. The work was supported by the grant of the Russian Federation Government for scientific research under the supervision of leading scientist at the Siberian Federal University, contract N. 14.Y26.31.0006, and by RFBR grant 14-01-00544.

References

[1] M.Sh Birman, The spectrum of singular boundary problems, (Russian) Mat. Sb. (N.S.) 55 (97) (1961), 125–174. (English) Amer. Math. Soc. Transl. 53 (1966), 23–80. [2] F. Berezin, Convex functions of operators, Mat.sb. 88 (1972), 268–278. [3] M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. Math. 106 (1977), 93–102. [4] L. Erdös, M. Loss and V. Vougalter, Diamagnetic behavior of sums of Dirichlet eigenvalues, Ann. Inst. Fourier (Grenoble) 50 (2000), 891–907. Bound States of operators with Heisenberg sub-Laplacian 387

[5] R. Frank and A. Laptev, Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group, IMRN 15 (2010), 2889–2902. [6] R. Frank, E.H. Lieb and R. Seiringer, Equivalence of Sobolev inequalities and Lieb–Thirring inequalities, in XVIth International Congress on Mathematical Physics, Proceedings of the ICMP held in Prague, August 3-8, 2009, P. Exner (ed.), 523–535, World Scientific, Singapore, [7] A.M. Hansson and A. Laptev, Sharp spectral inequalities for the Heisenberg Laplacian, In Groups and analysis 354, London Math. Soc. Lecture Note Ser., 100–115. Cambridge Univ. Press, Cambridge, 2008. [8] A. Laptev, On inequalities for the bound states of Schrödinger operators, In Partial differential operators and mathematical physics (Holzhau, 1994) 78, Oper. Theory Adv. Appl., Birkhäuser, Basel, (1995), 22–225. [9] A. Laptev and Yu. Safarov, A generalization of the Berezin–Lieb inequality, Amer. Math. Soc. Transl., (2) 175 (1996), 69–79. [10] D. Levin, M. Solomyak, The Rozenblum–Lieb–Cwikel inequality for Markov generators, J. Anal. Math. 71 (1997), 173–193. [11] P. Li, S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., no. 3 88 (1983), 309–318. [12] E. H. Lieb, Bounds on the eigenvalues of the Laplace and Schrödinger operators, Bull. Amer. Math. Soc. 82 (1976), 751–752. The number of bound states of one body Schrödinger operators and the Weyl problem, Proc. A.M.S. Symp. Pure Math. 36 (1980), 241–252. [13] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics. American Mathe- matical Society 14, Providence, RI, second edition, 2001. [14] E. H. Lieb and R. Seiringer, The stability of matter in quantum mechanics, Cambridge University Press, Cambridge, 2010. [15] E.H. Lieb and W.E. Thirring, Inequalities for the moments of the eigenvalues of Schrödinger equation and their relation to Sobolev inequalites, Studies in Mathematical Physics, editors E.H.Lieb, B.Simon and A.Wightman, Princeton Univ. Press (1976), 269–303. [16] G.V. Rozenblum, Distribution of the discrete spectrum of singular differential operators, Soviet Math. Dokl. 13 (1972), 245–249, and Soviet Math., (Iz. VUZ) 20 (1976), 63–71. [17] G.V. Rozenblum, M. Solomyak, The Cwikel–Lieb–Rozenblum estimator for generators of positive semigroups and semigroups dominated by positive semigroups, St. Petersburg Math. J., no. 6 9 (1998), 1195–1211. [18] L. Roncal and S. Thangavelu, Hardy’s inequality for fractional powers of the sub- Laplacian on the Heisenberg group Adv. Math. 302 (2016), 106–158. [19] J. Schwinger, On the bound states of a given potential, Proc. Nat. Acad. Sci. 47 (1961), 122–129.

On Holden’s seven guidelines for scientific computing and development of open-source community software

Knut-Andreas Lie

Dedicated to Helge Holden on the occasion of his 60th birthday

Abstract. Two decades ago, Helge Holden proposed seven guidelines to improve the way new achievements and results in scientific computing were presented, evaluated, and compared in contemporary scientific literature. In this essay, written as a tribute to Helge onhis 60th birthday, I revisit the guidelines and point out why they are still valid today seen from my perspective, working as a contract researcher at the interface between mathematics and applications in industry. Developing new computational methods usually involves a lot of experimental programming. Over the past decade, my research group has developed an open-source community code that today has hundreds of users worldwide. I discuss some considerations that have gone into this development and present a few lessons learned. Moreover, based on this experience, as well as from development of professional software for our clients, I present advice on how you can be more productive in your experimental programming and increase the impact of your scientific results. Science is what we understand well enough to explain to a computer. Art is everything else we do. – Donald Knuth, Foreword to the book A=B (1996)

1. Introduction

Throughout the 1980 and 90s, numerical computation established itself as a third way to science in complement to the classical duality of experiments and theoretical models. These were vigorous times for scientific computing. Major advancements in numerical discretization methods and iterative linear solvers, combined with a continuous and rapid growth in computing power, enabled highly resolved numerical simulations to be adopted in many new scientific disciplines. Growing maturity of third-generation programming languages like FORTRAN, C, and C++ enabled scientists to write simulation programs of unprecedented complexity, and color monitors with powerful computer graphics spawned the development of advanced and powerful visualization techniques that increased our ability to visually interpret and understand the results of advanced simulations. During the 390 K.-A. Lie same period, LATEX became widely adopted among scientists, which together with the emergence of the world wide web in the early 1990s, dramatically changed the way science was communicated. All of a sudden, it was quite simple to include both vector and raster graphics in your scientific papers, make very impressive presentations, and quickly share these with your colleagues around the world. Altogether, this presented unparalleled opportunities for members of the relatively young scientific-computing community, which grew rapidly in numbers. Being part of a revolution, it is easy to become too eager in your quest for progress and forget or disregard wisdom and well-established practices developed by previous generations. Helge Holden was among those who saw this, and during my first years as his student, he wrote a paper [11] in an attempt to influence the way computer simulations were performed and presented to the scientific community: “[…] some words of warning may be appropriate at this moment as we are easily becoming victims of ever more impressive presentations. It is easy both as spectators and performers of the art of scientific computing to forget the critical eye of the scholar and the rigorous requirements of modern science. It is becoming all too common to present results of simulations lacking sufficient documentation to allow the repetition or reproduction of the results.” To amend what he perceived as a serious deficiency, Helge proposed seven guidelines1:

1. Your results should always be reproducible.

2. Test the stability of your method with respect to variation of parameters.

3. Compare your method to other methods.

4. Report cases where your method fails.

5. When possible, compare the computer simulations to real experiments.

6. Establish standard test cases in your field.

7. Make your own code available to your colleagues.

Some of the observations presented in the manuscript were quite controversial at that time, and the paper was never accepted for publication. However, being controversial does not mean you are wrong, and in this essay, written on the occasion of Helge’s 60th birthday, I try to give him the credit he deserves by providing a complementary discussion of the ideas put forth in his original paper. In particular, I will try to relate part of the discussion to research activity over the past two decades, involving joint supervision of a number of master and doctoral

1The paper actually started out as “Ten commandments on scientific computing”, but was toned down during the process towards potential publication. Holden’s seven guidelines for scientific computing 391 students. A main achievement of this research is the development of MRST, a comprehensive toolbox for rapid prototyping of new computational methods for subsurface flow modelling (https://www.sintef.no/mrst), and OPM, an open innovation platform for industry-grade simulations (https://opm-project.org). In the last part of the essay, I discuss some of the considerations that have gone into the development of these softwares and summarize some lessons learned. Like many other scientists who spend a major portion of their time writing software, members of my research team are self-taught. However, we have been exposed to best practices for professional software development as part of our contract research. I summarize what I have observed to be good practices if you want to be a productive developer of computational methods, write reliable codes, and increase the impact of your work.

2. Reproducible research for scientific computing

Replication of experiments is usually considered the golden standard in science and should not be confused with the principle of reproducibility, which Helge suggested as a necessity if scientific computing was to be considered as a serious science. I like the explanation of the difference between the two concepts given by the editor of the Biostatistics journal [28]: The replication of scientific findings using independent investigators, methods, data, equipment, and protocols has long been, and will continue to be, the standard by which scientific claims are evaluated. However, in many fields of study there are examples of scientific investigations that cannot be fully replicated because of a lack of time or resources. In such a situation, there is a need for a minimum standard that can fill the void between full replication and nothing. One candidate for this minimum standard is “reproducible research”, which requires that data sets and computer code be made available to others for verifying published results and conducting alternative analyses. Although offered in a different scientific field, it applies equally well toscientific computing. The notion of reproducible research in scientific computing is usually attributed to Jon Claerbout [9, 10]. In 1990, he set a goal of reproducibility for his research group at Stanford University. The goal was that not only should anybody be able to recompute the group’s research results on any computer, but they should also be able to reproduce documents the group had published to present this research. At that time, this was a quite ambitious undertaking. Today, it somewhat simpler if you use notebook facilities in a scripted language. One good example is the Jupyter Notebook for Python (https://jupyter.org/), which enables you to mix computer code with rich text, mathematical formulas, plots, and rich media. 392 K.-A. Lie

Similar functionality was recently introduced through so-called Live Scripts in MATLAB, which in many aspects supersedes the useful, albeit less powerful publish function. Likewise, use of virtual machines or container systems like Docker can be good ways to disseminate research on computational methods. A virtual machine emulates a computer system and enables others to rerun the software and data used by the authors of a scientific paper, without having to download and set up the necessary software libraries used for the simulations. Software containers are more lightweight systems that only bundle the libraries and settings necessary to make your code run on any system. Notice, however, that virtual machines and container systems only ensure a very limited type of reproducibility since all you can do is rerun numerical experiments and change input parameters. Without access to source code, you really cannot dig into the code to understand how it works and verify that it indeed implements what is written in the scientific paper. Access to complete source code, as suggested in Holden’s last guideline, is therefore an important ingredient to reproducibility and the higher goal of replicability. We will come back to this later. During the last two decades, the idea of reproducible research in computational science has picked up significant momentum and has been voiced by a large group of well-respected and influential computational scientists, see e.g., [19]. However, if one chooses to look critically at scientific publishing, it has largely remained in the same sorry state as observed by Helge Holden in 1994 [11]. Here, I have chosen to include two quotations by other scientists. The first is from 1995 by Buckheit and Donoho [6]:

An article about computational science in a scientific publication is not the scholarship itself, it is merely advertising of the scholarship. The actual scholarship is the complete software development environment and the complete set of instructions which generated the figures.

One decade later, the unfortunate situation was expressed even more pointedly by LeVeque [17]:

Within the world of science, computation is now rightly seen as a third vertex of a triangle complementing experiment and theory. However, as it is now often practiced, one can make a good case that computing is the last refuge of the scientific scoundrel […] Where else in science can one get away with publishing observations that are claimed to prove a theory or illustrate the success of a technique without having to give a careful description of the methods used, in sufficient detail that others can attempt to repeat the experiment? […] Scientific and mathematical journals are filled with pretty pictures these days of computational experiments that the reader has no hope of repeating. Even Holden’s seven guidelines for scientific computing 393

brilliant and well intentioned computational scientists often do a poor job of presenting their work in a reproducible manner. The methods are often very vaguely defined, and even if they are carefully defined, they would normally have to be implemented from scratch by the reader in order to test them.

Even now, ten years after, much of the same observations hold true. As referee and editor, I have never to date been offered the possibility to look at any authors’ source code or use their software to rerun and verify numerical experiments reported in the paper. There are journals that require software to be published alongside papers, but these are few. Fortunately, there are indications that for many scientific journals it is more a question of when and how the requirement for reproducibility will be mandated. An increasing number of journals are offering authors the possibility to upload their computer code and input data, so that others can download and experiment with these on their own computer. Never- theless, even though readers tend to access scientific publications electronically, the standard is still a static text document in most journals, and review of software and interactive, notebook-type presentation of numerical experiments have yet to permeate scholarly publishing. In the future, one can only hope that the growing demand for open-access publishing and the general competition within scientific publishing will induce a much needed innovation toward more interactive formats that better support the principle of reproducibility.

3. From proof-of-concept towards widespread adoption

Many researchers develop new methods to satisfy their own curiosity, or because it is great fun, but I still believe that most of us do it because we want to make something useful and have a lasting impact on the scientific community and/or society. In this, academia and contract research organizations, like the one I have worked in for the past two decades, are not very different. The difference lies more in how we measure impact and success. In academia, the apparent success criteria are theories and scientific papers, whereas impact can be measured in terms of citations, invitations to conferences, etc., which are superficial indications of the more vague concept scientific quality. Publications and citations are also important in contract research organizations, but creating values for your clients and acquiring new research contracts generally rank higher. In this section, I discuss how Holden’s Guidelines 2 to 6 can be used to help you succeed and ensure that the methods you develop have an impact, regardless of whether you work in academia or closer to industry and commerce. My focus will primarily be on the experimental process leading up to new computer codes 394 K.-A. Lie whose aims are to verify that new computational methods work as claimed, verify and validate physical models, and/or provide proof-of-concepts for new computational workflows. High scientific quality in this process is utterly important ifthe computational methods developed should later enter large-scale community codes used for scientific discoveries in other parts or science, or professional production codes developed to support (critical) decisions in the private and public sector.

3.1. Verification and comparison with other methods. To justify the development of a new computational method and entice the interest of others, two approaches are common to use, possibly in combination:

• You can either demonstrate that your method solves a new problem not yet solved by other methods, or that is solves a class of problems that so far has only been partially solved; or

• You can demonstrate that your method solves a known class of problems better than existing methods, e.g., by comparing with these methods and/or pointing out deficiencies that your method does not have(Guideline #3).

Providing honest and fair comparisons with existing methods is more difficult than it may sound. In well-established applications of various forms of fluid or solid dynamics, there is usually a plenitude of computational methods to compare with. It is therefore tempting to pick a standard textbook method, which is simple to implement and whose limitations and deficiencies are widely accepted. While such comparisons can be informative to a certain point, they do not carry the same value as comparisons with a state-of-the-art method. The best is, of course, to collaborate directly with the developers of the method you wish to compare with, since they have intimate knowledge of the inner workings of the method and know how to tune (undocumented) features to insure optimal performance. Such collaborations are sometimes out of the question if commercial interests are involved or the goal of your research is to defeat the other method. Unless implemented as open source, it is therefore often difficult to get your hands on a functional and efficient implementation of the methods you should compare with, especially if they are from recent literature; I will get back to this later when discussing Guideline #7 in Section 4. Your only option is then to implement the methods yourself. This is in many cases a significant undertaking and you easily end up spending a considerable time reinventing or reverse-engineering crucial algorithmic features that are not well documented for the reasons discussed in the previous section. Let me take one of my own papers [14], which compared and contrasted various upscaling and multiscale methods for simulating two-phase flow in porous media, Holden’s seven guidelines for scientific computing 395 as an example. Writing this paper required almost a half-year of concentrated effort to bring our implementation of methods not developed by ourselves to a maturity level where we could trust them to provide fair and unbiased comparisons with our own multiscale method. This, despite the fact that the first author is an unusually smart and capable programmer. In our case, this exercise proved to be worthwhile since it gave us a lot of insight that could be used in subsequent research. Around that period, we had what I would describe more as a friendly competition than a direct cooperation with the developers of one of the contending multiscale method. Whenever they published a refined version of their multiscale method, we tried to come up with test cases that showed deficiencies in their method and rendered ours in a good light, and vice versa. My impression is that the overall development of multiscale methods benefited from such a healthy competition, and Iwould generally recommend it as a means to bring your research rapidly forward. This brings me to the choice of the cases you use to verify, validate, and assess the performance of your method. If your method solves partial or ordinary differential equations, the first thing to do, is to verify that it is able to reproduce analytical solutions on simplified problems. If possible, these solutions should verify correct behavior of as many as possible of the terms entering your model equations. Secondly, you should verify that your method converges and/or scales as anticipated. Once this is done, you should look at the robustness and versatility of your method. Slightly paraphrasing Guideline #2, this means that you should stress test your method with respect to assumptions and variation of parameters so that you know how robust the method is, what the limitations are, and so on. Looking at Guideline #4, the results of your tests should be reported regardless of whether they are positive or not. This will give your more scientific credibility in the long run. Looking at it from a purely selfish perspective, it is better that you discover and disseminate weaknesses in your own method, rather than having others pointing them out in subsequent publications. Unfortunately, in performing extensive and objective testing of your method, you have several mechanisms working against you. First of all, humans are inherently lazy, and if we can get away with only investigating and presenting a restricted range of numerical tests, we will almost inevitably do so. In particular, the publish or perish syndrome tends to leave us all with little time to conduct thorough tests of new methods. Once we have found a small series of cases showing the superiority of our new method, we seek to publish. In doing so, it is very easy to unconsciously bring the competition to your own home ground and design biased test cases focusing on the aspects for which your method is particularly good. Likewise, when running large series of test cases, it is very easy – despite our best intentions – to subconsciously only pick those instances that show our method in a good light. This is a known fallacy in experimental science, but 396 K.-A. Lie best practices that address this are, to the best of my knowledge, seldom discussed when teaching courses in computational science. The mechanisms discussed above are strengthened by contemporary publishing culture, which tends to focus on success rather than failure and rarely allows you to publish research on methods that fail to work. This is a pity, since it may often be more interesting for others to learn about well-conceived approaches, or approaches that suggest themselves naturally based on current knowledge, that turn out to not work as anticipated. Publishing negative results will not only prevent others from wasting precious time chasing dead ends, but if you also provide an explanation why your method did not work, others may gain significant new insight that can help them to come up with alternative solutions.

3.2. Benchmark problems and standard test cases. Holden’s Guideline #6 rightfully suggests that you should establish test cases that can work as a standard in your field. Setting up good test cases is not a simple undertaking, but usuallyhas great value for the scientific community, provided that the test case is well designed. (It is also smart from a bibliometric point of view, since papers introducing standard test cases typically generate a large number of citations.) Test cases come in many variants, from standard benchmark problems that are run to measure the computational performance of processors, (iterative) linear solvers, and nonlinear solvers, to more open-ended setups, where the challenge is to compute as accurate or optimal solution as possible. Benchmarks can also pose problems that do not yet have any known or well-established solution. Test cases that contain observed behavior of a physical system (as emphasized in Guideline #5) are particularly useful, since the ultimate goal of many computer simulations is to predict the results of actual physical processes. For most researchers, the use of standard test cases is a simple, yet effective way to compare different computational methods. Given that the test case is utilized for the same purpose, and results are reported in a consistent manner, it is in principle easy to compare results reported by different researchers. To a certain extent, this alleviates the need to compare with other state-of-the-art methods as long as these have been validated on the same test cases. Oftentimes, researchers will also modify standard test cases and (ab)use them for different purposes than what they were designed for. This can be quite useful, since the research community may have developed a familiarity with the original setup that enables your peers to quickly interpret and assess results also on a modified setup, as long the original case is not obfuscated beyond recognition. In his paper [11], Helge Holden pointed out that test cases should be set up based on a consensus process in the scientific community, not as a static decision, but in accordance to scientific progress and development of computers. Carefully Holden’s seven guidelines for scientific computing 397 designed test problems have the power to drive research in certain directions, and can be a useful way to align activity in a research community with certain business interests and societal needs. The danger is, of course, that test cases may have a too strong influence on the focus of a scientific community. This was also notedby Helge, who pointed out the danger that researchers may be tempted to tailor-make their methods to benchmark tests. I have seen this tendency in my own field of research: computational methods for subsurface multiphase flow. Simulation models of real petroleum reservoirs take a long time to make and contain a lot of information about company assets. These models are hence considered business critical and are seldom shared openly with the research community. For many researchers, the most obvious alternative when looking for realistic data has been the (in)famous SPE 10 test case [8]. This synthetic model was originally posed as a hard test case for numerical homogenization methods (referred to as upscaling methods in petroleum engineering) and has an exaggerated variation in petrophysical parameters, but a much simpler grid geometry and fluid model than what is common in models of real assets. This is often overlooked, and there are many examples of over-fitted methods that show excellent performance on SPE 10 and similar cases, but fail to provide solutions on problems of practical value. Once such a test case comes into widespread use, it has a self-reinforcing effect. Even though you realize that it not necessarily is a representative testcase, you have to use it because this is what your peers expect you to do. When posing a test case, there are many practical issues to consider, and these will obviously vary from one field of science to another. As a minimum requirement, the purpose of the test case should be clearly specified along with the set of assumptions that restrict the problem and/or leave room for the user to make his/her own choices. These should be stated in a document that can be referred to by a persistent and unique identifier such as a Digital Object Identifier (DOI) or alike. In many cases it is natural to identify output parameters to be measured, or offer a set of reference and/or user-generated solutions or output parameters for comparison. If the test case involves input (or output) data, one should make sure that these are published along with clear specification of legal rights, preferably under a permissive license that enables users to freely interact with the data. For test cases involving several subproblems, it is also important that each subproblem is clearly labeled so that users later can refer to it in a unique manner. Within subsurface flow modelling, a common approach to compare methods or modelling approaches is to invite participants to make their best attempt to reproduce a certain physical scenario. After a certain period, results are collected, compared and contrasted, and reported in a publication. (SPE 10 mentioned above was one such benchmark). In several cases, the data of the test case has only been 398 K.-A. Lie made available to registered participants, and after the study has finished, the data are no longer available. This is a short-sighted practice that should be avoided. Not only should the data offered as part of the original setup remain available, but results reported by different participants should be made openly available sothat researchers later can use them to make independent comparisons. The same goes for any truth model involved in the study. Last, but not least, let me point out that standard test cases can easily be created somewhat unintentionally. Numerical examples reported in the first papers discussing a certain class of methods have a tendency of later becoming de facto test cases. This means that rather than reporting somewhat haphazardly generated examples highlighting salient features of your method in a graceful view, you should always consider to what extent these examples can be reused by others and put extra effort into designing test cases that can be used to stress-test not only your method, but a wide class of methods designed for similar purpose.

3.3. Evolution of computational methods viewed using Gartner’s Hype Cycles. Over the years, I have watched the evolution of several computational technologies; some quite close, like GPU computing and multiscale methods, and others more from afar; some have become widely adopted, whereas others have dwindled into obscurity. If we disregard the rare and ingenious ideas that get widely adopted in almost no time, evolution of computational methods follow a very similar pattern, shown in Figure 1. This curve, called Gartner’s Hype Cycles, was developed to interpret technology hypes and enable industries to assess their risk when investing in emerging technologies. Let us see how it can be applied to describe research and dissemination of new methods in scientific computing. In part, this curve is the result of a divide in focus among most mathematicians and researchers in the applied sciences. Mathematicians tend to develop advanced theories and rather sophisticated methods for idealized problems, whereas researchers in applied sciences and industry tend to work on problems that are outside the bounds of contemporary theories using somewhat less sophisticated methods. Let me exemplify: Whereas a lot of theory for nonlinear PDEs is developed in unbounded domains, models for real physical processes are usually posed on bounded domains and are to a large degree determined by their boundary conditions. Likewise, mathematicians tend to express their results using functional spaces and study PDEs of a certain type (hyperbolic, parabolic, elliptic), whereas models of many physical processes often exhibit mixed characteristics or involve a mixture of differential equations, empirical laws, and tabulated relationships for which it is often not clear what the appropriate functional spaces are. Early reports on new methods arising from mathematics or computer science have a certain tendency of being overly optimistic with regard to generality and Holden’s seven guidelines for scientific computing 399

Peak of inflated expectations Plateau of In fo productivity r m

e d m s p i e m s i t s i p m o i m s is d m Slope of e im opt m ed enlightenment r orm o Inf f

n i

n U Trough of disillusionment Crash & burn Technology trigger My job: shorten this period

maturity

Figure 1. Development of computational methods following the Gartner Hype Cycle. An important aspect of my job as contract researcher is to identify promising technologies and make the transition from uniformed optimism to the plateau of productivity as short as possible.

application potential. There are at least two reasons for this. First of all, you need to have a certain missionary streak, or alternatively be a salesman to entice the interest of others. Secondly, new methods are generally not as well tested as one might hope because of the mechanisms discussed in the previous section, or because testing on state-of-the-art descriptions of physical phenomena is way too complicated to be contemplated within a realistic time-frame by anybody without expert application knowledge. However, once a new idea picks up momentum, it is bound to be tested for a wide variety of models and parameters as researchers try to adapt the idea to their own problems. This will generate many success stories, but also a lot of failures when the new method is applied outside its scope or range of validity. Sooner or later, the initial interest starts to wane when it becomes clear that the method is not as suitable as initially suggested or significant more research is required to introduce improvements or do necessary adaptions. Continuing to research a method down the slope of informed pessimism and through the trough of disillusionment can be a very frustrating exercise. However, as pointed out above, a lot of new insight can often be gained through failures, and if you manage to grit your teeth and stay focused, you may eventually be able to start climbing the slope of enlightenment and push your method up to the plateau of productivity, where 400 K.-A. Lie widespread adoption takes place. How do you accelerate the time to informed optimism? Here, good test problems play an essential part. These should not only be the kind that stress-test your method within its current scope, but it is also important to have a succession of increasingly challenging test cases that can be used as milestones to continuously drive your methods towards a wider scope. Equally important, you need to have flexible prototyping tools to support the necessary experimental programming. I will get back to this in the next section.

