Analysis & PDE Vol. 5 (2012)

ANALYSIS & PDE Volume 5 No. 4 2012

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EDITOR-IN-CHIEF Maciej Zworski University of California Berkeley, USA

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Michael Aizenman Princeton University, USA Nicolas Burq Université Paris-Sud 11, France [email protected] [email protected] Luis A. Caffarelli University of Texas, USA Sun-Yung Alice Chang Princeton University, USA [email protected] [email protected] Michael Christ University of California, Berkeley, USA Charles Fefferman Princeton University, USA [email protected] [email protected] Ursula Hamenstaedt Universität Bonn, Germany Nigel Higson Pennsylvania State Univesity, USA [email protected] [email protected] Vaughan Jones University of California, Berkeley, USA Herbert Koch Universität Bonn, Germany [email protected] [email protected] Izabella Laba University of British Columbia, Canada Gilles Lebeau Université de Nice Sophia Antipolis, France [email protected] [email protected] László Lempert Purdue University, USA Richard B. Melrose Massachussets Institute of Technology, USA [email protected] [email protected] Frank Merle Université de Cergy-Pontoise, France William Minicozzi II Johns Hopkins University, USA [email protected] [email protected] Werner Müller Universität Bonn, Germany Yuval Peres University of California, Berkeley, USA [email protected] [email protected] Gilles Pisier Texas A&M University, and Paris 6 Tristan Rivière ETH, Switzerland [email protected] [email protected] Igor Rodnianski Princeton University, USA Wilhelm Schlag University of Chicago, USA [email protected] [email protected] Sylvia Serfaty New York University, USA Yum-Tong Siu Harvard University, USA [email protected] [email protected] Terence Tao University of California, Los Angeles, USA Michael E. Taylor Univ. of North Carolina, Chapel Hill, USA [email protected] [email protected] Gunther Uhlmann University of Washington, USA András Vasy Stanford University, USA [email protected] [email protected] Dan Virgil Voiculescu University of California, Berkeley, USA Steven Zelditch Northwestern University, USA [email protected] [email protected]

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APDE peer review and production are managed by EditFLOW™ from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers http://msp.org/ A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2012 by Mathematical Sciences Publishers ANALYSIS AND PDE Vol. 5, No. 4, 2012 dx.doi.org/10.2140/apde.2012.5.705 msp

ON THE GLOBAL WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES

ALEXANDRU D.IONESCU, BENOIT PAUSADER AND GIGLIOLA STAFFILANI

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrödinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global Morawetz inequality adapted to the geometry of the manifold (in other words we adapt the method of Kenig and Merle to the variable coefﬁcient case), and a good understanding of the corresponding Euclidean problem (a theorem of Colliander, Keel, Stafﬁlani, Takaoka and Tao). As an application we prove global well-posedness and scattering in H 1 for the energy-critical defocusing initial-value problem 4 .i@t g /u u u ; u.0/ ; C D j j D on hyperbolic space ވ3.

1. Introduction 705 2. Preliminaries 709 3. Proof of the main theorem 718 4. Euclidean approximations 721 5. Proﬁle decomposition in hyperbolic spaces 725 6. Proof of Proposition 3.4 735 References 744

1. Introduction

The goal of this paper is to present a somewhat general method to prove global well-posedness of critical1 nonlinear Schrödinger initial-value problems of the form

.i@t g/u ᏺ.u/; u.0/ ; (1-1) C D D ij k on certain noncompact Riemannian manifolds .M; g/. Here g g .@ij @ / is the (negative) D ij k Laplace–Beltrami operator of .M; g/. In Euclidean spaces, the subcritical theory of such nonlinear Schrödinger equations is well established; see for example the books [Cazenave 2003; Tao 2006] for many references. Many of the subcritical methods extend also to the study of critical equations with small

Ionescu was supported in part by a Packard Fellowship, and Stafﬁlani by NSF Grant DMS 0602678. Pausader and Stafﬁlani thank the MIT/France program during which this work was initiated. MSC2000: 35Q55. Keywords: global well-posedness, energy-critical defocusing NLS, nonlinear Schrödinger equation, induction on energy. 1 3 Here critical refers to the fact that when .M; g/ .ޒ ; ıij /, the equation and the control (here the energy) are invariant D under the rescaling u.x; t/ 1=2u.x; 2t/. !

705 706 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI data. The case of large-data critical Schrödinger equations is more delicate, and was ﬁrst considered in [Bourgain 1999] and [Grillakis 2000] for defocusing Schrödinger equations with pure power nonlinearities and spherically symmetric data. The spherical symmetry assumption was removed in dimension d 3 D in [Colliander et al. 2008]; global well-posedness was then extended to higher dimensions d 4 in [Ryckman and Visan 2007; Visan 2007]. A key development in the theory of large-data critical dispersive problems was the article [Kenig and Merle 2006], on spherically symmetric solutions of the energy-critical focusing NLS in ޒ3. The methods developed in this paper found applications in many other large-data critical dispersive problems, leading to complete solutions or partial results. We adapt this point of view in our variable coefﬁcient setting as well. To keep things as simple as possible on a technical level, in this paper we consider only the energy- critical defocusing Schrödinger equation

4 .i@t g/u u u (1-2) C D j j 3 in hyperbolic space ވ . Suitable solutions of (1-2) on the time interval .T1; T2/ satisfy mass and energy conservation, in the sense that the functions Z Z Z 0 2 1 1 2 1 6 E .u/.t/ u.t/ d; E .u/.t/ gu.t/ d u.t/ d (1-3) WD ވ3 j j WD 2 ވ3 jr j C 6 ވ3 j j are constant on the interval .T1; T2/. Our main theorem concerns global well-posedness and scattering in H 1.ވ3/ for the initial-value problem associated to (1-2). Theorem 1.1. (a) (Global well-posedness.) If H 1.ވ3/2 then there exists a unique global solution 2 u C.ޒ H 1.ވ3// of the initial-value problem 2 W 4 .i@t g/u u u ; u.0/ : (1-4) C D j j D In addition, the mapping u is a continuous mapping from H 1.ވ3/ to C.ޒ H 1.ވ3//, and the ! W quantities E0.u/ and E1.u/ deﬁned in (1-3) are conserved. (b) (Scattering.) We have the bound

u L10.ވ3 ޒ/ C. H 1.ވ3//: (1-5) k k Ä k k As a consequence, there exist unique u H 1.ވ3/ such that ˙ 2 itg u.t/ e u H 1.ވ3/ 0 as t : (1-6) k ˙k D ! ˙1 It was observed by Banica [2007] that hyperbolic geometry cooperates well with the dispersive nature of Schrödinger equations, at least in the case of subcritical problems. In fact the long time dispersion of solutions is stronger in hyperbolic geometry than in Euclidean geometry. Intuitively, this is due to the fact that the volume of a ball of radius R 1 in hyperbolic spaces is about twice as large as the volume C 2 d R 2 R 2 Unlike in Euclidean spaces, in hyperbolic spaces ވ one has the uniform inequality d f d . d f d for any ވ j j ވ jr j f C .ވd /. In other words H 1.ވd /, L2.ވd /. 2 01 P ! WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 707 of a ball of radius R, if R 1; therefore, as outgoing waves advance one unit in the geodesic direction they have about twice as much volume to disperse into. This heuristic can be made precise; see [Anker and Pierfelice 2009; Banica 2007; Banica et al. 2008; 2009; Banica and Duyckaerts 2007; Bouclet 2011; Christianson and Marzuola 2010; Ionescu and Stafﬁlani 2009; Pierfelice 2008] for theorems concerning subcritical nonlinear Schrödinger equations in hyperbolic spaces (or other spaces that interpolate between Euclidean and hyperbolic spaces). The theorems proved in these papers are stronger than the corresponding theorems in Euclidean spaces, in the sense that one obtains better scattering and dispersive properties of the nonlinear solutions. We remark, however, that the global geometry of the manifold cannot bring any improvements in the case of critical problems. To see this, consider only the case of data of the form

1=2 1 N .x/ N .N ‰ .x//; (1-7) D 3 3 3 where C01.ޒ / and ‰ ޒ ވ is a suitable local system of coordinates. Assuming that is fixed 2 W ! 3 1 and letting N , the functions N C .ވ / have uniformly bounded H norm. For any T 0 and ! 1 2 01 fixed, one can prove that the nonlinear solution of (1-4) corresponding to data N is well approximated by 1=2 1 2 N v.N ‰ .x/; N t/ on the time interval . TN 2; TN 2/, for N sufficiently large (depending on T and ), where v is the solution on the time interval . T; T / of the Euclidean nonlinear Schrödinger equation 4 .i@t /v v v ; v.0/ : (1-8) C D j j D See Section 4 for precise statements. In other words, the solution of the hyperbolic NLS (1-4) with data 2 2 N can be regular on the time interval . TN ; TN / only if the solution of the Euclidean NLS (1-8) is regular on the interval . T; T /. This shows that understanding the Euclidean scale invariant problem is a prerequisite for understanding the problem on any other manifold. Fortunately, we are able to use the main theorem of Colliander et al. [2008] as a black box (see the proof of Lemma 4.2). The previous heuristic shows that understanding the scaling limit problem (1-8) is part of understanding the full nonlinear evolution (1-4), at least if one is looking for uniform control on all solutions below a certain energy level. This approach was already used in the study of elliptic equations, first in the subcritical case (where the scaling limits are easier) by Gidas and Spruck [1981] and also in the H 1 critical setting, see for example Druet, Hebey and Robert [Druet et al. 2004], Hebey and Vaugon [1995], Schoen [1989] and (many) references therein. Note however that in the dispersive case, we have to contend with the fact that we are looking at perturbations of a linear operator i@t g whose kernel is C infinite dimensional. Other critical dispersive models, such as large-data critical wave equations or the Klein–Gordon equation have also been studied extensively, both in the case of the Minkowski space and in other Lorentz manifolds. See, for example, [Bahouri and Gérard 1999; Bahouri and Shatah 1998; Burq et al. 2008; Burq and Planchon 2009; Grillakis 1990; 1992; Ibrahim and Majdoub 2003; Ibrahim et al. 2009; 2011; Kapitanski 1994; Kenig and Merle 2008; Killip et al. 2012; Laurent 2011; Shatah and Struwe 1993; 1994; 708 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

Struwe 1988; Tao 2006] for further discussion and references. In the case of the wave equation, passing to the variable coefﬁcient setting is somewhat easier due the ﬁnite speed of propagation of solutions. Nonlinear Schrödinger equations such as (1-1) have also been considered in the setting of compact Riemannian manifolds .M; g/; see [Bourgain 1993a; 1993b; Burq et al. 2004; 2005; Colliander et al. 2010; Gérard and Pierfelice 2010]. In this case the conclusions are generally weaker than in Euclidean spaces: there is no scattering to linear solutions, or some other type of asymptotic control of the nonlinear evolution as t . We note however the recent result of Herr, Tataru and Tzvetkov [Herr et al. 2011] ! 1 on the global well-posedness of the energy critical NLS with small initial data in H 1.ޔ3/. To simplify the exposition, we use some of the structure of hyperbolic spaces; in particular we exploit the existence of a large group of isometries that acts transitively on ވd . However the main ingredients in the proof are more basic, and can probably be extended to more general settings3. These main ingredients are:

(1) a dispersive estimate such as (2-24), which gives a good large-data local well-posedness/stability theory (Propositions 3.1 and 3.2); (2) a good Morawetz-type inequality (Proposition 3.3) to exploit the global defocusing character of the equation; (3) a good understanding of the Euclidean problem, provided in this case by a result of Colliander, Keel, Stafﬁlani, Takaoka and Tao [Colliander et al. 2008, Theorem 4.1]; (4) some uniform control of the geometry of the manifold at inﬁnity.

The rest of the paper is organized as follows: in Section 2 we set up the notations, and record the main dispersive estimates on the linear Schrödinger flow on hyperbolic spaces. We prove also several lemmas that are used later. In Section 3 we collect all the necessary ingredients described above, and outline the proof of the main theorem. The only component of the proof that is not known is Proposition 3.4 on the existence of a suitable minimal energy blow-up solution. In Section 4 we consider nonlinear solutions of (1-4) corresponding to data that contract at a point, as in (1-7). Using the main theorem in [Colliander et al. 2008] we prove that such nonlinear solutions extend globally in time and satisfy suitable dispersive bounds. In Section 5 we prove our main profile decomposition of H 1-bounded sequences of functions in hyperbolic spaces. This is the analogue of Keraani’s theorem [2001] in Euclidean spaces. In hyperbolic spaces we have to distinguish between two types of profiles: Euclidean profiles which may contract at a point, after time and space translations, and hyperbolic profiles which live essentially at frequency4 N 1. D Hyperbolic geometry guarantees that profiles of low frequency N 1 can be treated as perturbations.

3Two of the authors have applied a similar strategy to prove global regularity of the defocusing energy-critical NLS in other settings, such as ޔ3 [Ionescu and Pausader 2012a] and ޒ ޔ3 [Ionescu and Pausader 2012b], where other issues arise due to the presence of trapped geodesics or the lower power in the nonlinearity. 4Here we deﬁne the notion of frequency through the heat kernel, see (2-28). WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 709

Finally, in Section 6 we use our proﬁle decomposition and orthogonality arguments to complete the proof of Proposition 3.4.

2. Preliminaries

In this subsection we review some aspects of the harmonic analysis and the geometry of hyperbolic spaces, and summarize our notations. For simplicity, we will use the conventions in [Bray 1994], but one should keep in mind that hyperbolic spaces are the simplest examples of symmetric spaces of the noncompact type, and most of the analysis on hyperbolic spaces can be generalized to this setting (see, for example, [Helgason 1994]).

Hyperbolic spaces: Riemannian structure and isometries. For integers d 2 we consider the Minkowski space ޒd 1 with the standard Minkowski metric .dx0/2 .dx1/2 ::: .dxd /2 and deﬁne the bilinear C C C C form on ޒd 1 ޒd 1, C C Œx; y x0y0 x1y1 xd yd : D Hyperbolic space ވd is deﬁned as

ވd x ޒd 1 Œx; x 1 and x0 > 0 : D f 2 C W D g Let 0 .1; 0;:::; 0/ denote the origin of ވd . The Minkowski metric on ޒd 1 induces a Riemannian D C metric g on ވd , with covariant derivative D and induced measure d. We define އ SO.d; 1/ SOe.d; 1/ as the connected Lie group of .d 1/ .d 1/ matrices that WD D C C leave the form Œ ; invariant. Clearly, X SO.d; 1/ if and only if 2 tr X I X I ; det X 1; X00 > 0; d;1 D d;1 D where I is the diagonal matrix diagŒ 1; 1;:::; 1 (since Œx; y t x I y). Let ދ SO.d/ denote d;1 D d;1 D the subgroup of SO.d; 1/ that fixes the origin 0. Clearly, SO.d/ is the compact rotation group acting on the variables .x1;:::; xd /. We define also the commutative subgroup ށ of އ,

8 2 ch s sh s 0 3 9 < = ށ as 4 sh s ch s 0 5 s ޒ ; (2-1) WD : D W 2 ; 0 0 Id 1 and recall the Cartan decomposition

އ ދށ ދ; ށ as s Œ0; / : (2-2) D C C WD f W 2 1 g The semisimple Lie group އ acts transitively on ވd and hyperbolic space ވd can be identified with the homogeneous space އ=ދ SO.d; 1/= SO.d/. Moreover, for any h SO.d; 1/ the mapping D 2 L ވd ވd , L .x/ h x, defines an isometry of ވd . Therefore, for any h އ, we define the h W ! h D 2 isometries L2.ވd / L2.ވd /; .f /.x/ f .h 1 x/: (2-3) h W ! h D 710 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

We fix normalized coordinate charts which allow us to pass in a suitable way between functions defined on hyperbolic spaces and functions defined on Euclidean spaces. More precisely, for any h SO.d; 1/ 2 we define the diffeomorphism p ‰ ޒd ވd ;‰ .v1; : : : ; vd / h . 1 v 2; v1; : : : ; vd /: (2-4) h W ! h D C j j Using these diffeomorphisms we define, for any h އ, 2 C.ޒd / C.ވd /; .f /.x/ f .‰ 1.x//: (2-5) zh W ! zh D h d We will use the diffeomorphism ‰I as a global coordinate chart on ވ , where I is the identity element of އ. We record the integration formula Z Z 2 1=2 f .x/ d.x/ f .‰I .v//.1 v / dv (2-6) ވd D ޒd C j j d for any f C0.ވ /. 2 The Fourier transform on hyperbolic spaces. The Fourier transform (as defined by Helgason [1965] in the more general setting of symmetric spaces) takes suitable functions defined on ވd to functions defined on ޒ ޓd 1. For ! ޓd 1 and ރ, let b.!/ .1;!/ ޒd 1 and 2 2 D 2 C h ވd ރ; h .x/ Œx; b.!/i ; ;! W ! ;! D where .d 1/=2: D It is known that 2 2 gh . /h ; (2-7) ;! D C ;! d d where g is the Laplace–Beltrami operator on ވ . The Fourier transform of f C0.ވ / is defined by 2 the formula Z Z i f .; !/ f .x/h;!.x/ d f .x/Œx; b.!/ d: (2-8) Q D ވd D ވd This transformation admits a Fourier inversion formula: if f C .ވd / then 2 01 Z Z 1 i 2 f .x/ f .; !/Œx; b.!/ c./ d d!; (2-9) D d 1 Q j j 0 ޓ where, for a suitable constant C , .i/ c./ C D . i/ C d d 1 is the Harish-Chandra c-function corresponding to ވ , and the invariant measure of ޓ is normalized to 1. It follows from (2-7) that

