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Contemporary Mathematics 526 CONTEMPORARY MATHEMATICS 526 Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena The 2008–2009 Research Program on Nonlinear Partial Differential Equations Centre for Advanced Study at the Norwegian Academy of Science and Letters Oslo, Norway Helge Holden Kenneth H. Karlsen Editors American Mathematical Society http://dx.doi.org/10.1090/conm/526 Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena CONTEMPORARY MATHEMATICS 526 Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena The 2008–2009 Research Program on Nonlinear Partial Differential Equations Centre for Advanced Study at the Norwegian Academy of Science and Letters Oslo, Norway Helge Holden Kenneth H. Karlsen Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary 35L65, 35Q30, 35G25, 35J70, 35K65, 35B65, 39A14, 35B25, 35L05, 65M08. Library of Congress Cataloging-in-Publication Data Det Norske Videnskaps-Akademi. Research Program on Nonlinear Partial Differential Equations (2008–2009 : Centre for Advanced Study at the Norwegian Academy of Science and Letters) Nonlinear partial differential equations and hyperbolic wave phenomena : 2008–2009 Research Program on Nonlinear Partial Differential Equations, Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway / [edited by] Helge Holden, Kenneth H. Karlsen. p. cm. — (Contemporary mathematics ; v. 526) Includes bibliographical references. ISBN 978-0-8218-4976-7 (alk. paper) 1. Differential equations, Nonlinear—Congresses. 2. Differential equations, Hyperbolic— Congresses. 3. Differential equations, Partial—Congresses. I. Holden, H. (Helge), 1956– II. Karlsen, Kenneth Hvistendahl. III. Title. QA377.N6416 2008 515.353—dc22 2010014967 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 Contents Preface vii A hyperbolic model of granular flow Debora Amadori and Wen Shen 1 Hilbertian approaches to some non-linear conservation laws Yann Brenier 19 On the asymptotic behavior of the gradient flow of a polyconvex functional Jose´ A. Carrillo and Stefano Lisini 37 On degenerate partial differential equations Gui-Qiang G. Chen 53 Symmetric solutions to multi-dimensional conservation laws Nicola Costanzino and Helge Kristian Jenssen 91 Product estimates for wave-Sobolev spaces in 2 + 1 and 1 + 1 dimensions Piero D’Ancona, Damiano Foschi, and Sigmund Selberg 125 On the Cauchy problem for the modified Korteweg-de Vries equation with steplike finite-gap initial data Iryna Egorova and Gerald Teschl 151 Asymptotic analysis in thermodynamics of viscous fluids Eduard Feireisl 159 Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation Chunxia Guan, Kenneth H. Karlsen, and Zhaoyang Yin 199 Instability of solitary waves for a nonlinearly dispersive equation Henrik Kalisch and Nguyet Thanh Nguyen 221 Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues Philippe G. LeFloch 237 Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations Hailiang Liu and Zhaoyang Yin 273 Potential based, constraint preserving, genuinely multi-dimensional schemes for systems of conservation laws Siddhartha Mishra and Eitan Tadmor 295 v vi CONTENTS A local and low-order Navier-Stokes-Korteweg system Christian Rohde 315 Local existence for viscous system of conservation laws: Hs-data with s>1+d/2 Denis Serre 339 Finite difference methods for discretizing singular source terms in a Poisson interface problem John D. Towers 359 Preface This volume contains research and expository articles from the research pro- gram on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo, Norway during the aca- demic year 2008–09. The program was one of three programs hosted at the Centre for Advanced Study during 2008–09; in addition there was one program in social sciences and one in humanities, thereby creating a unique atmosphere for cross disciplinary discussions. Within the program on nonlinear partial