A personal story. A few years after the turn of the century, my research group started to develop multiscale methods for reservoir simulation. From my perspective, this process has followed the Gartner Hype Cycle. Helge has not been directly involved as a publishing author in this research, but has acted as co-supervisor for many master and doctoral students. I had first encountered the idea in 1997, when Helge and I met TomHou– who that year published the first paper on multiscale methods [12] – during a sabbatical at the Mittag–Leffler Institute in Stockholm. I remember that Helge asked me whether we should start working on this, but I found the idea to be somewhat contrived and doubted it would find widespread application. Quite ironic, since I later have spent more than a decade developing multiscale methods towards industry adoption. For completeness, let me briefly describe the multiscale idea. Assume that you have a variable-coefficient Poisson equation, in which the coefficient exhibits variations over many orders of magnitude and thatthe spatial variation of the coefficient takes place over a broad range of length scales with no clear scale separation. (Poisson’s equation arises in porous media if we combine mass conservation ∇ ⋅ ⃗푣= 푞 for an incompressible fluid with Darcy’s law ⃗푣= −퐊∇푝.) Discretizing the equation, e.g., with a standard first-order finite- volume method, we get the following

− ∇ ⋅ (퐊∇푝) = 푞 ⟶ 퐀퐩 = 퐪. (1)

Here, 푝 is the pressure (or more generally the potential) of a single-phase fluid, 퐊 is the permeability of the rock (i.e., the ability to transmit fluids), and 푞 is a volumetric source term. The key idea of multiscale methods is to partition the fine grid used to discretize the Poisson equation into a coarse grid, towhichwe associate a vector of unknowns 퐩푐. For each coarse grid block, we define and solve a variable-coefficient Poisson equation with zero right-hand side on thefine grid. The fine-scale solution is restricted so that the resulting solution is one atthe center of the block and zero at the centers of all the other blocks. By specifying appropriate boundary conditions, the local solution, which we will refer to as a Holden’s seven guidelines for scientific computing 401 multiscale basis function, will have compact support restricted inside the nearest neighbors of our block. Collecting these basis functions as columns in a matrix 퐏, we have derived a prolongation operator that maps unknowns on the coarse grid to unknowns on the fine grid. If we also define a restriction operator 퐑 that sums entities defined over all cells inside each block, we have a systematic method for forming a reduced flow problem on the coarse grid that is consistent with the differential operator ∇ ⋅ 퐊∇ on the fine grid,

퐑퐀퐏퐩푐 = 퐑퐪 ⟶ 퐀푚푠퐩푐 = 퐪푐. (2) What I have described is an algebraic formulation of the multiscale finite-volume method [13]. When I started working on these methods, some bold claims had already been made that multiscale methods would give three orders of magnitude computational speedup. Over the succeeding years, we used an alternative mixed formulation [7] to extend multiscale methods to the complex grid formats used in industry, which have unstructured topology and polyhedral cell geometries with bilinear non-matching faces and up to three orders-of-magnitude aspect ratios. By and large, we succeeded in adapting the method to these grids and developing automated coarsening methods that robustly could handle the many intricate special cases arising for such grids [3, 4]. However, our somewhat cyclopean path of development reached its peak of inflated expectations when we tried to extend the method from slightly compressible flow to models with the full complexity seen in industry-standard applications. It turned out that the mixed formulation of complex flow models was not as robust as existing literature had seemedto indicate. After several futile attempts, we abandoned the multiscale mixed finite- element method and let it slide down the slope of informed pessimism towards obscurity. During the same period, the development of the multiscale finite-volume method had followed an equally cyclopean path towards realistic flow physics. Useful developments included a fully algebraic formulation and reformulation of the method as an iterative method. In the iterative formulation, the multiscale matrix 퐀푚푠 is used as a global preconditioner to eliminate low-frequency error components in combination with a standard local smoother that effectively elimi- nates high-frequency error components. This makes the method quite similar to a multigrid method, but has the advantage of exposing parallelism and enabling users to stop the iteration at any tolerance and still obtain mass-conservative fluxes. On the other hand, the development had been quite unsuccessful in extending the method to unstructured grids and the special grid formats uses in the petroleum industry. Being able to handle such grids is a prerequisite for industrial adoption. On an offhand chance, I suggested to Olav Møyner, one of Helge’s andmy students, that he could write his master thesis on this problem. This turned out 402 K.-A. Lie to be a stroke of luck. Over the past 4–5 years, we have managed to develop a new and very robust formulation for fully unstructured and stratigraphic grids [24, 25], by combining original ideas from Olav with insight obtained working on the mixed method. Our new method has been implemented by Schlumberger and is a cornerstone of what today is considered as next-generation technology for reservoir simulation. The interested reader can find a more thorough discussion of this method, and the various technical developments that lead up to it over the past decade, in Lie et al. [22]. The research described above has had two unintended side effects. First of all, it has inevitably given members of my research group a lot of insight into multiphase flow in porous rocks and induced a shift in our research focus from mathematics towards reservoir engineering. More important, it has lead to the development of an open-source community code for rapid prototyping of new computational methods for subsurface flow simulation that currently is used by many hundred researchers, students, and engineers all over the world. More details about this software are given in Section 5, whereas a summary of the lessons learned during its development will be presented in the next section.

4. Development of open-source community code

In his seventh and last guideline, Helge recommended that academic computer codes used in simple numerical experiments should be made available to others. This is probably the guideline I have personally taken most to heart. Today, the arguments for Guideline # 7 probably seem overly cautious. In particular, I tend to differ on Helge’s observation that public release should be restricted tocodes of little commercial value. As you will see later in this section, it is possible to publicly release codes with significant commercial value; you only have to usea different business model than selling software licences if you want to use thecode to earn money. The idea of giving away your code for free was, as far as I understand from Helge, considered by many to be an almost ridiculous idea, which explains why the accompanying arguments were toned down during the unsuccessful review process. Almost twenty years later, LeVeque presented very compelling arguments for why you should release your code publicly. His short and humorous article [18] describes an alternative universe in which mathematical proofs are not required and presents ten reasons why papers should not contain proofs. Through this simple thought experiment, LeVeque shows how absurd it is that computational sciences does not live by the same standard as mathematics. If his arguments do not convince you, I do not have much new to add, except to say that it has worked Holden’s seven guidelines for scientific computing 403 marvels for my research group. If you are already convinced, I urge you to set a good example in your own research, and request others to follow your lead when you act as supervisor, as reviewer or editor for scientific journals and conferences, or as evaluator on grant proposals. My aim herein is to explain how you can bring methods from the peak of inflated expectations and onto the slope of enlightenment. My focus willthus be on somewhat larger codes than what Helge originally suggested to release. Based on our experience in developing and maintaining what has become two community codes, I can make a few simple observations of why this may be highly useful for a research group: • Publishing and maintaining an open-source code is an efficient (albeit not always simple) way of coordinating activities within a research group or among cooperating scientists. Developing a common code base ensures that results can be leveraged between different activities. • Releasing your code to the public does not jeopardize your intellectual property rights as long as you are careful in your choice of licensing policy and combine it with scientific publications. • Release of open-source code is an efficient means of attracting interest to your research and getting in connection with potential collaborators. I think that some of these observations also hold for less comprehensive codes, and if this is what is most relevant in your case, you may still gain useful insight from the following discussion.

4.1. Choice of language for experimental programming. Unlike professional programmers, who typically have a detailed specification of software requirements from end users, developers of new numerical methods and computational algorithms rarely know exactly what their programs should do. Obviously, you will know what problem you aim to solve and have an idea of how to do it, but generally you will not know whether your approach will work until you have tested. The first attempt seldom does, and getting a working algorithm usually involves alot of test and trials. Hence, the computer is your laboratory and should be treated as one, meaning that you should try to make your experimental programming as productive and reliable as possible, and that numerical experiments should be subject to the same standard as physical experiments (se Section 4.5). One important choice you have to make is what language to use. This choice will obviously depend on what part of computational science you work in, what computer languages you have been exposed to, and the level of your programming skills. However, for the type of work I am doing (developing simulation technology 404 K.-A. Lie

Table 1. List of factors that contribute to slowing down the development cycle of experimental programming in a third-generation compiled language compared to a fourth-generation scripted language. 3rd generation 4th generation Fortran, C, C++ MATLAB, Python Syntax complicated intuitive Cross-platform challenging Build process Linking of external libraries Type checking static dynamic Mathematical abstractions user-defined built-in Numerical computations libraries built-in Data analysis and visualization libraries/external built-in Debugger, profiling, etc external/IDE built-in Traversing data structures loops, iterators vectorization† † also: indirection maps and logical indices to describe physical processes), my recommendation is crystal clear: Unless you are really fluent in a compiled language like C, C++, or FORTRAN, as much possible of your initial experimental programming should be done in a scripting language like Python, Julia, or MATLAB/Octave with extensive support for numerical algorithms. The resulting code may not be as efficient as in a compiled language, but the development process is so much simpler. Once you get your ideas to work, you can always replace parts of your code by a compiled back-end code or rewrite everything from scratch in a compiled language. In my research group, we primarily use MATLAB for prototyping and C/C++ when developing production codes for our clients. Even with several very capable programmers in the group, it is my consistent experience that developing ideas in MATLAB and later reimplementing in C/C++ is more efficient than doing everything in C/C++ from the start. Table 1 lists a number of factors I believe contribute to slowing down experimental programming in a compiled language compared with a scripting language. The basic (imperative) syntax in a scripting language like MATLAB or its open-source clone Octave is fairly simple and will generally be intuitive to any mathematician with a basic course or two in programming. The language has many built-in mathematical abstractions, which together with numerical functions and routines for data analysis and visualization enable you to write quite compact programs that are close to the underlying mathematics. This is, of course, also possible in C++, provided that you have the right user-defined abstractions and suitable libraries for numerical computations, data analysis, and visualization. C++ is a multi- Holden’s seven guidelines for scientific computing 405 paradigm programming language that gives you the choice between procedural (imperative), object-oriented, generative, and template meta programming. Not only is the language wordier and less expressive than MATLAB/Octave, but it is easy to write quite obscure programs by utilizing features of the language that are alien to those who are less diligent in their search for sublime computer codes. The build process and linking of external libraries are two other factors that not only contribute to slow down the development cycle, but also severely challenge the portability of your code. For small, standalone codes, this is not a big problem, but for medium to large-scale codes, it can be a very time-consuming task to set up an appropriate build system, make sure that all necessary libraries and software modules are in sync, and insure cross-platform compatibility. These issues are largely non-existent in MATLAB since it is an interpreted and not a compiled language. There can obviously be some problems with backward and forward compatibility as a result of existing functionality being improved and new functionality being introduced, but by and large this has not been an issue for us. (The only exception might be the 3D graphics, which not only is surprisingly slow in MATLAB, but also has issues with cross-platform compatibility.) Altogether, the development process tends to be quite different in MATLAB/ Octave than in C++. Experimental programming is at its best when you can gradually make small changes to an existing and functional code. By using the built-in debugger, you can prototype while testing an existing program. As in any debugger, you can run code line by line, and stop and inspect variables at any point. However, since MATLAB/Octave is interpreted and has dynamic type checking, you can at any point not only change the content of your variables and data structures, but also modify them completely by changing their type, introducing new data members, etc. You can also introduce new variables, data structures, and (anonymous) functions, or go back and rerun parts of the code with changed parameters. This way, you can try out each operation and build your program as you go. In my opinion, this is one of the primary reasons why prototyping in MATLAB/Octave is so efficient. On the other hand, knowing the type of each variable at compile time (like in C/C++) ensures a certain consistency and can be very helpful in catching errors. Let me end the section with a few words about homespun versus commercial or community codes. This question is particularly relevant when working at the interface between mathematics and an applied science; in my case, simulation of CO2 storage and hydrocarbon recovery. It is very hard to make any general recommendation, but let me observe the following: To avoid the danger of spending a lot of time reinventing the wheel, you should know what is current state-of-the- art and know both the capabilities and limitations of contemporary commercial and community codes. This is generally no little undertaking, but should at least 406 K.-A. Lie be attempted to prevent you from falling prey to the infamous not-invented-here syndrome. On the other hand, reimplementing well-established functionality is a good way to increase your expertise and understand tacit assumptions and limitations in exiting software. As computational software matures, it tends to include an increasing number of (undocumented) safeguards ensuring robust behavior even when the software is used with inconsistent input and outside its normal range of validity. Simulators of physical processes should never be used as black boxes, and it is of uttermost importance to have a number of experts that understand their inner workings.

4.2. Advice for good development practices. At the end, I will share some advice based on my experience as manager for the development of MRST [20, 26] as well as various research projects that have involved a significant amount of software development. My advice are not necessarily particularly original (see, e.g., [30]), nor as focused on the actual coding and software tools as die-hard programmers would have liked, but hopefully they may still be useful:

• Learn standard tools for efficient software development like (distributed) version-control systems, issue trackers, unit testing, and task automation (i.e., systems that enable automatic acceptance and regression testing of your code) and use them to your advantage.

• Write for humans, not computers. In experimental programming, researchers spend the majority of their time reading/writing code and not waiting for computer runs to finish. Using consistent style and formatting makes it easier for others to read and understand your code.

• Write your software incrementally, using an agile approach. Get a first (simplified) version to work as early as possible, test it, and use the results to improve and expand your implementation. Be prepared to make substantial changes as you gain more insight into the problem.

• Break your code up into easy understandable functions, document what the functions do, their input and output arguments, and include, if possible, small examples of their typical and intended use.

• Be lazy! If the computer can do a task for you, let it do it. (One example: use automatic differentiation as discussed on page 416 to avoid the error-prone process of deriving and implementing derivatives of functions).

• Document functionality, not semantics and mechanics. If nothing else, ensure that your future self is able to understand the code. Holden’s seven guidelines for scientific computing 407

• Avoid premature optimization. Once the code is working as anticipated, you can always profile it and try to remove bottlenecks. If this obfuscates the code, you should consider keeping the original version as part of the documentation.

4.3. Maintaining integrity of your code. It is challenging to maintain integrity of your software under the (frequent) restructuring of code that inevitably follows from an agile approach; in particular for complex code features that are not well covered by e.g., unit tests. My best advice to maintain integrity is: by thinking about it all the time2. That said, you can make life easier for yourself if changes are committed to the version-control system and tested as frequently as possible. Best practices for multiple developers working on distributed software repositories suggest that commits should be merged into baseline every day. My experience is that this rule is difficult to enforce strictly in research projects, butit is seldom a good idea to work for more than a day without testing your code, or to let your private development branches deviate too far from baseline. I also recommend that you use a tool for software self-testing like Jenkins, which automates the task of pulling code from your repository and running a set of predefined tests. Ideally, automated tests should incorporate as many ofthe cases you use for validation and verification as possible and should not only check that the code runs through without errors, but also verify that results are correct and monitor performance measures such as iteration counts, convergence rates, computational time, etc. To design meaningful tests, you should keep in mind that computed results are rarely bitwise identical so that suitable mathematical norms should be used when comparing against analytical solutions and previously stored computations. In OPM, for instance, we distinguish between acceptance tests, which check that you are within a prescribed tolerance of an analytical or numerical reference solution, and regression tests, which check that you reproduce previous results within zero or a tiny tolerance. Often, results of these tests need to be manually interpreted, since it is generally difficult to design fully automated tests of results that keep changing as your algorithms and methods get better and better. To avoid becoming a drag, the self tests should neither be too extensive nor run too frequently (e.g., once per day), and possibly be split up into multiple levels that are run at different time intervals. Code review, or peer review of source code, is another recommended best practice to maintain integrity and ensure correctness of new code. We have used this with some degree of success in OPM. However, formal code reviews can easily

2Supposedly, this is what Sir Isaac Newton answered when asked how he discovered the law of gravity. I have not been able to verify the truth of it, but I still think it is a good explanation that characterizes a lot of scientific work. 408 K.-A. Lie degenerate to counterproductive discussions about semantics and mechanics more than assumptions and functionality, and I am personally more fond of the informal peer review that arises naturally when multiple developers collaborate to test and maintain the same code.

4.4. Choosing the right development model. When setting up a new open- source project, there are several choices you need to make. First of all, you need to decide where to host your software repository and how you wish to distribute your code. Should you place your code in a public repository on a centralized service like GitHub and Bitbucket so that anybody who wants can have access at any time, or should you place it in a private repository and only provide periodic releases or only release it when you have finished the development? The fact that our work can/will be viewed and evaluated by our peers tends to keep most of us on our toes, and any of the first two alternatives is thus preferable. The third alternative is an ensnaring invitation to procrastinate important activities such as cleaning up code, documenting it, writing examples/tutorials, and so on, and should thus be avoided. Whether you should choose the first or the second alternative depends on the commercial setting; how your research is funded; licensing and copyright questions; how you, your organization, or your project wish to collaborate or cooperate with external contributors to the software; and to what extent you want to retain control of future developments. As explained already, my research group has been involved in two larger open- source projects over several years. The Open Porous Media (OPM) initiative follows an open and fully transparent innovation model. The code, which is hosted on a public repository on GitHub, has been jointly developed by researchers from several different organizations, funded through contract research. Copyright is held jointly by the developers’ organizations and commercial companies funding the development. Contributions from third-party developers who wish to retain copyright are welcome and encouraged, which is a major advantage and incentive to contribute. However, contributions will be reviewed by a meritocratic group of official maintainers, who are appointed partly by affiliation, and contributions are requested to follow the GNU Public License (GPL), which is a copyleft license. The overall model ensures a certain negative control for the funding partners, and is designed to prevent undue commercial utilization or hijacking of the project. The disadvantage is that this limits the incentives of non-funding partners to develop and maintain the code on their own. In my experience, it is also relatively costly to provide strategic direction and ensure that necessary consensus is reach among the developers. Several large open-source projects, like FEniCS, use an alternative model in which the project is managed by a non-profit foundation. Some open-source projects are characterized by the fact that full copyright is Holden’s seven guidelines for scientific computing 409 owned exclusively by a single entity. This model is partially used for MRST [20, 26], which I will discuss in more detail in Section 5. MRST is an important part of our research infrastructure, but is developed as a shared public utility and not as a product to be sold. The central software modules that make up the biannual releases are owned by SINTEF (which is a non-profit research foundation) and are kept in private repositories on Bitbucket. Code contributions to the fundamental parts of the software are only accepted if the contributor transfers copyright of the code to SINTEF. On the other hand, we both encourage and help our collaborators and third-party developers to create and release add-on modules to the software as long as these follow the GPL license used for the rest of the software. In principle, these modules can modify or replace any part of the basic functionality in MRST. The reason we have chosen this particular model is that it gives us strategic control of the basic functionality, freedom to use the software we developed as we wish in contract research and if necessary release it to our clients under a different license, as well as clear incentives to maintain and improve the software.

4.5. Maintaining a lab journal. The discussion in the previous sections focused mainly on software engineering, which for many scientific purposes is secondary to the development of algorithms and methods. In this latter process, the computer is essentially your laboratory. Within experimental sciences, it is basic knowledge that all experiments should be documented in a lab journal. The same should go without saying for numerical experiments: • State hypotheses and cases you want to test, report your results, and discuss how you interpret them, potential causes for incorrect behavior, ideas for future improvements, etc. • If possible, use a notebook format (like in Jupyter or Live Scripts in MATLAB) to set up your test cases, in which you can mix text, plots, and computer code. • Save the exact input parameters and your results; disk space is much less expensive than the time you spend, should you later need to go back and recreate your results. Use a version-control system for all input data that have been manually created. • Mark entries in the lab journal with a unique label from your version control system identifying the exact code and inputs you used to run the tests. With a version-control system like Git or alike, this would be the hexadecimal commit number. This will save you a lot of time when (and not if) you have to go back and redo some of your experiments (e.g., when getting a paper back from review). It will 410 K.-A. Lie also make it much simpler to communicate your preliminary results to colleagues or your supervisor/students.

4.6. Personal attitude. Last, but not least, I would like to point out that open- source projects have a huge affinity on software nerds/geeks. To avoid becoming one, I suggest:

• Don’t use your programming skills to show off! Try to aid others rather than alienating them.

• Keep it simple, stupid! If you feel pride in having managed to condense a complicated computational construct to a few code lines, chances are pretty high that you will be the only one understanding it.

• Try not to become a religious fanatic who quarrels or fights turf wars over standardization, and whose opinions get stronger the less important the issue discussed is.

• If using object orientation, avoid becoming an onion producer who makes layers upon layers of abstractions (with no core) in an attempt to be generic.

With this, I wish you good luck in your experimental programming. I look forward to see your source code on the net as an integer part of your next paper.

5. The MATLAB Reservoir Simulation Toolbox

In the last section of this essay, I try to make the ideas discussed above more concrete and demonstrate that it is indeed possible to also publish codes that have a significant commercial potential. To this end, I will describe a comprehensive open-source software developed by my research group over the past decade. The discussion is admittedly detailed at times, but I still hope that readers outside of the reservoir simulation community may find it inspiring and possibly learn something from our use of the MATLAB (or Octave) language. I believe, in particular, that our close relation between mathematical operators and their numerical implementation can be useful for others working with low-order finite-volume discretizations of flow equations within other fields of science and engineering.

5.1. A brief history of the software and why it was developed. What is today the MATLAB Reservoir Simulation Toolbox (MRST) [21, 20, 15, 5, 26] grew out of research on mimetic discretizations and multiscale methods for reservoir simulation on complex grids, as outlined in Section 3.3. It was decided early on to Holden’s seven guidelines for scientific computing 411 use MATLAB as our primary development platform, in part because of an idea that a scripting language would be more efficient, inspired by the late Hans Petter Langtangen’s pioneering work [16], and in part because we happened to know MATLAB quite well. At first, our development was poorly coordinated. Asan example, writing one of our earliest papers [1] involved three different codes, each written by only one of the authors. Obviously, this was no viable path, and hence MRST was born. A few years earlier, we had published an educational paper that essentially explained how to implement a simple reservoir simulator in less than 50 lines of MATLAB code [2]. This paper and the accompanying code had attracted much more interest than anticipated. Inspired by this, and with Helge’s last guideline at the back of my mind, I decided we should release our new code under a free software license. We chose to use the GNU General Public license, since this would prevent others from simply picking up our software and use it in commercial products. Pushing multiscale and mimetic discretization methods toward realistic applications meant that we had to develop a lot of general infrastructure for multiphase flow simulations on unstructured grids [21]. This made the new software quite attractive also in other projects, and MRST grew gradually into a general prototyping framework that was used in more or less all of our research. All the way, our development policy has been that generic ideas from contract research is put into MRST and released publicly. Code which is decided to have business-critical value to our clients or ourselves, is isolated in separate branches or modules and is never published. Since the software serves many different purposes, we have, after a bit back and forth, come to the conclusion that it is best to organize MRST so that it consists of a small core module offering basic functionality, and a large set of add-on modules that each implements specific computational methods or mathematical models. Many of these modules can be combined to support more comprehensive workflows, but there are also cases in which two modules offer functionality that makes them mutually exclusive. Continuing to release research results in the form of open-source software was not uncontroversial within my organization, but somehow I managed to convince my superiors that the marketing effect would far out-weight the loss of potential license fees. My winning argument was that when your market is monopolized by a few software providers, you need to use another mechanism to attract potential clients. With our industry clients, my argument is that allowing us to release generic parts of new functionality we develop for them, is the price they pay for being able to leverage functionality developed for other clients. Initially, MRST was written using a procedural (imperative) programming paradigm and focused almost exclusively on incompressible flow [21]. We made a few attempts at extending the capabilities to contemporary flow physics, but 412 K.-A. Lie were not really successful until one of my colleagues, Stein Krogstad, decided to implement automatic differentiation [27]. (I will come back to this in more detail below.) This opened unparalleled capabilities for rapid prototyping – and within a few weeks, we had developed our first compressible, three-phase solver and verified that it gave satisfactory match with the market-leading commercial simulator simplified test cases. Ensuring robust and accurate simulations on models of real hydrocarbon assets is far more challenging, and it took us several months to figure out the correct way to interpolate tabulated fluid data3, reverse-engineer undocumented features in models of near-horizontal wells, etc. This is generally where the science stops and the art or tricks-of-the-trade starts. A full-fledged reservoir simulator contains a lot of intricate functionality, like well modelling, nonlinear solvers with time- step control, (multilevel) iterative linear solvers with appropriate preconditioning methods, and so on. This means that codes written with a procedural approach gradually become quite unwieldy, unless these are meticulously designed, which seldom is the case in experimental programming. To amend this, Olav Møyner (who was doctoral student of Helge and me at the time) developed a new object- oriented framework. Combined with automatic differentiation, this framework offers very powerful support for rapid prototyping [15, 5, 23]. At this point, you may ask how efficient MATLAB is for reservoir simulation. The incompressible simulators written using a procedural approach are quite efficient, typically a factor 3–5 times slower than commercial solvers, and wehave been able to simulate two-phase flow on models having up to 60 million grid cells on a standard workstation. Industry-standard models for three-phase compressible flow are significantly more computationally demanding. Moreover, our automatic differentiation approach has primarily been written to be as flexible as possible and incurs a certain overhead, but we believe that this can be significantly reduced through a more careful implementation. In sum, I currently would not recommend simulation of models containing more than a few hundred thousand cells, which in most cases should be more than sufficient when developing proof-of-concept simulators and workflows for models with realistic complexity. Looking at the large user community that the software has attracted, it seems that a somewhat suboptimal computational performance is by far out-weighted by the flexibility that MRST offers. Each of the past eight biannual releases havebeen downloaded from 1000 to 2000 unique computers (according to Google Analytics), and at the time of writing, the software has been used in 110 master and doctoral theses, and in more than 170 journal and proceedings papers by authors not

3As an example: If two parameters 휇 and 퐵 that enter your flow equations as 1/휇퐵, should you interpolate 휇 and 퐵, 휇퐵, 1/휇 and 1/퐵, or 1/휇퐵? It turned out that the latter choice was the correct. Holden’s seven guidelines for scientific computing 413 affiliated with SINTEF. What are the points that make the software attractive to such a large audience? First of all, it is because the software is free, in a high-level language like MATLAB, and offers full access to source code. However, I also believe that the fact thatwe have been quite diligent in documenting the code and developing tutorials and examples that highlight salient features has contributed to make it more attractive. Last, but not least, we have put significant effort into developing routines for reading and processing input data on industry-standard format, which significantly simplifies the process of testing new methods on realistic scenarios. Let me also add that major parts of MRST can also be run in the latest version of Octave, as a completely free alternative to MATLAB, provided a number of changes are made to account for minor differences between Octave and MATLAB. The main exception is graphical user interfaces, which are written quite differently in the two languages. OPM Flow (https://opm-project.org) was originally developed as a C++- cousin of MRST, intended for full-scale commercial simulations. The two have many similarities in the underlying design, which simplifies the process of moving methods prototyped in MRST into industrial adoption. Lately, however, the OPM project has focused more on optimizing computational performance and this has resulted in larger and increasing differences in the two codes.