2 2 gf .; !/ . /f .; !/: (2-10) D C Q

A WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 711

We record also the nontrivial identity Z Z i i f .; !/Œx; b.!/ d! f . ; !/Œx; b.!/ d! d 1 Q D d 1 Q ޓ ޓ d d for any f C01.ވ /, ރ, and x ވ . 2 2 2 2 d According to the Plancherel theorem, the Fourier transform f fQ extends to an isometry of L .ވ / 2 d 1 2 ! onto L .ޒ ޓ ; c./ d d!/; moreover C j j Z Z 1 2 f1.x/f2.x/ d f1.; !/f2.; !/ c./ d d!; (2-11) ވd D 2 ޒ ޓd 1 z z j j 2 d for any f1; f2 L .ވ /. As a consequence, any bounded multiplier m ޒ ރ deﬁnes a bounded 2 2 d W C ! operator Tm on L .ވ / by the formula

Tm.f /.; !/ m./ f .; !/: (2-12) D Q The question of Lp boundedness of operators deﬁned by multipliers as in (2-12) is more delicate if p d p 2. A necessary condition for boundedness on L .ވ / of the operator Tm is that the multiplier m ¤ extends to an even analytic function in the interior of the region ᐀p ރ < 2=p 1 [Clerc D f 2 W j= j j j g and Stein 1974]. Conversely, if p .1; / and m ᐀p ރ is an even analytic function which satisﬁes 2B1 W ! the symbol-type bounds

˛ ˛ @ m./ C.1 / for any ˛ Œ0; d 2 ޚ and ᐀p; (2-13) j j Ä C j j 2 C \ 2 p d then Tm extends to a bounded operator on L .ވ / [Stanton and Tomas 1978]. As in Euclidean spaces, there is a connection between convolution operators in hyperbolic spaces and multiplication operators in the Fourier space. To state this connection precisely, we normalize first the Haar measures on ދ and އ such that R 1 dk 1 and ދ D Z Z f .g 0/ dg f .x/ d އ D ވd d for any f C0.ވ /. Given two functions f1; f2 C0.އ/ we define the convolution 2 2 Z 1 .f1 f2/.h/ f1.g/f2.g h/ dg: (2-14) D އ A function K އ ރ is called ދ-biinvariant if W ! K.k1gk2/ K.g/ for any k1; k2 ދ: (2-15) D 2 Similarly, a function K ވd ރ is called ދ-invariant (or radial) if W ! K.k x/ K.x/ for any k ދ and x ވd : (2-16) D 2 2 d If f; K C0.ވ / and K is ދ-invariant then we define (compare to (2-14)) 2 Z 1 .f K/.x/ f .g 0/K.g x/ dg: (2-17) D އ 712 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

If K is ދ-invariant then the Fourier transform formula (2-8) becomes Z K.; !/ K./ K.x/ˆ .x/ d; (2-18) z D z D ވd where Z i ˆ.x/ Œx; b.!/ d! (2-19) D d 1 ޓ is the elementary spherical function. The Fourier inversion formula (2-9) becomes Z 1 2 K.x/ K./ˆ.x/ c./ d; (2-20) D 0 z j j for any ދ-invariant function K C .ވd /. With the convolution defined as in (2-17), we have the 2 01 important identity .f K/.; !/ f .; !/ K./ (2-21) D Q z d 5 for any f; K C0.ވ /, provided that K is ދ-invariant . 2 We define now the inhomogeneous Sobolev spaces on ވd . There are two possible definitions: using the Riemannian structure g or using the Fourier transform. These two definitions agree. In view of (2-10), for s ރ we define the operator . /s=2 as given by the Fourier multiplier .2 2/s=2. For 2 B ! C p .1; / and s ޒ we define the Sobolev space W p;s.ވd / as the closure of C .ވd / under the norm 2 1 2 01 s=2 f p;s d . / f p d : k kW .ވ / D k kL .ވ / For s ޒ let H s W 2;s. This definition is equivalent to the usual definition of the Sobolev spaces 2 D s=2 on Riemannian manifolds (this is a consequence of the fact that the operator . g/ is bounded on Lp.ވd / for any s ރ, s 0, since its symbol satisfies the differential inequalities (2-13)). In particular, 2 < Ä for s 1 and p .1; /, D 2 1 Â Z Ã1=p 1=2 p f W p;1.ވd / . / f Lp.ވd / p gf d ; (2-22) k k D k k ވd jr j where ˛ 1=2 gf D f D˛f : jr j WD j Nj We record also the Sobolev embedding theorem

W p;s , Lq if 1 < p q < and s d=p d=q: (2-23) ! Ä 1 D Dispersive estimates. Most of our perturbative analysis in the paper is based on the Strichartz estimates for the linear Schrödinger ﬂow. For any H s.ވd /, s ޒ, let eitg C.ޒ H s.ވd // denote the 2 2 2 W solution of the free Schrödinger evolution with data , i.e.,

2 2 eitg .; !/ .; !/ e it. /: D z C 5Unlike in Euclidean Fourier analysis, there is no simple identity of this type without the assumption that K is ދ-invariant.

B WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 713

The main inequality we need is the dispersive estimate6 (see [Anker and Pierfelice 2009; Banica 2007; Banica et al. 2008; Ionescu and Stafﬁlani 2009; Pierfelice 2008])

itg d.1=p 1=2/ e p p . t ; p Œ2d=.d 2/; 2; p p=.p 1/; (2-24) L L 0 0 k k ! j j 2 C D for any t ޒ 0 . The Strichartz estimates below then follow from a general theorem from [Keel and 2 n f g Tao 1998]. Proposition 2.1 (Strichartz estimates). Assume that d 3 and I .a; b/ ޒ is a bounded open interval. D Â (i) If L2.ވd / then 2 itg . e 2 2 2d=.d 2/ d L2 : (2-25) .L1t Lx Lt Lx /.ވ I/ k k \ k k 1 2 2 2d=.d 2/ d (ii) If F .L L L Lx C /.ވ I/ then 2 t x C t Z t i.t s/g . e F.s/ ds F .L1L2 L2L2d=.d 2//.ވd I/: (2-26) 2 2 2d=.d 2/ d k k t x t x C a .L1t Lx Lt Lx /.ވ I/ C \ To exploit these estimates in dimension d 3, for any interval I ޒ and f C.I H 1.ވ3// we D Â 2 W deﬁne f Z.I/ f 10 3 ; Lt;x .ވ I/ k k WD k k k=2 f k . / f 2 2 6 3 ; k Œ0; /; (2-27) S .I/ .L1t Lx Lt Lx /.ވ I/ k k WD k k \ 2 1 k=2 f N k .I/ . / f 1 2 2 6=5 3 ; k Œ0; /: .Lt Lx Lt Lx /.ވ I/ k k WD k k C 2 1 We use the S 1 norms to estimate solutions of linear and nonlinear Schrödinger equations. Nonlinearities are estimated using the N 1 norms. The L10 norm is the “scattering” norm, which controls the existence of strong solutions of the nonlinear Schrödinger equation, see Proposition 3.1 and Proposition 3.2 below.

Some lemmas. In this subsection we collect and prove several lemmas that will be used later in the paper. 2 3 2 3 For N > 0 we deﬁne the operator PN L .ވ / L .ވ /, W ! 2 2 N g PN N ge ; WD (2-28) 2 2 N 2.2 1/ PN f .; !/ N . 1/e f .; !/: D C C Q One should think of PN as a substitute for the usual Littlewood–Paley projection operator in Euclidean spaces that restricts to frequencies of size N ; this substitution is necessary in order to have a suitable Lp theory for these operators, since only real-analytic multipliers can deﬁne bounded operators on Lp.ވ3/ [Clerc and Stein 1974]. InA view of the Fourier inversion formula we have Z PN f .x/ f .y/PN .d.x; y// d.y/; D ވ3

6In fact this estimate can be improved if t 1, see [Ionescu and Staffilani 2009, Lemma 3.3]. This leads to better control j j of the longtime behavior of solutions of subcritical Schrödinger equations in hyperbolic spaces, compared to the behavior of solutions of the same equations in Euclidean spaces (see [Banica 2007; Banica et al. 2008; Ionescu and Staffilani 2009; Anker and Pierfelice 2009]). 714 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI where 3 5 4r PN .r/ . N .1 N r/ e : (2-29) j j C The estimates in the following lemma will be used in Section 5. 3 1 3 Lemma 2.2. (i) Given .0; 1 there is R 1 such that for any x ވ , N 1, and f H .ވ /, 2 2 2 1=2 PN f .x/ . N . f 1B.x;R N 1/ L6.ވ3/ f L6.ވ3// j j k k C k k where B.x; r/ denotes the ball B.x; r/ y ވ3 d.x; y/ < r . D f 2 W g (ii) For any f H 1.ވ3/, 2 1=3 1=2 2=3 f 6 3 . f sup N P f .x/ : L .ވ / L2.ވ3/ N k k kr k N 1 j j x ވ3 2 Proof. (i) The inequality follows directly from (2-29): Z Z PN f .x/ . f .y/ PN .d.x; y// d.y/ f .y/ PN .d.x; y// d.y/ j j 1 j j j j C c 1 j j j j B.x;RN / B.x;RN / . f 1B.x;R N 1/ L6.ވ3/ AN;0;6=5 f L6.ވ3/ AN;R;6=5; k k C k k where, for R Œ0; /, N Œ1; / and p Œ1; 2 2 1 2 1 2 ÂZ Ã1=p ÂZ Ã1=p p 1 p 2 AN;R;p PN .d.0; y// d.y/ . PN .r/ .sh r/ dr WD d.0;y/ RN 1 j j RN 1 j j ÂZ Ã1=p 3 1 5p 2 3 3=p 1 . N .1 N r/ r dr . N .1 R/ : 1 C C RN The inequality follows if R 1=. D (ii) Such improved Sobolev embeddings in various settings have been used before, for example, in [Bahouri and Gérard 1999; Keraani 2001]. For any f H 1.ވ3/ we have the identity 2 Z 1 1 f c N PN .f / dN: (2-30) D N 0 D 1=2 Thus, with A supN 0 N PN f L .ވ3/ WD k k 1 Z Z Z dN dN 6 . 1 6 f d PN1 f ::: PN6 f ::: d 3 3 ވ j j ވ 0 N1 ::: N6 j j j j N1 N6 Ä Ä Ä Z Z dN dN . 4 2 5 6 A N5 PN5 f PN6 f d 3 ވ 0 N5 N6 j jj j N5 N6 Z Z Ä Ä 4 1 2 . A N PN f dNd; ވ3 0 j j where the last inequality follows by Schur’s lemma. The claim follows since Z Z 1 2 1=2 2 N PN f dNd c . / f L2.ވ3/; ވ3 0 j j D k k WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 715 as a consequence of the Plancherel theorem and the definition of the operators PN , and, for any N Œ0; 1/, 2

1=2 N PN f 3 . P2f 3 : (2-31) L1.ވ / L1.ވ / k k k k We will also need the following technical estimate: Lemma 2.3. Assume H 1.ވ3/ satisﬁes 2 1=2 itg H 1.ވ3/ 1; sup K PK e .x/ ı; (2-32) k k Ä K 1 j j Ä t ޒ x 2ވ3 2 for some ı .0; 1. Then, for any R > 0 there is C.R/ 1 such that 2 1=2 itg 1=20 N ge 5 15=8 1 2 2 2 2 C.R/ı (2-33) Lt Lx .B.x0;RN / .t0 R N ;t0 R N // kr k C Ä 3 for any N 1, any t0 ޒ, and any x0 ވ . 2 2 Proof. We may assume R 1, x0 0, t0 0. It follows from (2-32) that for any K > 0 and t ޒ D D D 2 itg 1=2 itg PK e L .ވ3/ . ıK ; PK e L6.ވ3/ . 1 k k 1 k k I therefore, by interpolation, itg 1=2 1=4 PK e 12 3 . ı K : k kL .ވ / Thus, for any K > 0 and t ޒ, 2 itg 1=2 1=4 g.PK e / 12 3 . ı K .K 1/; kr kL .ވ / C which shows that, for any K > 0 and N 1, 1=2 itg 1=2 1=4 5=4 N g.PK e / 5 15=8 1 2 2 . ı K .K 1/N : (2-34) Lt Lx .B.0;N / . N ;N // kr k C We will prove below that, for any N 1 and K N , itg 1=2 g.PK e / 2 1 2 2 . .NK/ : (2-35) Lx;t .B.0;N / . N ;N // kr k Assuming this and using the energy estimate

itg g.PK e / 2 3 . 1; L1t Lx .ވ ޒ/ kr k we have, by interpolation,

itg 1=5 g.PK e / 5 2 1 2 2 . .NK/ : Lt Lx .B.0;N / . N ;N // kr k Therefore, for any N 1 and K N 1=2 itg 1=5 1=5 N g.PK e / 5 15=8 1 2 2 . N K : (2-36) Lt Lx .B.0;N / . N ;N // kr k The desired bound (2-33) follows from (2-34), (2-36), and the identity (2-30). It remains to prove the local smoothing bound (2-35). Many such estimates are known in more general settings; see, for example, [Doi 1996]. We provide below a simple self-contained proof specialized to 716 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

3 our case. Assuming N 1 ﬁxed, we will construct a real-valued function a aN C .ވ / with the D 2 1 properties

˛ 3 D aD˛a . 1 in ވ ; j j 3 3 g.ga/ . N in ވ ; (2-37) j j ˛ ˛ ˇ 3 3 X X˛ N 1B.0;N 1/ . X X D˛Dˇa in ވ for any vector-ﬁeld X T .ވ /: 2 Assuming such a function is constructed, we deﬁne the Morawetz action Z ˛ Ma.t/ 2 D a.x/ u.x/D˛u.x/ d.x/; D = ވ3 N

itg where u PK e . A formal computation (see [Ionescu and Staffilani 2009, Proposition 4.1] for a WD complete justification) shows that Z Z ˛ ˇ 2 @t Ma.t/ 4 D D a D˛uDˇu d g.ga/ u d: D < ވ3 N ވ3 j j Therefore, by integrating on the time interval Œ N 2; N 2 and using the first two properties in (2-37), 2 Z N Z ˛ ˇ 4 .D D a D˛uDˇu/ d dt N 2 ވ3 < N 2 Z N Z 2 2 sup Ma.t/ g.ga/ u d dt Ä 2 j j C N 2 ވ3 j j j j t N j jÄ 2 Z N . 3 2 . 1 2 sup u.t/ L2.ވ3/ u.t/ H 1.ވ3/ N u.t/ L2.ވ3/ dt K NK : 2 k k k k C 2 k k C t N N j jÄ The desired bound (2-35) follows, in view of the inequality in the last line of (2-37) and the assumption K N since a is real valued. Finally, it remains to construct a real-valued function a C .ވ3/ satisfying (2-37). We are looking 2 1 for a function of the form

a.x/ a.ch r.x//; r d.0; x/; a C .Œ1; //: (2-38) WD Q D Q 2 1 1 To prove the inequalities in (2-37) it is convenient to use coordinates induced by the Iwasawa decomposition of the group އ: we deﬁne the global diffeomorphism

ˆ ޒ2 ޒ ވ3; ˆ.v1; v2; s/ tr.ch s e s v 2=2; sh s e s v 2=2; e sv1; e sv2/; W ! D C j j C j j and ﬁx the global orthonormal frame

s s e3 @s; e1 e @ 1 ; e2 e @ 2 : WD WD v WD v With respect to this frame, the covariant derivatives are

De e ı e3; De e3 e˛; De e˛ De e3 0 for ˛; ˇ 1; 2: ˛ ˇ D ˛ˇ ˛ D 3 D 3 D D WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 717

See [Ionescu and Stafﬁlani 2009, Section 2] for these calculations. In this system of coordinates we have

ch r ch s e s v 2=2: (2-39) D C j j Therefore, for a as in (2-38), we have

s 2 1 2 D3a .sh s e v =2/ a .ch r/; D1a v a .ch r/; D2a v a .ch r/: D j j Q0 D Q0 D Q0 Using the formula