differential equa- tions 40 researchers from all over the world participated. Our program focused on nonlinear partial differential equations modeling various wave phenomena. There was a certain emphasis on nonlinear hyperbolic waves which is reflected in the cur- rent volume. Each one of the articles in this volume represents current advances in the area of nonlinear partial differential equations. A range of topics is covered, including sys- tems of conservation laws, Navier–Stokes equations of compressible viscous fluids, Navier–Stokes–Korteweg type systems and models for phase transitions, nonlin- ear evolution equations, degenerate/mixed type nonlinear partial differential equa- tions arising in fluid mechanics and differential geometry, nonlinear dispersive wave equations (Korteweg–de Vries, Camassa–Holm, etc.), and Poisson interface prob- lems and level set formulations. The articles reflect recent developments in nonlin- ear analysis, and touch upon many methods from functional analysis and calculus of variations, including entropy analysis, renormalization, weak convergence tech- niques, kinetic formulations, non-classical shocks and kinetic relations, asymptotic analysis, gradient flows, and numerical methods. The volume is suitable for gradu- ate students and researchers interested in nonlinear analysis and partial differential equations. Our research was generously funded by the Centre for Advanced Study, and all our activities at the Centre were greatly facilitated by the very able and friendly staff at the Centre, namely Willy Østreng, Trude Gran Peters, Maria Sætre, and Marit Finnemyhr Strøm. Trondheim and Oslo, June 3, 2010 Helge Holden and Kenneth H. Karlsen vii Titles in This Series 527 Ricardo Casta˜no-Bernard, Yan Soibelman, and Ilia Zharkov, Editors, Mirror symmetry and tropical geometry, 2010 526 Helge Holden and Kenneth H. Karlsen, Editors, Nonlinear partial differential equations and hyperbolic wave phenomena, 2010 525 Manuel D. Contreras and Santiago D´ıaz-Madrigal, Editors, Five lectures in complex analysis, 2010 524 Mark L. Lewis, Gabriel Navarro, Donald S. Passman, and Thomas R. Wolf, Editors, Character theory of finite groups, 2010 523 Aiden A. Bruen and David L. Wehlau, Editors, Error-correcting codes, finite geometries and cryptography, 2010 522 Oscar Garc´ıa-Prada, Peter E. Newstead, Luis Alverez-C´´ onsul, Indranil Biswas, Steven B. Bradlow, and Tom´asL. G´omez, Editors, Vector bundles and complex geometry, 2010 521 David Kohel and Robert Rolland, Editors, Arithmetic, geometry, cryptography and coding theory 2009, 2010 520 Manuel E. Lladser, Robert S. Maier, Marni Mishna, and Andrew Rechnitzer, Editors, Algorithmic probability and combinatorics, 2010 519 Yves F´elix, Gregory Lupton, and Samuel B. Smith, Editors, Homotopy theory of function spaces and related topics, 2010 518 Gary McGuire, Gary L. Mullen, Daniel Panario, and Igor E. Shparlinski, Editors, Finite fields: Theory and applications, 2010 517 Tewodros Amdeberhan, Luis A. Medina, and Victor H. Moll, Editors, Gems in experimental mathematics, 2010 516 Marlos A.G. Viana and Henry P. Wynn, Editors, Algebraic methods in statistics and probability II, 2010 515 Santiago Carrillo Men´endez and Jos´eLuisFern´andez P´erez, Editors, Mathematics in finance, 2010 514 Arie Leizarowitz, Boris S. Mordukhovich, Itai Shafrir, and Alexander J. Zaslavski, Editors, Nonlinear analysis and optimization II, 2010 513 Arie Leizarowitz, Boris S. Mordukhovich, Itai Shafrir, and Alexander J. Zaslavski, Editors, Nonlinear analysis and optimization I, 2010 512 Albert Fathi, Yong-Geun Oh, and Claude Viterbo, Editors, Symplectic topology and measure preserving dynamical systems, 2010 511 Luise-Charlotte Kappe, Arturo Magidin, and Robert Fitzgerald Morse, Editors, Computational group theory and the theory of groups, II, 2010 510 Mario
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