5.2. Key ideas for rapid prototyping. In this section, I will try to briefly explain some of the principles we have used in MRST to support rapid prototyping. Our choices are admittedly strongly influenced by the type of problems we study and the low-order finite-volume methods we use. Still, there might be some insight here that also applies to other types of problems and numerical methods. In developing the toolbox, we have tried to make functionality that enables clean and simple implementation of flow equations as close to the underlying mathematical models as possible. This way, we seek to ensure less error-prone coding and create quite compact codes that are relatively simple to maintain and extend. Key ideas to this end include:

• Hide specific details of grid, discretizations, constitutive laws, and parameters describing geologic and petrophysical properties.

• Always use a fully unstructured grid format to represent all types of grids so that algorithms can be implemented without knowing the specifics of the grid.

• Define abstract discretization and averaging/mapping operators that are not tied to specific flow equations and can be precomputed independently. 414 K.-A. Lie

• Use vectorization to ensure an almost 1-to-1 correspondence between continuous and discrete variables to avoid visible loops and use as few indices as possible. • Use automatic differentiation to avoid having to explicitly linearize flow equations, analytically compute and implement derivatives, gradients and Jacobians, which generally is a time-consuming and error-prone process.

Vectorization, logical indexing, and summation techniques. The language of MATLAB/Octave is quite expressive and has many different constructions that help to make your code shorter and hence easier to read and maintain. Two relative simple techniques are commonly used to avoid looping through arrays, as one typically would do in C++ and similar compiled languages. Vectorization lets you operate directly on the matrix level and write code almost as if you were working with scalar variables % Vectorization % For - loop f = s i n ( y ). exp (- x . ^ 2 / 2 ) ; f = z e r o s ( s i z e ( x )); f o r i=1:numel ( x ) f ( i ) = s i n ( y ( i )) exp (- x ( i ) ^ 2 / 2 ) ; end Because of MATLAB’s Just-in-time (JIT) compiler, the vectorized code can be slower than the for loop when the arrays 퐱 and 퐲 have few elements. On the other hand, the vectorized code is much closer to the mathematics and significantly more efficient on large arrays. This was a trivial example, but the principle applies to more complex cases. Another nice feature is logical indexing. To exemplify, we can set all negative elements of a vector to zero v ( v<0) = 0 ; or compute the average of negative and positive values i = v>=0; avg = [sum( v(~i )) sum( v ( i ))]./[ sum(~i ) sum( i )]; Another useful construct, is the accumarray(p,v) function, which collects all elements of v that have identical subscripts in p, sums them, and stores in the location given by p. As an example, let p be a partition vector defining a coarse grid so that p(i)=j if cell i belongs to block j. The average of a scalar quantity v defined in each cell can be computed as avg = accumarray ( p , v )./ accumarray ( p , 1 ) ; To compute the average over a vector quantity with 푚 elements per cell, defined as an 푛 × 푚 array v, we can use a sparse matrix to sum the elements and bsxfun for element-by-element division, Holden’s seven guidelines for scientific computing 415

Idealized models Grid structure in MRST Industry models

c F(c) f C1 C2 111 3 1 7 8 122 1 2

8 133 1 8 3 144 9 1 7 255 4 2 6 266 2 5 9 4 277 2 6 1 2 2 288 2 7 . . . 22. . . 6 . . . 31. . . 1 5 . . 5 ...... 3 4 Map: cell → faces Map: face → cells

Figure 2. Illustration of the grid structure in MRST and typical grids used in subsurface flow simulation. The two tables show the mappings used to define discrete differentiation operators; for the face-to-cell mapping, only the last two columns are actually stored.

tmp = sparse( p , n , 1) [ v , ones( n ,1)]; avg = bsxfun( @rdivide , tmp (: ,1:end - 1) ,tmp (: ,end )) The last two constructs are powerful, although not as neat as logical indexing, and are used a lot in MRST for computational efficiency and to generate compact codes devoid of for loops.

Grids and discretizations. The most fundamental quantity in MRST is the grid, which generally will be a collection of 3D polyhedral cells having an unstructured topology. To ensure maximum flexibility in developing new computational algorithms, all grids are represented in a relatively verbose format containing geometric properties such as vertices; face centroids, normals, and areas; and cell centroids and volumes. The grid topology is described in terms of mappings between cells and faces, and between faces and the cells they separate, as shown in Figure 2. Using these mappings, we can define discrete divergence and gradient operators. The div operator is a linear mapping from faces to cells. Let 퐯[푓] denote a discrete flux over face 푓 with orientation from cell 퐶1(푓) to cell 퐶2(푓). Then

1, if 푐=퐶(푓), div(퐯)[푐] = ∑ sgn(푓)퐯[푓], sgn(푓) = { 1 (3) 푓∈퐹(푐) −1, if 푐=퐶2(푓).

Likewise, the grad operator maps from cell pairs 퐶1(푓), 퐶2(푓) to faces 푓

grad(퐩)[푓] = 퐩[퐶2(푓)] − 퐩[퐶1(푓)], (4) where 퐩[푐] is the pressure associated with cell 푐. Since div and grad are linear operators, they can be represented by a sparse matrix 퐃 so that grad(퐱) = 퐃퐱. 416 K.-A. Lie

Continuous Discrete in MATLAB

Incompressible ﬂow: Incompressible ﬂow:

∇ · (K∇p) + q = 0 eq = div(T .* grad(p)) + q;

Compressible ﬂow: Compressible ﬂow:

∂(φρ) eq = (pv(p).*rho(p)-pv(p0).*rho(p0))/dt ... + ∇ · (ρK∇p) + q = 0 ∂t + div(avg(rho(p)).*T.*grad(p))+q;

Figure 3. Correspondence between how flow equations are specified mathematically and implemented in MRST using the discrete operators. Here, pv and rho are functions evaluating porosity 휙 and density 휌 as function of pressure, and avg is a mapping from cells to 1 faces, avg(휌)[푓] = 2 (휌[퐶1(푓)] + 휌[퐶2(푓)]).

If we assume zero flux across the boundary, the discrete gradient operator is the adjoint of the divergence operator, as in the continuous case, i.e., div(퐱) = −퐃푇퐱. To discretize Poisson’s equation (1), we also need to represent the operator ∇ ⋅ 퐊∇ by defining a transmissibility 퐓[푓], so that 퐯[푓] = −퐓[푓]grad(퐩)[푓]. To derive a concrete expression for 퐓, we change notation slightly and let 퐯푖,푗 denote the flux from cell 푖 to cell 푗. Using Darcy’s law and a standard finite-difference approximation, we have that

(푝푖 − 휋푖,푗)푖,푗 ⃗푐 퐯 = − ∫ 퐊∇푝 ⋅ ⃗푛 푑푠 ≈ 퐴 퐊 ⋅ ⃗푛 = 푇 (푝 − 휋 ), 푖,푗 푖푗 푖푗 푖 | ⃗푐 |2 푖,푗 푖,푗 푖 푖,푗 Γ푖푗 푖,푗 where the interface Γ푖푗 between cells 푖 and 푗 has area 퐴푖푗 and directional normal 푖,푗⃗푛 . Moreover, 퐊푖 is the constant value of 퐊 inside cell 푖, 휋푖푗 is the pressure at the centroid of Γ푖푗, and 푖,푗⃗푐 is the vector from the cell centroid to the face centroid. A similar expression holds for cell 푗. If we require continuity of fluxes, 퐯푖,푗 = −퐯푗,푖, −1 −1 −1 it follows that 푇푖푗 = [푇푖,푗 + 푇푗,푖 . Constructing the discrete operators and computing the transmissibility can be done in approximately twenty lines in MATLAB using the unstructured grid format, as we will see later. In practice, you will probably want to add a few safeguards as we have done in MRST, which make the code somewhat longer. Once the operators and 퐓 are computed, we do not need to know any detail of the grid to discretize our flow equations. This can be done quite compactly, as shown in Figure 3.

Automatic differentiation. The basic premise of automatic differentiation (AD), also called algorithmic differentiation, is that standard function evaluations in a computer code consists of a sequence of elementary unary and binary oper- ations, for which known differentiation rules exist. The key idea is now to keep Holden’s seven guidelines for scientific computing 417 track of variable values and their derivatives with respect to a set of independent variables. Consider a scalar independent variable 푥 and a dependent variable 푣 computed as a function of 푥, i.e., 푣 = 푓(푥). Automatic differentiation introduces a new extended pair ⟨푥, 1⟩, i.e., the value 푥 and its derivative 1. Using this extended pair, the computer can use elementary derivative rules for unary and binary op- erations together with the chain rule to mechanically accumulate derivatives of 푣 evaluated at the specific value 푥 represented as ⟨푓(푥), 푓′(푥)⟩. If, for instance, 푣 = sin(푥), then the corresponding AD-pair reads ⟨sin(푥), − cos(푥)⟩. Similarly, we have for binary operators

⟨푢, 푢푥⟩ + ⟨푣, 푣푥⟩ = ⟨푢 + 푣, 푢푥 + 푣푥⟩ , ⟨푢, 푢푥⟩ ∗ ⟨푣, 푣푥⟩ = ⟨푢푣, 푢푣푥 + 푢푥푣⟩ .

In MRST, these rules are implemented using operator overloading as suggested in [27], so that all function evaluations can be written exactly the same way regardless of whether AD is used or not.

Putting it all together. Now, let us see if we can put the pieces together and implement a flow solver that is applicable to both structured an unstructured grids. For the moment, I will skip details of how the grid G and the permeability K are generated. Simple grids can be generated by a few calls to grid-factory routines in MRST. We start by extracting grid information C = G . faces . neighbors ; C = C ( a l l ( C ~= 0 , 2 ) , : ) ; cn = gridCellNo ( G ); F = G . cells . faces ( : , 1 ) ; [ nf , nc ] = deal ( s i z e ( C , 1 ) , G . cells . num ); The first two lines extract the last two columns of the face-to-cell map fromFigure 2 and remove all external faces (indicated by one of the cell numbers being zero). The next two lines extract the two columns of the cell-to-face mapping, whereas the last line gets the number nf of internal face and the number of cells nc. Using this information, it is straightforward to construct the discrete operators D = s p a r s e ( [ ( 1 : nf ) ’ ; ( 1 : nf )’], C , ones ( nf , 1 ) [ - 1 1 ] , nf , nc ); grad = @ ( x ) D x ; div = @ ( x )- D ’ x ; To compute the transmissibility, we start by extracting the face normal and the matrix containing vectors from cell to face centroids sgn = 2 ( cn == G . faces . neighbors ( F , 1 ) ) - 1 ; c = G . faces . centroids ( F ,:)- G . cells . centroids ( cn ,:); n = bsxfun ( @times , sgn , G . faces . normals ( F ,:)); 418 K.-A. Lie

Here, the first line determines the correct sign of the face normal. Now, wehave all information we need to compute the transmissibility, [ i , j ] = deal ([1 1 2 2],[1 2 1 2]); hT = sum( c (:, i ). bsxfun ( @times , K ( cn ,:), n (:, j ) ) , 2 ) ; hT = hT ./ sum( c . c , 2 ) ; T = 1 . / accumarray ( F , 1 . / hT ,[ G . faces . num , 1 ] ) ; T = T ( a l l ( C ~ = 0 , 2 ) , : ) ; The first line sets up of the row and column numbers of the permeability tensor, which is stored as a vector of the form [퐾푥푥, 퐾푥푦, 퐾푦푥, 퐾푦푦] for each cell in a 2D grid. (For 3D grids, 퐊 has nine entries.) The next two lines compute the one-sided transmissibilities, the next line their harmonic average, and the last line extracts those corresponding to internal faces. In the actual prototyping framework, you would not have to implement all the generic code lines discussed above, but rather call a function that does this for you with a lot of safeguards S = setupOperatorsTPFA ( G , rock ); Here, rock is a structure containing petrophysical properties, including 퐊. Now, we are finally in a position to specify and solve our equations. Tothis end, we declare pressure as our primary variable, which hence will be considered the independent variable when linearizing the discrete equations q = . . . % this is case specific p = initVariablesADI ( z e r o s ( nc , 1 ) ) ; eq = div ( T . grad ( p))+q ; eq ( 1 ) = eq ( 1 ) + p ( 1 ) ; p = - eq . jac { 1 } \ eq . val ; The second line defines 푝 to be an AD-variable initialized with all zeros, the third line defines our discrete equation on residual form as shown in Figure 3. With zero Neumann conditions only, the solution is not unique and the fourth line modifies the first element of the system matrix to (somewhat arbitrarily) fix the pressure in the first cell to zero. Going back to (1), we have a residual equation on the form 퐑(퐩) = 퐀퐩 + 퐪 = ퟎ. The last line computes the solution as 퐩 = −(휕퐑/휕퐩)−1퐪 = −퐀−1퐪. Figure 4 shows the setup and solution of two specific problems. The only difference between these two cases is the specification of the grid G and the source term q. Notice also that the same code can be used to compute pressure on complex stratigraphic and unstructured grids in 3D after trivial modifications of the i and j arrays to span 3×3 tensors in the transmissibility calculation. To extend the code to the compressible, single-phase equation shown in Fig- ure 3, we need to define functions that compute 휌 and 휙 as functions of 푝, and add an outer loop for the time steps and an inner Newton iteration to solve what Holden’s seven guidelines for scientific computing 419

% Make grid % Make grid using external grid generator G = twister(cartGrid([8 8])); pv = [-1 -1; 0 -.5; 1 -1; 1 1; 0 .5; -1 1; -1 -1]; G = computeGeometry(G); fh = @(p,x) 0.025 + 0.375*sum(p.^2,2); [p,t] = distmesh2d(@dpoly, fh, 0.025, [-1 -1; 1 1], pv, pv); % Set source terms(flowSW ->NE) G = computeGeometry(pebi(triangleGrid(p, t))); q = zeros(G.cells.num,1); q ([1 end]) = [1 -1]; % Set source terms(flowSW ->NE) q = zeros(G.cells.num,1); % Unit insotropic permeability v = sum(G.cells.centroids ,2); K = ones(G.cells.num,4); K(:,[2 3]) = 0; [~,i1 ]=min(v); [ ~,i2 ]=max(v); q([i1 i2]) = [1 -1];

Figure 4. Poisson problems describing single-phase flow on a rectangular grid and onan unstructured Voronoi grid; the latter is constructed from a triangulation generated by an open-source mesh generator [29]. The color plots show pressure with red denoting high pressures near the fluid source and blue low pressures near the sink.The spy plots show the sparsity structure of the 퐃 matrix used to define the discrete div and grad operators. For 휕 휕 the rectangular grid, the upper block corresponds to 휕푥 and the lower block 휕푦 . Permeability is specified in exactly the same way for the unstructured and structured grids. is now a nonlinear residual equation; details are given in [20]. These single-phase problems are almost trivial, but should give you an idea of how to construct more advanced solvers.

Acknowledgments. First of all, I would like to thank Helge for the fruitful collaboration and cooperation we have had over the past 25 years. This has not only benefited the many students we have supervised together, but also my more senior colleagues, whom I continuously expose to the requirements for high scientific quality I have learned from you. By example, you have taught me that being a supervisor is similar to being a father; you do not stop caring for your children and helping just because they have left the nest. I have tried to pay this on as best as I could. This essay was written while participating in the long program on Compu- tational Issues in Oil Field Applications at the Institute for Pure and Applied Mathematics (IPAM) as UCLA. I thank IPAM for the invitation, the generous funding, and the great hospitality offered to me. 420 K.-A. Lie

References

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[30] G. Wilson, D. A. Aruliah, C. T. Brown, N. P. Chue Hong, M. Davis, R. T. Guy, S. H. D. Haddock, K. D. Huff, I. M. Mitchell, M. D. Plumbley, B. Waugh, E.P. White, and P. Wilson. Best practices for scientific computing. PLOS Biology, 12(1):1–7, 01 2014. doi:10.1371/journal.pbio.1001745. Sharp uniqueness results for discrete evolutions

Yurii Lyubarskii and Eugenia Malinnikova

To Helge Holden on the occasion of his 60th birthday

Abstract. We prove sharp uniqueness results for a wide class of one-dimensional discrete evolutions. The proof is based on a construction from the theory of complex Jacobi matrices combined with growth estimates of entire functions.

1. Introduction

We study solutions of discrete evolution equations of the form

휕푡퐮 = 퐴퐮, (1)

2 where 퐮∶ [0, 푇] → 푙 (푋) for some Hilbert space 푋, 퐮 = {푢푘}푘, 푢푘 ∶ [0, 푇] → 푋, and 퐴 is a bounded operator on 푙2(푋) of a special form. Namely, we assume that the matrix of 퐴 (its elements are operators in 푋) is banded, i.e., contains just a finite number of non-zero diagonals. We are looking for uniqueness result of the following type:

If a solution 퐮 = {푢푘}푘 of (1) decays sufficiently fast in spatial variable 푘 at two moments of time 푡 = 0, 푇, then 퐮 ≡ 0.

The model example of such evolution is the discrete Schrödinger equation 휕푡퐮 = 푑 2 푑−1 −푖(Δ푑 + 푉)퐮 on the standard lattice ℤ . For this case we set 푋 = 푙 (ℤ ), i.e., the space 푙2(ℤ푑) is considered as 푙2(푙2(ℤ푑−1)), and the discrete Laplace operator 2 푑 2 푑 on 푑-dimensional lattice, 훿푑 ∶ 푙 (ℤ ) → 푙 (ℤ ) is defined inductively,

2 (Δ1퐮)푘 = 푢푘+1 + 푢푘−1 − 2푢푘 for 퐮 = {푢푘} ∈ 푙 (ℤ) and (2) 2 2 푑−1 (Δ푑퐮)푘 = 푢푘+1 + 푢푘−1 − 2푢푘 + Δ푑−1푢푘 for 퐮 = {푢푘} ∈ 푙 (푙 (ℤ )).

2 푑−1 Further, the potential part is (푉퐮)푘 = 푉푘푢푘, with 푉 = {푉푘}, where 푉푘 ∶ 푙 (ℤ ) → 푙2(ℤ푑−1) are diagonal operators for 푘 ∈ ℤ. The uniqueness problem for this evolution has been considered in [11, 8, 9, 10, 1]. The research was supported by Grant 213638 of the Research Council of Norway 424 Y. Lyubarskii and E. Malinnikova

Our research is motivated by a remarkable series of papers [5, 6, 7] (see also references therein) which studied the continuous case. In these articles a sharp uniqueness statement is obtained for solutions of Schrödinger equations with time-dependent potentials; the result is applicable to some non-linear equations. For the potential-free Schrödinger evolution, the uniqueness statement can be considered as a version of the classical Hardy uncertainty principle. The Fourier transform applied to both the discrete and continuous Schrödinger evolutions transforms the uniqueness questions into those on growth of analytic functions. In [11] and [8] the theory of entire functions has been applied to the model case of free discrete evolution (퐴 = −푖Δ푑). It was proved that in dimension 푑 = 1 the inequality

1 푒 |푛| |푢푛(0)| + |푢푛(1)| < ( ) , 푛 ∈ ℤ ⧵ {0}, √|푛| 2|푛|

−푛 −2푖푡 implies 푢푛(푡) = 퐴푖 푒 퐽푛(1 − 2푡), where 퐽푛 is the Bessel function. In particular, a solution to the free Schrödinger evolution equation cannot decay faster than 퐽푛(1) simultaneously at 푡 = 0 and 푡 = 1. This result was also generalized to special classes of time-independent potentials, first those with compact supports [11] and then fast decaying [1]. General bounded potentials were considered in [11] (in dimension 푑 = 1) and [10] (in arbitrary dimension). For time-dependent potentials, the uniqueness results obtained in [11, 10] show that the inequality

|푢(푡, 푘)| ≤ 퐶 exp(−훾|푘| log |푘|) for some fixed 훾 > 훾0 implies 푢 ≡ 0; however, these results are not sharp. In this note we combine the entire function techniques developed in [11] with some ideas from the theory of complex Jacobi matrices in order to consider general discrete models with time-independent banded operator 퐴. Thus we cover for example one-dimensional heat and Schrödinger evolutions with bounded potentials as well as some discrete versions of higher order one-dimensional operators, and also some higher dimensional operators (with very specific potentials). The article is organized as follows. The next section contains preliminaries related to banded operators and generalized eigenvectors. We also consider some model examples of operator 퐴 where the problem (1) admits explicit solution. In section 3 we apply the theory of entire functions to show that any solution to general time-independent evolution which decays sufficiently fast at two times is orthogonal to all generalized eigenvectors of the adjoint operator 퐴∗; this argument holds for general banded operators on 푙2(푋). For the case of a selfadjoint operator 퐴 and 푋 = ℂ, one can apply general results on completeness of the set of generalized eigenvectors in order to see that this orthogonality implies that the solution is Discrete evolutions 425 trivial. At the end of section 3 the multidimensional selfadjoint case, i.e., when 퐴 = 퐴∗ and 푋 = 푙2(푍푑−1), is also considered. We demand additional decay of solution in complementary spatial variables. This decay is needed to include the space 푙2(ℤ푑) in a Gelfand triple, and to apply a general result on the completeness of the set of generalized eigenvectors. The more complicated non-selfadjoint case is presented in Section 4. The construction is inspired by a version of Shohat– Favard theorem for complex Jacobi matrices. We consider first the case 푋 = ℂ in order to show the main ideas without further technical details. For general 푋 we need an additional assumption. Namely, we assume that the matrix entries of the operator 퐴 commute with each other. We don’t know if this assumption is necessary. In Section 5 we consider a closely related question on decay of the solutions of the discrete stationary equation.

2. Preliminaries

2.1. Banded operators. We consider operators 퐴∶ 푙2(푋) → 푙2(푋), where 푋 is a Hilbert space,

2 2 2 푙 (푋) = { 퐱 = {푥푗}푗∈ℤ, 푥푗 ∈ 푋, ‖퐱‖ = ∑ ‖푥푗‖푋 < ∞ }. 푗

This includes operators on 푙2 sequences over ℤ푑, we identify this space with 푙2(푙2(ℤ푑−1)). We assume that 퐴∶ 푙2(푋) → 푙2(푋) is a banded operator, i.e., for some integer 푠 푗+푠 2 (퐴퐱)푗 = ∑ 퐴푗,푘푥푘, 퐱 ∈ 푙 (푋), (3) 푘=푗−푠 where 퐴푗,푘 ∶ 푋 → 푋 are bounded operators. We will refer to these operators as to entries of 퐴. The number 2푠 plays the role of order of 퐴; it will define the order of decay in the corresponding uniqueness statement.

In addition we assume that the “external” entries 퐴푗,푗±푠 are invertible and

−1 −1 ‖퐴푗,푗±푠‖ ≤ 훿 , ‖퐴푗,푘‖ ≤ 푎, (4) for some 푎, 훿 > 0, independent of 푗. Clearly, the adjoint operator 퐴∗ is also banded and satisfies the same conditions (4).

2.2. Generalized eigenvectors. We consider generalised eigenvectors of 퐴∗. Since 퐴∗ is a banded operator, the expression 퐴∗퐞 makes sense for any sequence 426 Y. Lyubarskii and E. Malinnikova

∗ 퐞 = {푒푗}푗∈ℤ with 푒푗 ∈ 푋. We say that 퐞 is a generalized eigenvector if 퐴 퐞 = 휆0퐞 for some 휆0 ∈ ℂ. For any 휆 ∈ ℂ and any vectors 푒−푠, 푒−푠−1, … , 푒푠−1 ∈ 푋 there exists a unique vector 퐞(휆) = {푒푗(휆)}푗∈ℤ with 푒푗(휆) ∈ 푋 such that ∗ 푒푗(휆) = 푒푗, 푗 = −푠, … , 푠 − 1, and 퐴 퐞(휆) = 휆퐞(휆). It is defined by

푒푗(휆) = 푒푗, 푗 = −푠, … , 푠 − 1, (5) 푠−1 ∗ −1 ∗ 푒푠+푘(휆) = (퐴푠+푘,푘) ( ∑ 퐴푚+푘,푘푒푚+푘(휆) − 휆푒푘(휆)) , 푘 ≥ 0, (6) 푚=−푠 푠 ∗ −1 ∗ 푒−푠−푘(휆) = (퐴−푠−푘,−푘) ( ∑ 퐴푚−푘,−푘푒푚−푘(휆) − 휆푒−푘(휆)) , 푘 ≥ 1. 푚=−푠+1 (7)

The vectors 푒푗(휆) are polynomials in 휆 (with values in 푋) of degree less than [|푗|/푠] + 1. Let 푀 = max−푠≤푗<푠 ‖푒푗‖. Then an induction argument yields 푛+푠 ‖푒푛(휆)‖ ≤ 푀푦 , 푛 ≥ −푠, for all 푦 > 1 such that 푦2푠 ≥ 훿−1(푎(푦2푠−1 +푦2푠−2 +…+푦 +1)+|휆|푦푠). We multiply the last inequality by (푦 − 1), and see that it holds if 푦2푠+1 ≥ (푎훿−1 + 1)푦2푠 + 훿−1|휆|푦푠+1, which is in turn satisfied if we choose 푦 ≥ 훿−1/푠|휆|1/푠 + 푎훿−1 + 1. Similar estimates can be repeated for negative 푛. We obtain

−푘 푘+2 ‖푒푘푠+푟(휆)‖, ‖푒−푘푠−푟−1(휆)‖ ≤ 퐶푀훿 (|휆| + 푏) , 푘 ≥ 1, 0 < 푟 ≤ 푠, (8) for some 푏 = 푏(푠, 푎, 훿).