D˛D a e˛.e .a// .De e /.a/; ˛; ˇ 1; 2; 3; ˇ D ˇ ˛ ˇ D we compute the Hessian:

1 2 2 2 D1D1a .v / a .ch r/ ch ra .ch r/; D2D2a .v / a .ch r/ ch ra .ch r/; D Q00 C Q0 D Q00 C Q0 1 2 s 2 2 D1D2a D2D1a v v a .ch r/; D3D3f .sh s e v =2/ a .ch r/ ch ra .ch r/; D D Q00 D j j Q00 C Q0 1 s 2 D1D3a D3D1a v .sh s e v =2/a .ch r/; D D j j Q00 2 s 2 D2D3a D3D2a v .sh s e v =2/a .ch r/: D D j j Q00 Therefore, using again (2-39),

˛ 2 2 2 D aD˛a .sh r/ .a .ch r// ; ga ..ch r/ 1/a .ch r/ 3.ch r/a .ch r/; (2-40) D Q0 D Q00 C Q0 and

˛ ˇ 2 1 1 2 2 3 s 2 2 X X D˛D a ch ra .ch r/ X a .ch r/.X v X v X .sh s e v =2// : (2-41) ˇ D Q0 j j CQ00 C C j j We ﬁx now a such that Q a .y/ .y2 1 N 2/ 1=2; y Œ1; /: Q0 WD C 2 1 The ﬁrst identity in (2-37) follows easily from (2-40). To prove the second identity in (2-37), we use again (2-40) to derive

2 2 1=2 2 2 2 3=2 ga b.ch r/; where b.y/ 3y.y 1 N / y.y 1/.y 1 N / : D D C C Using (2-40) again, it follows that

2 2 2 3=2 g.ga/ . y .y 1 N / where y ch r; j j C D which proves the second inequality in (2-37). Finally, using (2-41),

˛ ˇ 2 2 2 X X D˛D a ch ra .ch r/ X ..ch r/ 1/ a .ch r/ X ˇ Q0 j j jQ00 j j j N 2 ch r..ch r/2 1 N 2/ 3=2 X 2; D C j j which proves the last inequality in (2-37). This completes the proof of the lemma. 718 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

3. Proof of the main theorem

In this section we outline the proof of Theorem 1.1. The main ingredients are a local well-posedness and stability theory for the initial-value problem, which in our case relies only on the Strichartz estimates in Proposition 2.1, a global Morawetz inequality, which exploits the defocusing nature of the problem, and a compactness argument, which depends on the Euclidean analogue of Theorem 1.1 proved in [Colliander et al. 2008]. We start with the local well-posedness theory. Let

ᏼ .I; u/ I ޒ is an open interval and u C.I H 1.ވ3// D f W Â 2 W g with the natural partial order

.I; u/ .I ; u / if and only if I I and u .t/ u.t/ for any t I: Ä 0 0 Â 0 0 D 2 Proposition 3.1 (local well-posedness). Assume H 1.ވ3/. Then there is a unique maximal solution 2 .I; u/ .I./; u.// ᏼ, 0 I, of the initial-value problem D 2 2 4 .i@t g/u u u ; u.0/ (3-1) C D j j D 3 0 1 on ވ I. The mass E .u/ and the energy E .u/ deﬁned in (1-3) are constant on I, and u 1 < k kS .J / 1 for any compact interval J I. In addition, Â u Z.I / if I I Œ0; / is bounded; k k C D 1 C WD \ 1 (3-2) u Z.I / if I I . ; 0 is bounded: k k D 1 WD \ 1 In other words, local-in-time solutions of the equation exist and extend as strong solutions as long as 10 their spacetime Lx;t norm does not blow up. We complement this with a stability result. Proposition 3.2 (stability). Assume I is an open interval, Œ 1; 1, and u C.I H 1.ވ3// satisﬁes 2 Q 2 W the approximate Schrödinger equation

4 3 .i@t g/u u u e on ވ I: C Q D Qj Qj C Assume in addition that

u 10 3 sup u.t/ 1 3 M; (3-3) Lt;x .ވ I/ H .ވ / k Qk C t I k Q k Ä 2 1 3 for some M Œ1; /. Assume t0 I and u.t0/ H .ވ / is such that the smallness condition 2 1 2 2 u.t0/ u.t0/ 1 3 e 1 (3-4) k Q kH .ވ / C k kN .I/ Ä holds for some 0 < < 1, where 1 1 is a small constant 1 1.M / > 0. Ä D Then there exists a solution u C.I H 1.ވ3// of the Schrödinger equation 2 W 4 3 .i@t g/u u u on ވ I; C D j j and

u S 1.ވ3 I/ u S 1.ވ3 I/ C.M /; u u S 1.ވ3 I/ C.M /: (3-5) k k C k Qk Ä k Qk Ä WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 719

Both Proposition 3.1 and Proposition 3.2 are standard consequences of the Strichartz estimates and Sobolev embedding theorem (2-23); see, for example, [Colliander et al. 2008, Section 3]. We will use Proposition 3.2 with 0 and with 1 to estimate linear and nonlinear solutions on hyperbolic spaces. D D We next state the global Morawetz estimate: Proposition 3.3 [Ionescu and Staffilani 2009, Proposition 4.1]. Assume that I ޒ is an open interval, Â and u C.I H 1.ވ3// is a solution of the equation 2 W 4 3 .i@t g/u u u on ވ I: C D j j Then, for any t1; t2 I, 2 6 u 6 3 . sup u.t/ 2 3 u.t/ 1 3 : (3-6) L .ވ Œt1;t2/ L .ވ / H .ވ / k k t Œt1;t2 k k k k 2 Next, recall the conserved energy E1.u/ defined in (1-3). For any E Œ0; / let S.E/ be defined by 2 1 S.E/ sup u ; E1.u/ E ; D fk kZ.I/ Ä g where the supremum is taken over all solutions u C.I H 1.ވ3// defined on an interval I and of energy 2 W less than E. We also define

Emax sup E; S.E/ < : D f 1g Using Proposition 3.2 with u 0; e 0, I ޒ, M 1, 1, one checks that Emax > 0. It follows Q Á Á D D from Proposition 3.1 that if u is a solution of (1-2) and E.u/ < Emax, then u can be extended to a globally deﬁned solution which scatters.

If Emax , then Theorem 1.1 is proved, as a consequence of Propositions 3.1 and 3.2. If we D C1 assume that Emax < , then, there exists a sequence of solutions satisfying the hypothesis of the C1 following key proposition, to be proved later. Proposition 3.4. Let u C.. T ; T k / H 1.ވ3//, k 1; 2;::: , be a sequence of nonlinear solutions k 2 k W D of the equation 4 .i@t g/u u u ; C D j j k k deﬁned on open intervals . T ; T / such that E.u / Emax. Let t . T ; T / be a sequence of k k ! k 2 k times with

k lim uk Z. Tk ;tk / lim uk Z.tk ;T / : (3-7) k k k D k k k D C1 !1 !1 1 3 Then there exists w0 H .ވ / and a sequence of isometries hk އ such that, up to passing to a 21 1 2 subsequence, u .t ; h x/ w0.x/ H strongly. k k k ! 2 Using these propositions we can now prove our main theorem.

Proof of Theorem 1.1. Assume for contradiction that Emax < . Then, we ﬁrst claim that there exists a 1 C1 solution u C.. T ; T / H / of (1-2) such that 2 W E.u/ E and u u : (3-8) max Z. T ;0/ Z.0;T / D k k D k k D C1 720 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

Indeed, by hypothesis, there exists a sequence of solutions u deﬁned on intervals I . T ; T k / k k D k satisfying E.u / Emax and k Ä u : k k kZ.Ik / ! C1 But this is exactly the hypothesis of Proposition 3.4, for suitable points t . T ; T k /. Hence, up to k 2 k a subsequence, we get that there exists a sequence of isometries hk އ such that hk .uk .tk // w0 1 1 3 2 ! strongly in H . Now, let u C.. T ; T / H .ވ // be the maximal solution of (3-1) with initial 2 W data w0, in the sense of Proposition 3.1. By the stability theory Proposition 3.2, if u Z.0;T / < , k k C1 then T and uk Z.tk ; / C. u Z.0; // which is impossible. Similarly, we see that D C1 k k C1 Ä k k C1 u Z. T ;0/ , which completes the proof of (3-8). k k D C1 We now claim that the solution u obtained in the previous step can be extended to a global solution.

Indeed, using Proposition 3.1, it sufﬁces to see that there exists ı > 0 such that, for all times t . T ; T /, 2 u 1: Z..t ı;t ı/ . T ;T // k k C \ Ä

If this were not true, there would exist a sequence ık 0 and a sequence of times tk . T ık ; T ık / ! 2 C such that

u Z.tk ık ;tk ık / 1: (3-9) k k C Applying Proposition 3.4 with u u, we see that, up to a subsequence, .u .t // w strongly in k D hk k k ! H 1 for some translations h އ. We consider z the maximal nonlinear solution with initial data w, then k 2 by the local theory Proposition 3.1, there exists ı > 0 such that

1 z Z. ı;ı/ 2 : k k Ä

Proposition 3.2 gives that u Z.tk ı;tk ı/ 1=2 ok .1/, which again contradicts our hypothesis (3-9). k k C Ä C 1 In other words, we proved that if Emax < then there is a global solution u C.ޒ H / of (1-2) such 1 2 W that

E.u/ Emax and u Z. ;0/ u Z.0; / : D k k 1 D k k 1 D C1 We claim now that there exists ı > 0 such that for all times,

u.t/ 6 ı: (3-10) k kL Indeed, otherwise, we can ﬁnd a sequence of times t .0; / such that u.t / 0 in L6. Applying k 2 1 k ! again Proposition 3.4 to this sequence, we see that, up to a subsequence, there exist h އ such that k 2 .u.t // w in H 1 with w 0. But this contradicts conservation of energy. hk k ! D But now we have a contradiction with the Morawetz estimate (3-6), which shows that Emax as D C1 desired. Propositions 3.1 and 3.2 are standard consequences of the Strichartz estimates, while Proposition 3.3 was proved in [Ionescu and Stafﬁlani 2009]. Therefore it only remains to prove Proposition 3.4. We collect the main ingredients in the next two sections and complete the proof of Proposition 3.4 in Section 6. WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 721

4. Euclidean approximations

In this section we prove precise estimates showing how to compare Euclidean and hyperbolic solutions of both linear and nonlinear Schrödinger equations. Since the global Euclidean geometry and the global hyperbolic geometry are quite different, such a comparison is meaningful only in the case of rescaled data that concentrate at a point. We fix a spherically symmetric function C .ޒ3/ supported in the ball of radius 2 and equal to 1 2 01 in the ball of radius 1. Given H 1.ޒ3/ and a real number N 1 we define 2 P 3 1=2 =N QN C .ޒ /; .QN /.x/ .x=N / .e /.x/; 2 01 D 3 1=2 N C .ޒ /; N .x/ N .QN /.N x/; (4-1) 2 01 D 3 1 fN C .ވ /; fN .y/ N .‰ .y//; 2 01 D I 7 where ‰I is defined in (2-4). Thus QN is a regularized, compactly supported modification of the profile , is an H 1-invariant rescaling of Q , and f is the function obtained by transferring N P N N N to a neighborhood of 0 in ވ3. We define also Z Z 1 1 2 1 6 Eޒ3 ./ dx dx: D 2 ޒ3 jr j C 6 ޒ3 j j We will use the main theorem of [Colliander et al. 2008], in the following form. Theorem 4.1. Assume H 1.ޒ3/. Then there is a unique global solution v C.ޒ H 1.ޒ3// of the 2 P 2 W P initial-value problem 4 .i@t /v v v ; v.0/ ; (4-2) C D j j D and 1 v 2 2 6 3 C .E 3 . //: (4-3) L1t Lx Lt Lx .ޒ ޒ/ z ޒ jr j \ Ä This solution scatters in the sense that there exists H 1.ޒ3/ such that ˙1 2 P it v.t/ e ˙1 H 1.ޒ3/ 0 (4-4) k k P ! 5 3 5 3 as t . If H .ޒ /, then v C.ޒ H .ޒ // and sup v.t/ 5 3 . 1: H .ޒ / H 5.ޒ3/ ! ˙1 2 2 W t ޒ k k k k 2 The main result in this section is the following lemma:

1 3 Lemma 4.2. Assume H .ޒ /, T0 .0; /, and 0; 1 are given, and define fN as in (4-1). 2 P 2 1 2 f g (i) There is N0 N0.; T0/ sufficiently large such that for any N N0 there is a unique solution D 2 2 1 3 UN C.. T0N ; T0N / H .ވ // of the initial-value problem 2 W 4 .i@t g/UN UN UN ; UN .0/ fN : (4-5) C D j j D 7This modification is useful to avoid the contribution of coming from the Euclidean infinity, in a uniform way depending on the scale N . 722 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

Moreover, for any N N0,

1 2 2 . 1 UN S . T0N ;T0N / E ./ 1: (4-6) k k ޒ3 (ii) Assume " .0; 1 is sufﬁciently small (depending only on E1 ./), and let H 5.ޒ3/ satisfy 1 ޒ3 0 2 5 2 0 H 1.ޒ3/ "1. Let v0 C.ޒ H / denote the solution of the initial-value problem k k P Ä 2 W 4 .i@t /v v v ; v .0/ : C 0 D 0j 0j 0 D 0 For R; N 1 we deﬁne 3 v .x; t/ .x=R/v .x; t/; .x; t/ ޒ . T0; T0/; R0 D 0 2 1=2 2 3 2 2 v .x; t/ N v .N x; N t/; .x; t/ ޒ . T0N ; T0N /; (4-7) R0 ;N D R0 2 1 3 2 2 VR;N .y; t/ v .‰ .y/; t/.y; t/ ވ . T0N ; T0N /: D R0 ;N I 2 Then there is R0 1 (depending on T0 and and "1) such that, for any R R0, 0

1 2 2 . 1 lim sup UN VR;N S . T0N ;T0N / E ./ "1: (4-8) N k k ޒ3 !1 Proof. All of the constants in this proof are allowed to depend on E1 ./; for simplicity of notation we ޒ3 will not track this dependence explicitly. Using Theorem 4.1 we have

v0 2 2 6 3 . 1; sup v0.t/ 5 3 . 1: (4-9) .L1t Lx Lt Lx /.ޒ ޒ/ H .ޒ / 0 H 5.ޒ3/ kr k \ t ޒ k k k k 2

We will prove that for any R0 sufﬁciently large there is N0 such that VR0;N is an almost-solution of (4-5), for any N N0. We will then apply Proposition 3.2 to upgrade this to an exact solution of the initial-value problem (4-5) and prove the lemma. Let x Á x Á5Á e .x; t/ .i@ /v v v 4.x; t/ v .x; t/ v .x; t/ 4 R t R0 R0 R0 R R 0 0 WD C j j D j 3 j 2 x Á 1 X x Á R v .x; t/./ 2R @j v .x; t/@j : C 0 R C 0 R j 1 D . Since v0.x; t/ 5 3 1, see (4-9), it follows that j j k 0kH .ޒ / 3 Â 3 3 Ã X . X X @k eR.x; t/ 3 3 1ŒR;2R. x / v0.x; t/ @k v0.x; t/ @k @j v0.x; t/ : j j k 0kH .ޒ / j j j j C j j C j j k 1 k 1 k;j 1 D D D Therefore

lim eR 2 2 3 0: (4-10) R jr j Lt Lx .ޒ . T0;T0// D !1 Letting 4 5=2 2 eR;N .x; t/ .i@t /v v v .x; t/ N eR.N x; N t/; WD C R0 ;N R0 ;N j R0 ;N j D WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 723 it follows from (4-10) that there is R0 1 such that, for any R R0 and N 1,

eR;N 1 2 3 2 2 "1: (4-11) Lt Lx .ޒ . T0N ;T0N // jr j Ä

1 With VR;N .y; t/ v .‰ .y/; t/ as in (4-7), let D R0 ;N I

4 ER;N .y; t/ .i@t g/VR;N VR;N VR;N .y; t/ WD C j j (4-12) 1 1 eR;N .‰ .y/; t/ gVR;N .y; t/ .v /.‰ .y/; t/: D I C R0 ;N I

To estimate the difference in the formula above, let @j , j 1; 2; 3, denote the standard vector-fields on 3 3 D ޒ and @j .‰I / .@j / and induced vector-fields on ވ . Using the definition (2-4) we compute z WD

vivj gij .y/ gy.@i; @j / ıij ; y ‰I .v/: WD z z D 1 v 2 D C j j Using the standard formula for the Laplace–Beltrami operator in local coordinates