2.3. Model examples. Our main example is 퐴 = 훼Δ푑, where Δ푑 is the discrete lattice Laplacian given by (2) and 훼 ∈ ℂ. Clearly, this is an operator of the form (3) 2 푑−1 with 푋 = 푙 (ℤ ), 푠 = 1, 퐴푗,푗±1 = 훼퐼, and 퐴푗,푗 = 훼(Δ푑−1 − 2퐼). For 푑 = 1 solutions to the corresponding evolution problem can be expressed in terms of the Bessel functions of the second kind; one of them is

−2훼(푡−푡0) 푢푛(푡) = 퐼푛(2훼(푡 − 푡0))푒 . In higher dimension we have solutions of the form

푑−1 ∏ −2푑훼(푡−푡0) 푢푛(푡) = { 퐼푛(2훼(푡 − 푡0))( 퐼푛푙(2훼(푡 − 푡0)))푒 } . 푑−1 푙=1 (푛1,…,푛푑−1)∈ℤ Discrete evolutions 427

The powers of the discrete Laplacian provide examples of higher order operators that satisfies our assumptions. However, a simpler model is given by the operator with 퐴푗,푗±푠 = 퐼, 퐴푗,푗 = −2퐼 and 퐴푗,푘 = 0 otherwise. Then a solution is given by

푢푛(푡) = 퐶푟퐼푞(2(푡 − 푡0)), 푛=푞푠+푟,0≤푟<푠.

For 푡0 = 푇/2 this solution indicates the critical speed of decay in spatial variables:

푒푇 |푞| |푢 (0)| + |푢 (푇)| ≍ |푞|−1/2 ( ) . 푛 푛 2|푞|

3. Orthogonality to generalized eigenfunctions and self-adjoint operators

3.1. Controlled decay. We need the following auxiliary statement.

Lemma 3.1. Suppose that 퐮∶ [0, 푇] → 푙2(푋) is a solution to (1) and 퐴 satisfies conditions (3) and (4). Suppose further that

푘 −푘/2 ‖푢푗(0)‖푋 ≤ 퐶0 푘 , 푘 = [|푗|/푠] + 1. (9)

Then for each 푡 ∈ [0, 푇] there exists 퐶푡 such that

푘 −푘/2 ‖푢푗(푡)‖푋 ≤ 퐶푡 푘 , 푘 = [|푗|/푠] + 1, 푡 ∈ [0, 푇]. (10)

|푗| 2 Proof. Consider the function 푓퐵(푡) = ∑푗 퐵 ‖푢푗(푡)‖푋. It satisfies the differential ′ 푠 inequality 푓퐵(푡) ≤ 퐶1퐵 푓퐵(푡), where 퐶1 does not depend on 퐵. Therefore

푠 퐶1퐵 푡 푓퐵(푡) ≤ 푒 푓퐵(0). (11)

푠 푠 퐶2퐵 퐶3퐵 In addition, (9) implies that 푓퐵(0) ≤ 푒 . Then 푓퐵(푡) ≤ 푒 with 퐶3 = 퐶3(푡) 푠 and, in particular, ‖푢(푗, 푡)‖2 ≤ 퐵−|푗|푒퐶3퐵 . We optimize the last inequality by choosing 퐵 ≍ 푘 and get the required estimate (10).

In this argument we assumed that 푓퐵(푡) is well-defined for all 퐵. To justify this one can first consider the functions

̃ |푗| 2 푓,퐵 (푡) = ∑ min{퐵 , 퐵 }‖푢(푗, 푡)‖푋, 푗 obtain estimate (11) for these functions with constants independent of 푁, and then pass to the limit as 푁 → ∞. 428 Y. Lyubarskii and E. Malinnikova

Corollary 3.2. Let the function 퐮∶ [0, 푇] → 푙2(푋) satisfy the hypothesis of Lemma 3.1 and 퐞 be a generalized eigenvector of 퐴∗. Then the inner product

⟨퐮(푡), 퐞⟩ = ∑ ⟨푢푗(0), 푒푗⟩푋 푗∈ℤ is well-defined.

This statement follows from the lemma and the fact that ‖푒푗‖ grows in 푗 not faster than exponentially, see (8).

3.2. Orthogonality. We now prove that any solution to (1) which decays at two moments faster than the model one is orthogonal to all generalized eigenvectors of 퐴∗. Proposition 3.3. Suppose that 퐴∶ 푙2(푋) → 푙2(푋) is a banded operator satisfying (3) and (4). Suppose that 퐞 is a generalized eigenvector of 퐴∗. Let further 퐮∶ [0, 푇] → 2 푙 (푋) satisfy 휕푡퐮 = 퐴퐮, and

|푘| −|푘| −|푘| |푘| |푘| ‖푢푗(푡)‖푋 ≤ 퐶푒 (2 + 휀) |푘| 푇 훿 , 푘 = [푗/푠], when 푡 = 0, 푇. (12) Then ⟨푢(0), 퐞⟩ = 0.

∗ Proof. Let 퐴 퐞 = 휆0퐞, with 퐞 = {푒푗}푗. We define a family 퐞(휆) of generalized eigenvectors by (5)–(7). In this way the eigenvector 퐞 is included into an analytic family of eigenvectors 퐞(휆), 휆 ∈ ℂ. We consider the family of entire functions

휙(푡, 휆) = ⟨퐞(휆), 퐮(푡, )⟩푙2(푋) = ∑⟨푒푗(휆), 푢푗(푡)⟩푋. 푗 Differentiating with respect to 푡, we obtain

∗ 휕푡휙(푡, 휆) = ⟨퐞(휆), 퐴퐮⟩ = ⟨퐴 퐞(휆), 퐮⟩ = 휆휙(푡, 휆). Then for each 휆 we have 휙(푡, 휆) = 푒휆푡휙(0, 휆). (13) At the same time estimates (12) and (8) give |휙(0, 휆)|, |휙(푇, 휆)| ≤ 퐶푒푇|휆|/(2+휀). (14) The proof can be now completed in the same spirit as Theorem 2.3 in [11]. We include a brief argument in order to make the presentation mainly self-contained, and refer the reader to monograph [14] for definitions and basic facts related to entire functions. Let ln |휙(0, 푟푒푖휃)| ln |휙(푇, 푟푒푖휃)| ℎ0(휃) = lim sup , ℎ푇(휃) = lim sup , 휃 ∈ [0, 2휋] 푟→∞ 푟 푟→∞ 푟 Discrete evolutions 429 be the indicator functions of the entire functions 휙(0, 휆) and 휙(푇, 휆). Relation (13) for 휃 = 0 and 푡 = 푇 yields ℎ푇(0) = 푇 + ℎ0(0). (15) On the other hand it follows from (14) that 푇 ℎ (휃), ℎ (휃) < , 휃 ∈ [0, 2휋], 0 푇 2 + 휀 and, by (5) in [14, Lecture 8] (for our case 휌 = 1 in this relation), 푇 |ℎ (휃)|, |ℎ (휃)| < , 휃 ∈ [0, 2휋]. 0 푇 2 + 휀 The latter inequality is incompatible with (15) unless 휙(0, 휆) = 0.

3.3. Selfadjoint case. In this subsection 푋 = 푙2(ℤ푑−1), and 퐴 = 퐴∗ or 퐴 = 푐퐴∗ for some 푐 ∈ ℂ. This happens for example in the model cases of heat or Schrödinger evolutions with real potentials. 2 푑 2 푑−1 The elements in 푙 (ℤ ) are denoted by 퐱 = {푥푘}푘, 푥푘 ∈ 푙 (ℤ ). We say that 푘 is the main variable and call the 푑 − 1 arguments of 푥푘 complementary spatial variables. In order to obtain the completeness of the generalized eigenvectors, and thus prove the uniqueness theorem applying the results of the previous sub- sections, we include 푙2(ℤ푑) into an appropriate Gelfand triple Φ ↪ 푙2(ℤ푑) ↪ Φ′, see, e.g., [4, 12, 13]. This can be done by demanding some decay of solution in complementary variables. Given 훼 ∈ ℝ we consider the weighted space

2 푑−1 2 훼 2 푙훼(ℤ ) = { 퐜 = {푐푚}푚∈ℤ푑−1 ∶ ‖퐜‖훼 = ∑ (1 + |푚|) |푐푚| < ∞ }. 푚∈ℤ푑−1 Theorem 3.4. Suppose that 훼 > 푑 − 1 and 퐴∶ 푙2(푙2(ℤ푑−1)) → 푙2(푙2(ℤ푑−1)),

푗+푠

(퐴퐮)푗 = ∑ 퐴푗,푘푢푘, 푘=푗−푠

2 푑−1 2 푑−1 is a banded operator, where 퐴푗,푘 are bounded in 푙 (ℤ ) as well as in 푙훼(ℤ ). Let 2 푑−1 further the external operators 퐴푗,푗±푠 be invertible in 푙훼(ℤ ) and −1 −1 ‖퐴 ‖ 2 2 ≤ 훿 , ‖퐴 ‖ 2 2 ≤푀, 푘=푗−푠,…,푗+푠. 푗,푗±푠 푙훼→푙훼 푗,푘 푙훼→푙훼 2 2 푑−1 If 퐮∶ [0, 푇] → 푙 (푙훼(ℤ )) satisfies 휕푡퐮 = 퐴퐮, and the decay condition in main spatial variable

|푘| −|푘| −|푘| |푘| |푘| ‖푢(푡, 푗)‖ 2 푑−1 ≤ 퐶푒 (2 + 휀) |푘| 푇 훿 , 푘 = [푗/푠], for 푡 = 0, 푇. 푙훼(ℤ ) Then 푢 ≡ 0. 430 Y. Lyubarskii and E. Malinnikova

Remark. In the model case, when 퐴 is a the sum of the Laplace operator and a real bounded potential (up to a unimodular factor), the operators 퐴푗,푘 are band- limited themselves and bounded in weighted spaces. Moreover 퐴푗,푗±푠 are identity operators and the norm estimate holds with 훿 = 1.

Proof. We consider the space

2 푑−1 2 |푘|1/2 2 Φ = { 퐂 = {퐜푘}푘∈ℤ, 퐜퐤 ∈ 푙훼(ℤ ) ∶ ‖퐂‖Φ = ∑ 푒 ‖퐜퐤‖훼 < ∞ }. 푘∈ℤ

Then the dual space (with respect to pairing in 푙2(ℤ푑) is

′ 2 푑−1 2 −|푘|1/2 2 Φ = { 퐂 = {퐜푘}푘∈ℤ, 퐜퐤 ∈ 푙훼(ℤ ) ∶ ‖퐂‖Φ′ = ∑ 푒 ‖퐜퐤‖−훼 < ∞ }. 푘∈ℤ

We have Φ ↪ 푙2(ℤ푑) ↪ Φ′ and the inclusion is a Hilbert–Schmidt operator since 훼 > 푑 − 1. We observe also that 퐴∶ Φ → Φ and hence 퐴∶ Φ′ → Φ′ are bounded operators. By repeating the arguments of the previous section, we obtain that 퐮(0) ∈ Φ is orthogonal to all generalized eigenvectors of 퐴 in Φ′. Then by general result, see for example [4, Chapter V,Theorem 1.4], we obtain that 퐮(0) = 0.

4. A sharp uniqueness result for bounded evolutions

4.1. Main result. We are now ready to prove our main result.

Theorem 4.1. Suppose that 퐴∶ 푙2(푋) → 푙2(푋),

푗+푠

(퐴퐮)푗 = ∑ 퐴푗,푘푢푘, 푘=푗−푠 is a banded operator satisfying (3) and (4). Further, assume that all operators 퐴푗,푘 2 commute. Let 퐮∶ [0, 푇] → 푙 (푋) satisfy 휕푡퐮 = 퐴퐮 and the decay condition (12):

|푢(푡, 푗)| ≤ 퐶푒|푘|(2 + 휀)−|푘||푘|−|푘|푇|푘|훿|푘|, 푘 = [푗/푠], for 푡 = 0, 푇.

Then 푢 ≡ 0.

The theorem follows from Proposition 3.3 and the proposition below. In dimension one our result can be applied to both heat and Schrödinger evolutions with bounded time-independent potentials as well as to evolutions defined by higher order difference operators. In higher dimension this approach allows us to work only with potentials depending on the variable in the direction of decay. Discrete evolutions 431

2 Proposition 4.2. Let 퐮 = {푢푗}푗∈ℤ ∈ 푙 (푋) be such that

|푗| ∑ 퐶 ‖푢푗‖ < ∞ 푗∈ℤ for every 퐶. Let also ⟨퐞, 퐮⟩ = 0 for each generalized eigenvector 퐞 of a banded operator 퐴∗. Then 퐮 = ퟎ. Our proof of the above proposition is inspired by a well known construction, sometimes referred to as the Shohat–Favard theorem for complex Jacobi matrices. We refer the reader to the survey articles [2, 3] and references therein.

4.2. Dimension one. To avoid extra technical details and explain the idea we first assume that 푋 = ℂ and write 퐴푗,푘 = 푎푗,푘 ∈ ℂ Proof of Proposition 4.2, 푋 = ℂ. Consider the families of polynomials

(푟) 푃푗 (휆), 푟 = −푠, −푠 + 1, … , 0, … , 푠 − 1, 푗 ∈ ℤ defined by the relations

(푟) 푃푗 (휆) = 훿푗,푟, 푗 = −푠, −푠 + 1, … , 0, … , 푠 − 1, 푗+푠 (푟) (푟) 휆푃푗 (휆) = ∑ 푘,푗̄푎 푃푘 (휆). (16) 푘=푗−푠

(푟) (푟) For each 휆 ∈ ℂ and 푟 = −푠, … , 푠 − 1 the vector 퐯 (휆) = {푃푗 (휆)}푗 is a generalized eigenvector of 퐴∗ with eigenvalue 휆.̄ Therefore

(푟) ∑ 푢푗푃푗 (휆) = 0. (17) 푗 Let 퐴∶̄ 푙2(ℂ) → 푙2(ℂ) denote the “complex conjugate” of 퐴:

푗+푠

(퐴퐮)̄ 푗 = ∑ 푗,푘̄푎 푢푘. 푘=푗−푠

(푟) 2 2 We consider 푃푛 (퐴)∶̄ 푙 (ℤ) → 푙 (ℤ). The scalar relation (16) now yields

푗+푠 ̄ (푟) ̄ (푟) ̄ 퐴푃푗 (퐴) = ∑ 푘,푗̄푎 푃푘 (퐴). 푘=푗−푠 This in particular implies that

(푟) |푛| ‖푃푛 (퐴)‖̄ ≤ 퐶 for some 퐶 > 0. (18) 432 Y. Lyubarskii and E. Malinnikova similar to (8). We claim that (17) implies

(푟) ∑ 푢푛푃푛 (퐴)̄ = 0, 푛 and due to (18) the series converges absolutely. Let further 흈(푛) be the 푛-th coordinate vector in 푙2(ℤ). An induction argument shows that 푠−1 (푟) (푟) (푛) ∑ 푃푛 (퐴)흈̄ = 흈 . 푟=−푠 Then 푠−1 (푟) (푟) (푛) 0 = ∑ ∑ 푢푛푃푛 (퐴)흈̄ = ∑ 푢푛휎 . 푟=−푠 푛 푛 Hence 푢 ≡ 0.

4.3. General case. We extend the above construction to banded operators on 푙2(푋) with commuting entries.

Proof of Proposition 4.2, General case. We split the proof into several steps.

(푟) Step 1. We define families of operator-polynomials {푃푗 (휆)}푗, −푠 ≤ 푟 < 푠, 휆 ∈ ℂ by

(푟) (푟) 푃푟 = 퐼, 푃푗 = 0, 푗 ≠ 푟 and −푠 ≤ 푟 < 푠, 푗+푠 (푟) ∗ (푟) 휆푃푗 (휆) = ∑ 퐴푘,푗푃푘 (휆). (19) 푘=푗−푠

(푟) For any 푥 ∈ 푋 the sequence 퐯 = {푣푗}푗 = {푃푗 (휆)푥}푗 is a generalized eigenvector of 퐴∗, 퐴∗퐯 = 휆퐯. We have (푟) 푚 (푟) 푃푗 (휆) = ∑ 휆 퐶푗,푚 , 푚≥0

(푟) (푟) where 퐶푗,푚 ∶ 푋 → 푋 and the sum is finite. Moreover, all coefficients 퐶푗,푚 are ∗ products of the operators 퐴푘,푙 and their inverses (we will use this fact to interchange the order of operators). (푟) Now the orthogonality relation 퐮 ⟂ {푃푗 (휆)푥}푗 implies

(푟) 푚 (푟) 0 = ∑⟨푢푗, 푃푗 (휆)푥⟩푋 = ∑ 휆 ∑⟨푢푗, 퐶푗,푚푥⟩푋. 푗 푚 푗 Discrete evolutions 433

The series converges since we assume that ‖푢푗‖푋 decays fast in 푗. We conclude (푟) that each coefficient ∑푗⟨푢푗, 퐶푗,푚푥⟩푋 vanishes. Then

(푟) ∗ ∑(퐶푗,푚) 푢푗 = 0. (20) 푗

Step 2. Denote by 퐴̄ the “conjugate” operator

푗+푠 ̄ ̄ ̄ ̄ ∗ 퐴퐯 = 퐴{푣푗} = {(퐴퐯)푗}, (퐴퐯)푗 = ∑ 퐴푗,푘푣푘. 푘=푗−푠

2 By 푖푚 we denote the embedding 푋 ↪ 푙 (푋) that places a given vector 푥 ∈ 푋 into 푚-th position and zeros in all other positions:

(푖푚푥)푘 = 훿푚,푘푥.

Define further

(푟) 푚̄ (푟) (푟) 2 풫푗 푢 = ∑ 퐴 푖푟퐶푗,푚푢, 푢 ∈ 푋, 풫푗 ∶ 푋 → 푙 (푋). (21) 푚≥0

∗ (푟) (푟) ∗ Then (21), (19), and the commutation relation 퐴푘,푗퐶푘,푚 = 퐶푘,푚퐴푘,푗 imply

푗+푠 ̄ (푟) (푟) ∗ 퐴풫푗 푢 = ∑ 풫푘 퐴푘,푗푢. 푘=푗−푠 We show by induction that for any 푣 ∈ 푋

푠−1 (푟) ∑ 풫푛 푣 = 푖푛푣. (22) 푟=−푠

(푟) Indeed, for 푛 = −푠, …, 푠 − 1 this follows from the definition of 풫푛 . Further by the recurrence formula

푛−1 (푟) ∗ ̄ (푟) (푟) ∗ 풫푛 퐴푛,푛−푠푣 = 퐴풫푛−푠(푣) − ∑ 풫푘 (퐴푘,푛−푠푣) 푘=푛−2푠 Taking the sum with respect to 푟 and using the induction hypothesis, we obtain

푠−1 푛−1 (푟) ∗ ̄ ∗ ∗ ∑ 풫푛 퐴푛,푛−푠푣 = 퐴푖푛−푠푣 − ∑ 푖푘퐴푘,푛−푠푣 = 푖푛(퐴푛,푛−푠푣). 푟=−푠 푘=푛−2푠

∗ Now (22) follows since 퐴푛,푛−푠 is invertible. 434 Y. Lyubarskii and E. Malinnikova

2 Step 3. We denote by 휋푘 the 푘th projection of 푙 (푋) to 푋, 휋푘퐯 = 푣푘. Now we fix some 푥 ∈ 푋 and for each 푗 ∈ ℤ and 푟 = −푠, …, 푠 − 1 consider a sequence (푟,푗) (푟,푗) 2 훼 = {훼푘 }푘 ∈ 푙 (ℂ) defined by 훼(푟,푗) = ⟨푢 , 휋 풫(푟)푥⟩ . 푘 푗 푘 푗 푋 (푟) (푟,푗) 2 Let 훼 = ∑푗 훼 ∈ 푙 , we have

훼(푟) = ∑⟨푢 , 휋 풫(푟)푥⟩ = ∑ ∑⟨푢 , 휋 퐴푚̄ 푖 퐶(푟) 푥⟩ . 푘 푗 푘 푗 푋 푗 푘 푟 푗,푚 푋 푗 푚 푗 The coefficients of operators 퐴푚̄ are operators from 푋 to 푋, they are products ∗ 푚̄ (푟) of operators 퐴푙,푘. Clearly, 휋푘퐴 푖푟 is such a coefficient, it commutes with 퐶푗,푚. Therefore (푟) (푟) ∗ 푚̄ 훼푘 = ∑ ⟨∑(퐶푗,푚) 푢푗, 휋푘퐴 푖푟푥⟩ = 0, 푚 푗 푋 the last identity follows from (20). On the other hand, by (22)

푠−1 (푟,푗) (푟) ⟨푢푗, 푥⟩, 푘 = 푗 ∑ 훼푘 = ⟨푢푗, 휋푘(∑ 풫푗 푥)⟩ = ⟨푢푗, 휋푘푖푗푥⟩푋 = { 푟=−푠 푟 푋 0, 푘 ≠ 푗

(푟) (푟,푗) Finally, 0 = ∑푟 훼푘 = ∑푗 ∑푟 훼푘 = ⟨푢푘, 푥⟩. Thus 푢 = 0.

4.4. Decay of stationary solutions. It was mentioned in [10] that uniqueness results imply some estimates on the possible decay of stationary solutions of discrete Schrödinger operators. We suggest two elementary but reasonably sharp results. Proposition 4.3. Suppose that 퐴 is a banded operator on 푙2(푋) satisfying (3) and (4). There exists a constant 푐 = 푐(퐴) such that if a solution 퐮 ∈ 푙2(퐴) of 퐴퐮 = 0 −푐푗 satisfies ‖퐮푗‖푋 ≤ 퐶푒 , then 푢 ≡ 0. Proof. The recurrence formula implies 푠 −1 푢푛−푠(휆) = 퐴푛−푠,푛 ( ∑ 퐴푚+푛,푛푢푚+푛) . 푚=−푠+1

−1 푠 Clearly, ‖푢푛−푠‖푋 ≤ 훿 푎 ∑푚=−푠+1 ‖푢푛+푚‖푋. If 푀푗 = max−푠<푚≤푠 ‖푢푗+푚‖푋 then −1 −1 푀푗 ≥ (2푠) 훿푎 푀푗−1. This actually implies that if

푗 −1 −1 ‖푢(푗)‖푋 ≤ 퐶푞 , 푞 < 훿푎 (2푠) , then 푢 ≡ 0. Discrete evolutions 435

We could formulate a slightly more general result, saying that

ln 푀푗 lim inf ≥ 푐, 푗→∞ 푗 for any non-trivial solution of the stationary equation. A similar approach can be applied to the case of the discreet Schrödinger operator with a bounded potential 푉∶ ℤ푑 → ℂ, a straightforward computation 푑 shows that if 푢∶ ℤ → ℂ satisfies Δ푑푢 + 푉푢 = 0 and

ln(max|푛|∞∈{,+1} |푢(푛)|) lim inf < −‖푉‖∞ − 4푑 + 1 (23) →∞ 푁

푑 where |푛|∞ = max{푛1, … 푛푑} for 푛 ∈ ℤ , then 푢 ≡ 0. Indeed, the equation implies

max |푢(푛)| ≤ (4푑 − 2 + ‖푉‖∞) max |푢(푛)| + max |푢(푛)|, |푛|∞=−1 |푛|∞= |푛|∞=+1 and (23) follows.

Acknowledgments. This work has been done while the authors were visiting Department of Mathematics at Purdue University. It is our pleasure to thank the department for hospitality. We also want to thank A. Pushnitski for a useful discussion.

References

[1] I. Alvarez-Romero, G. Teschl, A Dynamic Uncertainty Principle for Jacobi Opera- tors, arXiv:1608.04244 [2] A. Beardon, The theorems of Stieltjes and Favard, Comp. Math. Funct. Theory, 11 (2011), 247–262. [3] B. Beckermann, Complex Jacobi matrices, Journ. Comp. Appl. Math, 127 (2001), 17–65. [4] Yu. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Transla- tions of Mathematical Monographs, 17. American Mathematical Society, Providence, RI, 1968. [5] M. Cowling, L. Escauriaza, C. E. Kenig, G. Ponce & L. Vega, The Hardy uncertainty principle revisited. Indiana Univ. Math. J. 59 (2010), 2007–2025. [6] L. Escauriaza, C. E. Kenig, G. Ponce & L. Vega The sharp Hardy uncertainty principle for Schrödinger evolutions. Duke Math. J. 155 (2010), 163–187. [7] L. Escauriaza, C. E. Kenig, G. Ponce & L. Vega Uniqueness properties of solutions to Schrödinger equations. Bull. Amer. Math. Soc. 49 (2012), 415-422. 436 Y. Lyubarskii and E. Malinnikova

[8] A. Fernández-Bertolin, A Discrete Hardy’s uncertainty principle and Discrete evolutions, arXiv:1506.00119. [9] A. Fernández-Bertolin, Convexity properties of Discrete Schrödinger evolutions and Hardy’s uncertainty principles, arXiv:1506.03717. [10] A. Fernández-Bertolin, L. Vega, Uniqueness properties for Discrete equations and Carleman estimates, arXiv:1509.08545 [11] Ph. Jaming, Yu. Lyubarskii, E. Malinnikova, K.-M. Perfekt, Uniqueness for discrete Schrödinger evolutions, arXiv:1505.05398, to appear in Rev. Mat. Iberoam. [12] I. M. Gelfand, G. E. Shilov, Generalized functions, Vol. 3:Theory of differential equations. Translated from the Russian by Meinhard E. Mayer Academic Press, 1967. [13] I. M. Gelfand, N. Ya.Vilenkin, Generalized functions, vol. 4: Applications of harmonic analysis. Translated by Amiel Feinstein Academic Press, 1964. [14] B. Ya. Levin, Lectures on entire functions. Translations of Mathematical Monographs, AMS, 1996. Spatial analyticity of solutions to nonlinear dispersive PDE

Sigmund Selberg

Dedicated to Helge Holden on the occasion of his 60th birthday

Abstract. Given a nonlinear dispersive PDE, for example the KdV equation, we consider the Cauchy problem with real analytic initial data. For data with a uniform radius of analyticity, we are interested in obtaining lower bounds on the radius of analyticity at later times. A rather general approach to this problem is presented, based on Bourgain’s Fourier restriction norm method. Applications to the KdV equation (periodic and non-periodic) and the Dirac–Klein–Gordon equations are discussed.

1. Introduction

Given a nonlinear evolutionary PDE in the independent variables (푡, 푥) ∈ ℝ × ℝ, consider the Cauchy problem with real analytic initial data at 푡 = 0. If these data have a uniform radius of analyticity 휎0 > 0, in the sense that there exists a holomorphic extension to the complex strip

푆휍 = { 푥 + 푖푦 ∈ ℂ ∶ 푥, 푦 ∈ ℝ, |푦| < 휎 }, with 휎 = 휎0, then we ask whether the solution at some later time 푡 > 0 also has a uniform radius of analyticity 휎 = 휎(푡) > 0, in which case we would, moreover, like to have an explicit lower bound on 휎(푡). Heuristically, the picture one should have in mind is that 휎(푡) is the distance from the 푥-axis to the nearest complex singularity of the holomorphic extension of the solution at time 푡. If at some time 푡 this singularity actually hits the 푥-axis, then the solution itself suffers a breakdown of regularity. This point of view is the basis for the widely used singularity tracking method [16] in numerical analysis, where a spectral method is used to obtain a numerical estimate of 휎(푡). This estimate can then be used to predict either the formation of a singularity in finite time or alternatively global regularity. Even in cases where singularity formation does not occur (as is the case for our main examples, the Korteweg–de Vries equation and the 1d and 2d Dirac–Klein– 438 S. Selberg

Gordon equations), it is still of interest to obtain lower bounds on 휎(푡), as this has implications for the rate of convergence of spectral methods for the equation one is looking at (see [1] for an example of this). We will describe here a method for obtaining lower bounds on 휎(푡), based on Bourgain’s Fourier restriction norm method [3, 4]. The method will be illustrated on the Korteweg–de Vries equation (KdV)

푢푡 + 푢푥푥푥 + 푢푢푥 = 0 (1) and the Dirac–Klein–Gordon equations (DKG)

0 1 (−푖훾 휕푡 − 푖훾 휕푥 + 푀) 휓 = 휙휓, { 2 2 2 ∗ 0 (2) (휕푡 − 휕푥 + 푚 ) 휙 = 휓 훾 휓, written here for the 1d case, so 푥 ∈ ℝ. More generally, the method applies to a class of Cauchy problems for nonlinear 푑 dispersive PDE on ℝ푡 × ℝ푥, of the form

푑 푢푡 = 푖ℎ(퐷)푢 + 푁[푢] (푡 ∈ ℝ, 푥 ∈ ℝ ) , 푢(0, 푥) = 푢0(푥), (3) and for which local well-posedness for initial data 푢0 in a range of the Sobolev spaces 퐻푠(ℝ푑) = 푊 푠,2(ℝ푑) can be proved using a contraction mapping argument based on estimates for the nonlinear operator 푁[ ⋅ ] in the Bourgain spaces 푋푠,푏 (this will be made precise below). Here we denote ∇ 퐷 = 푥 , 푖 and ℎ(퐷) is the Fourier multiplier given by

ℎ(퐷)푓 = ℱ−1 [ℎ(휉) ℱ푓(휉)] , where ℎ(휉) is a given function and

푓(휉)ˆ = ℱ푓(휉) = ∫ 푒−푖푥⋅휉푓(푥) 푑푥 (휉 ∈ ℝ푑) ℝ푑 is the Fourier transform on ℝ푑. For example, the KdV equation is of the form (3) with 푑 = 1, ℎ(휉) = 휉3 and 1 2 푁[푢] = −푢휕푥푢 = − 2 휕푥(푢 ). The DKG system can also be written in the form (3), with 푢 then being vector-valued and ℎ matrix-valued, but this reformulation is a bit more involved and we do not include it here. We limit attention to nonlinear operators 푁[ ⋅ ] containing second order and higher order terms and satisfying the following assumption. Nonlinear dispersive PDE 439

(A) 푁[푢] is a finite linear combination of 푘-linear operators 푁푘[푢, … , 푢] for 푘 ≥ 2, where 푁푘 is of the form

푘

ℱ푁푘[푢1, … , 푢푘](휉) = ∫ 푚푘(휉1, … , 휉푘) ∏ 푗̂푢 (휉푗) (4) 휉1+⋯+휉푘=휉 푗=1

for a given symbol 푚푘. Here we use the shorthand

푘−1

∫ 푓(휉1, … , 휉푘) = ∫ 푓 (휉1, … , 휉푘−1, 휉 − ∑ 휉푗) 푑휉1 … 푑휉푘−1. 푑 푘−1 휉1+⋯+휉푘=휉 (ℝ ) 푗=1

For example, for the KdV equation we have 푁[푢] = 푁2[푢, 푢] with 푚2(휉1, 휉2) = 1 − 2 푖(휉1 + 휉2) and

1 ℱ푁 [푢 , 푢 ](휉) = − 푖휉 ∫ ̂푢 (휉 − 휂) ̂푢 (휂) 푑휂. 2 1 2 2 1 2

Remark 1. Assumption (A) implies, in particular, that (3) is time-translation invariant. That is, if 푢(푡, 푥) is a solution, then so is 푣(푡, 푥) ≔ 푢(푡 + 푡0, 푥) for any 푡0, with initial condition 푣(푡 = 푡0) = 푢0. 휍,푠 푑 Now consider (3) with data 푢0 in the Gevrey space 퐺 (ℝ ) defined, for 휎 > 0 and 푠 ∈ ℝ, by 휍,푠 푑 2 푑 퐺 (ℝ ) = { 푓 ∈ 퐿 (ℝ ) ∶ ‖푓‖퐺휍,푠 < ∞ }, where ‖ 휍‖휉‖ 푠 ˆ ‖ ‖푓‖퐺휍,푠 = ‖푒 ⟨휉⟩ 푓(휉)‖ 2 퐿휉

푑 and for 휉 = (휉1, … , 휉푑) ∈ ℝ we denote

‖휉‖ = |휉1| + ⋯ + |휉푑|, 2 2 1/2 |휉| = (|휉1| + ⋯ + |휉푑| ) , ⟨휉⟩ = (1 + |휉|2)1/2.

Note that 퐺휍,푠 = ℱ−1(푒−휍| ⋅ |⟨ ⋅ ⟩−푠퐿2(ℝ푑)) is isometrically isomorphic to 퐿2(ℝ푑). We record the fact that any 푓 ∈ 퐺휍,푠 has a uniform radius of analyticity 휎.

Lemma 1. Every 푓 ∈ 퐺휍,푠(ℝ푑) has a holomorphic extension to the strip

푑 푑 푆휍 = { 푥 + 푖푦 ∈ ℂ ∶ 푥, 푦 ∈ ℝ and |푦푗| < 휎 for 푗 = 1, … , 푑 }. 440 S. Selberg

휍,푠 푑 Proof. Let 푓 ∈ 퐺 . For each 푎 ∈ ℝ we must find a holomorphic extension 퐹푎 to 푑 the polydisc 퐷(푎1, 휎) × ⋯ × 퐷(푎푑, 휎) ⊂ ℂ . By uniqueness, two such extensions agree on the intersection of their domains. Moreover, the union of the polydiscs 휍,푠 equals 푆휍. By invariance under translation (that is, 푓 ∈ 퐺 implies 푓( ⋅ − 푎) ∈ 퐺휍,푠), it suffices to do the case 푎 = 0. Fix 휎′ ∈ (0, 휎). Using multi-index notation we have, by Fourier inversion,

sup |휕훼푓(푥)| ≤ 푐 ∫ |휉훼||푓(휉)|ˆ 푑휉, 푥∈ℝ푑 ℝ푑 so by Taylor’s theorem it is easy to see that 푓 is given by its Taylor series,

휕훼푓(0) 푓(푥) = ∑ 푥훼 for 푥 with |푥 | < 휎′, 푗 = 1, … , 푑, 훼! 푗 훼 and that this series converges absolutely. Indeed,

|휕훼푓(0)| ∑ |푥훼| 훼! 훼 ′ 훼1 ′ 훼푑 훼1 훼푑 (휎 |휉1|) (휎 |휉푑|) ˆ |푥1| |푥푑| ≤ 푐 ∑ ⋯ ∑ (∫ ⋯ |푓(휉)| 푑휉) ( ′ ) ⋯ ( ′ ) 푑 훼1! 훼푑! 휎 휎 훼1 훼푑 ℝ 훼1 훼푑 ′ |푥 | |푥 | 휍 ‖휉‖ ˆ 1 푑 ≤ 푐 ∑ ⋯ ∑ (∫ 푒 |푓(휉)| 푑휉) ( ′ ) ⋯ ( ′ ) 푑 휎 휎 훼1 훼푑 ℝ 1/2 훼1 훼푑 2(휍′−휍)‖휉‖ −2푠 |푥1| |푥푑| ≤ 푐 (∫ 푒 ⟨휉⟩ 푑휉) ‖푓‖퐺휍′,푠 ∑ ⋯ ∑ ( ′ ) ⋯ ( ′ ) 푑 휎 휎 ℝ 훼1 훼푑

′ ′ is finite provided |푥푗| < 휎 . Since 휎 < 휎 was arbitrary, we conclude that

휕훼푓(0) 퐹 (푧) ≔ ∑ 푧훼 0 훼! 훼

푑 converges absolutely for 푧 ∈ ℂ with |푧푗| < 휎 for 푗 = 1, … , 푑, and this is the holomorphic extension we seek.

Observe that the norm ‖푓‖퐺휍,푠 is obtained from the standard Sobolev norm

‖ 푠 ˆ ‖ ‖푓‖퐻푠 = ‖⟨휉⟩ 푓(휉)‖ 2 퐿휉 by the substitution 푓 ⟶ 푒휍‖퐷‖푓. Nonlinear dispersive PDE 441

Indeed, ‖푓‖ = ‖푒휍‖퐷‖푓‖ . 퐺휍,푠 ‖ ‖퐻푠 The same substitution can be used in the setting of Bourgain’s Fourier restriction norm method. The Bourgain space 푋푠,푏 (defined below) then yields a Gevrey-modified space 푋휍,푠,푏. This was done by Bourgain [5, Theorem 8.12] for the Kadomtsev–Petviashvili equation, but the argument applies to dispersive PDE of the form (3) in general. In brief summary, the consequences that can be abstracted from Bourgain’s argument are as follows:

(B1) If local well-posedness of (3) can be proved for 퐻푠 initial data by a contraction argument in 푋푠,푏, then the same argument works with the Gevrey modification, hence for short times the radius of analyticity will notdecay.

(B2) If, moreover, the solution extends globally (so the 퐻푠 norm does not blow up in finite time), then the solution remains analytic for all time, but nolower bound is obtained on 휎(푡) > 0 as 푡 → ∞.

(B3) Finally, if the 퐻푠 norm is conserved, then an exponential lower bound on 휎(푡) is obtained. That is, 휎(푡) ≥ 푐 exp(−퐴푡) for some positive constants 푐 and 퐴 depending on the initial data.

The final assertion is not included in[5], but is proved in the next section, where we also discuss and briefly outline the proofs of the first two assertions. Our main aim here is to present a refinement of Bourgain’s method, yielding an improvement of (B3). First, however, let us see how the KdV and DKG equations fit into the preceding discussion.

• For the KdV equation, (B1), (B2) and (B3) all apply for data in 퐿2 = 퐻0. Indeed, Bourgain [4] proved local well-posedness for such data. Moreover, the 퐿2 norm is conserved.

• For the 1d and 2d DKG equations, (B1) and (B2) apply, but not (B3), since there is no conservation law for the field 휙. Local well-posedness for initial 푠 푟 푟−1 data (휓, 휙, 휕푡휙)(푡 = 0) ∈ 퐻 × 퐻 × 퐻 has been extensively studied. See [12] and the references therein for the 1d case, and [8] for the 2d case. In the 1d case, it is relatively straightforward to extend the local result globally in time when 푠 ≥ 0, by using the conservation of the 퐿2 norm of 휓(푡, ⋅ ). This is much harder to do in the 2d case, but was achieved in [9].

Thus, for KdV one obtains by (B3) an exponential lower bound 휎(푡) ≥ 푐 exp(−퐴푡) for all 푡 > 0. It turns out that this can be improved to an algebraic lower bound −푝 4 휎(푡) ≥ 푐푡 . This was first proved in [2] for 푝 = 12 and improved to 푝 = 3 + 휀 442 S. Selberg in [14] using a refinement of Bourgain’s method, relying on an almost conservation law in 퐺휍,0, converging to the 퐿2 conservation as 휎 → 0.

3 휍0,푠 Theorem 1 ([14]). Let 휎0 > 0 and 푠 > − 4 . Let 푢0 ∈ 퐺 (ℝ). Consider the KdV equation (1) with initial condition 푢(푡 = 0) = 푢0. The solution 푢 satisfies 푢(푡) ∈ 퐺휍(푡),푠 for all 푡 ∈ ℝ, with −(4/3+휀) 휎(푡) = min (휎0, 푐|푡| ) , where 휀 > 0 can be taken arbitrarily small and 푐 > 0 is a constant depending on 푢0, 휎0, 푠 and 휀. We remark that the method can also be used to handle KdV in the periodic case, where the result 휎(푡) ≥ 푐푡−2 has been obtained [10]. The idea of using an almost conservation law in the context of spatial analyticity first appeared in [15] for the 1d DKG equations, where the following result was obtained.

Theorem 2 ([15]). Let 휎0 > 0 and

휍0,0 2 휍0,1 휍0,0 (휓0, 휙0, 휙1) ∈ 퐺 (ℝ; ℂ ) × 퐺 (ℝ; ℝ) × 퐺 (ℝ; ℝ). Then for the solution (휓, 휙) of the 1d DKG equations (2) with initial condition

휓(0, 푥) = 휓0(푥), 휙(0, 푥) = 휙0(푥), 휕푡휙(0, 푥) = 휙1(0, 푥), we have

휍(푡),0 휍(푡),1 휍(푡),0 (휓, 휙, 휕푡휙)(푡) ∈ 퐺 × 퐺 × 퐺 for all 푡 ∈ ℝ, where −4 휎(푡) ≥ min (휎0, 푐푡 ) with a constant 푐 > 0 depending on 푚, 푀, 휎0, 푟, 푠, and the norm of the data. This is of course a huge improvement over (B2) (recall that (B3) does not apply for DKG), as (B2) gives no explicit lower bound, only positivity. For the much more involved 2d case of DKG we have obtained [13] a lower bound 휎(푡) ≥ 휎0 exp(−퐴푡). We remark also that DKG can be written as a nonlinear symmetric hyperbolic system, hence the general results from [6] concerning persistence of spatial analyticity for such systems apply, yielding a lower bound

푡

휎(푡) ≥ 휎0 exp (−퐴 ∫ (1 + ‖휓(푠)‖퐿∞ + ‖휙(푠)‖퐿∞ + ‖휕휙(푠)‖퐿∞) 푑푠) , 0 Nonlinear dispersive PDE 443 but this is weaker than our results for both 1d and 2d DKG mentioned above. In fact, the best estimate known on the 퐿∞ norm of the solutions of 1d and 2d DKG seems to be 푂(exp(퐶푡)), hence one would get 휎(푡) ≥ 휎0 exp(−퐴 exp(퐶푡)). In the next section we introduce some function spaces. In section 3 we discuss (B1), (B2) and (B3) further and outline their proofs. Then in section 4 we refine Bourgain’s method, showing that if an “almost conservation law” holds in the Gevrey space, then the exponential lower bound in (B3) can be improved to an algebraic lower bound. Finally, in section 5 we illustrate the general method in the case of the KdV equation, recalling the key steps in the proof of the almost conservation law for KdV, which then yields the result in Theorem 1.

2. Function spaces

푠,푏 푠,푏 푑 For 푠, 푏 ∈ ℝ, the Bourgain space 푋 = 푋휏=ℎ(휉)(ℝ푡 × ℝ푥) associated to the 푑 dispersive operator 휕푡 − 푖ℎ(퐷) is defined to be the completion of 풮(ℝ푡 × ℝ푥) with respect to the norm ‖ 푠 푏 ‖ ‖푢‖푋푠,푏 = ‖⟨휉⟩ ⟨휏 − ℎ(휉)⟩ ̂푢(휏, 휉)‖ 2 , 퐿휏,휉 where

̂푢(휏, 휉) = ∫ 푒−푖(푡휏+푥⋅휉)푢(푡, 푥) 푑푡 푑푥 (휏 ∈ ℝ, 휉 ∈ ℝ푑) ℝ×ℝ푑 is the space-time Fourier transform. The space 푋푠,푏 is well-suited for capturing the dispersive smoothing effect of the operator 휕푡 − 푖ℎ(퐷) away from the characteristic hypersurface 휏 = ℎ(휉) (see section 2.6 of [17]). By analogy with the relationship 퐺휍,푠 = 푒−휍‖퐷‖(퐻푠), we define the Gevrey- modified Bourgain space 푋휍,푠,푏, for 휎 > 0, by

푋휍,푠,푏 = 푒−휍‖퐷‖ (푋푠,푏) , with norm ‖ 휍‖휉‖ 푠 푏 ‖ ‖푢‖푋휍,푠,푏 = ‖푒 ⟨휉⟩ ⟨휏 − ℎ(휉)⟩ ̂푢(휏, 휉)‖ 2 . 퐿휏,휉 Note that 푋휍,푠,푏 is well-defined, since 푒−휍‖퐷‖ = ℱ−1푒−휍‖ ⋅ ‖ ℱ clearly maps 푋푠,푏 into itself, for 휎 ≥ 0. The restriction of 푋푠,푏 to a time-slab (−훿, 훿) × ℝ푑 is denoted 푋푠,푏(훿). This is a Banach space when equipped with the norm

푠,푏 푑 ‖푢‖푋푠,푏(훿) = inf { ‖푣‖푋푠,푏 ∶ 푣 ∈ 푋 and 푢 = 푣 on (−훿, 훿) × ℝ }. 444 S. Selberg

The restriction 푋휍,푠,푏(훿) is similarly defined, and then we clearly have 푋휍,푠,푏(훿) = 푒−휍‖퐷‖ (푋푠,푏(훿)) , hence the well-known properties of 푋푠,푏 and its restrictions carry over to 푋휍,푠,푏 simply by the substitution 푢 → 푒휍‖퐷‖푢. The properties we require here are contained in the next four lemmas. Proofs of Lemmas 2, 3 and 5 can be found, for example, in section 2.6 of [17]. Lemma 4 follows by the argument used to prove Lemma 3.2 of [7]. 1 푠,푏 푠 Lemma 2. If 푏 > 2 , then 푋 ⊂ 퐶(ℝ, 퐻 ) and

sup‖푢(푡)‖퐻푠 ≤ 푐푏‖푢‖푋푠,푏. 푡∈ℝ 1 ′ 1 Lemma 3. Assume − 2 < 푏 < 푏 < 2 and 훿 > 0. Then 푏′−푏 ‖푢‖푋푠,푏(훿) ≤ 푐푏,푏′훿 ‖푢‖푋푠,푏′(훿). 1 1 Lemma 4. Assume − 2 < 푏 < 2 and 훿 > 0. For any time interval 퐼 ⊂ [−훿, 훿] we then have

‖휒퐼푢‖푋푠,푏 ≤ 푐푏‖푢‖푋푠,푏(훿).

Next, consider the Cauchy problem, for given 퐹(푡, 푥) and 푢0(푥),

푢푡 = 푖ℎ(퐷)푢 + 퐹, 푢(0) = 푢0, whose solution is given by the Duhamel formula 푡 ′ ′ ′ 푢(푡) = 푈(푡)푢0 + ∫ 푈(푡 − 푡 )퐹(푡 ) 푑푡 , 0 푖푡ℎ(퐷) where 푈(푡) = 푒 is the free propagator of 휕푡 − 푖ℎ(퐷). 1 푠 Lemma 5. Assume 2 < 푏 ≤ 1 and 0 < 훿 ≤ 1. Then for all 푢0 ∈ 퐻 and 퐹 ∈ 푋푠,푏−1(훿), we have the estimates

‖푈(푡)푢0‖푋푠,푏(훿) ≤ 푐푏‖푢0‖퐻푠, ‖ 푡 ‖ ‖∫ 푈(푡 − 푡′)퐹(푡′) 푑푡′‖ ≤ 푐 ‖퐹‖ . ‖ ‖ 푏 푋푠,푏−1(훿) 0 푋푠,푏(훿)

3. Bourgain’s observations, and the new observation (B3)

In this section we discuss in more detail (B1), (B2) and (B3), and outline their proofs. Nonlinear dispersive PDE 445

3.1. Observation (B1). We start with a definition. By local well-posedness (LWP) of (3) in 퐻푠, we mean the following. 푠 LWP. For any 푅 > 0 there exists 훿 = 훿(푅) > 0, such that for any 푢0 ∈ 퐻 with ‖푢0‖퐻푠 ≤ 푅 there exists 푢 ∈ 퐶([−훿, 훿]; 퐻푠) solving (3) on (−훿, 훿) × ℝ with initial condition 푢(0) = 푢0, and satisfying

sup ‖푢(푡)‖퐻푠 ≤ 푐푅. 푡∈[−훿,훿] Moreover, 푢 is unique in some subspace of 퐶([−훿, 훿]; 퐻푠). By a standard contraction argument in 푋푠,푏, which we outline in an appendix for 1 the convenience of the reader, LWP is easily seen to hold if there exist 2 < 푏 < 푏′ < 1 such that

‖푁[푢]‖푋푠,푏′−1 ≤ 푝 (‖푢‖푋푠,푏) ‖푢‖푋푠,푏, (5)

‖푁[푢] − 푁[푣]‖푋푠,푏′−1 ≤ 푝 (‖푢‖푋푠,푏 + ‖푣‖푋푠,푏) ‖푢 − 푣‖푋푠,푏, (6) where 푝 is an increasing polynomial. 1 ′ Proposition 1. Let 푠 ∈ ℝ. If (5) and (6) hold for some choice of 2 < 푏 < 푏 < 1, then LWP holds. Moreover, 푢 ∈ 푋푠,푏(훿) and is unique in that space. Recalling assumption (A), we note that the estimates (5) and (6) will hold if 푘

‖푁푘[푢1, … , 푢푘]‖푋푠,푏′−1 ≤ 푐푘 ∏‖푢푗‖푋푠,푏 (7) 푗=1 for each of the finitely many 푘 ≥ 2 involved in the linear combination of 푁푘’s constituting 푁. We shall in fact assume something stronger than this, namely that the sign of the symbol 푚푘 does not matter, so that we can take absolute values inside the integral (4). That is, we will assume ‖ 푘 ‖ 푘 푠 푏′−1 ‖⟨휉⟩ ⟨휏 − ℎ(휉)⟩ ∫ |푚푘(휉1, … , 휉푘)| ∏|푗 ̂푢 (휉푗)|‖ ≤ 푐푘 ∏‖푢푗‖푋푠,푏. (8) ‖ ‖ 2 ∑ 휉푗=휉 푗=1 퐿휏,휉 푗=1 3 For example, for the KdV equation, this estimate with 푠 > − 4 is a consequence of the following. 3 Theorem 3 (Kenig, Ponce and Vega [11, Thm. 2.2]). Given 푠 > − 4 , there exist 1 푏 ∈ ( 2 , 1) and 휀 > 0 such that such that the following estimate holds for any 푏′ ∈ [푏, 푏 + 휀):

‖휕푥(푢푣)‖푋푠,푏′−1 ≤ 푐‖푢‖푋푠,푏‖푣‖푋푠,푏. Here 푐 > 0 is a constant depending only on 푠, 푏 and 푏′. 446 S. Selberg

Next we define local well-posedness in the Gevrey space.

휍0,푠 LWP’. For any 푅 > 0 there exists 훿 = 훿(푅) > 0, such that for any 푢0 ∈ 퐺 with ‖푢0‖퐺휍0,푠 ≤ 푅 there exists 푢 ∈ 퐶([−훿, 훿]; 퐺휍0,푠) solving (3) on (−훿, 훿) × ℝ with initial condition 푢(0) = 푢0, and satisfying

sup ‖푢(푡)‖퐺휍0,푠 ≤ 푐푅. 푡∈[−훿,훿]

We remark that if LWP’ holds, then the initial radius of analyticity 휎0 persists throughout the (short) time interval (−훿, 훿). We can now state precisely the assertion made in (B1).