1=2 1=2 ij gf g @i. g g @j f / D j j z j j z we derive the pointwise bound

3 ˇ 1 1 ˇ X 1 k 1 k ˇ gf .y/ .f ‰I /.‰ .y// ˇ . ‰ .y/ f .y/ ; rz ı I j I j jrz j k 1 D for any C 3 function f ވ3 ރ supported in the ball of radius 1 around 0, where, by deﬁnition, for W ! k 1; 2; 3 D k X ˇ k1 k2 k3 ˇ h.y/ ˇ@ @ @ h.y/ˇ: jrz j WD z1 z2 z3 k1 k2 k3 k C C D Therefore the identity (4-12) gives the pointwise bound

3 1 1 X X 1 k 1ˇ k1 k2 k3 1 ˇ ER;N .y; t/ . eR;N .‰ .y/; t/ ‰ .y/ ˇ@ @ @ v0 .‰ .y/; t/ˇ jrz j jr j I C j I j 1 2 3 R;N I k 1 k1 k2 k3 k D C C D 1 3 3=2 X ˇ k1 k2 k3 1 ˇ . eR;N .‰ .y/; t/ R N ˇ@ @ @ v0 .N.‰ .y/; t/ˇ: jr j I C 1 2 3 R I k1 k2 k3 1;2;3 C C 2f g

Using also (4-11), it follows that for any R0 sufﬁciently large there is N0 such that for any N N0

gER0;N 1 2 3 2 2 2"1: (4-13) Lt Lx .ވ . T0N ;T0N // jr j Ä 724 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

To verify the hypothesis (3-3) of Proposition 3.2, we use (4-9) and the integral formula (2-6) to estimate, for N large enough,

V 10 3 2 2 sup V .t/ 1 3 R0;N L .ވ . T0N ;T0N // R0;N H .ވ / x;t 2 2 k k C t . T0N ;T0N / k k 2 . v 10 3 2 2 sup v .t/ 2 3 R0 0;N L .ޒ . T0N ;T0N // R0 0;N L .ޒ / x;t 2 2 k k C t . T0N ;T0N / kr k 2 (4-14) v0 10 3 sup v0 .t/ 2 3 R0 Lx;t .ޒ . T0;T0// R0 L .ޒ / D k k C t . T0;T0/ kr k 2 . 1:

Finally, to verify the inequality on the first term in (3-4) we estimate, for R0; N large enough, . fN VR0;N .0/ H 1.ވ3/ N vR0 ;N .0/ H 1.ޒ3/ QN vR0 .0/ H 1.ޒ3/ k k k 0 k P D k 0 k P (4-15) QN H 1.ޒ3/ 0 H 1.ޒ3/ 0 vR0 .0/ H 1 3"1: Ä k k P C k k P C k 0 k P Ä The conclusion of the lemma follows from Proposition 3.2, provided that "1 is fixed sufficiently small depending on E1 ./. ޒ3 As a consequence, we have: Corollary 4.3. Assume H 1.ޒ3/, " > 0, I ޒ is an interval, and 2 P Â it .e / p q 3 "; (4-16) Lt Lx .ޒ I/ jr j Ä where 2=p 3=q 3=2, q .2; 6. For N 1 we define, as before, C D 2 1=2 =N 1=2 1 .QN /.x/ .x=N / .e /.x/; N .x/ N .QN /.N x/; N .y/ N .‰ .y//: D D z D I Then there is N1 N1. ; "/ such that, for any N N1, D itg g.e N / p q 3 2 .q ": (4-17) z Lt Lx .ވ N I/ jr j Proof. As before, the implicit constants may depend on E1 . /. We may assume that C .ޒ3/. ޒ3 2 01 Using the dispersive estimate (2-24), for any t 0, ¤ 1=2 it 3=q 3=2 1=2 3=q 3=2 g q . . . g/ .e N / L .ވ3/ t . g/ N q 3 t N q 3 k z k x j j k z kLx0 .ވ / j j jr j Lx0 .ޒ / . t 3=q 3=2N 3=q 3=2: j j Thus, for T1 > 0,

itg 1=p g.e N / p q 3 2 2 . T : z Lt Lx .ވ Œޒ . T1N ;T1N // 1 jr j n Therefore we can ﬁx T1 T1. ; "/ such that, for any N 1, D itg g.e N / p q 3 2 2 .q ": z Lt Lx .ވ Œޒ . T1N ;T1N // jr j n 2 2 2 The desired bound on the remaining interval N I . T1N ; T1N / follows from Lemma 4.2(ii) \ with 0. D WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 725

5. Proﬁle decomposition in hyperbolic spaces

In this section we show that given a bounded sequence of functions f H 1.ވ3/ we can construct certain k 2 profiles and express the functions fk in terms of these profiles. In other words, we prove the analogue of Keraani’s theorem [2001] in hyperbolic geometry. 2 3 Given .f; t0; h0/ L .ވ / ޒ އ we define 2 … f .x/ .e it0g f /.h 1x/ . e it0g f /.x/: (5-1) t0;h0 D 0 D h0 As in Section 4 — see (4-1) — given H 1.ޒ3/ and N 1, we define 2 P 1=2 1 1=2 =N TN .x/ N .N ‰ .x//; where .y/ .y=N / .e /.y/; (5-2) WD z I z WD and observe that

1 3 1 3 . TN H .ޒ / H .ވ / is a bounded linear operator with TN H 1.ވ3/ H 1.ޒ3/ : (5-3) W P ! k k k k P Deﬁnition 5.1. (1) We deﬁne a frame to be a sequence ᏻ .N ; t ; h / Œ1; / ޒ އ, k 1; 2;::: , k D k k k 2 1 D where N 1 is a scale, t ޒ is a time, and h އ is a translation element. We also assume that k k 2 k 2 either Nk 1 for all k (in which case we call ᏻk k 1 a hyperbolic frame) or that Nk (in D f g % 1 which case we call ᏻk k 1 a Euclidean frame). Let Ᏺe denote the set of Euclidean frames, f g ˚ « Ᏺe ᏻ .Nk ; tk ; hk / k 1 Nk Œ1; /; tk ޒ; hk އ; Nk ; D D f g W 2 1 2 2 % 1

and let Ᏺh denote the set of hyperbolic frames, ˚ « Ᏺh ᏻ .1; tk ; hk / k 1 tk ޒ; hk އ : D z D f g W 2 2

(2) We say that two frames .Nk ; tk ; hk / k 1 and .Nk0 ; tk0 ; hk0 / k 1 are orthogonal if f g f g 2 lim ln.Nk =Nk / Nk tk tk0 Nk d.hk 0; hk0 0/ : (5-4) k j 0 j C j j C D C1 !1 Two frames that are not orthogonal are called equivalent. 1 3 (3) Given H .ޒ / and a Euclidean frame ᏻ ᏻk k 1 .Nk ; tk ; hk / k 1 Ᏺe, we deﬁne the 2 P D f g D f g 2 Euclidean proﬁle associated with .; ᏻ/ as the sequence , where zᏻk

… .TN /; (5-5) zᏻk WD tk ;hk k The operators … and T are defined in (5-1) and (5-2). 1 3 (4) Given H .ވ / and a hyperbolic frame ᏻ ᏻk k 1 .1; tk ; hk / k 1 Ᏺh we define the 2 z D fz g D f g 2 hyperbolic profile associated with . ; ᏻ/ as the sequence ᏻ , where z zzk

ᏻ …tk ;hk : (5-6) zzk WD 726 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

Deﬁnition 5.2. We say a sequence .f / bounded in H 1.ވ3/ is absent from a frame ᏻ .N ; t ; h / k k D f k k k gk if its localization to ᏻ converges weakly to 0, i.e., if for all proﬁles associated to ᏻ, we have zᏻk

lim fk ; ᏻk H 1 H 1.ވ3/ 0: (5-7) k h z i D !1 Remark 5.3. (i) If ᏻ .1; t ; h / is a hyperbolic frame, this is equivalent to saying that D k k k … 1 fk * 0 tk ;h k as k in H 1.ވ3/. ! 1 (ii) If ᏻ is a Euclidean frame, this is equivalent to saying that for all R > 0

R 1=2 g .v/ .v=R/N … 1 fk .‰I .v=Nk // * 0 k k tk ;h D k as k in H 1.ޒ3/. ! 1 P We prove ﬁrst some basic properties of proﬁles associated to equivalent/orthogonal frames.

Lemma 5.4. (i) Assume ᏻk k 1 .Nk ; tk ; hk / k 1 and ᏻk0 k 1 .Nk0 ; tk0 ; hk0 / k 1 are two f g D f g f g D f g equivalent Euclidean frames (or hyperbolic frames), and H 1.ޒ3/ (or H 1.ވ3/). Then 2 P 2 there is H 1.ޒ3/ (or H 1.ވ3/) such that, up to a subsequence, 0 2 P 0 2

lim ᏻk 0 H 1.ވ3/ 0; (5-8) k kz zᏻk0 k D !1

where ᏻk ; 0 are as in Deﬁnition 5.1. z zᏻk0

(ii) Assume ᏻk k 1 .Nk ; tk ; hk / k 1 and ᏻk0 k 1 .Nk0 ; tk0 ; hk0 / k 1 are two orthogonal frames f g D f g f g D f g (either Euclidean or hyperbolic) and ᏻ ; are associated proﬁles. Then z k zᏻk0 ˇZ ˇ ˇ ˛ ˇ lim D D˛ d lim 3 3 0: (5-9) ˇ ᏻk ᏻk0 ˇ ᏻk ᏻk0 L .ވ / k ˇ ވ3 z z ˇ C k z z D !1 !1 (iii) If and are two Euclidean proﬁles associated to the same frame, then zᏻk zᏻk Z ˛ lim gᏻk ; g ᏻk L2 L2.ވ3/ lim D ᏻk D˛ ᏻk d k hr z r z i D k ވ3 z z !1 Z!1 .x/ .x/dx ; L2 L2.ޒ3/ D ޒ3 r r D hr r i

Proof. (i) The proof follows from the deﬁnitions if ᏻk k 1; ᏻk0 k 1 are hyperbolic frames: by passing f g f g 1 to a subsequence we may assume limk tk0 tk t and limk hk0 hk h, and deﬁne !1 C D N !1 D N

0 …t;h: WD N N

To prove the claim if ᏻk k 1; ᏻk0 k 1 are equivalent Euclidean frames, we decompose ﬁrst, using f g f g the Cartan decomposition (2-2)

1 h h m as n ; m ; n ދ; s Œ0; /: (5-10) k0 k D k k k k k 2 k 2 1 WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 727

Therefore, using the compactness of the subgroup ދ and the deﬁnition (5-4), after passing to a subsequence, we may assume that

2 lim Nk =Nk0 N ; lim Nk .tk tk0 / t; lim mk m; lim nk n; lim Nk sk s: (5-11) k D k D N k D k D k DN !1 !1 !1 !1 !1 We observe that for any N 1, H 1.ޒ3/, t ޒ, g އ, and q ދ 2 P 2 2 2 1 …t;gq.TN / …t;g.TN q/; where q.x/ .q x/: D D Therefore, in (5-10) we may assume that 1 m n I; h h as : k D k D k0 k D k With x .s; 0; 0/, we deﬁne N D N 1=2 it 1 3 .x/ N .e N /.N x x/; H .ޒ /; 0 WD N 0 2 P and deﬁne 0, 0 , and 0 as in (5-5). The identity (5-8) is equivalent to z zNk0 zᏻk0

i.t tk /g lim T 1 e k0 .T / 1 3 0: (5-12) N 0 0 h h Nk H .ވ / k k k k0 k k D !1 To prove (5-12) we may assume that C .ޒ3/, H 5.ޒ3/, and apply Lemma 4.2(ii) with 0. 0 2 01 2 D Let v.x; t/ .eit/.x/ and, for R 1, D 1=2 2 1 vR.x; t/ .x=R/v.x; t/; vR;N .x; t/ N vR.N x; N t/; VR;N .y; t/ vR;N .‰ .y/; t/: D k D k k k k D k I It follows from Lemma 4.2(ii) that for any " > 0 sufficiently small there is R0 sufficiently large such that, for any R R0, i.tk0 tk /g lim sup e .TNk / VR;Nk .tk0 tk / H 1.ވ3/ ": (5-13) k Ä !1 Therefore, to prove (5-12) it suffices to show that, for R large enough,

lim sup 1 .T 0/ VR;N .t 0 tk / 1 3 . "; hk h Nk0 k k H .ވ / k k0 !1 which, after examining the definitions and recalling that C .ޒ3/, is equivalent to 0 2 01 1=2 1 1 1=2 1 2 lim sup N .N ‰ .h h y// N vR.N ‰ .y/; N .t t // 1 3 . ": k0 0 k0 I k0 k k k I k k0 k Hy .ވ / k !1 After changing variables y ‰I .x/ this is equivalent to D 1=2 1 1 1=2 2 lim sup N .N ‰ .h h ‰I .x/// N vR.N x; N .t t // 1 3 . ": k0 0 k0 I k0 k k k k k0 k Hx .ޒ / k P !1 Since, by definition, .z/ N 1=2v.N z x; t/, this follows provided that 0 D N N 1 1 3 lim Nk ‰I .hk0 hk ‰I .x=Nk // x x for any x ޒ : k DN 2 !1 This last claim follows by explicit computations using (5-11) and the definition (2-4). 728 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

(ii) It sufﬁces to prove that one can extract a subsequence such that (5-9) holds. We analyze three cases: Case 1: ᏻ; ᏻ Ᏺ . We may assume that ; C .ވ3/ and select a subsequence such that either 0 2 h 2 01

lim tk tk0 (5-14) k j j D 1 or !1

lim tk tk0 t ޒ; lim d.hk 0; hk0 0/ : (5-15) k D N 2 k D 1 !1 !1 Using (2-24) it follows that

1 … 6 3 … .g/ 6 3 . .1 t / k t;h kL .ވ / C k t;h kL .ވ / C j j 1 … 6 3 … .g / 6 3 . .1 t / ; k t;h kL .ވ / C k t;h kL .ވ / C j j for any t ޒ and h އ. Thus 2 2 . 1 1 ᏻk 3 3 …t ;h 6 3 …t ;h 6 3 ; .1 tk / .1 t 0 / ; (5-16) z zᏻk0 L .ވ / Ä k k k kL .ވ / k k0 k0 kL .ވ / C j j C j k j and ˇZ ˇ ˇZ ˇ ˇZ ˇ ˇ ˛ ˇ ˇ ˇ ˇ i.tk t /g ˇ D D˛ d g d 1 e k0 .g/ d ˇ ᏻk ᏻk0 ˇ ˇ ᏻk ᏻk0 ˇ ˇ h h ˇ ˇ ވ3 z z ˇ D ˇ ވ3 z z ˇ D ˇ ވ3 k0 k ˇ

. i.tk tk0 /g . 1 h 1h e .g/ L6.ވ3/ L6=5.ވ3/ ; .1 tk tk0 / : k k0 k k k k C j j The claim (5-9) follows if the selected subsequence satisfies (5-14). If the selected subsequence satisfies (5-15) then, as before, ˇ Z ˇ ˇ ˛ ˇ D D˛ d ˇ ᏻk ᏻk0 ˇ ˇ ވ3 z z ˇ ˇZ ˇ ˇ i.tk tk0 /g ˇ ˇ h 1h e g dˇ D ˇ ވ3 k0 k ˇ Z . itg i.tk tk0 /g itg g L2.ވ3/ e N e L2.ވ3/ e N h 1h g d: k k k k C ވ3 j j j k k0 j The first limit in (5-9) follows. Using the bound (5-16), the second limit in (5-9) also follows, up to a subsequence, if lim supk tk . Otherwise, we may assume that limk tk T , limk tk0 j j D 1 !1 D !1 D T T t and estimate!1 0 D N itk g itk0 g ᏻk 3 3 e h e h 3 3 z zᏻk0 L .ވ / D k k k0 kL .ވ / itk g iTg . e e 6 3 ; k kL .ވ / itk0 g iT 0g iTg iT 0g e e L6.ވ3/ e h 1h .e / L3.ވ3/: C k k C k k k0 k The second limit in (5-9) follows in this case as well. 3 3 Case 2: ᏻ Ᏺ , ᏻ Ᏺe. We may assume that C .ވ / and C .ޒ /. We estimate 2 h 0 2 2 01 2 01 ˇ Z ˇ ˇZ ˇ ˇ ˛ ˇ ˇ ˇ 1 D D˛ d … .g/ … .T / d . T 2 3 . N ˇ ᏻk ᏻk0 ˇ ˇ tk ;hk tk0 ;hk0 Nk0 ˇ Nk0 L .ވ / ; k0 ˇ ވ3 z z ˇ D ˇ ވ3 ˇ k k WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 729 and