Proposition 2 (Observation (B1)). Let 푠 ∈ ℝ. Assume that (8) holds for some 1 ′ choice of 2 < 푏 < 푏 < 1, so that in particular (5) and (6) hold and hence LWP holds by Proposition 1. Then also LWP’ holds for all 휎0 > 0, with the same 훿( ⋅ ) and 푐 as in LWP (hence independent of 휎0). Proof. It suffices to check that (5) and (6) also hold (with the same 푝) with the Gevrey-modification, that is, when the 푋푠,푏 norms on both sides are replaced by the corresponding 푋휍0,푠,푏 norms. Indeed, the standard argument (see the appendix) that is used to prove Proposition 1 then yields LWP’. Thus, it suffices to prove the Gevrey-modification of(7), namely

푘

′ ‖푁푘[푢1, … , 푢푘]‖푋휍0,푠,푏 −1 ≤ 푐푘 ∏‖푢푗‖푋휍0,푠,푏, 푗=1

푑 but this is immediate from the assumption (8) and the fact that for 휉1, … , 휉푘 ∈ ℝ ,

푘 ‖ ‖ 푒휍0‖휉1+⋯+휉푘‖ ≤ ∏ 푒휍0‖휉푗‖, 푗=1 where we simply used the triangle inequality.

Remark 2. Since (3) is invariant under time-translation (see Remark 1), it is seen that when LWP and LWP’ hold, then they hold more generally with the initial condition taken at 푡 = 푡0 for any 푡0, that is, 푢(푡 = 푡0) = 푢0.

3.2. Observation (B2). Again we start with a definition. By global well-posedness (GWP) of (3) for 퐻푠 data, we mean that LWP holds and that the solution extends globally in time: Nonlinear dispersive PDE 447

GWP. LWP holds, the solution 푢 extends globally in time, and for any 푇 > 0 we have 푢 ∈ 퐶([−푇, 푇]; 퐻푠). We define, analogously, the notion of global well-posedness in the Gevrey space as follows. GWP’. LWP’ holds, the solution 푢 extends globally in time, and for any 푇 > 0 there 휍(푇),푠 exists 휎(푇) ∈ (0, 휎0] such that 푢 ∈ 퐶([−푇, 푇]; 퐺 ). With these definitions, we now prove the following.

Proposition 3 (Observation (B2)). Let 푠 ∈ ℝ and 휎0 > 0. Assume that (8) holds 1 ′ for some 2 < 푏 < 푏 < 1, so that LWP and LWP’ hold by Propositions 1 and 2. If moreover GWP holds, then so does GWP’. Proof. We restrict to positive times (the argument for negative times is similar). Set 푋 = { 푇 > 0 ∶ there exists 휎(푇) > 0 such that 푢 ∈ 퐶([0, 푇]; 퐺휍(푇),푠) }. We need to show that 푋 = (0, ∞). First observe that LWP’ implies that 푋 is non- empty and open.1 It then only remains to prove that if (0, 푇) ⊂ 푋, then 푇 ∈ 푋. To this end, we first use the elementary inequality 푒휀푥 ≤ 1 + 휀푒푥 for all 휀 ∈ (0, 1] and 푥 ≥ 0, to get the key estimate

‖푢(푡)‖퐺휀휍,푠 ≤ ‖푢(푡)‖퐻푠 + 휀‖푢(푡)‖퐺휍,푠. (9) Now fix 푇 > 0 with (0, 푇) ⊂ 푋. From GWP we have

sup ‖푢(푡)‖퐻푠 ≤ 퐶푇 < ∞. 0≤푡≤푇

Set 훿 = 훿(퐶푇 + 1) (with 훿( ⋅ ) as in LWP and LWP’) and 푇1 = 푇 − 훿/2. Then 푇1 ∈ 푋, hence there exists 휎1 > 0 such that

sup ‖푢(푡)‖퐺휍1,푠 ≤ 퐴푇 < ∞. 0≤푡≤푇1 Applying (9) we then obtain, for any 휀 ∈ (0, 1],

sup ‖푢(푡)‖퐺휀휍1,푠 ≤ 퐶푇 + 휀퐴푇. 0≤푡≤푇1

Choosing 휀 so small that 휀퐴푇 ≤ 1, we now conclude from LWP’ (applied with 휀휍1,푠 initial condition at time 푇1) that 푢(푡) can be continued in 퐺 until the time 푇1 + 훿, which exceeds 푇. It follows that 푇 ∈ 푋, and this concludes the proof. 1See Remark 2. 448 S. Selberg

3.3. Observation (B3). As remarked, (B1) and (B2) are abstracted from arguments due to Bourgain, whereas (B3) is a new observation, to the best of our knowledge.

Proposition 4 (Observation (B3)). Let 푠 ∈ ℝ and 휎0 > 0. Assume that 1 ′ •(8) holds for some choice of 2 < 푏 < 푏 < 1, hence LWP and LWP’ hold, by Propositions 1 and 2.

푠 • The 퐻 norm is conserved, that is, ‖푢(푡)‖퐻푠 = ‖푢0‖퐻푠 for all 푡 ∈ ℝ. Then GWP’ holds and we have the lower bound

−퐴|푡| 휎(푡) ≥ 휎0푒 for all 푡 ∈ ℝ, where the constant 퐴 > 0 depends on 휎0, 푠 and 푢0. In fact, we can take

log 푐(‖푢 ‖ 휍 ,푠 + 1) 퐴 = 0 퐺 0 , 훿 (‖푢0‖퐺휍0,푠 + 1) where 훿( ⋅ ) and 푐 are as in LWP LWP’. Proof. First note that by the conservation assumption, GWP follows from LWP. Then GWP’ holds by Proposition 3. We now prove the lower bound, restricting to positive times 푡, without loss of generality. Set

푅 = ‖푢0‖퐺휍0,푠 + 1, 푀 = 푐푅, 훿 = 훿(푅), with 푐 and 훿( ⋅ ) as in LWP’. We will prove that, for all 푛 ∈ ℕ,

휎0 sup ‖푢(푡)‖퐺휍(푛훿),푠 ≤ 푀, where 휎(푛훿) = 푛−1 . (10) 0≤푡≤푛훿 푀 This implies the claimed lower bound on 휎(푡). Indeed, given 푡 > 0, choose 푛 ∈ ℕ so that (푛 − 1)훿 ≤ 푡 ≤ 푛훿. Then

−(푛−1) −푡/훿 휎(푡) ≥ 휎0푀 ≥ 휎0푀 , so writing 푀 = 푒 퐴훿 −퐴푡 we obtain 휎(푡) ≥ 휎0푒 , as desired. Now let us prove (10). The case 푛 = 1 holds by LWP’. Next, assuming that (10) holds for some 푛, we prove it for 푛 + 1. Applying (9) with 휀 = 1/푀 and using conservation of 퐻푠, we get

휍 ,푠 sup ‖푢(푡)‖퐺휀휍(푛훿),푠 ≤ ‖푢0‖퐻푠 + 휀푀 ≤ ‖푢0‖퐺 0 + 1. 0≤푡≤푛훿 Then LWP’ (with initial condition at time 푡 = 푛훿) implies (10) for 푛 + 1, noting that 휀휎(푛훿) = 휎((푛 + 1)훿). Nonlinear dispersive PDE 449

4. A further refinement of Bourgain’s method

Finally, we present a refinement of the idea behind (B3), yielding an improved lower bound. As in the proof of (B3), we apply LWP’ repeatedly to cover an arbitrarily large time interval [0, 푇] by moving in short time steps 훿. But instead of using the elementary inequality (9) to estimate the growth of ‖푢(푡)‖퐺휍,푠 in each time step, we use now an “almost conservation law”, which contains 휎 as a parameter and which reduces to the 퐻푠 conservation law in the limit 휎 → 0. To be precise, by almost conservation law (ACL), we mean the following property. ACL. For a given 푠 ∈ ℝ, there exist 휅, 휀 > 0 and a positive, increasing function 푝, such that with 훿 = 훿(‖푢0‖퐺휍0,푠) as in LWP’ we have the estimate

2 2 휀 2 sup‖푢(푡)‖퐺휍,푠 ≤ ‖푢0‖퐺휍,푠 + 휅휎 푝 (‖푢0‖퐺휍,푠) |푡|≤훿 for all 휎 ∈ (0, 휎0]. We can then prove the following refinement of (B3).

Theorem 4. Let 푠 ∈ ℝ. Assume that LWP’ holds for all 휎0 > 0. Assume further that ACL holds. Then GWP’ holds with

−1/휀 휎(푇) ≥ min (휎0, 푐푇 ) , where 푐 depends on 푠, 휎0 and 푢0.

휍0,푠 Proof. Fix 휎0 ∈ ℝ and 푢0 ∈ 퐺 . Regarding 휎 ∈ (0, 휎0] as a parameter, define 2 푁휍(푡) = ‖푢(푡)‖퐺휍,푠. Set 1/2 훿 = 훿 ([2푁휍0(0)] ) , with 훿( ⋅ ) as in LWP’. Now suppose that for given 휎 > 0 and 푡0 ≥ 0 we have

sup 푁휍(푡) ≤ 2푁휍0(0). 푡∈[0,푡0]

Then we can apply LWP’, with initial time 푡 = 푡0, to extend the solution to [푡0, 푡0+훿]. Moreover, by ACL,

휀 sup 푁휍(푡) ≤ 푁휍(푡0) + 휅휎 푝 (2푁휍0(0)) . 푡∈[푡0,푡0+훿] 450 S. Selberg

In this way, we cover time intervals [0, 훿], [훿, 2훿] etc., and obtain

휀 푁휍(훿) ≤ 푁휍(0) + 휅휎 푝 (2푁휍0(0)) , 푁 (2훿) ≤ 푁 (훿) + 휅휎휀푝 (2푁 (0)) ≤ 푁 (0) + 2휅휎휀푝 (2푁 (0)) , 휍 휍 휍0 휍 휍0 (11) … 휀 푁휍(푛훿) ≤ 푁휍(0) + 푛휅휎 푝 (2푁휍0(0)) . This continues as long as

휀 푛휅휎 푝 (2푁휍0(0)) ≤ 푁휍0(0),

since then the last line of (11) is bounded by 2푁휍0(0), so we can take one more step. Thus, the induction stops at the first integer 푛 for which

휀 푛휅휎 푝 (2푁휍0(0)) > 푁휍0(0), and then we have reached a final time

푇 = 푛훿, so 푇 휅휎휀푝 (2푁 (0)) > 푁 (0). 훿 휍0 휍0 This shows, firstly, that 푇 will be arbitrarily large for 휎 > 0 small enough. Moreover, it shows that 1/2 푁휍 (0) ⋅ 훿 ([2푁휍 (0)] ) 휎휀 > 0 0 , 푇휅푝 (2푁휍0(0)) proving 휎 > 푐푇−1/휀 as claimed.

5. Almost conservation law for KdV

To illustrate on a concrete example the general method developed in the preceding sections, we now briefly recall from [14] the key steps in the proof of the almost conservation law (ACL) for the KdV equation. Theorem 1 (from [14]) can then obtained as a consequence of Theorem 4. Nonlinear dispersive PDE 451

Let 푢 be the solution of the KdV equation (1), and set

푈 = 푒휍|퐷|푢, which is real-valued since 푢 is real-valued. We mimic the proof of the conservation of 2 2 ∫ 푢(푡, 푥) 푑푥 = ‖푢(푡)‖퐿2 ℝ to get an almost conservation of

2 푠 ∫ 푈(푡, 푥) 푑푥 = ‖푢(푡)‖퐺휍,0. ℝ The point of departure is the equation, implied by (1),

푈푡 + 푈푥푥푥 + 푈푈푥 = 퐹, where 1 퐹 = 휕 (푒휍|퐷|푢 ⋅ 푒휍|퐷|푢 − 푒휍|퐷|(푢 ⋅ 푢)) . 2 푥 Multiplying by 푈 and integrating yields

2 ∫ 푈푈푡 푑푥 + ∫ 푈푈푥푥푥 푑푥 + ∫ 푈 푈푥 푑푥 = ∫ 푈퐹 푑푥.

After an integration by parts this becomes

1 푑 1 1 ∫ 푈2 푑푥 − ∫ 휕 (푈 푈 ) 푑푥 + ∫ 휕 (푈3) 푑푥 = ∫ 푈퐹 푑푥. 2 푑푡 2 푥 푥 푥 3 푥 ℝ The second and third terms on the left vanish, and integrating in time we obtain

| | ‖푢(훿)‖2 ≤ ‖푢(0)‖2 + 2|∫ 휒 (푡) ⋅ 푈퐹 푑푥 푑푡|. 퐺휍,0 퐺휍,0 | [0,훿] |

By applying Parseval’s identity and Hölder’s inequality, we can estimate the integral on the right side by | | |∫ 휒[0,훿](푡) ⋅ 푈퐹 푑푥푑푡| ≤ ‖휒[0,훿](푡)푈‖푋0,1−푏‖휒[0,훿](푡)퐹‖푋0,푏−1 | ℝ2 |

≤ 퐶‖푈‖푋0,1−푏(훿)‖퐹‖푋0,푏−1(훿), where we applied Lemma 4 in the last step. Using now the crucial estimate

휀 2 ‖퐹‖푋0,푏−1 ≤ 푐휎 ‖푢‖푋휍,0,푏, (12) 452 S. Selberg

1 3 which holds for some 푏 ∈ ( 2 , 1) and all 휀 ∈ (0, 4 ), we obtain

2 2 휀 3 ‖푢(훿)‖퐺휍,0 ≤ ‖푢(0)‖퐺휍,0 + 푐휎 ‖푢‖푋휍,0,푏(훿).

But from the proof of LWP’ (see the appendix), we have ‖푢‖푋휍,0,푏(훿) ≤ 푐‖푢(0)‖퐺휍,0, hence we finally obtain the almost conservation law for KdV:

2 2 휀 3 ‖푢(훿)‖퐺휍,0 ≤ ‖푢(0)‖퐺휍,0 + 푐휎 ‖푢(0)‖퐺휍,0. Thus, Theorem 4 implies Theorem 1. The proof of the key estimate (12) can be found in [14]. It relies on the cancel- lation estimate

휀 푒휍|휉|푒휍|휂| − 푒휍|휉+휂| ≤ [2휎 min(|휉|, |휂|) 푒휍|휉|푒휍|휂| (휀 ∈ [0, 1], 휉, 휂 ∈ ℝ), and the bilinear estimate from Theorem 3.

Appendix: Proof of Proposition 1

(푛) (푛) (푛−1) To solve (3) we use the iteration scheme 푢푡 = 푖ℎ(퐷)푢 + 푁[푢 ] with initial (푛) (−1) (0) condition 푢 (0) = 푢0. Here 푛 ∈ ℕ0 and we set 푢 = 0, so 푢 (푡) = 푈(푡)푢0 is the homogeneous part and for 푛 ∈ ℕ we have

푡 푢(푛)(푡) = 푢(0)(푡) + ∫ 푈(푡 − 푡′)푁[푢(푛−1)(푡′)] 푑푡′. 0

We now prove, assuming (5) and (6) hold, that 푢(푛) is a Cauchy sequence in 푠,푏 푋 (훿) for 훿 = 훿(푅) > 0 sufficiently small, assuming ‖푢0‖퐻푠 ≤ 푅. We will use the fact that, by taking infimums over extensions, the estimates (5) and (6) hold also for the restricted spaces 푋푠,푏(훿). Set (푛) (푛) (푛−1) 퐴푛 = ‖푢 ‖푋푠,푏(훿), 퐵푛 = ‖푢 − 푢 ‖푋푠,푏(훿).

Note first that 퐴0 ≤ 푐푅 by Lemma 5. Moreover, if 퐴푛−1 ≤ 2푐푅 for some 푛 ∈ ℕ, then by Lemmas 3 and 5 and the estimate (5) we obtain

퐴 ≤ 푐푅 + 푐′훿푏′−푏‖푁[푢(푛−1)]‖ ≤ 푐푅 + 푐″훿푏′−푏푝(2푐푅)2푐푅, 푛 ‖ ‖푋푠,푏′−1(훿) so by induction we get 퐴푛 ≤ 2푐푅 for all 푛 provided 훿 > 0 is so small that

2푐″훿푏′−푏푝(2푐푅) ≤ 1. (13) Nonlinear dispersive PDE 453

Using (6) we then similarly obtain

′ 1 퐵 ≤ 푐″훿푏 −푏푝(4푐푅)퐵 ≤ 퐵 푛 푛−1 2 푛−1 provided 2푐″훿푏′−푏푝(4푐푅) ≤ 1. (14) Define 훿(푅) as the largest 훿 satisfying the two conditions (13) and (14). It now follows that 푢(푛) is a Cauchy sequence in 푋푠,푏(훿), hence it has a limit 푢 in that space, solving (3). Note that ‖푢‖푋푠,푏(훿) ≤ 2푐푅, and by Lemma 2 we have 푢 ∈ 푠 ′ 퐶([−훿, 훿]; 퐻 ) and sup푡∈[−훿,훿]‖푢(푡)‖퐻푠 ≤ 푐 푅. Finally, to prove uniqueness in 푋푠,푏(훿), assume 푢, 푣 both belong to that space and satisfy (3). Then for 0 < 휀 < 훿 we have by Lemmas 3 and 5 and the estimate (6), ″ 푏′−푏 ‖푢 − 푣‖푋푠,푏(휀) ≤ 푐 휀 푝 (‖푢‖푋푠,푏(훿) + ‖푣‖푋푠,푏(훿)) ‖푢 − 푣‖푋푠,푏(휀), so for 휀 > 0 small enough we get ‖푢 − 푣‖푋푠,푏(휀) = 0. Thus 푢(푡) = 푣(푡) for 푡 ∈ [−휀, 휀], and by a continuity argument it now follows that equality holds for all 푡 ∈ (−훿, 훿).

References

[1] Magnar Bjørkavåg and Henrik Kalisch, Exponential convergence of a spectral projection of the KdV equation, Phys. Lett. A 365 (2007), no. 4, 278–283. [2] Jerry L. Bona, Zoran Gruji, and Henrik Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 6, 783–797. MR 2172859 (2006e:35282) [3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299 [4] , Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209–262. MR 1215780 (95d:35160b) [5] , On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal. 3 (1993), no. 4, 315–341. MR 1223434 (94d:35142) [6] Marco Cappiello, Piero D’Ancona, and Fabio Nicola, On the radius of spatial analyticity for semilinear symmetric hyperbolic systems, J. Differential Equations 256 (2014), no. 7, 2603–2618. MR 3160455 [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal. 211 (2004), no. 1, 173–218. MR 2054622 (2005a:35241) 454 S. Selberg

[8] Piero D’Ancona, Damiano Foschi, and Sigmund Selberg, Local well-posedness below the charge norm for the Dirac-Klein-Gordon system in two space dimensions, J. Hyperbolic Differ. Equ. 4 (2007), no. 2, 295–330. MR 2329387 [9] Axel Grünrock and Hartmut Pecher, Global solutions for the Dirac-Klein-Gordon system in two space dimensions, Comm. Partial Differential Equations 35 (2010), no. 1, 89–112. MR 2748619 [10] Alex Himonas, Henrik Kalisch, and Sigmund Selberg, On persistence of spatial analyticity for the dispersion-generalized periodic KdV equation, Preprint 2016. [11] Carlos E. Kenig, Gustavo Ponce, and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573–603. MR 1329387 (96k:35159) [12] Shuji Machihara, Kenji Nakanishi, and Kotaro Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math. 50 (2010), no. 2, 403–451. MR 2666663 (2011d:35435) [13] Sigmund Selberg, On persistence of spatial analyticity for solutions of the Dirac-Klein- Gordon equations in two space dimensions, Preprint 2016. [14] Sigmund Selberg and Daniel Oliveira da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, To appear in Ann. Henri Poincaré. [15] Sigmund Selberg and Achenef Tesfahun, On the radius of spatial analyticity for the 1d Dirac-Klein-Gordon equations, Journal of Differential Equations 259 (2015), 4732–4744. [16] Catherine Sulem, Pierre-Louis Sulem, and Hélène Frisch, Tracing complex singularities with spectral methods, J. Comp. Phys. 50 (1983), no. 8, 138–161. [17] Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Math- ematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006, Local and global analysis. MR 2233925 (2008i:35211) Publications by Helge Holden

Theses

θ[1] Konvergens mot punkt-interaksjoner (In Norwegian), Cand. real. thesis, University of Oslo 1981

θ[2] Point interactions and the short-range expansion. A solvable model in quantum mechanics and its approximation, Dr. Philos. Dissertation, University of Oslo 1985

Books

β[1] Solvable Models in Quantum Mechanics Texts and Monographs in Physics Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo 1988, 452 pp., (with S. Albeverio, F. Gesztesy, R. Høegh-Krohn), Translation into the Russian, Mir, Moscow 1991, (Translated by Yu. A. Kuperin, K. A. Makarov, V. A. Geiler), Second edition with an Appendix by P. Exner, AMS Chelsea Publishing, volume 350, Chelsea Publishing, American Mathematical Society, Providence, 2005

β[2] Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach Birkhäuser Verlag, Basel, 1996, 231 pp., Second edition, Universitext, Springer-Verlag, 2010, 305 pp., (with J. Ubøe, B. Øksendal, T. Zhang)

β[3] Sturm–Liouville Operators and Hilbert Spaces: A Brief Introduction Tapir forlag, Trondheim, 2000, 90 pp., Second edition, 2001

β[4] Front Tracking for Hyperbolic Conservation Laws 456 Publications by Helge Holden

Applied Mathematical Sciences, volume 152, Springer-Verlag, New York, 2002, 380 pp., Second corrected printing, 2007, Softcover and eBook, 2011, Second edition (Hard- and softcover, eBook, “MyCopy”), 2015, 516 pp., (with N. H. Risebro)

β[5] Soliton Equations and Their Algebro-Geometric Solutions I: (1 + 1)-Dimensional Continuous Models Cambridge Studies in Advanced Mathematics, volume 79, Cambridge University Press, Cambridge, 2003, 530 pp., (with F. Gesztesy)

β[6] Soliton Equations and Their Algebro-Geometric Solutions Volume II: (ퟏ + ퟏ)-Dimensional Discrete Models Cambridge Studies in Advanced Mathematics, volume 114, Cambridge University Press, Cambridge, 2008, 452 pp., (with F. Gesztesy, J. Michor, and G. Teschl)

β[7] Operator Splitting for Nonlinear Partial Differential Equations with Rough Solutions Analysis and Matlab Programs EMS Series of Lectures in Mathematics, EMS Publishing House, Zurich, 2010, 226 pp., (with K. H. Karlsen, K.-A. Lie, N. H. Risebro)

Publications in international, refereed journals

[1] The spectrum of defect periodic point interactions, Letters in Mathematical Physics 7 (1983) 221–228, (with R. Høegh-Krohn, F. Martinelli)

[2] The short range expansion, Advances in Applied Mathematics 4 (1983) 402–421, (with R. Høegh-Krohn, S. Johannesen)

[3] On absence of diffusion near the bottom of the spectrum for arandom Schrödinger operator on 퐿2(퐑휈), Communications in Mathematical Physics 93 (1984) 197–217, (with F. Martinelli)

[4] The short-range expansion in solid state physics, Annales de l’Institut Henri Poincaré, Section A, Physique Théorique 41 (1984) 335–362, (with R. Høegh-Krohn, S. Johannesen) Publications by Helge Holden 457

[5] The short-range expansion for multiple well scattering theory, Journal of Mathematical Physics 26 (1985) 145–151, (with R. Høegh-Krohn, M. Mebkhout)

[6] The Fermi surface for point interactions, Journal of Mathematical Physics 27 (1986) 385–405, (with R. Høegh-Krohn, S. Johannesen, T. Wentzel-Larsen)

[7] On coupling constant thresholds in two dimensions, Journal of Operator Theory 14 (1985) 263–276

[8] A unified approach to eigenvalues and resonances of Schrödinger operators using Fredholm determinants, Journal of Mathematical Analysis and Applications 123 (1987) 181–198, Addendum 132 (1988) 309, (with F. Gesztesy)

[9] Point interactions in two dimensions. Basic properties, approximations and applications to solid state physics, Journal für die reine und angewandte Mathematik 380 (1987) 87–107, (with S. Albeverio, F. Gesztesy, R. Høegh-Krohn)

[10] Stochastic multiplicative measures, generalized Markov semigroups and group valued stochastic processes and fields, Journal of Functional Analysis 78 (1988) 154–184, (with S. Albeverio, R. Høegh-Krohn)

[11] On energy gaps in a new type of analytically solvable models in quantum mechanics, Journal of Mathematical Analysis and Applications 134 (1988) 9–29, (with F. Gesztesy, W. Kirsch)

[12] On the Riemann problem for a prototype of mixed type conservation law, Communications on Pure and Applied Mathematics 20 (1987) 229–264

[13] A new class of analytically solvable models in quantum mechanics on the line, Journal of Physics A: Mathematical and General 20 (1987) 5157–5177, (with F. Gesztesy)

[14] A numerical method for first order nonlinear scalar hyperbolic conservation laws in one dimension, Computers and Mathematics with Applications 15 (1988) 595–602, (with L. Holden, R. Høegh-Krohn) 458 Publications by Helge Holden

[15] A law of large numbers and a central limit theorem for the Schrödinger operator with zero range potentials, Journal of Statistical Physics 51 (1988) 206–214, (with R. Figari, A. Teta)

[16] Representation and construction of multiplicative noise, Journal of Functional Analysis 87 (1989) 250–272, (with S. Albeverio, R. Høegh-Krohn, T. Kolsrud)

[17] Trapping and cascading of eigenvalues in the large coupling limit, Communications in Mathematical Physics 118 (1988) 597–634, (with F. Gesztesy, D. Gurarie, M. Klaus, L. Sadun, B. Simon, P. Vogl)

[18] Construction of quantized Higgs-like fields in two dimensions, Physics Letters 222B (1989) 263–268, (with S. Albeverio, R. Høegh-Krohn, T. Kolsrud)