ᏻk ᏻ 3 3 …t ;h L .ވ3/ …t ;h .TN / 3 3 z z k0 L .ވ / Ä k k k k 1 k0 k0 k0 L .ވ / 1=4 1=2 . g 2 3 . g/ .TN / 2 3 .; N 0 : k kL .ވ / k0 L .ވ / k The limits in (5-9) follow. 3 Case 3: ᏻ; ᏻ Ᏺe. We may assume that ; C .ޒ / and select a subsequence such that either 0 2 2 01

lim Nk =Nk0 0; (5-17) k D !1 or 2 lim Nk =Nk0 N .0; /; lim Nk tk tk0 ; (5-18) k D 2 1 k j j D 1 !1 !1 or

2 lim Nk =Nk0 N .0; /; lim Nk .tk tk0 / t ޒ; lim Nk d.hk 0; hk0 0/ : (5-19) k D 2 1 k D N 2 k D 1 !1 !1 !1 Assuming (5-17) we estimate, as in Case 2, ˇZ ˇ ˇZ ˇ ˇ ˛ ˇ ˇ ˇ D D˛ d … .g.T // … .T / d ˇ ᏻk ᏻk0 ˇ ˇ tk ;hk Nk tk0 ;hk0 Nk0 ˇ ˇ ވ3 z z ˇ D ˇ ވ3 ˇ 1 . g.TN / 2 3 TN 2 3 .; Nk N 0 k k kL .ވ /k k0 kL .ވ / k and

ᏻk 3 3 …t ;h .TN / 9 3 …t ;h .TN / 9=2 3 z zᏻk0 L .ވ / Ä k k k L .ވ / k0 k0 k0 L .ވ / 7=12 5=12 1=6 1=6 . . g/ .TN / 2 3 . g/ .TN / 2 3 .; N N 0 : k L .ވ / k0 L .ވ / k k The limits in (5-9) follow in this case. To prove the limit (5-9) assuming (5-18), we estimate ﬁrst, using (2-24),

2 1 … .TN f / 6 3 . .1 N t / ; (5-20) k t;h kL .ވ / f C j j for any t ޒ, h އ, N Œ0; /, and f C .ޒ3/. Thus 2 2 2 1 2 01

ᏻk 3 3 …t ;h .TN / 6 3 …t ;h .TN / 6 3 z zᏻk0 L .ވ / Ä k k k L .ވ / k0 k0 k0 L .ވ / . .1 N 2 t / 1.1 N 2 t / 1; ; C k j k j C k0 j k0 j and ˇZ ˇ ˇZ ˇ ˇ ˛ ˇ ˇ ˇ D D˛ d g d ˇ ᏻk ᏻk0 ˇ ˇ ᏻk ᏻk0 ˇ ˇ ވ3 z z ˇ D ˇ ވ3 z z ˇ ˇZ ˇ ˇ i.tk t /g ˇ 1 e k0 .T / g.T / d ˇ h h Nk Nk0 ˇ D ˇ ވ3 k0 k ˇ

. i.tk tk0 /g 1 e .TNk / 6 3 g.TN / 6=5 3 hk0 hk L .ވ / k0 L .ވ / . .1 N 2 t t / 1: ; C k j k k0 j 730 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

The claim (5-9) follows if the selected subsequence veriﬁes (5-18). Finally, it remains to prove the limit (5-9) if the selected subsequence veriﬁes (5-19). For this we will use the following claim: if .gk ; Mk /k 1 އ Œ1; /, limk Mk , limk Mk d.gk 0; 0/ , 2 1 !1 D 1 !1 D 1 and f; g H 1.ޒ3/ then 2 P ˇZ ˇ ˇ 1=2 1=2 ˇ lim ˇ gk . g/ .TMk f / . g/ .TMk g/ dˇ gk .TMk f / .TMk g/ L3.ވ3/ 0: (5-21) k ˇ ވ3 ˇCk k D !1 Assuming this, we can complete the proof of (5-9). It follows from (5-12) that if f H 1.ޒ3/ and P 2 2 sk k 1 is a sequence with the property that limk Nk sk s ޒ then f g !1 DN 2 isk g lim e .TNk f / TN f 0 H 1.ވ3/ 0; (5-22) k k k0 k D !1 where f .x/ N 1=2.e isf /.N x/. We estimate 0 D N ˇZ ˇ ˇZ ˇ ˇ ˛ ˇ ˇ 1=2 i.tk t /g 1=2 ˇ D D˛ d . g/ 1 e k0 .T / . g/ .T / d ˇ ᏻk ᏻk0 ˇ ˇ h h Nk Nk0 ˇ ˇ ވ3 z z ˇ D ˇ ވ3 k0 k ˇ ˇZ ˇ ˇ 1=2 1=2 ˇ . . g/ 1 .T / . g/ .T / d ˇ h h Nk0 0 Nk0 ˇ ˇ ވ3 k0 k ˇ

i.tk t /g 1 e k0 .T / 1 .T / : H 1.ޒ3/ h h Nk h h N 0 0 H 1.ވ3/ C k k P k0 k k0 k k In view of (5-21) and (5-22), both terms in the expression above converge to 0 as k , as desired. If 2 ! 1 limk Nk tk then, using (5-20), we estimate !1 j j D 1 . 2 1 ᏻk 3 3 …t ;h .TN / 6 3 …t ;h .TN / 6 3 ; .1 N tk / ; kz zᏻk0 kL .ވ / Ä k k k k kL .ވ /k k0 k0 k0 kL .ވ / C k j j 2 which converges to 0 as k . Otherwise, up to a subsequence, we may assume that limk Nk tk ! 1 !1 D T ޒ, limk and write 2 !1 itk g it g 1 e .T / e k0 .T / : ᏻk ᏻ0 L3.ވ3/ h h Nk N 0 L3.ވ3/ z z k D k0 k k This converges to 0 as k , using (5-21) and (5-22), as desired. ! 1 It remains to prove the claim (5-21). In view of the H 1.ޒ3/ H 1.ވ3/ boundedness of the operators P 3 ! 1=2 1 TN , we may assume that f; g C01.ޒ / and replace TMk f and TMk g by Mk f .Mk ‰I .x// and 1=2 1 2 Mk g.Mk ‰I .x// respectively, up to small errors. Then we notice that the supports of these functions become disjoint for k sufficiently large (due to the assumption limk Mk d.gk 0; 0/ ). The limit !1 D 1 (5-21) follows. 3 (iii) By the boundedness of TN , it suffices to consider the case when ; C .ޒ /. In this case, we k 2 01 have 1=2 1 g TN N .Nk ‰ / 2 3 0 r k k I L .ވ / ! as k . Hence, by the unitarity of …tk ;hk , it suffices to compute ! 1 Z ˝ 1 1 ˛ lim Nk g .Nk ‰ / ; g .Nk ‰I / L2 L2.ވ3/ .x/ .x/dx; k r r D ޒ3 r r !1 which follows after a change of variables and use of the dominated convergence theorem. WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 731

Our main result in this section is the following.

1 3 Proposition 5.5. Assume that .fk /k 1 is a bounded sequence in H .ވ /. Then there are sequences 1 3 1 3 of pairs . ; ᏻ / H .ޒ / Ᏺe and . ; ᏻ / H .ވ / Ᏺ , ; 1; 2;::: , such that, up to a 2 P z 2 h D subsequence, for any J 1, X X J fk rk ; (5-23) D zᏻk C zᏻk C 1 J 1 J z Ä Ä Ä Ä 8 where and are the associated profiles in Definition 5.1, and zᏻ zᏻ k zk 1=2 itg J lim lim sup sup N PN e rk .x/ 0: (5-24) J k N 1 j j D !1 !1 t ޒ x 2ވ3 2 Moreover the frames ᏻ 1 and ᏻ 1 are pairwise orthogonal. Finally, the decomposition is f g fz g asymptotically orthogonal in the sense that ˇ ˇ ˇ 1 X 1 X 1 1 J ˇ lim lim sup E .f / E . / E . / E .r / 0; ˇ k zᏻ zᏻ k ˇ (5-25) J k ˇ k zk ˇ D !1 1 J 1 J !1 Ä Ä Ä Ä where E1 is the energy defined in (1-3). The profile decomposition in Proposition 5.5 is a consequence of the following finitary decomposition.

1 3 Lemma 5.6. Let .fk /k 1 be a bounded sequence of functions in H .ވ / and let ı .0; ı0 be sufficiently 2 2 small. Up to passing to a subsequence, the sequence .fk /k 1 can be decomposed into 2J 1 O.ı / C D terms X X fk rk ; (5-26) D zᏻk C zᏻk C 1 J 1 J z Ä Ä Ä Ä where and are Euclidean and hyperbolic profiles, respectively, associated to the sequences zᏻk zᏻ 1 3zk 1 3 . ; ᏻ / H .ޒ / Ᏺe and . ; ᏻ / H .ވ / Ᏺ as in Definition 5.1. 2 P z 2 h Moreover the remainder r is absent from all the frames ᏻ, ᏻ, 1 ; J and k z Ä Ä 1=2 itg lim sup sup N e PN rk .x/ ı: (5-27) k N 1 j j Ä !1 t ޒ x 2ވ3 2 In addition, the frames and are pairwise orthogonal, and the decomposition is asymptotically ᏻ ᏻz orthogonal in the sense that 2 X 2 X 2 2 gfk L2 g L2 g L2 grk L2 ok .1/ (5-28) kr k D kr zᏻk k C kr zᏻk k C kr k C 1 J 1 J z Ä Ä Ä Ä where o .1/ 0 as k . k ! ! 1 8 1=2 it It is convenient to use the critical norm N P e g f L to measure smallness of the remainder in (5-24), as it k N k N1;x;t already selects the parameters of the frames. Other critical norms have been used as well; see, for example, [Keraani 2001] and [Laurent 2011]. In any case, by Sobolev and Strichartz estimates, one obtains full control of the Z norm of the remainders, see (6-1). 732 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

We show first how to prove Proposition 5.5 assuming the finitary decomposition of Lemma 5.6. Proof of Proposition 5.5. We apply Lemma 5.6 repeatedly for ı 2 l , l 1; 2;::: and we obtain the D D result except for (5-25). To prove this, it suffices from (5-28) to prove the addition of the L6-norms. But from Lemma 2.2 and (5-24), we see that

J lim sup lim sup rk L6.ވ3/ 0 J k k k D !1 !1 so that ˇ 6 J 6 ˇ J 6 lim sup lim sup ˇ fk L6 fk rk L6 ˇ rk L6 0: (5-29) J k k k k k C k k D !1 !1 Now, for fixed J, we see that ˇ ˇ ˇˇ J ˇ6 X ˇ ˇ6 X ˇ ˇ6ˇ ˇˇfk rk ˇ ˇ ˇ ˇ ˇ ˇ ˇ zᏻk zᏻk ˇ 1 J 1 J z Ä Ä Ä Ä X ˇ ˛ ˇ ˇ ˇ ˇ5 X ˇ ˛ ˇ ˇ ˇ ˇ5 X ˇ ˇ ˇ ˇ5 ˇ ˇ5ˇ ˇÁ . ˛ J ˇ ᏻ ˇ ˇ ˇ ˇ ˇ ᏻ˛ ˇ ˇ ˇ ˇ ˇ ᏻ ˇ ˇ ᏻ ˇ ˇ ᏻ ˇ ˇ ᏻ ˇ z k zᏻk C z k zᏻk C z k z k C z k z k 1 ˛ ˇ J 1 ˛ ˇ J z z 1 ; J z z Ä ¤ Ä Ä ¤ Ä Ä Ä so that ˇ ˇ ˇ J 6 X 6 X 6 ˇ . X ˛ ˇ ˇ fk rk L6 L6 L6 ˇ J fk fk L3 ˇ zᏻk zᏻk ˇ 1 J 1 J z ˛;ˇ Ä Ä Ä Ä where the summation ranges over all pairs .f ˛; f ˇ/ of profiles such that f ˛ f ˇ and where we have k k k ¤ k used the fact that the L6 norm of each profile is bounded uniformly. From Lemma 5.4(ii), we see that this converges to 0 as k . The identity (5-25) follows using also (5-29). ! 1 1 3 Proof of Lemma 5.6. For .gk /k a bounded sequence in H .ވ /, we let

1 ˇ itg ˇ ı..gk /k / lim sup sup N 2 ˇPN e gk .h 0//ˇ: (5-30) D k N 1 !1 t ޒ h2އ 2 If ı..f / / ı, then we let J 0 and f r and Lemma 5.6 follows. Otherwise, we use inductively k k Ä D k D k the following:

1 3 ˛ Claim. Assume .gk /k is a bounded sequence in H .ވ / which is absent from a family of frames .ᏻ /˛ A Ä and such that ı..g / / ı. Then, after passing to a subsequence, there exists a new frame ᏻ which is k k 0 orthogonal to ᏻ˛ for all ˛ A and a proﬁle of free energy Ä zᏻk0

lim gᏻ L2 & ı (5-31) k kr z k0 k !1 ˛ such that gk is absent from the frames ᏻ0 and ᏻ , ˛ A. zᏻk0 Ä Once we have proved the claim, Lemma 5.6 follows by applying repeatedly the above procedure. Indeed, we let .f ˛/ be deﬁned as follows: .f 0/ .f / and if ı..f ˛/ / ı, then apply the above k k k k D k k k k WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 733

˛ claim to .fk /k to get a new sequence

˛ 1 ˛ fk C fk ˛ 1 : D zᏻkC ˛ ˇ By induction, .fk /k is absent from all the frames ᏻ , ˇ ˛. This procedure stops after a ﬁnite number 2 ˛ ˛ 1 Ä ˛ (O.ı /) of steps. Indeed, since f f ˛ is absent from ᏻ , we get from (5-7) that k D k zᏻk k ˛ 1 2 ˛ 2 2 ˛ ˛ ˛ gfk L2 gfk L2 gᏻ L2 2 fk ; ᏻ H 1 H 1.ވ3/ kr k D kr k C kr z k k C h z k i ˛ 2 2 gf 2 g ˛ 2 ok .1/ D kr k kL C kr zᏻk kL C and therefore by induction,

2 X 2 A 2 gf 2 g ˛ 2 gf 2 o .1/: kr k kL D kr zᏻ kL C kr k kL C k 1 ˛ A Ä Ä Since each proﬁle has a free energy & ı, this is a ﬁnite process and Lemma 5.6 follows.

Now we prove the claim. By hypothesis, there exists a sequence ᏻ .N ; t ; h / such that the zk D k k k k lim supk in (5-30) is greater than ı=2. If lim supk Nk , then, up to passing to a subsequence, !1 !1 D 1 we may assume that ᏻk k 1 ᏻ0 is a Euclidean frame. Otherwise, up to passing to a subsequence, we fz g D may assume that Nk N 1 and we let ᏻ0 .1; tk ; hk /k k 1 be a hyperbolic frame. In all cases, we ! D f g get a frame ᏻ0 .Mk ; tk ; hk /k k 1 such that D f g 1 1 2 ˇ itk g ˇ ˇ˝ 2 ˛ ˇ ı=2 lim N PN .e /gk .hk 0/ lim … 1 gk ; N PN .ı0/ 2 2 3 (5-32) k ˇ k ˇ ˇ tk ;h k k L L .ވ /ˇ Ä k D k k !1 !1 for some sequence Nk comparable to Mk . Now, we claim that there exists a proﬁle f associated to the frame ᏻ such that Qᏻk0 0

lim sup gf 2 . 1 Qᏻk0 L k kr k !1 and 5 2 2 Nk g … 1 fᏻ N e .ı0/ 0 tk ;hk Q k0 k ! 5 2 1 3 N g strongly in H .ވ /. Indeed, if ᏻ0 is a hyperbolic frame, then f N 2 e ı0. If Nk , we let 3 x 2=4 WD ! 1 f .x/ .4/ 2 e e ı0. By the unitarity of … it sufﬁces to see that WD j j D 5 2 2 N g N e k ı0 TN f 1 3 0 (5-33) k k k kH .ވ / ! which follows by inspection of the explicit formula

1 r r 2 zg z 4z .e ı0/.P/ 3 e e D .4z/ 2 sinh r for r dg.0; P/. D 734 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

Since g is absent from the frames ᏻ˛, ˛ A, and we have a nonzero scalar product in (5-32), we see k Ä from the discussion after Deﬁnition 5.2 that ᏻ0 is orthogonal to these frames.