[19] A new front-tracking method for reservoir simulation, SPE Reservoir Engineering 7 (1992) 107–116, (with F. Bratvedt, K. Bratvedt, C. Buchholz, L. Holden, N. H. Risebro)

[20] Explicit construction of solutions of the modified Kadomtsev–Petviashvili equation, Journal of Functional Analysis 98 (1991) 211–228, (with F. Gesztesy, E. Saab, B. Simon)

[21] On the stochastic Buckley–Leverett equation, SIAM Journal of Applied Mathematics 51 (1991) 1472–1488, (with N. H. Risebro)

[22] On the Toda and Kac–van Moerbeke systems, Transactions of the American Mathematical Society 339 (1993) 849–868, (with F. Gesztesy, B. Simon, Z. Zhao)

[23] A method of fractional steps for scalar conservation laws without the CFL condition, Mathematics of Computation 60 (1993) 221–232, (with N. H. Risebro)

[24] Stochastic boundary value problems. A white noise functional approach, Probability Theory and Related Fields 95 (1993) 39–419, (with T. Lindstrøm, B. Øksendal, J. Ubøe, T.-S. Zhang)

[25] Discrete Wick calculus and stochastics functional equations, Potential Analysis 1 (1992) 291–306, (with T. Lindstrøm, B. Øksendal, J. Ubøe) Publications by Helge Holden 459

[26] Frontline and Frontsim; Two full scale, two-phase, black oil reservoir simulators based on front tracking, Surveys on Mathematics in Industry 3 (1993) 185–215, (with F. Bratvedt, K. Bratvedt, C. F. Buchholz, T. Gimse, L. Holden, N. H. Risebro)

[27] Comment on a recent note on the Schrödinger equation with a 훿′-interaction, Journal of Physics A: Mathematical and General 26 (1993) 3903–3904, (with S. Albeverio, F. Gesztesy)

[28] The Burgers equation with a noisy force, Communications in Partial Differential Equations 19 (1994) 119–142, (with T. Lindstrøm, B. Øksendal, J. Ubøe, T.-S. Zhang)

[29] Trace formulae and inverse scattering for Schrödinger operators, Bulletin of the American Mathematical Society 29 (1993) 250–255, (with F. Gesztesy, B. Simon, Z. Zhao)

[30] Trace formulas and conservation laws for nonlinear evolution equations, Reviews in Mathematical Physics 6 (1994) 51–95, Errata, ibid. 673, (with F. Gesztesy)

[31] A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis, 26 (1995) 999–1017, (with N. H. Risebro)

[32] The pressure equation for fluid flow in a stochastic medium, Potential Analysis 4 (1995) 655–674, (with T. Lindstrøm, B. Øksendal, J. Ubøe, T.-S. Zhang)

[33] Maximum principles for a class of conservation laws, SIAM Journal of Applied Mathematics 55 (1995) 651–661, (with N. H. Risebro, A. Tveito)

[34] Absolute summability of the trace relation for certain Schrödinger operators, Communications in Mathematical Physics 168 (1995) 137–168, (with F. Gesztesy, B. Simon)

[35] Higher order trace relations for Schrödinger operators, Reviews in Mathematical Physics 7 (1995) 893–922, (with F. Gesztesy, B. Simon, Z. Zhao)

[36] Conservation laws with a random source, Applied Mathematics & Optimization 36 (1997) 229–241, (with N. H. Risebro) 460 Publications by Helge Holden

[37] Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac–van Moerbeke hierarchy, Memoirs of the American Mathematical Society 135 (1998), no. 641, (with W. Bulla, F. Gesztesy, G. Teschl)

[38] Finite difference approximation of the pressure equation for fluid flowina stochastic medium, Communications in Partial Differential Equations 21 (1996) 1367–1388, (with Y. Hu)

[39] A trace formula for multidimensional Schrödinger operators, Journal of Functional Analysis 141 (1996) 449–465, (with F. Gesztesy, B. Simon, Z. Zhao)

[40] Riemann problems with a kink, SIAM Journal of Mathematical Analysis 30 (1999) 497–515, (with N. H. Risebro)

[41] An unconditionally stable method for the Euler equations, Journal of Computational Physics 150 (1999) 76–96, (with K.-A. Lie, N. H. Risebro)

[42] Unconditionally stable splitting methods for the shallow water equations, BIT Numerical Mathematics 39 (1999) 451–472, (with R. Holdahl, K.-A. Lie)

[43] Dubrovin equations and integrable systems on hyperelliptic curves, Mathematica Scandinavica, 91 (2002) 91–126, (with F. Gesztesy)

[44] Operator splitting methods for generalized Korteweg–de Vries equations, Journal of Computational Physics 153 (1999) 203–222, (with K. H. Karlsen, N. H. Risebro)

[45] The classical Boussinesq hierarchy revisited, Det Kongelige Norske Videnskabers Selskabs Skrifter, (Transactions of the Royal Norwegian Society of Sciences and Letters) 1 (2000), (with F. Gesztesy)

[46] Darboux-type transformations and hyperelliptic curves, Journal für die reine und angewandte Mathematik, 527 (2000) 151–183, (with F. Gesztesy)

[47] Borg-type theorems for matrix-valued Schrödinger operators, Journal of Differential Equations 167 (2000) 181–210, (with S. Clark, F. Gesztesy, B. Levitan)

[48] The Riemann problem for an elastic string with a linear Hooke’s law, Quarterly of Applied Mathematics 60 (2002) 695–705, (with H. Hanche-Olsen, N. H. Risebro) Publications by Helge Holden 461

[49] Operator splitting methods for degenerate convection-diffusion equations II:, Numerical examples with emphasis on reservoir simulation and sedimentation, Computational Geosciences 4 (2000) 287–322, (with K. H. Karlsen, K.-A. Lie)

[50] Algebro-geometric solutions of Camassa–Holm hierarchy, Revista Matemática Iberoamericana 19 (2003) 73–142, (with F. Gesztesy)

[51] Real-valued algebro-geometric solutions of the Camassa–Holm hierarchy, Philosophical Transactions of the Royal Society (London) A 366 (2008) 1025–1054, (with F. Gesztesy)

[52] The hyperelliptic 휁-function and the integrable massive Thirring equation, Proceedings of the Royal Society (London) 459A (2003) 1581–1610, (with J. C. Eilbeck and V. Z. Enolskii)

[53] On uniqueness and existence of entropy solutions of weakly coupled systems of nonlinear degenerate parabolic systems, Electronic Journal of Differential Equations, 2003 (2003), no. 46, 1–31, (with K. H. Karlsen and N. H. Risebro)

[54] Spectral analysis of Darboux transformations for the focusing NLS hierarchy, Journal d’Analyse Mathématique 93 (2004) 139–197, (with R. C. Cascaval, F. Gesztesy, and Y. Latushkin)

[55] Stability of solutions of quasilinear parabolic equations, Journal of Mathematical Analysis and Applications 308 (2005) 221–239, (with G. M. Coclite)

[56] Algebro-geometric solutions of a discrete system related to the trigonometric moment problem, Communications in Mathematical Physics 258 (2005) 149–177, (with J. Geronimo, F. Gesztesy)

[57] Convergence of a finite difference scheme for the Camassa–Holm equation, SIAM Journal of Numerical Analysis 44 (2006) 1655–1680, (with X. Raynaud)

[58] Contract adjustment under uncertainty, Journal of Economic Dynamics and Control 34 (2010) 657–680, (with L. and S. Holden) 462 Publications by Helge Holden

[59] Wellposedness for a parabolic-elliptic system, Discrete and Continuous Dynamical Systems 13 (2005) 659–682, (with G. M. Coclite and K. H. Karlsen)

[60] Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM Journal of Mathematical Analysis 37 (2005) 1044–1069, (with G. M. Coclite and K. H. Karlsen)

[61] A convergent numerical scheme for the Camassa–Holm equation based on multipeakons, Discrete and Continuous Dynamical Systems 14 (2006) 505–523, (with X. Raynaud)

[62] Convergent difference schemes for the Hunter–Saxton equation, Mathematics of Computation 76 (2007) 699–744, (with K. H. Karlsen and N. H. Risebro)

[63] The Schrödinger–Maxwell system with Dirac mass, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 24 (2007) 773–793, Erratum 25 (2008) 833–836, (with G. M. Coclite)

[64] Global conservative solutions of the Camassa–Holm equation—a Lagrangian point of view, Communications in Partial Differential Equations 32 (2007) 1511–1549, (with X. Raynaud)

[65] The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data, Revista Matemática Iberoamericana 24 (2008) 117–182, (with F. Gesztesy and G. Teschl)

[66] Global conservative multipeakon solutions of the Camassa–Holm equation, Journal of Hyperbolic Differential Equations 4 (2007) 39–64, (with X. Raynaud)

[67] Global conservative solutions of the generalized hyperelastic-rod wave equation, Journal of Differential Equations 233 (2007) 448–484, (with X. Raynaud)

[68] Local conservation laws and the Hamiltonian formalism for the Toda hierarchy revisited, Det Kongelige Norske Videnskabers Selskabs Skrifter, (Transactions of the Royal Norwegian Society of Sciences and Letters) 2006(3) 1–30, (with F. Gesztesy) Publications by Helge Holden 463

[69] Periodic conservative solutions of the Camassa–Holm equation, Annales de l’Institut Fourier (Grenoble) 58 (2008) 945–988, (with X. Raynaud)

[70] Well-posedness of higher-order Camassa–Holm equations, Journal of Differential Equations 246 (2009) 929–963, (with G. M. Coclite and K. H. Karlsen)

[71] Optimal rebalancing of portfolios with transaction costs, Stochastics 85 (2013) 371–394, doi:10.1080/17442508.2011.651219, (with L. Holden)

[72] Algebro-geometric finite-band solutions of the Ablowitz–Ladik hierarchy, International Mathematics Research Notices 2007, Article ID rnm082, 55 pp., (with F. Gesztesy, J. Michor, G. Teschl)

[73] The algebro-geometric initial value problem for the Ablowitz–Ladik hierarchy, Discrete and Continuous Dynamical Systems 26 (2010) 151–196, (with F. Gesztesy, J. Michor, G. Teschl)

[74] A convergent finite difference method for a nonlinear variational wave equation, IMA Journal of Numerical Analysis 29 (2009) 539–572, (with K. H. Karlsen and N. H. Risebro)

[75] The solution of the Cauchy problem with large data for a model of a mixture of gases, Journal of Hyperbolic Differential Equations 6 (2009) 25–106, (with N. H. Risebro and H. Sande)

[76] Local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik hierarchy, Studies in Applied Mathematics 120 (2008) 361–423, (with F. Gesztesy, J. Michor, G. Teschl)

[77] Dissipative solutions for the Camassa–Holm equation, Discrete and Continuous Dynamical Systems 24 (2009) 1047–1112, (with X. Raynaud)

[78] Ground states of the Schrödinger–Maxwell system with Dirac mass: Existence and asymptotics, Discrete and Continuous Dynamical Systems, Series A 27 (2010) 117–132, (with G. M. Coclite)

[79] Global dissipative multipeakon solutions for the Camassa–Holm equation, Communications in Partial Differential Equations 33 (2008) 2040–2063, (with X. Raynaud) 464 Publications by Helge Holden

[80] Front tracking for a model of immiscible gas flow with large data, BIT Numerical Mathematics 50 (2010) 331–376, (with N. H. Risebro and H. Sande)

[81] Symmetric waves are traveling waves, International Mathematics Research Notices 2009, Article ID rnp100, 19 pp., doi:10.1093/imrn/rnp100, (with M. Ehrnström and X. Raynaud)

[82] Zero diffusion-dispersion-smoothing limits for a scalar conservation law with discontinuous flux function, International Journal of Differential Equations 2009 (2009), Article ID 279818, pp. 33, doi:10.1155/2009/279818, (with K. H. Karlsen and D. Mitrovic)

[83] Lipschitz metric for the Hunter–Saxton equation, Journal de Mathématiques Pures et Appliquées 94 (2010) 68–92, (with A. Bressan and X. Raynaud)

[84] The Kolmogorov–Riesz compactness theorem, Expositiones Mathematicae 28 (2010) 385–394, Addendum, ibid. 34 (2016) 243–245, doi:10.1016/j.exmath.2015.12.003, (with H. Hanche-Olsen)

[85] Operator splitting for the KdV equation, Mathematics of Computation 80 (2011) 821–846, (with K. H. Karlsen, N. H. Risebro, and T. Tao)

[86] Global semigroup of conservative solutions of the nonlinear variational wave equation, Archive for Rational Mechanics and Analysis 201 (2011) 871–964, (with X. Raynaud)

[87] Strong compactness of approximated solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function, Acta Mathematica Scientia 29B (2009) 1573–1612, (with K. H. Karlsen, D. Mitrovic, and E. Yu. Panov)

[88] Lipschitz metric for the periodic Camassa–Holm equation, Journal of Differential Equations 250 (2011) 1460–1492, (with K. Grunert and X. Raynaud)

[89] 퐿∞ solutions for a model of polytropic gas flow with diffusive entropy, SIAM Journal of Mathematical Analysis 43 (2011) 2253–2274, (with H. Frid and K. H. Karlsen) Publications by Helge Holden 465

[90] The damped string problem revisited, Journal of Differential Equations 251 (2011) 1086–1127, (with F. Gesztesy)

[91] Lipschitz metric for the Camassa–Holm equation on the line, Discrete and Continuous Dynamical Systems, Series A 33 (2013) 2809–2827, (with K. Grunert and X. Raynaud)

[92] Abstract wave equations and associated Dirac-type operators, Annali di Matematica Pura ed Applicata 191 (2012) 631–676, (with F. Gesztesy, J. M. Goldstein, and G. Teschl)

[93] Operator splitting for two-dimensional incompressible fluid equations, Mathematics of Computation 82 (2013) 719–748, (with K. H. Karlsen and T. Karper)

[94] Operator splitting for partial differential equations with Burgers nonlinearity, Mathematics of Computation, 82 (2013) 173–185, (with C. Lubich and N. H. Risebro)

[95] Global conservative solutions of the Camassa–Holm equation for initial data, with nonvanishing asymptotics, Discrete and Continuous Dynamical Systems, Series A 32 (2012) 4209–4227, (with K. Grunert and X. Raynaud)

[96] Global solutions for the two-component Camassa–Holm system, Communications in Partial Differential Equations 37 (2012) 2245–2271, (with K. Grunert and X. Raynaud)

[97] Operator splitting for well-posed active scalar equations, SIAM Journal of Mathematical Analysis 45 (2013) 152–180, (with K. H. Karlsen and T. Karper)

[98] Convergence of a fully discrete finite difference scheme for the Korteweg–de Vries equation, IMA Journal of Numerical Analysis 35 (2015) 1047–1077, doi:10.1093/imanum/dru040, (with U. Koley and N. H. Risebro)

[99] On the inverse problem for scalar conservation laws, Inverse Problems 30 (2014) 035015 (35 pp.), (with F. S. Priuli and N. H. Risebro)

[100] Global dissipative solutions of the two-component Camassa–Holm system, for initial data with nonvanishing asymptotics, Nonlinear Analysis: Real World Applications 17 (2014) 203–244, (with K. Grunert and X. Raynaud) 466 Publications by Helge Holden

[101] A continuous interpolation between conservative and dissipative solutions for the two-component Camassa–Holm system, Forum of Mathematics, Sigma, (2015) vol. 3, e1, 73 pp., doi:10.1017/fms.2014.29, (with K. Grunert and X. Raynaud)

[102] On factorizations of analytic operator-valued functions and eigenvalue multiplicity questions, Integral Equations and Operator Theory 82 (2015) 61–94, doi:10.1007/s00020-014-2200-7, Erratum loc. sit. 85 (2016) 301–302, doi:10.1007/s00020-016-2290-5, (with F. Gesztesy and R. Nichols)

[103] On the Braess paradox with nonlinear dynamics and control theory, Journal of Optimization Theory and Applications 168 (2016) 216–230, doi:10.1007/s10957-015-0729-5, (with R. Colombo)

[104] Convergence of finite difference schemes for the Benjamin–Ono equation, Numerische Mathematik 134 (2016) 249–274, doi:10.1007/s00211-015-0778-6, (with R. Dutta, U. Koley, N. H. Risebro)

[105] The general peakon-antipeakon solution for the Camassa–Holm equation, Journal of Hyperbolic Differential Equations 13 (2016) 353–380, (with K. Grunert)

[106] Operator splitting for the Benjamin–Ono equation, Journal of Differential Equations 259 (2015) 6694–6717, (with R. Dutta, U. Koley, N. H. Risebro)

[107] Isentropic fluid dynamics in a curved pipe, Zeitschrift für angewandte Mathematik und Physik, 67:131 (2016) 10 pp., doi:10.1007/s00033-016-0725-0, (with R. Colombo)

[108] Real-valued algebro-geometric solutions of the two-component Camassa–Holm hierarchy, Annales de l’Institut Fourier (Grenoble) 67 (2017) 1185–1230, doi:10.1007/s00033-016-0725-0, (with J. Eckhardt, F. Gesztesy, A. Kostenko, G.Teschl)

[109] Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh 푀-functions, and a generalized index of unbounded meromorphic operator-valued functions, Journal of Differential Equations 261 (2016) 3551–3587, doi:10.1016/j.de.2016.05-033, (with J. Behrndt, F. Gesztesy, R. Nichols)

[110] A Lipschitz metric for the Hunter–Saxton equation, arXiv:1612.02961v1, submitted, (with J. A. Carrillo, K. Grunert) Publications by Helge Holden 467

[111] Continuum limit of Follow-the-Leader models — a short proof, Discrete and Continuous Dynamical Systems 38(2) (2018) 715–722 doi:10.3934/dcds.2018031, (with N. H. Risebro)

[112] An improvement of the Kolmogorov–Riesz compactness theorem, Expositiones Mathematicae, to appear, doi:10.1016/j.exmath.2018.03.002, (with H. Hanche-Olsen and E. Malinnikova)

[113] Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill–Whitham–Richards model for traffic flow, Networks & Heterogeneous Media, to appear, (with N. H. Risebro)

Publications in proceedings of conferences

π[1] On absence of diffusion for low energy for a random Schrödinger operator on 퐿2(퐑휈), Physica 124A (1984) 413–418, (with F. Martinelli)

π[2] Some exactly solvable models in quantum mechanics and the low energy expansion, In Proceedings of the Second International Conference on Operator Algebras, Ideals,, and Their Applications in Theoretical Physics, Leipzig 1983, Edited by H. Baumgärtel, G. Laßner, A. Pietsch, A. Uhlmann, Teubner, Leipzig 1984, pp. 12–28, (with S. Albeverio, F. Gesztesy, R. Høegh-Krohn)

π[3] Lifshitz singularity of the integrated density of states and absence of diffusion near the bottom of the spectrum for a random Hamiltonian, In Chaotic Behavior in Quantum Systems: Theory and Applications, Edited by G. Casati, Plenum Press, New York-London 1985, pp. 77–83, (with F. Martinelli)

π[4] Markov cosurfaces and gauge fields, In Stochastic Methods and Computer Techniques in Quantum Dynamics, Acta Physica Austriaca, Supplementum XXVI, Edited by H. Mitter, L. Pittner, Springer-Verlag, Wien-New York 1984, pp. 211–231, (with S. Albeverio, R. Høegh-Krohn)

π[5] Markov processes on infinite dimensional spaces, Markov fields and Markov cosurfaces, In Stochastic Space-Time Models and Limit Theorems, Edited by L. Arnold, P. Kotelenez, Reidel, Dordrecht-Boston-Lancaster 1984, pp. 11–40, (with S. Albeverio, R. Høegh-Krohn) 468 Publications by Helge Holden

π[6] Stochastic Lie group-valued measures and their relations to stochastic curve integrals, gauge fields and Markov cosurfaces, In Stochastic Processes — Mathematics and Physics, Proceedings Bielefeld 1984, Edited by S. Albeverio, P. Blanchard, L. Streit, Lecture Notes in Mathematics, Volume 1158, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo 1986, pp. 1–24, (with S. Albeverio, R. Høegh-Krohn)

π[7] Random fields with values in Lie groups and Higgs fields,In Stochastic Processes in Classical and Quantum Systems. Proceedings,, Ascona, Switzerland 1985, Edited by S. Albeverio, G. Casati, D. Merlini, Lecture Notes in Physics, Volume 262, Springer-Verlag, Berlin-Heidelberg-New York 1986, pp. 1–13, (with S. Albeverio, R. Høegh-Krohn)

π[8] The Schrödinger operator for a particle in a solid with deterministic and, stochastic point interactions, In Schrödinger Operators, Aarhus 1985, Edited by E. Balslev, Lecture Notes in Mathematics, Volume 1218, Springer-Verlag, Berlin-Heidelberg-New York 1986, pp. 1–38, (with S. Albeverio, F. Gesztesy, R. Høegh-Krohn, W. Kirsch)

π[9] On some recent results for conservation laws in one dimension, In Recent Developments in Mathematical Physics, Edited by H. Mitter, L. Pittner, Springer Proceedings in Physics, Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo 1987, pp. 240–244

π[10] On the Riemann problem for a prototype of mixed type conservation law. II, In Current Progress in Hyperbolic Systems: Riemann Problems and Computations, Contemporary Mathematics, Volume 100, Edited by W. B. Lindquist, American Mathematical Society, Providence 1989, pp. 331–367, (with L. Holden)

π[11] A remark on the formation of crystals at zero temperature, In Stochastic Methods in Mathematical Physics., Proceedings of the XXIV Karpacz Winter School on Theoretical, Physics, Karpacz, Poland, Edited by R. Gielerak, W. Karwowski, World Scientific, Singapore-New Jersey-London-Hong Kong 1989, pp. 211–220, (with S. Albeverio, R. Høegh-Krohn, T. Kolsrud, M. Mebkhout) Publications by Helge Holden 469

π[12] Some recent results for an explicit conservation law in one dimension, In Nonlinear Hyperbolic Equations - Theory, Numerical Methods and Applications, Proceedings of the Second International Conference on Hyperbolic, Problems, Aachen, 1988, Edited by J. Ballmann, R. Jeltsch, Notes on Numerical Fluid Mechanics 24 (1989) 238–245, Vieweg, Braunschweig, (with L. Holden)

π[13] A covariant Feynman-Kac formula for unitary bundles over Euclidean space, In Stochastic Partial Differential Equations and Applications II. Proceedings, Trento 1988, Edited by G. Da Prato, L. Tubaro, Lecture Notes in Mathematics, Volume 1390, Springer-Verlag, Berlin-Heidelberg-New York 1989, pp. 1–12, (with S. Albeverio, R. Høegh-Krohn, T. Kolsrud)

π[14] Point interaction Hamiltonians for crystals with random defects, In Applications of Self-Adjoint Extensions in Quantum Physics,, Proceedings, Dubna, USSR, 1987, Edited by P. Exner, P. S̆eba, Lecture Notes in Physics, Volume 324, Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo 1989, pp. 87–99, (with S. Albeverio, R. Figari, F. Gesztesy, R. Høegh-Krohn, W. Kirsch)

π[15] On point interactions in magnetic field systems, In Schrödinger Operators, Standard and Non-Standard, Edited by P. Exner, P. S̆eba, World Scientific, Singapore-New Jersey-London-Hong Kong, 1989, pp. 147–164, (with F. Gesztesy, P. S̆eba)

π[16] Some qualitative properties of 2 × 2 systems of conservations laws of mixed type, In Nonlinear Evolution Equations, Edited by B.L. Keyfitz, M. Shearer, The IMA Volumes in Mathematics and Its Applications, Volume 27, Springer-Verlag, New York-Berlin-Heidelberg-Vienna-Paris-Tokyo 1990, pp. 67–78, (with L. Holden, N. H. Risebro)

π[17] A stochastic approach to conservation laws, In Third International Conference on Hyperbolic Problems., Theory, Numerical Methods and Applications, Uppsala, 1990, Edited by B. Engquist, B. Gustafsson, Studentlitteratur/Chartwell-Bratt, Lund-Bromley 1991, pp. 575–587, (with N. H. Risebro) 470 Publications by Helge Holden

π[18] A new representation of soliton solutions of the Kadomtsev–Petviashvili equation, In Ideas and Methods in Mathematical Analysis, Stochastics,, and Applications. In Memory of Raphael Høegh-Krohn (1938-1988), Edited by S. Albeverio, J. E. Fenstad, H. Holden, T. Lindstrøm, Cambridge University Press, Cambridge, 1992, pp. 472–479, (with F. Gesztesy)

π[19] First order nonlinear scalar hyperbolic conservations laws in one dimension, In Ideas and Methods in Mathematical Analysis, Stochastics, and Applications. In Memory of Raphael Høegh-Krohn (1938-1988), Edited by S. Albeverio, J. E. Fenstad, H. Holden, T. Lindstrøm, Cambridge University Press, Cambridge, 1992, pp. 480–510, (with L. Holden)

π[20] Front tracking for petroleum reservoirs, In Ideas and Methods in Mathematical Analysis, Stochastics, and Applications. In Memory of Raphael Høegh-Krohn (1938-1988), Edited by S. Albeverio, J. E. Fenstad, H. Holden, T. Lindstrøm, Cambridge University Press, Cambridge, 1992, pp. 409–427, (with F. Bratvedt, K. Bratvedt, C. F. Buchholz, T. Gimse, L. Holden, N. H. Risebro)

π[21] Front tracking for groundwater simulations, In Computational Methods in Water Resources IX., Vol. 1: Numerical Methods in Water Resources, Edited by T. F. Russell, R. E. Ewing, C. A. Brebbia, W. G. Gray, G. F. Pinder, Elsevier Applied Science, London–New York, 1992, pp. 97–104, (with F. Bratvedt, K. Bratvedt, C. F. Buchholz, T. Gimse, N. H. Risebro)