1 3 Now, in the case ᏻ0 is a hyperbolic frame, we let H .ވ / be any weak limit of … t ;h 1 gk . Then, 2 k k passing to a subsequence, we may assume that for any ' H 1.ވ3/, 2 ˝ ˛ ˝ ˛ g.… 1 gk /; g' 2 2 g.gk …t ;h /; g…t ;h ' 2 2 0; tk ;h L L k k k k L L r k r D r r ! so that g g … is absent from ᏻ . In particular, we see from (5-32) that k0 D k tk ;hk 0 5 2 ˇ ˝ 2 N g ˛ ˇ ı=2 lim … 1 gk ; gN .e ı0/ 2 2 ˇ tk ;h L L ˇ Ä k k !1 5 2 ˇ˝ 2 N g ˛ ˇ . ˇ ; gN .e ı0/ L2 L2 ˇ g L2.ވ3/ Ä kr k so that (5-31) holds. Finally, to prove that g is also absent from the frames ᏻ˛, 1 ˛ A it suffices by k0 Ä Ä hypothesis to prove this for , but this follows from Lemma 5.4(ii). zᏻk0 In the case N , we first choose R > 0 and we define k ! 1 1 R 2 .v/ .v=R/N .… 1 gk /.‰I .v=Nk //; (5-34) k k tk ;h D k where is a smooth cut-off function as in (4-1). This sequence satisfies R . lim sup k L2.ޒ3/ lim sup ggk L2.ވ3/ k kr k k kr k !1 !1 and therefore has a subsequence which is bounded in H 1.ޒ3/ uniformly in R > 0. Passing to a P subsequence, we can find a weak limit R H 1.ޒ3/. Since the bound is uniform in R > 0, we can let 2 P R and find a weak limit such that ! 1 R * in H 1 and H 1.ޒ3/. Now, for ' C .ޒ3/, we have loc 2 P 2 01 1 2 1 TN ' N '.Nk ‰ / 1 3 0 k k I H .ވ / ! as k and with Lemma 5.4(iii), we compute that ! 1 gk ; g' 2 2 3 … 1 gk ; gTN ' 2 2 3 ᏻk0 L L .ވ / tk ;h k L L .ވ / h z i D h k i 1 ˝ 2 1 ˛ … 1 gk ; gN '.Nk ‰ / 2 2 3 ok .1/ tk ;h k I L L .ވ / D k C (5-35) ; ' L2 L2.ޒ3/ ok .1/ D h i C ; ' 1 1 3 o .1/: ᏻ0 ᏻ0 H H .ވ / k D hz k z k i C

In particular, g0 gk is absent from ᏻ0 and from (5-32), we see that (5-31) holds. Finally, from k D zᏻk0 Lemma 5.4(ii) again, gk0 is absent from all the previous frames. This ﬁnishes the proof of the claim and hence the proof of the ﬁnitary statement. WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 735

6. Proof of Proposition 3.4

In this section, we ﬁrst give the proof of Proposition 3.4 assuming a few lemmas that we prove at the end.

Proof of Proposition 3.4. Using the time translation symmetry, we may assume that tk 0 for all k 1. 1 3 D We apply Proposition 5.5 to the sequence .uk .0//k which is bounded in H .ވ / and we get sequences of 1 3 1 3 pairs . ; ᏻ / H .ޒ / Ᏺe and . ; ᏻ / H .ވ / Ᏺ , ; 1; 2;::: , such that the conclusion 2 P z 2 h D of Proposition 5.5 holds. Up to using Lemma 5.4(i), we may assume that for all , either t 0 for all k k D or .N /2 t and similarly, for all , either t 0 for all k or t . k j k j ! 1 k D j k j ! 1 Case I: all profiles are trivial, 0, 0 for all ; . In this case, we get from Strichartz estimates, D D (5-24) and Lemma 2.2(ii) that u .0/ r J satisfies k D k 3 2 itg itg 5 itg 5 e .uk .0// Z.ޒ/ . e .uk .0// 6 18 e .uk .0// 6 k k k kLt Lx k kL1t Lx 11 4 (6-1) 1 Á 15 . 15 2 itg uk .0/ L2 sup N e PN .uk .0// .x/ 0 kr k N 1;t;x j j ! as k . Applying Lemma 6.1, we see that ! 1 itg itg uk Z.ޒ/ e uk .0/ 10 3 uk e uk .0/ 1 0 Lt;x .ވ ޒ/ S .ޒ/ k k Ä k k C k k ! as k , which contradicts (3-7). ! 1 Now, for every linear profile (resp. ), define the associated nonlinear profile Ue;k (resp. Uh;k ) zᏻk zᏻk z as the maximal solution of (1-2) with initial data U .0/ (resp. U .0/ ). We may write e;k zᏻ h;k zᏻ D k D zk Uk if we do not want to discriminate between Euclidean and hyperbolic profiles. We can give a more precise description of each nonlinear profile. (1) If ᏻ Ᏺe is a Euclidean frame, this is given in Lemma 6.2. 2 (2) If t 0, letting .I ; W / be the maximal solution of (1-2) with initial data W .0/ , we see k D D that for any interval J b I ,

U .t/ h W .t t / 1 0 (6-2) k h;k k k kS .J / ! as k (indeed, this is identically 0 in this case). ! 1 (3) If t , then we deﬁne .I ; W / to be the maximal solution of (1-2) satisfying9 k ! C1 itg W .t/ e 1 3 0 k kH .ވ / ! as t . Then, applying Proposition 3.2, we see that on any interval J . ; T / b I , we ! 1 D 1 have (6-2). Using the time reversal symmetry u.t; x/ u. t; x/, we obtain a similar description !N when t . k ! 1 9Note that .I ; W / exists by Strichartz estimates and Lemma 6.1. 736 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

1 3 Case IIa: there is only one Euclidean proﬁle, i.e., there exists such that uk .0/ ok .1/ in H .ވ /. D zᏻk C 1 Applying Lemma 6.2, we see that Ue;k is global with uniformly bounded S -norm for k large enough. Then, using the stability Proposition 3.2 with u U , we see that for all k large enough, Q D e;k u .E 1 k k kZ.I/ max which contradicts (3-7).

1 3 Case IIb: there is only one hyperbolic proﬁle, i.e., there is such that uk .0/ ok .1/ in H .ވ /. D zᏻk C If t , then, using Strichartz estimates, we see that z k ! C1 itg itg ge …t ;h 30 ge 30 0 k k 10 13 3 10 13 3 kr kLt Lx .ވ . ;0// D kr kLt Lx .ވ . ; t // ! 1 1 k itg as k , which implies that e uk .0/ Z. ;0/ 0 as k . Using again Lemma 6.1, we see ! 1 k k 1 ! ! 1 that, for k large enough, uk is deﬁned on . ; 0/ and uk Z. ;0/ 0 as k , which contradicts 1 k k 1 ! ! 1 (3-7). Similarly, t yields a contradiction. Finally, if t 0, we get that k ! 1 k D .h / 1 uk .0/ k ! converges strongly in H 1.ވ3/, which is the desired conclusion of the proposition.

Case III: there exists or and > 0 such that

1 1 2 < lim sup E . /; lim sup E . / < E 2: (6-3) zᏻ zᏻ max k k k zk !1 !1 Taking k sufficiently large and maybe replacing by =2, we may assume that (6-3) holds for all k. In this case, we claim that, for J sufficiently large, app X X U U U eitg r J U J eitg r J k D e;k C h;k C k D prof;k C k 1 J 1 J Ä Ä Ä Ä is a global approximate solution with bounded Z norm for all k sufficiently large. First, by Lemma 6.2, all the Euclidean profiles are global. Using (5-25), we see that for all and all k 1 1 sufficiently large, E .Uh;k / < Emax . By (6-2), this implies that E .W / < Emax so that by the definition of Emax, W is global and by Proposition 3.2, Uh;k is global for k large enough and U .t/ W .t t / 1 0 (6-4) k h;k hk k kS .ޒ/ ! as k . ! 1 Now we claim that 1 app 2 lim sup gU 2 4Emax (6-5) k L1t Lx k kr k Ä !1 is bounded uniformly in J. Indeed, we first observe using (5-25) that app J J gU 2 gU 2 gr 2 kr k kL1t Lx Ä kr prof;k kL1t Lx C kr k kLx J 1 gU 2 .2Emax/ 2 : Ä kr prof;k kL1t Lx C WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 737

Using Lemma 6.3, we get that for ﬁxed t and J,

J 2 X 2 X 0 U .t/ U 2 U .t/; U .t/ 2 2 g prof;k L2 g k L L2 g k g k L L kr k x Ä kr k 1t x C hr r i 1 2J Ä Ä ¤ 0 X 1 2 E .U / o .1/ 2Emax o .1/; Ä k C k Ä C k 1 2J Ä Ä where o .1/ 0 as k for fixed J. k ! ! 1 We also have app lim sup gU 30 .E ; 1 (6-6) k 10 13 max k kr kLt Lx !1 is bounded uniformly in J . Indeed, from (6-3) and (5-25), we see that for all and all k sufficiently large 1 (depending maybe on J), E .U / < Emax and from the definition of Emax, we conclude that k . sup Uk Z.ޒ/ Emax; 1: k k Using Proposition 3.2, we see that this implies that

sup gU 10 .E ; 1: k 3 max kr kLt;x Besides, using Lemma 6.1, we obtain that 2 . 1 gUk 10 E .Uk / kr k 3 Lt;x 1 if E .U / ı0 is sufﬁciently small. Hence there exists a constant C C.Emax; / such that, for all , k Ä D and all k large enough (depending on ), 2 1 . gUk 10 CE .Uk / Emax; 1; kr kL 3 Ä t;x (6-7) 2 . 2 1 . Uk 10 gUk 30 CE .Uk / Emax; 1; k kLt;x kr k 10 13 Ä Lt Lx the second inequality following from Hölder’s inequality between the ﬁrst and the trivial bound

1 gU 2 2E .U /: kr k kL1t Lx Ä k Now, using (6-7) and Lemma 6.3, we see that

ˇ 10 10 ˇ X X 7 ˇ ˇ J 3 ˛ 3 ˇ ˛ 3 ˇ gUprof;k 10 gUk 10 ˇ . gUk / gUk L1 ˇkr k 3 kr k 3 ˇ Ä k r r k t;x Lt;x 1 ˛ 2J Lt;x 1 ˛ ˇ 2J Ä Ä Ä ¤ Ä X ˛ ˇ . 5 . Emax; . gUk / gUk Emax; ok .1/: k r r kL 3 1 ˛ ˇ 2J t;x Ä ¤ Ä Consequently, 10 10 J 3 X ˛ 3 gUprof;k 10 gUk 10 ok .1/ kr k 3 Ä kr k 3 C Lt;x 1 ˛ 2J Lt;x Ä Ä X 1 ˛ .E ; C E .U / o .1/ .E ; 1 max k C k max 1 ˛ 2J Ä Ä 738 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI and using Hölder’s inequality and (6-5), we get (6-6). Using (6-5) and (6-6) we can apply Proposition 3.2 to get ı >0 such that the conclusion of Proposition 3.2 holds. Now, for F.x/ x 4x, we have D j j app app app 4 X ˛ ˛ X ˛ app e i@t g U U U .i@t g/U F.U / F.U / F.U /: D C k k j k j D C k k C k k 1 ˛ 2J 1 ˛ 2J Ä Ä Ä Ä The ﬁrst term is identically 0, while using Lemma 6.4, we see that taking J large enough, we can ensure 6 2 1; 5 that the second is smaller than ı given above in Lt Hx -norm for all k large enough. Then, since u .0/ U app.0/, Sobolev’s inequality and the conclusion of Proposition 3.2 imply that for all k large, k D k and all interval J

app app u . u 1 u U 1 U 1 .E ; 1 k k kZ.J / k k kS .J / Ä k k k kS .J / C k k kS .ޒ/ max where we have used (6-6). Then, we see that uk is global for all k large enough and that uk has uniformly bounded Z-norm, which contradicts (3-7). This ends the proof.

Criterion for linear evolution.

Lemma 6.1. For any M > 0, there exists ı > 0 such that for any interval J ޒ, if

itg g 2 3 M and e ı; kr kL .ވ / Ä k kZ.J / Ä

it0g then for any t0 J, the maximal solution .I; u/ of (1-2) satisfying u.t0/ e satisﬁes J I and 2 D itg 3 u e S 1.J / ı ; k k Ä (6-8) u 1 C.M; ı/: k kS .J / Ä Besides, if J . ; T /, then there exists a unique maximal solution .I; u/, J I of (1-2) such that D 1

itg lim g.u.t/ e / 2 3 0 (6-9) t kr kL .ވ / D !1 and (6-8) holds in this case too. The same statement holds in the Euclidean case when .ވ3; g/ is replaced 3 by .ޒ ; ıij /.

Proof of Lemma 6.1. The ﬁrst part is a direct consequence of Proposition 3.2. Indeed, let v eitg . D Then clearly (3-3) is satisﬁed while using Strichartz estimates,

4 4 itg 4 gv v 6 v ge 30 . M ı ; 2 5 3 Z.J / 10 13 3 kr j j kLt Lx .J ވ / Ä k k kr kLt Lx .J ވ / thus we get (3-4). Then we can apply Proposition 3.2 with 1 to conclude. The second claim is D classical and follows from a ﬁxed point argument. WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 739

Description of a Euclidean nonlinear proﬁle. Let ˚ 2 « Ᏺe .Nk ; tk ; hk /k Ᏺe tk 0 for all k or lim Nk tk ; z D 2 W D k j j D 1 ˚ !1 « Ᏺh .1; tk ; hk /k Ᏺh tk 0 for all k or lim tk : z D 2 W D k j j D 1 !1 1 3 Lemma 6.2. Assume H .ޒ / and .N ; t ; h / Ᏺe. Let U be the solution of (1-2) such that 2 P k k k k 2 z k U .0/ … .TN /. k D tk ;hk k (i) For k large enough, U C.ޒ H 1/ is globally deﬁned, and k 2 W U 2C .E1 .//: (6-10) k k kZ.ޒ/ Ä Q ޒ3 (ii) There exists a Euclidean solution u C.ޒ H 1.ޒ3// of 2 W P 4 .i@t / u u u (6-11) C D j j

with scattering data ˙1 deﬁned as in (4-4) such that the following holds, up to a subsequence: for any " > 0, there exists T .; "/ such that for all T T .; "/ there exists R.; "; T / such that for all R R.; "; T /, we have Uk uk 1 2 "; (6-12) S . t tk TN / k Q k j jÄ k Ä for k large enough, where

1=2 1 1 2 .h 1 uk /.t; x/ N .Nk ‰I .x/=R/u.Nk ‰I .x/; Nk .t tk //: k Q D k In addition, up to a subsequence,

Uk 1; 30 10 1; 10 " (6-13) 10 13 3 3 3 2 k kLt Hx Lt Hx .ވ N t tk T / Ä \ f k j j g and for any .t t / TN 2, ˙ k k 2 g Uk .t/ …tk t;hk TNk ˙1 L "; (6-14) kr k Ä for k large enough (depending on ; "; T; R). Proof. We may assume that h I for any k. k D If t 0 for any k then the lemma follows from Lemma 4.2 and Corollary 4.3: we let u be the k D nonlinear Euclidean solution of (6-11) with u.0/ and notice that for any ı > 0 there is T .; ı/ such D that

u 10=3 3 ı: Lx;t .ޒ t T .;ı/ / kr k fj j g Ä The bound (6-12) follows for any fixed T T .; ı/ from Lemma 4.2. Assuming ı is sufficiently small and T is sufficiently large (both depending on and "), the bounds (6-13) and (6-14) then follow from itg 2 10=3 1;10=3 3 Corollary 4.3 (which guarantees smallness of 1 .t/ e Uk . Nk T .; ı// in Lt Hx .ވ ޒ/) ˙ ˙ and Lemma 6.1. 740 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

2 2 Otherwise, if limk Nk tk , we may assume by symmetry that Nk tk . Then we let u !1 j j D 1 ! C1 be the solution of (6-11) such that

it .u.t/ e / 2 3 0 r L .ޒ / ! as t (thus ). We let u.0/ and apply the conclusions of the lemma to the frame ! 1 1 D Q D .N ; 0; h / Ᏺe and V .s/, the solution of (1-2) with initial data V .0/ TN . In particular, we k k k 2 k k D hk k Q see from the fact that N 2t and (6-14) that k k ! C1

V . t / … TN 1 3 0 k k k tk ;hk k kH .ވ / ! as k . Then, using Proposition 3.2, we see that ! 1

U V . t / 1 0 k k k k kS .ޒ/ ! as k , and we can conclude by inspecting the behavior of V . This ends the proof. ! 1 k Noninteraction of nonlinear proﬁles.

Lemma 6.3. Let ᏻk and be two profiles associated to orthogonal frames ᏻ and ᏻ0 in Ᏺe Ᏺh. Let z zᏻk0 z [ z Uk and U be the solutions of the nonlinear equation (1-2) such that Uk .0/ ᏻ and U .0/ . k0 z k k0 zᏻk0 1 1 D D Suppose also that E .ᏻk / < Emax (resp. E . / < Emax ) if ᏻ Ᏺh (resp. ᏻ0 Ᏺh). Then z zᏻk0 2 2 ˇ ˇ supˇ gUk .T /; gU 0 .T / L2 L2.ވ3/ˇ Uk gU 0 15 . gUk / gU 0 5 k k 5 8 3 k 3 3 T ޒ hr r i Ck r kLt Lx .ވ ޒ/Ck r r kLt;x .ވ ޒ/ 2 0 (6-15) ! as k . ! 1 Proof. It suffices to prove (6-15) up to extracting a subsequence, and fix " > 0 sufficiently small. We only provide the proof that the second norm in (6-15) decays; the other two claims are similar.

Applying Lemma 6.2 if Uk is a proﬁle associated to a Euclidean frame (respectively (6-4) if Uk is a proﬁle associated to a hyperbolic frame), we see that

U 1 U 1 M < k k kS C k k0 kS Ä C1 and that there exist R and ı such that

gUk 10 30=13 10=3 3 R Uk 10 3 R "; Lt Lx Lx;t ..ވ ޒ/ ᏿ / Lx;t ..ވ ޒ/ ᏿N ;t ;h / kr k \ n Nk ;tk ;hk C k k n k k k Ä (6-16) sup gUk 10 30=13 10=3 ı Uk 10 ı "; Lt Lx Lx;t .᏿ / Lx;t .᏿N ;S;h/ S;h kr k \ Nk ;S;h C k k k Ä where a ˚ 3 1 1 2 2 « ᏿ .x; t/ ވ ޒ dg.h x; 0/ aN and t T a N : (6-17) N;T;h WD 2 W Ä j j Ä

A similar claim holds for Uk0 with the same values of R, ı. WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 741

If N =N , then for k large enough we estimate k k0 ! 1

Uk gU 0 30 Uk gU 0 30 Uk gU 0 30 k 5 16 k 5 16 R k 5 16 3 R k r kLt Lx Ä k r kLt Lx .᏿ / C k r kLt Lx ..ވ ޒ/ ᏿ / Nk ;tk ;hk n Nk ;tk ;hk

10 30 10 R 30 Uk gU 0 10 ı Uk ވ3 ޒ gU 0 Lt;x k L L 13 .᏿ / Lt;x .. / ᏿N ;t ;h / k 10 13 Ä k k kr k t N ;t ;h C k k n k k k kr kL Lx k0 k k t

.M ":

The case when Nk0 =Nk is similar. ! 1 1 Otherwise, we can assume that C Nk =Nk0 C for all k, and then ﬁnd k sufﬁciently large that R R Ä Ä ᏿N ;t ;h ᏿N ;t ;h ∅. Using (6-16) it follows as before that k k k \ k0 k0 k0 D

Uk gU 0 30 .M ": k 5 16 k r kLt Lx Hence, in all cases,

lim sup Uk gU 0 15 .M ": k 5 8 k k r kLt Lx !1 The convergence to 0 of the second term in (6-15) follows.

Control of the error term.

Lemma 6.4. With the notations in the proof of Proposition 3.4,

app X ˛ lim lim sup g F.Uk / F.Uk / 6 0: (6-18) J k r 2 5 D !1 1 ˛ 2J Lt Lx !1 Ä Ä Proof. Fix "0 > 0. For ﬁxed J, we let

X X X U J U U U prof;k D e;k C h;k D k 1 J 1 J 1 2J Ä Ä Ä Ä Ä Ä be the sum of the proﬁles. Then we separate

app X ˛ g F.Uk / F.Uk / 6 r 2 5 1 ˛ 2J Lt Lx Ä Ä app J J X ˛ g F.Uk / F.Uprof;k / 6 g F.Uprof;k / F.Uk / 6 : Ä r 2 5 C r 2 5 Lt Lx 1 ˛ 2J Lt Lx Ä Ä We ﬁrst claim that, for ﬁxed J,

J X ˛ lim sup g.F.Uprof;k / F.Uk // 6 0: (6-19) r 2 5 D k 1 ˛ 2J Lt Lx !1 Ä Ä Indeed, using that ˇ Â Â Ã Ãˇ ˇ X ˛ X ˛ ˇ X 3 ˛ ˇ ˇ g F U F.U / ˇ . U U gU ; ˇr k k ˇ j k j j k r k j 1 ˛ 2J 1 ˛ 2J ˛ ˇ; Ä Ä Ä Ä ¤ 742 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI we see that Â Ã J X ˛ . X 3 ˛ ˇ g F.Uprof;k / F.Uk / 6 Uk 10 Uk gUk 15 : r 2 5 k kLt;x k r kL5L 8 1 ˛ 2J Lt Lx ˛ ˇ; t x Ä Ä ¤ 10 Therefore (6-19) follows from (6-15) since the sum is over a ﬁnite set and each proﬁle is bounded in Lt;x by (6-7).

Now we prove that, for any given "0 > 0,

app J lim sup lim sup g F.U / F.U / 6 . "0: (6-20) k prof;k 2 5 J k r Lt Lx !1 !1 30 J 10 1; 13 This would complete the proof of (6-18). We ﬁrst remark that, from (6-6), Uprof;k has bounded Lt Hx - 10 norm, uniformly in J for k sufﬁciently large. We also let j0 j0."0/ independent of J be such that D

˛ sup lim sup U 10 . "0: (6-21) k Lt;x ˛ j0 k k k !1 Now we compute

5 1 J itg J J . X X p itg J j 1 p J 5 j g.F.Uprof;k e rk / F.Uprof;k // 6 g .e rk / g .Uprof;k / 6 : r C L2L 5 r r L2L 5 t x j 1p 0 t x D D 1; 30 J itg J 10 13 Since both Uprof;k and e rk are bounded in Lt Hx uniformly in J, if there is at least one term itg J e rk with no derivative, we can bound the norm in the expression above by

p itg J j 1 p J 5 j itg J g .e r / g .U / 6 .E ; e r 10 k prof;k 2 5 max k Lt;x r r Lt Lx k k uniformly in J , so that taking the limit k and then J , we get 0. Hence we need only consider ! 1 ! 1 the term

J 4 itg J .U / g.e r / 6 : prof;k k 2 5 r Lt Lx J 4 Expanding further .Uprof;k / and using Lemma 6.3 and (6-7), we see that

J 4 itg J X ˛ 4 itg J lim sup .Uprof;k / g.e rk / 6 lim sup .Uk / g.e rk / 6 k r kL2L 5 D k r kL2L 5 k t x k 1 ˛ J t x !1 !1 Ä Ä . X ˛ 3 ˛ itg J lim sup Uk 10 Uk g.e rk / 15 k kLt;x k r kL5L 8 k 1 ˛ J t x !1 Ä Ä X 1 ˛ ˛ itg J .E ; lim sup E .U / U g.e r / 15 max k k k 5 8 k k r kLt Lx 1 ˛ j0 !1 Ä Ä X 1 ˛ ˛ itg J lim sup E .U / U 10 g.e r / 30 k k Lt;x k 10 13 C k k k kr kLt Lx j0 ˛ J !1 Ä Ä

10 The fact that j0 exists follows from (5-25) and (6-7). WELL-POSEDNESS OF ENERGY-CRITICAL SCHRÖDINGER EQUATIONS IN CURVED SPACES 743 where j0 is chosen in (6-21). Consequently, using the summation formula for the energies (5-25), we get

J 4 itg J ˛ itg J lim sup .U / g.e r / 6 .E ; "0 sup lim sup U g.e r / 15 : prof;k k 2 5 max k k 5 8 k r Lt Lt C 1 ˛ j0 k r Lt Lx !1 Ä Ä !1 ˛ Finally, we obtain from Lemma 2.3 that for any proﬁle Uk ,

˛ itg J lim lim sup U g.e r / 15 0: (6-22) k k 5 8 3 J k k r kLt Lx .ވ ޒ/ D !1 !1 ˛ This would imply (6-20) and hence complete the proof of Lemma 6.4. To prove (6-22), ﬁx " > 0. For Uk a given, we consider the sets ᏿N;T;h as deﬁned in (6-17). For R large enough we have, using (6-16),

˛ itg J U g.e r / 15 k k 5 8 3 R k r kLt Lx ..ވ ޒ/ ᏿ / n Nk ;tk ;hk ˛ itg J 10 R 30 . U 3 g.e r / Emax; ": k Lx;t ..ވ ޒ/ ᏿N ;t ;h / k 10 13 Ä k k n k k k kr kLt Lx Now in the case of a hyperbolic proﬁle U , we know that W as in (6-2) satisﬁes W L10 .ވ3 ޒ/. h;k 2 x;t We choose W ; C .ވ3 ޒ/ such that 0 2 c1 ; W W 0 10 3 ": Lx;t .ވ ޒ/ k k Ä Using (6-4) we see that there exists a constant C;" such that

itg J ; itg J U g.e r / 15 .U h W 0. t // g.e r / 15 h;k k 5 8 R h;k k k k 5 8 R k r kL Lx .᏿ / Ä k r kL Lx .᏿ / t Nk ;tk ;hk t Nk ;tk ;hk ; itg J W 0 L g.e r / 15 1t;x k 5 8 R C k k kr kL Lx .᏿ / t Nk ;tk ;hk itg J .E ; " C;" g.e r / 15 : max k 5 8 R C kr kL Lx .᏿ / t Nk ;tk ;hk In the case of a Euclidean proﬁle, we choose v C .ޒ3 ޒ/ such that 2 c1

u v 10 3 "; Lt;x .ޒ ޒ/ k k Ä for u given in Lemma 6.2. Then, using (6-12), we estimate as before

1 itg J itg J U g.e r / 15 .E ; " C;".N / 2 g.e r / 15 : e;k k 5 8 R max k k 5 8 R k r kL Lx .᏿ / C kr kL Lx .᏿ / t Nk ;tk ;hk t Nk ;tk ;hk Therefore, we conclude that in all cases,

1 ˛ itg J ˛ itg J U g.e r / 15 .E ; " C˛;".N / 2 g.e r / 15 : k k 5 8 R max k k 5 8 R k r kL Lx .᏿ / C kr kL Lx .᏿ / t Nk ;tk ;hk t Nk ;tk ;hk Finally we use Lemma 2.3 and (5-24) to conclude that

˛ itg J 15 . lim lim sup Uk g.e rk / Emax; ": J 5 8 R k k r kLt Lx .᏿N ;t ;h / !1 !1 k k k Since " was arbitrary, we obtain (6-22) and hence ﬁnish the proof. 744 ALEXANDRUD.IONESCU,BENOITPAUSADER AND GIGLIOLA STAFFILANI

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Received 5 Aug 2010. Accepted 1 Apr 2011.

ALEXANDRU D.IONESCU: [email protected] Department of Mathematics, Princeton University, Washington Road, Princeton, NJ 08544, United States

BENOIT PAUSADER: [email protected] Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI 02912, United States

GIGLIOLA STAFFILANI: [email protected] Department of Mathematics, Massachusetts Institue of Technology, 77 Massachusetts Avenue, 2-246, Cambridge, MA 02139, United States

mathematical sciences publishers msp ANALYSIS AND PDE Vol. 5, No. 4, 2012 dx.doi.org/10.2140/apde.2012.5.747 msp

GENERALIZED RICCI FLOW I: HIGHER-DERIVATIVE ESTIMATES FOR COMPACT MANIFOLDS

YI LI

We consider a generalized Ricci flow with a given (not necessarily closed) three-form and establish higher-derivative estimates for compact manifolds. As an application, we prove the compactness theorem for this generalized Ricci flow. Similar results still hold for a more generalized Ricci flow.

1. Introduction 747 2. Short-time existence of GRF 752 3. Evolution of curvatures 754 4. Derivative estimates 756 5. Compactness theorem 764 6. Generalization 771 Acknowledgment 774 References 774

1. Introduction

Throughout this paper manifolds always mean smooth and closed (compact and without boundary) manifolds. Let Met.M / denote the space of smooth metrics on a manifold M , and C 1.M / the set of all smooth functions on M . We denote by C the universal constants depending only on the dimension of M , which may take different values at different places. An important and natural problem in differential geometry is to ﬁnd a canonical metric on a given manifold. A classical example is the uniformization theorem (e.g., [Chow and Knopf 2004]), which says that every smooth surface admits a unique conformal metric of constant curvature. To generalize to higher dimensional manifolds, Hamilton [1982] introduced a system of equations

@gij 2Rij ; (1-1) @t D now called the Ricci flow, an analogue of the heat equation for metrics. There are two ways to understand the Ricci flow: one way comes from the two-dimensional sigma model (see [Bakas 2007]), while another comes from Perelman’s energy functional [Perelman 2002] defined by Z 2Á f Ᏺ.g; f / R f e dVg;.g; f / Met.M / C 1.M /; (1-2) D M C jr j 2

MSC2010: 53C44, 35K55. Keywords: Ricci ﬂow, Generalized Ricci ﬂow, BBS derivative estimates, compactness theorems, energy functionals.

747 748 YI LI where R, , and dVg, is the scalar curvature, Levi-Civita connection, and volume form of g, respectively. r He showed that the Ricci flow is the gradient flow of (1-2) and the functional Ᏺ is monotonic along this gradient flow. Precisely, under the following system

@gij @f 2 2Rij ; R f f ; (1-3) @t D @t D C jr j we have Z d ˇ ˇ2 f Ᏺ.g; f / 2 ˇRij i j f ˇ e dVg 0: (1-4) dt D M C r r Perelman’s energy functional plays an essential role in determining the structures of singularities of the Ricci flow and then the proof of Poincaré conjecture and Thurston’s generalization conjecture; for more details we refer readers to [Cao and Zhu 2006; Chow et al. 2006; 2007; 2008; 2010; Kleiner and Lott 2008; Morgan and Tian 2007; Perelman 2002]. Ricci flow coupled with a one-form or a two-form. If we consider the two-dimensional nonlinear sigma model [Bakas 2007; Oliynyk et al. 2006], then we obtain a generalized Ricci flow that is the Ricci flow coupled with the evolution equation for a two-form. This flow can be also obtained from the point of view of Perelman-type energy functional. Denoting by Ꮽp.M / the space of p-forms on M , we consider the energy functional

Ᏺ.1/ Met.M / Ꮽ2.M / C .M / ޒ W 1 ! defined by Z .1/ 2 1 2Á f Ᏺ .g; B; f / R f 12 H e dVg; (1-5) D M C jr j j j where H dB. As showed in [Oliynyk et al. 2006], the gradient flow of Ᏺ.1/ satisfies D @gij 1 k` 2Rij 2 i j f Hi H ; (1-6) @t D r r C 2 jk` @Bij k k 3 H ij 3H ij f; (1-7) @t D rk rk @f R f 1 H 2; (1-8) @t D C 4 j j and under a family of diffeomorphisms the system (1-6)–(1-8) is equivalent to

@gij 1 k` 2Rij Hi H ; (1-9) @t D C 2 jk` @Bij k 3 H ij ; (1-10) @t D rk @f R f f 2 1 H 2: (1-11) @t D C jr j C 4 j j

Using the adjoint operator d , Equation (1-10) can be written as

@Bij .d H /ij ; (1-12) @t D GENERALIZED RICCI FLOW, I 749 and therefore (because of H dB) D @H dd H HLH; (1-13) @t D D where HL .dd d d/ denotes the Hodge–Laplace operator. D C The flow (1-9)–(1-10) can be interpreted as the connection Ricci flow [Streets 2008]. If we replace H dB by F dA, i.e., replace a two-form by a one-form, then the flow (1-6)–(1-7) or (1-9)–(1-10) is D D exactly the Ricci Yang–Mills flow studied by Streets [2007] and Young [2008].

Ricci flow coupled with a one-form and a two-form. There is another generalized Ricci flow which connects to Thurston’s conjecture — roughly stating that a three-dimensional manifold with a given topology has a canonical decomposition into simple three-dimensional manifolds, each of which admits one, and only one, of eight homogeneous geometries: ޓ3, the round three-sphere; ޒ3, the Euclidean space; ވ3, the standard hyperbolic space; ޓ2 ޒ; ވ2 ޒ; Nil, the three-dimensional nilpotent Heisenberg group; SLf.2; ޒ/; Sol, the three -dimensional solvable Lie group. The proof of Thurston’s conjecture can be found in [Cao and Zhu 2006; Kleiner and Lott 2008; Morgan and Tian 2007; Perelman 2002]. To better understanding Thurston’s conjecture, Gegenberg and Kunstatter [2004] proposed a generalized flow by considering the modified 3D stringy theory. This flow is the Ricci flow coupled with evolution equations for a one-form and a two-form. As in (1-5), we define an energy functional

Ᏺ.2/ Met.M / Ꮽ1.M / Ꮽ2.M / C .M / ޒ W 1 ! by Z .2/ 2 1 2 1 2Á f Ᏺ .g; A; B; f / R f 12 H 2 F e dVg; (1-14) D M C jr j j j j j where H dB, and F dA. In [He et al. 2008], the authors showed that the gradient ﬂow of Ᏺ.2/ D D satisﬁes

@gij 1 k` k 2Rij 2 i j f Hi H 2Fi F ; (1-15) @t D r r C 2 jk` C jk @Ai j j 2 j F i 2F i j f; (1-16) @t D r r @Bij k k 3 H ij 3H ij f; (1-17) @t D rk rk @f R f 1 H 2 F 2; (1-18) @t D C 4 j j C j j and under a family of diffeomorphisms the system (1-15)–(1-18) is equivalent to

@gij 1 k` k 2Rij Hi H 2Fi F ; (1-19) @t D C 2 jk` C jk @Ai j 2 j F i; (1-20) @t D r @Bij k 3 H ij ; (1-21) @t D rk 750 YI LI

@f R f f 2 1 H 2 F 2: (1-22) @t D C jr j C 4 j j C j j

Using again the adjoint operator d , we have @F @H HLF; HLH: (1-23) @t D @t D The flow (1-19)–(1-21) clearly contains the Ricci flow, the flow (1-9)–(1-10) or the connection Ricci flow, and the Ricci Yang–Mills flow; we expect this flow can give another proof of the Poincaré conjecture and Thurston’s generalization conjecture, with less analysis on singularities.