π[22] The Wick product, In Frontiers in Pure and Applied Probability, Volume I, Edited by H. Niemi, G. Högnas, A. N. Shiryaev, A. Melnikov, VSP and TVP Science Publishers, Utrecht/Moscow, 1993, pp. 29–67, (with H. Gjessing, T. Lindstrøm, B. Øksendal, J. Ubøe, T.-S. Zhang)

π[23] A review of stochastic methods applied to reservoir evaluation, In Stochastic Processes, Physics and Geometry II, Edited by S. Albeverio, U. Cattaneo, D. Merlini, World Scientific, Singapore, 1995, pp. 364–388, (with L. Holden)

π[24] Low temperature expansions around classical crystalline ground states, In Stochastic Processes, Physics and Geometry II, Edited by S. Albeverio, U. Cattaneo, D. Merlini, World Scientific, Singapore, 1995, pp. 29–38, (with S. Albeverio, R. Gielerak, T. Kolsrud, M. Mebkhout) Publications by Helge Holden 471

π[25] Three-dimensional reservoir simulation based on front tracking, In North Sea Oil and Gas Reservoirs III, Edited by J. O. Aasen, E. Berg, A. T. Buller, O. Hjelmeland, R. M. Holt, J. Kleppe, O. Torsæter, Kluwer, Dordrecht, 1993, pp. 247–257, (with F. Bratvedt, K. Bratvedt, C. F. Buchholz, T. Gimse, L. Holden, R. Olufsen, N. H. Risebro)

π[26] A mathematical model of traffic flow on a network of roads, In Nonlinear Hyperbolic Equations — Theory, Numerical Methods and Applications, Proceedings of the Fourth International Conference on Hyperbolic Problems, Taormina, 1992, Edited by A. Donato, F. Oliveri, Notes on Numerical Fluid Mechanics 43 (1993) 329–335, (with N. H. Risebro)

π[27] Recent results for conservation laws — theory, numerics and applications, In Industrial Mathematics Week, Trondheim August 1992. Proceedings, Department of Mathematical Sciences, NTH, 1993, pp. 131–144, (with T. Gimse, N. H. Risebro)

π[28] Discrete Wick products, In Stochastic Analysis and Related Topics, Edited by T. Lindstrøm, B. Øksendal, A. S. Üstünel, Stochastic Monographs, Volume 8, Gordon & Breach Science Publ., Amsterdam, 1993, pp. 123–148, (with T. Lindstrøm, B. Øksendal, J. Ubøe)

π[29] A comparison experiment for Wick multiplication and ordinary multiplication, In Stochastic Analysis and Related Topics, Edited by T. Lindstrøm, B. Øksendal, A. S. Üstünel, Stochastic Monographs, Volume 8, Gordon & Breach Science Publ., Amsterdam, 1993, pp. 149–160, (with T. Lindstrøm, B. Øksendal, J. Ubøe, T. Zhang)

π[30] An equation modelling transport of a substance in a stochastic medium, In Seminar on Stochastic Analysis, Random Fields and Applications, Edited by E. Bolthausen, M. Dozzi, and F. Russo, Birkhäuser, Basel, 1995, pp. 123–134, (with J. Gjerde, B. Øksendal, J. Ubøe, T. Zhang)

π[31] On new trace formulae for Schrödinger operators, In Acta Applicandae Mathematicae 39 (1995) 315–333, (with F. Gesztesy)

π[32] The stochastic Wick-type Burgers equation, In Stochastic Partial Differential Equations (Edinburgh, 1994), London Mathematical Society Lecture Notes Series, Vol. 216, Edited by A. Etheridge, Cambridge University Press, Cambridge, 1995, pp. 141–161, (with T. Lindstrøm, B. Øksendal, J. Ubøe, T.-S. Zhang) 472 Publications by Helge Holden

π[33] Reservoir simulation by front tracking, In Hyperbolic problems: Theory, Numerics, Applications, Edited by J. Glimm, J. W. Grove, M. J. Graham, B. J. Plohr, World Scientific, Singapore, 1996, pp. 52–62, (with T. Gimse, N. H. Risebro)

π[34] On trace formulas for Schrödinger-type operators, In Multiparticle Quantum Scattering with Applications to Nuclear,, Atomic and Molecular Physics, Edited by D. G. Truhlar, B. Simon, IMA Volumes in Mathematics and its Applications, Springer, New York, pp. 121–145, (with F. Gesztesy)

π[35] Systems of conservation laws on networks — a model for traffic flow, Zeitschrift für Angewandte Mathematik und Mechanik 76 (1996) Suppl. 3, 295–298, (with N. H. Risebro, T. With Martinsen)

π[36] A white noise approach to stochastic differential equations driven by Wiener and Poisson processes, In Nonlinear Theory of Generalized Functions, Editors M. Grosser, G. Hörmann, M. Kunzinger, M. Oberguggenberger, Chapman&Hall/CRC, Boca Raton, 1999, pp. 293–314, (with B. Øksendal)

π[37] The Cole–Hopf and Miura transformations revisited, In Mathematical Physics and Stochastic Analysis. Essays in Honour of Ludwig Streit, Editors S. Albeverio, Ph. Blanchard, L. Ferreira, T. Hida, Y. Kondratiev, and R. Vilela Mendes, World Scientific, Singapore, 2000, pp. 198–214, (with F. Gesztesy)

π[38] Operator splitting methods for degenerate convection–diffusion equations I: Convergence and Entropy Estimates, In Stochastic Processes, Physics and Geometry: New Interplays. II., A Volume in Honor of Sergio Albeverio, Editors F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. Röckner, S. Scarlatti, CMS Conference Proceedings, Volume 29, Canadian Mathematical Society, Providence (USA), 2000, pp. 293–316, (with K. H. Karlsen, K.-A. Lie)

π[39] Operator splitting methods for convection–dominated nonlinear partial differential equations, In Godunov Methods. Theory and Applications, Editor E. F. Toro, Kluwer Academic Press/Plenum Publishers, 2001, pp. 469–475, (with K. H. Karlsen, K.-A. Lie, N. H. Risebro)

π[40] A white noise approach to stochastic Neumann boundary value problems, In Acta Applicandae Mathematicae 63 (2000) 141–150, (with B. Øksendal) Publications by Helge Holden 473

π[41] A combined sine-Gordon and modified Korteweg–de Vries hierarchy and its algebro-geometric solutions, In Differential Equations and Mathematical Physics, Editors R. Weikard, G. Weinstein, AMS/IP Studies in Advanced Mathematics, Vol. 16, American Mathematical Society/International Press, Providence, 2000, pp. 133–173, (with F. Gesztesy)

π[42] The classical massive Thirring system revisited, In Stochastic Processes, Physics and Geometry: New Interplays. I., A Volume in Honor of Sergio Albeverio, Editors F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. Röckner, S. Scarlatti, CMS Conference Proceedings, Volume 28, Canadian Mathematical Society, Providence (USA), 2000, pp. 163-200, (with V. Z. Enolskii, F. Gesztesy)

π[43] On the Camassa–Holm and the Hunter–Saxton equations, In European Conference of Mathematics. Stockholm, June 27–July 2, 2004, Editor A. Laptev, European Mathematical Society, Zurich, 2005, pp. 173–200.

π[44] Algebro-geometric solutions of the KdV and Camassa–Holm equation, Oberwolfach Reports 1 (2004), pp. 275–279, Editors A. Constantin, J. Escher, European Publishing House, Zürich, (with F. Gesztesy)

π[45] Global weak solutions for a shallow water equation, In Hyperbolic Problems: Theory, Numerics, Applications, Editors S. Benzoni-Gavage, D. Serre, Springer, Heidelberg, 2008, pp. 389–396, (with G. M. Coclite and K. H. Karlsen)

π[46] A numerical scheme based on multipeakons for conservative solutions of the Camassa–Holm equation, In Hyperbolic Problems: Theory, Numerics, Applications, Editors S. Benzoni-Gavage, D. Serre, Springer, Heidelberg, 2008, pp. 873–882, (with X. Raynaud)

π[47] The Ablowitz–Ladik hierarchy revisited, In Methods of Spectral Analysis in Mathematical Physics, Operator Theory: Advances and Applications. Vol. 186, 2009, pp. 139–190, Editors J. Janas, P. Kurasov, A. Laptev, S. Naboko, G. Stolz, (with F. Gesztesy, J. Michor, G. Teschl)

π[48] Convergence of front tracking and the Glimm scheme for a model of the flow of immiscible gases, In Hyperbolic Problems: Theory, Numerics and Applications. Part 2, Editors E. Tadmor, J.-G. Liu, A. Tzavaras, American Mathematical Society, Proc. of Symposia in Applied Mathematics, vol. 67.2, 2009, pp. 653–662, (with H. Sande and N. H. Risebro) 474 Publications by Helge Holden

π[49] Periodic conservative solutions for the two-component Camassa–Holm system, In Spectral Analysis, Differential Equations and Mathematical Physics, A Festschrift for Fritz Gesztesy on the Occasion of his 60th Birthday, Editors H. Holden, B. Simon, and G. Teschl, American Mathematical Society, Proceedings of Symposia in Mathematics, Vol. 87, 2013, pp. 165–182, (with K. Grunert and X. Raynaud)

π[50] Lipschitz metric for the two-component Camassa–Holm system, In Hyperbolic Problems: Theory, Numerics, Applications, Editors F. Ancona, A. Bressan, P. Marcati, A. Marson, American Institute for Mathematical Sciences, Series on Applied Mathematics, Vol. 8, 2014, pp. 193–207, (with K. Grunert and X. Raynaud)

π[51] On the index of meromorphic operator-valued functions and some applications, In Functional Analysis and Operator Theory for Quantum Physics, Editors J. Dittrich, H. Kovařık, and A. Laptev, EMS Publishing House, Zurich, 2017, pp. 95–128, (with J. Behrndt, F. Gesztesy, R. Nichols)

π[52] Burgers meets Braess, Oberwolfach Reports, vol. 13, issue 2, pp. 1715–1717, Editors R. M. Colombo, P. LeFloch, C. Rohde, European Publishing House, Zürich, 2016.

π[53] On the equivalence of Eulerian and Lagrangian variables for the two-component Camassa–Holm system, In Current Research in Nonlinear Analysis: In Honor of Haim Brezis and Louis Nirenberg, Editor Th. M. Rassias, Springer, to appear, (with M. Grasmaier and K. Grunert)

Books edited

ε[1] Schrödinger Operators, Proceedings of the Nordic Summer School in Mathematics. Sønderborg, Denmark 1988, Lecture Notes in Physics, Volume 345, Springer-Verlag, Berlin-Heidelberg-New York-London-, Paris-Tokyo-Hong Kong 1989, 458 pp., (jointly edited with A. Jensen)

ε[2] Ideas and Methods in Mathematical Analysis, Stochastics, and Applications. In Memory of Raphael Høegh-Krohn (1938-1988), Cambridge University Press, Cambridge 1992, 509 pp., (jointly edited with S. Albeverio, J. E. Fenstad, T. Lindstrøm) Publications by Helge Holden 475

ε[3] Ideas and Methods in Quantum and Statistical Physics. In Memory of Raphael Høegh-Krohn (1938-1988), Cambridge University Press, Cambridge 1992, 542 pp., (jointly edited with S. Albeverio, J. E. Fenstad, T. Lindstrøm)

ε[4] The Collected Works of Lars Onsager (With Commentary), World Scientific, Singapore, 1996, 1088 pp, (with P. C. Hemmer, S. K. Ratkje)

ε[5] Stochastic Processes, Physics and Geometry: New Interplays. I, A Volume in Honor of Sergio Albeverio, CMS Conference Proceedings, Volume 28, Canadian Mathematical Society, Providence (USA), 2000, 343 pp., (jointly edited with F. Gesztesy, J. Jost, S. Paycha, M. Röckner, S. Scarlatti)

ε[6] Stochastic Processes, Physics and Geometry: New Interplays. II, A Volume in Honor of Sergio Albeverio, CMS Conference Proceedings, Volume 29, Canadian Mathematical Society, Providence (USA), 2000, 645 pp., (jointly edited with F. Gesztesy, J. Jost, S. Paycha, M. Röckner, S. Scarlatti)

ε[7] The Abel Prize 2003–2007. The First Five Years, Springer, Heidelberg, 2010, 327 pp., (with R. Piene)

ε[8] Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, Contemporary Mathematics, American Mathematical Society, Providence, Vol. 526, 2010, 389 pp., (jointly edited with K. H. Karlsen)

ε[9] Nonlinear Partial Differential Equations. The Abel Symposium 2010, Abel Symposia, Vol. 7, Springer, Heidelberg, 2012, 360 pp., (jointly edited with K. H. Karlsen)

ε[10] Høydepunkter i Skrifter og Forhandlinger. Et utvalg artikler fra perioden 1761–2011, Skrifter fra Det Kongelige Norske Videnskabers Selskab, nr. 4, 2011, 256 pp., (jointly edited with K. Overskaug)

ε[11] Spectral Analysis, Differential Equations and Mathematical Physics., A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, Vol. 87, 2013, 376 pp., (jointly edited with B. Simon and G. Teschl) 476 Publications by Helge Holden

ε[12] The Abel Prize 2008–2012, Springer, Heidelberg, 2014, 571 pp., (with R. Piene)

ε[13] Hyperbolic Conservation Laws and Related Analysis with Applications., Edinburgh, September 2011., Springer Proceedings in Mathematics & Statistics, Volume 49, Springer, New York, 2014, 384 pp., (With G.-Q. G. Chen and K. H. Karlsen)

ε[14] The Abel Prize 2013–2017, Springer, Heidelberg, 2018, to appear, (with R. Piene)

Reports

ρ[1] Construction of Higgs fields in two dimensions. I, Preprint, Kungliga Tekniska Högskolan, 1990, (with S. Albeverio, R. Høegh-Krohn, T. Kolsrud)

ρ[2] Reservoir evaluation by stochastic differential equations, Report, Norwegian Computing Center, Oslo, 1991, (with L. Holden)

Miscellaneous

μ[1] Matematikkolympiaden i Australia, 1988 (In Norwegian), NORMAT 36, no 4, (1988) 160–161

μ[2] Matematikkolympiaden 1988 i Australia — løsninger (In Norwegian), NORMAT 37, no 2, (1989) 84–89

μ[3] The industrial mathematics curriculum at the Norwegian Institute of Technology — The first ten years, In Industrial Mathematics Week, Trondheim August 1992. Proceedings, Department of Mathematical Sciences, NTH, 1993, pp. 11–22, Also published in Société Europeenne pour la Formation, des Ingénieurs, SEFI Math 4 (1993) 14–19

μ[4] En trafikkmodell (In Norwegian), In Den levende matematikken, Edited by T. Sevje, Undervisningsforlaget, Sandefjord, 1994, pp. 64–85

μ[5] Seven guidelines for scientific computing, Manuscript, NTH, 1994 Publications by Helge Holden 477

μ[6] Buckley–Leverett equation, In Encyclopaedia of Mathematics, Supplement I, Edited by M. Hazewinkel, Kluwer, Dordrecht, 1997, p. 161

μ[7] Paradokser og rushtrafikk (In Norwegian), Elementa, 79 (1996) 191–193

μ[8] Vår tids matematikk – også som skolefag (In Norwegian), Kronikk, Aftenposten, November 13, 1998, (with I. Holden, K. Seip)

μ[9] Comment of Special Issue on Geostatistics with Guest Editor A. G. Journel, Mathematical Geology 30 (1998) 245, (with K. Bratvedt, E. Bølviken, T. Gimse, L. Holden, R. Knarud)

μ[10] Lars Onsager. Vår største naturvitenskapsmann (In Norwegian), Medlemsblad for Vindern historielag, 3/2000, pp. 23–27.

μ[11] eVITenskap og Anvendelser (eVITA). Forskning i en ny epoke (In Norwegian), Report submitted to the Ministry of Eduction and Research, May 2004, (with M. Dæhlen et al.)

μ[12] Peter D. Lax. Elements from his contributions to mathematics, Presentation on the occasion of the announcement of the Abel Prize Laureate 2005, English and Norwegian text to be found at https://www.abelprisen.no, Spanish translation appeared in Boletin del departamento de matemáticas,, Universidad Nacional Autonoma de Mexico, no 167–8, April 2005.

μ[13] Letter from the President, ECMI Newsletter, Numbers 35–37, (2004–2005)

μ[14] Matematikkens bidrag til Olje-Norge (In Norwegian), Kronikk, Aftenposten, May 24, 2005

μ[15] Peter D. Lax. Abelprisvinner 2005 (In Norwegian), Normat 53 (2005) 145–154.

μ[16] Abelprisen – en sunn og frisk treåring (In Norwegian), Infomat, oktober 2005; https://matematikkforeningen.files.wordpress.com/2016/06/0510.pdf, Also printed in Matilde, nr. 26, 2006, p. 23–24.

μ[17] Om Poincaré, Perelman og kuler (In Norwegian), Kronikk, Morgenbladet, October 13–19, 2006.

μ[18] Utenlandske doktorgrader — ja takk! (In Norwegian), Bladet Forskning, 1/2006. 478 Publications by Helge Holden

μ[19] Informatikk betyr mye, men ikke alt (In Norwegian), Bladet Forskning, December 2007, p. 29.

μ[20] The Abel Prize — the first five years, European Mathematical Society Newsletter, issue 64, 2007, p. 3, Chinese translation: Mathematical Advance in Translation (3) 2013, pp. 277, 194.

μ[21] Digitalisering av matematisk litteratur, og Abels samlede spesielt (In Norwegian), NB21, December 2007, p. 24–26.

μ[22] Verdens smarteste mann til Norge (In Norwegian), forskning.no, October 29, 2008, https://forskning.no/meninger/kronikk/2008/10/verdens-smarteste-mann-til-norge

μ[23] Terence Tao og undervisning av flinke elever (In Norwegian), Kronikk, Adresseavisen, December 8, 2008

μ[24] Big problems in mathematics — solved and unsolved, In Transference. Interdisciplinary Communications 20082009, (W. Østreng, editor), 7 pp., Appeared on: https://cas.oslo.no/about_cas/cas_reports/seminar_booklets/

μ[25] A survey of Peter D. Lax’s contributions to mathematics, In The Abel Prize 2003–2007. The First Five Years., Edited by H. Holden and R. Piene, Springer, Heidelberg, 2010, pp. 199–214, (with P. Sarnak)

μ[26] Kollisjonsfare? (In Norwegian), Dagens Næringsliv, October 9, 2010

μ[27] Vitenskapelige skrifter i 250 år (In Norwegian), Kronikk, Adresseavisen, March 9, 2012, (with A. Stubhaug)

μ[28] Matematikkens gave (In Norwegian), Kronikk, Aftenposten, March 21, 2012, (with R. Piene)

μ[29] The Camassa–Holm equation, In Encyclopedia of Applied and Computational Mathematics, Edited by B. Engquist, Springer Reference, doi:10.1007/978-3-540-70529-1_1

μ[30] Noen refleksjoner om søknadsprosessen for SFF (In Norwegian), Bladet Forskning, September 2012, p. 25.

μ[31] Would Abel have received the Abel Prize? On Niels Henrik Abel and his prize, Norges Tekniske Vitenskapsakademi, Årbok 2013, pp. 61–64. Publications by Helge Holden 479

μ[32] Ville Abel fått Abelprisen? Om Abel og prisen hans (In Norwegian), Det Norske Videnskaps-Akademi, Årbok 2013, pp. 151–163.

μ[33] Alle vil støtte et gjennombrudd (In Norwegian), Innlegg, Aftenposten, January 21, 2014, A longer version published in Universitetsavisa, Jan. 23, 2014, titled “Flaks favoriserer forberedte”, http://www.universitetsavisa.no/leserbrev/article19613.ece

μ[34] Matematikk gir økt produktivitet (In Norwegian), Innlegg, Dagens Næringsliv, March 1, 2014, (with J. E. Reinhardsen)

μ[35] Hvem skal vi gi fast ansettelse? (In Norwegian), Innlegg, Aftenposten, May 10, 2014, A longer version published in Universitetsavisa, May 15, 2014, http://www.universitetsavisa.no/leserbrev/article22141.ece

μ[36] Styringsmodeller (In Norwegian), Innlegg, Morgenbladet, November 14, 2014, Online only: https://morgenbladet.no/debatt/2014/styringsmodeller#.VGyQ24s73Zd

μ[37] Universitetsledere (In Norwegian), Innlegg, Morgenbladet, November 28, 2014, Also online: https://morgenbladet.no/debatt/2014/universitetsledere#.VHwWyos73Zc

μ[38] Forskning og det uventede (In Norwegian), Kronikk, Adresseavisen, September 7, 2015

μ[39] Ja takk til ansatt rektor! (In Norwegian), Innlegg, Aftenposten, June 20, 2016, (with G. Busterud, K. Melum Eide, B. Foss)

μ[40] Ansatt og ansvarlig ledelse (In Norwegian), Innlegg, Dagens Næringsliv, July 22, 2016

μ[41] Oljefondet kan spare penger (In Norwegian), “Forskning viser at …”, Dagens Næringsliv, July 23, 2016, (with L. Holden)

μ[42] Vi kan gje formidling av verdsklasse (In Norwegian), Kronikk, Adresseavisen, October 11, 2016, (with A. Stendahl Rokne, R. Andersen)

μ[43] Utfordringer ved evaluering av avsluttede ERC-prosjekter (In Norwegian), Innlegg, Forskningspolitikk 4/2016, p. 22–23.

μ[44] Forskerne må ta sin del av ansvaret (In Norwegian), Innlegg, Aftenposten, September 20, 2017. 480 Publications by Helge Holden

μ[45] Tellekanter er ikke så galt (In Norwegian), Innlegg, Aftenposten, September 24, 2017.

μ[46] DKNVS med i europeisk akademiorganisasjon (In Norwegian), Årbok, Det Kongelige Norske Videnskabers Selskab, pp. 37–41, 2017. List of Contributors

Nacira Agram 3 Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N–0316 Oslo, Norway [email protected]

Sergio Albeverio 37 Institut für Angewandte Mathematik, Endenicher Allee 60, 53115 Bonn, HCM; BIBOS; IZKS; Cerfim (Locarno). [email protected]

Alan R. Champneys 55 Department of Engineering Mathematics, University of Bristol [email protected]

Gui-Qiang G. Chen 73 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK [email protected]

Giuseppe Maria Coclite 97 Department of Mechanics, Mathematics and Management, Polytechnic University of Bari, Via E. Orabona 4, I–70125 Bari, Italy [email protected]

Rinaldo M. Colombo 111 INDAM Unit, c/o DII, University of Brescia, Via Branze 38, 25123 Brescia, Italy [email protected]

Félix del Teso 129 Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway [email protected] 482 List of Contributors

Lorenzo di Ruvo 97 Department of Mathematics, University of Bari, via E. Orabona 4, I–70125 Bari, Italy [email protected]

Jørgen Endal 129 Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway [email protected]

Pavel Exner 169 Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež near Prague Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Břehová 7, 11519 Prague, Czechia [email protected]

Hermano Frid 183 Instituto de Matemática Pura e Aplicada-IMPA, Estrada Dona Castorina, 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil [email protected]

Fritz Gesztesy 207 Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, USA [email protected]

Maria Gokieli 111 ICM, University of Warsaw, Pawińskiego 5a, 02-106 Warsaw, Poland [email protected]

Katrin Grunert 227 Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway [email protected] List of Contributors 483

Graziano Guerra 261 Department of Mathematics and Applications, Milano-Bicocca University, Italy [email protected]

Andreas Hiltebrand 287 ANSYS Switzerland, Zurich [email protected]

Poul G. Hjorth 1, 55 Department of Applied Mathematics and Computer Science, Technical University of Denmark [email protected]

Markus Holzleitner 319 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria [email protected]

Espen R. Jakobsen 129 Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway [email protected]

Kenneth Hvistendahl Karlsen 97 Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway [email protected]

Werner Kirsch 349 Fakultät für Mathematik und Informatik, FernUniversität Hagen, Germany [email protected]

Aleksey Kostenko 319 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria [email protected] 484 List of Contributors

Thomas Kriecherbauer 349 Mathematisches Institut, Universität Bayreuth, Germany [email protected] Ari Laptev 381 Ari Laptev: Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK [email protected] Siran Li 73 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK [email protected] Knut-Andreas Lie 389 SINTEF Digital, Mathematics and Cybernetics, Oslo, Norway, Also: Department of Mathematical Sciences, NTNU, Trondheim. [email protected] Lance Littlejohn 207 Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, USA [email protected] Yurii Lyubarskii 423 Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, 7491, Norway [email protected] Eugenia Malinnikova 423 Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, 7491, Norway [email protected] Harry Man 55 Department of English and Modern Languages, Oxford Brookes University [email protected] Sonia Mazzucchi 37 Dipartimento di Matematica, Università di Trento, via Sommarive 14 I-38123 Trento, Italy. [email protected] List of Contributors 485

Siddhartha Mishra 287 Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, HG G 57.2, Rämistrasse 101, Zürich -8092, SwitzerlandCenter of Mathematics for Applications (CMA) University of Oslo, P.O.Box -1053, Blindern, Oslo-0316, Norway [email protected]

Bernt Øksendal 3 Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N–0316 Oslo, Norway [email protected]

Xavier Raynaud 227 Applied Mathematics, SINTEF ICT, Oslo, Norway Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway [email protected]

Massimiliano D. Rosini 111 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, Plac Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland [email protected]

Sigmund Selberg 437 Department of Mathematics, University of Bergen, P.O. Box 7083, 5020 Bergen, Norway [email protected]

Wen Shen 261 Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA [email protected] 486 List of Contributors

Gerald Teschl 319 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria [email protected] Andrei Velicu 381 Andrei Velicu: Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK [email protected] Samia Yakhlef 3 Department of Mathematics, University of Biskra, Algeria [email protected] colophon

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