Main results. For convenience, we refer to GRF the generalized Ricci flow and RF.A; B/ the Ricci flow coupled with a one-form A and a two-form B. Let .M; g/ denote an n-dimensional closed Riemannian manifold with a three-form H H . In D f ijk g the first part of this paper we consider the following GRF on M :

@ 1 k` gij .x; t/ 2Rij .x; t/ H .x; t/Hj .x; t/; (1-24) @t D C 2 ik` @ H.x; t/ H.x; t/; H.x; 0/ H.x/; g.x; 0/ g.x/: (1-25) @t D HL;g.x;t/ D D It is clearly from (1-9) and (1-13) that the gradient flow of the energy functional Ᏺ.1/ is a special case of (1-24)–(1-25). The corresponding case that H is closed is called the refined generalized Ricci flow (RGRF):

@ 1 k` gij .x; t/ 2Rij .x; t/ H .x; t/Hj .x; t/; (1-26) @t D C 2 ik` @ H.x; t/ dd H.x; t/; H.x; 0/ H.x/; g.x; 0/ g.x/: (1-27) @t D g.x;t/ D D

Here dg.x;t/ is the dual operator of d with respect to the metric g.x; t/. Lemma 1.1. Under RGRF, H.x; t/ is closed if the initial value H.x/ is closed. Proof. Since the exterior derivative d is independent of the metric, we have @ @ dH.x; t/ d H.x; t/ d dd H.x; t/ 0: @t D @t D g.x;t/ D so dH.x; t/ dH.x/ 0. D D The closedness of H is very important and has physical interpretation [Bakas 2007; Oliynyk et al. 2006]. Streets [2008] considered the connection Ricci ﬂow in which H is the geometric torsion of connection. Proposition 1.2. If .g.x; t/; H.x; t// is a solution of RGRF and the initial value H.x/ is closed, then it is also a solution of GRF. Proof. From Lemma 1.1 and the assumption we know that H.x; t/ are all closed. Hence

H.x; t/ dd H.x; t/: HL;g.x;t/ D g.x;t/ GENERALIZED RICCI FLOW, I 751

For GRF, a basic and natural question is the existence. The short-time existence for RGRF has been established in [He et al. 2008], where the authors have already showed the short-time existence for RF.A; B/ obviously including RGRF. In this paper, we prove the short- time existence for RGF.

Theorem 1.3. There is a unique solution to GRF for a short time. More precisely, let .M; gij .x// be an n-dimensional closed Riemannian manifold with a three-form H H , then there exists a constant D f ijk g T T .n/ > 0 depending only on n such that the evolution system D @ 1 kp `q gij .x; t/ 2Rij .x; t/ g .x; t/g .x; t/H .x; t/Hjpq.x; t/; @t D C 2 ik` @ H.x; t/ H.x; t/; H.x; 0/ H.x/; g.x; 0/ g.x/; @t D HL;g.x;t/ D D has a unique solution .gij .x; t/; H .x; t// for a short time 0 t T . ijk Ä Ä After establishing the local existence, we are able to prove the higher derivatives estimates for GRF. Precisely, we have the following Theorem 1.4. Suppose that .g.x; t/; H.x; t// is a solution to GRF on a closed manifold M n and K is an arbitrary given positive constant. Then for each ˛ > 0 and each integer m 1 there exists a constant Cm depending on m; n; max ˛; 1 , and K such that if f g Rm.x; t/ K; H.x/ K j jg.x;t/ Ä j jg.x/ Ä for all x M and t Œ0; ˛=K, then 2 2 m 1 m Cm Rm.x; t/ g.x;t/ H.x; t/ g.x;t/ (1-28) jr j C jr j Ä t m=2 for all x M and t .0; ˛=K. 2 2 As an application, we can prove the compactness theorem for GRF.

Theorem 1.5 (compactness for GRF). Let .Mk ; gk .t/; Hk .t/; Ok / k ގ be a sequence of complete f g 2 pointed solutions to GRF for t Œ˛; !/ 0 such that: 2 3 (i) There is a constant C0 < independent of k such that 1 ˇ ˇ sup Rm C0; sup H .x; ˛/ C0: ˇ gk .x;t/ˇgk .x;t/ k gk .x;˛/ .x;t/ Mk .˛;!/ Ä x Mk j j Ä 2 2 (ii) There exists a constant 0 > 0 satisﬁes

inj .O / 0: gk .0/ k

Then there exists a subsequence jk k ގ such that f g 2

.Mjk ; gjk .t/; Hjk .t/; Ojk / .M ; g .t/; H .t/; O /; ! 1 1 1 1 converges to a complete pointed solution .M ; g .t/; H .t/; O /; t Œ˛; !/ to GRF as k . 1 1 1 1 2 ! 1 752 YI LI

In the second part of this paper, we consider the Ricci flow coupled with a one-form and a two-form. This flow is the gradient flow of Ᏺ.2/ and takes the form

@ 1 k` k gij .x; t/ 2Rij Hi .x; t/H .x; t/ 2Fi .x; t/F .x; t/; (1-29) @t D C 2 jk` C jk @ j Ai.x; t/ 2 j F i.x; t/; Ai.x; 0/ Ai.x/; gij .x; 0/ gij .x/; (1-30) @t D r D D @ k Bij .x; t/ 3 H ij .x; t/; Bij .x; 0/ Bij .x/: (1-31) @t D rk D

Here A Ai and B Bij is a one-form and a two-form on M , respectively, and F dA; H dB. D f g D f g D D For this ﬂow, we can also prove the short-time existence, higher derivative estimates, and the compactness theorem. The rest of this paper is organized as follows. In Section 2, we prove the short-time existence and uniqueness of the GRF for any given three-form H . In Section 3, we compute the evolution equations for the Levi-Civita connections, Riemann, Ricci, and scalar curvatures of a solution to the GRF. In Section 4, we establish higher derivative estimates for GRF, called Bernstein–Bando–Shi (BBS) derivative estimates (e.g., [Cao and Zhu 2006; Chow and Knopf 2004; Chow et al. 2007; 2008; 2010; Morgan and Tian 2007; Shi 1989]). In Section 5, we prove the compactness theorem for GRF by using BBS estimates. In Section 6, based on the work of [He et al. 2008], the similar results are established for RF.A; B/.

2. Short-time existence of GRF

In this section we establish the short-time existence for GRF. Our method is standard: we use the DeTurck trick in Ricci ﬂow to prove its short-time existence. We assume that M is an n-dimensional closed Riemannian manifold with metric 2 i j ds gij .x/ dx dx (2-1) z D z and with Riemannian curvature tensor R . We also assume that H H is a ﬁxed three-form f zijk`g z D f zijk g on M . In the following we put k` hij H Hj : (2-2) WD ik` Suppose the metrics 2 1 i j ds gij .x; t/ dx dx (2-3) yt D 2 y are the solutions of1 @ gij .x; t/ 2Rij .x; t/ hij .x; t/; gij .x; 0/ gij .x/ (2-4) @t y D y C y y D z for a short time 0 t T . Consider a family of smooth diffeomorphisms 't M M.0 t T / of Ä Ä W ! Ä Ä M . Let ds2 ' ds2; 0 t T (2-5) t WD t yt Ä Ä 1In the following computations we don’t need to use the evolution equation for H.x; t/, hence we only consider the evolution equation for metrics. GENERALIZED RICCI FLOW, I 753 be the pull-back metrics of ds2. For coordinates system x x1;:::; xn on M , let yt D f g 2 i j ds gij .x; t/ dx dx (2-6) t D and 1 n y.x; t/ 't .x/ y .x; t/; : : : ; y .x; t/ : (2-7) D D f g Then we have @y˛ @yˇ gij .x; t/ g .y; t/: (2-8) D @xi @xj y˛ˇ By the assumption g .x; t/ are the solutions of y˛ˇ @ g .x; t/ 2R .x; t/ h .x; t/; g .x; 0/ g .x/: (2-9) @t y˛ˇ D y˛ˇ C y˛ˇ y˛ˇ D z˛ˇ k k k We use Rij ; Rij ; Rij ; ; ; ; ; ; ; hij ; hij ; hij to denote the Ricci curvatures, Christoffel y z ij yij zij r ry rz y z symbols, covariant derivatives, and products of the three-form H with respect to gij ; gij ; gij respectively. z y Then @ @y˛ @yˇ Â @ Ã @ Â@y˛ Ã @yˇ @y˛ @ Â@yˇ Ã gij .x; t/ g .y; t/ g .y; t/ g .y; t/: @t D @xi @xj @t y˛ˇ C @xi @t @xj y˛ˇ C @xi @xj @t y˛ˇ From (2-9) we have @ @g @y g .y; t/ 2R .y; t/ h .y; t/ y˛ˇ ; @t y˛ˇ D y˛ˇ C y˛ˇ C @y @t and @ @y˛ @yˇ @y˛ @yˇ gij .x; t/ 2 R .y; t/ h .y; t/ @t D @xi @xj y˛ˇ C @xi @xj y˛ˇ @y˛ @yˇ @g @y @ Â@y˛ Ã @yˇ @y˛ @ Â@yˇ Ã y˛ˇ g .y; t/ g .y; t/: C @xi @xj @y @t C @xi @t @xj y˛ˇ C @xi @xj @t y˛ˇ Since @y˛ @yˇ @y˛ @yˇ Rij .x; t/ R .y; t/; hij .x; t/ h .y; t/; D @xi @xj y˛ˇ D @xi @xj y˛ˇ using [Shi 1989, §2, (29)], we obtain @ Â@y˛ @xk Ã Â@y˛ @xk Ã gij .x; t/ 2Rij .x; t/ hij .x; t/ i g j g : (2-10) @t D C C r @t @y˛ jk C r @t @y˛ ik

According to DeTurck trick, we deﬁne y.x; t/ 't .x/ by the equation D @y˛ @y˛ gˇ .k k /; y˛.x; 0/ x˛; (2-11) @t D @xk ˇ zˇ D then (2-10) becomes @ gij .x; t/ 2Rij .x; t/ hij .x; t/ iVj j Vi; gij .x; 0/ gij .x/; (2-12) @t D C C r C r D z 754 YI LI where ˇ k k Vi g g . /: (2-13) D ik ˇ zˇ Lemma 2.1. The evolution equation (2-12) is a strictly parabolic system. Moreover,

@ ˛ˇ ˛ˇ pq ˛ˇ pq gij g ˛ gij g gipg R g gjpg R @t D rz rzˇ z zj˛qˇ z zi˛qˇ 1 ˛ˇ pq g g igp˛ j g 2 gjp qg 2 ˛gjp giq 2 j gp˛ giq 2 igp˛ gjq C 2 rz rz qˇ C rz rz iˇ rz rzˇ rz rzˇ rz rzˇ 1 ˛ˇ pq g g Hi˛pH : C 2 jˇq Proof. It is an immediate consequence of Lemma 2.1 of [Shi 1989]. Now we can prove the short-time existence of GRF.

Theorem 2.2. There is a unique solution to GRF for a short time. More precisely, let .M; gij .x// be an n-dimensional closed Riemannian manifold with a three-form H H , then there exists a constant D f ijk g T T .n/ > 0 depending only on n such that the evolution system D @ 1 kp `q gij .x; t/ 2Rij .x; t/ g .x; t/g .x; t/H .x; t/Hjpq.x; t/; @t D C 2 ik` @ H.x; t/ H.x; t/; H.x; 0/ H.x/; g.x; 0/ g.x/; @t D HL;g.x;t/ D D has a unique solution .gij .x; t/; H .x; t// for a short time 0 t T . ijk Ä Ä Proof. We proved that the ﬁrst evolution equation is strictly parabolic by Lemma 2.1. Form the Ricci identity, we have H H Rm H which is also strictly parabolic. Hence from HL;g.x;t/ D LB;g.x;t/ C the standard theory of parabolic systems, the evolution system has a unique solution.

3. Evolution of curvatures

The evolution equation for the Riemann curvature tensors to the usual Ricci ﬂow (e.g., [Cao and Zhu 2006; Chow and Knopf 2004; Chow et al. 2007, 2008; 2010; Hamilton 1982; Morgan and Tian 2007; Shi 1989]) is given by @ R R ; (3-1) @t ijk` D ijk` C ijk` where

pq 2.B B B B / g .R R R Rqj R R R R /; ijk` D ijk` ij`k i`jk C ikj` pjk` q` C ipk` C ijp` qk C ijkp q` pr qs and B g g Rpiqj R . From this we can easily deduce the evolution equation for the Riemann ijk` D rks` curvature tensors to GRF. Let vij .x; t/ be any symmetric 2-tensor, we consider the ﬂow @ gij .x; t/ vij .x; t/: (3-2) @t D GENERALIZED RICCI FLOW, I 755

1 k` Applying a formula in [Chow and Knopf 2004] to our case vij 2Rij hij with hij H Hj , WD C 2 D ik` we obtain

@ 1 1 1 R 2 i R i h 2 i R i h @t ijk` D 2 r rk j` C 2 r rk j` C r r` jk 2 r r` jk 1 1 2 j R j h 2 j R j h C r rk i` 2 r rk i` r r` ik C 2 r r` ik 1 gpqR . 2R 1 h / R . 2R 1 h / C 2 ijkp q` C 2 q` C ijp` qk C 2 qk pq i R i R j R j R g .R R R R / D r rk j` r r` jk r rk i` C r r` ik ijkp q` C ijp` qk 1 1 pq i h i h j h j h g R h R h C 4 r rk j` C r r` jk C r rk i` r r` ik C 4 ijkp q` C ijp` qk R 2 B B B B D ijk` C ijk` ij`k iljk C ikj` pq g .R R R Rqj R R R R / pjk` q` C ipk` C ijp` qk C ijkp q` 1 1 pq i h i h j h j h g R h R h : C 4 r rk j` C r r` jk C r rk i` r r` ik C 4 ijkp q` C ijp` qk Proposition 3.1. For GRF we have @ R R 2 .B B B B / @t ijk` D ijk` C ijk` ij`k i`jk C ikj` pq g .R R R Rqj R R R R / pjk` q` C ipk` C ijp` qk C ijkp q` 1 1 pq . i h i h j h j h / g R h R h : C 4 r rk j` C r r` jk C r rk i` r r` ik C 4 ijkp q` C ijp` qk In particular: Corollary 3.2. For GRF we have

2 @ X Rm Rm Rm Rm H H Rm iH 2 iH: (3-3) @t D C C C r r i 0 D Proof. From Proposition 3.1, we obtain @ Rm Rm Rm Rm 2h h Rm : @t D C Cr C On the other hand, h H H and D 2h . .H H // . H H / 2H H H H: r D r r D r r D r C r r Combining these terms, we obtain the result. Proposition 3.3. For GRF we have

@ 1 R R 2 R ; Rpq 2 Rpi; R h ; R Rip; h @t ik D ik C h piqk i h pk i C 4 h `q i`kqi C h kpi 1 2 j` j` i H g i h g j h h : C 4 r rk j j C r r` jk C r rk i` ik Proof. Since @ j` @ jp `q R g R 2g g R Rpq @t ik D @t ijk` C ijk` 756 YI LI and ij ij pq ij pr qs 2 g hij g HipqHj g g g HipqHj rs H ; D D D j j it follows that

j` pq pq g Œ i h i h j h j h g h R g h R r rk j` C r r` jk C r rk i` r r` ik C q` ijkp C qk ijp` 2 j` j` j` pq pq i H g i h g j h h g g h R g h Rip: D r rk j j C r r` jk C r rk i` ik C q` ijkp C qk From these identities, we get the result.

As a consequence, we obtain the evolution equation for scalar curvature.

Proposition 3.4. For GRF we have

@ 2 1 2 1 1 ik j` R R 2 Ric H hij ; Rij g g i j h : @t D C j j 2 j j C 2 h i C 2 r r k` Proof. From the usual evolution equation for scalar curvature under the Ricci ﬂow, we have

@ 2 1 ik R R 2 Ric g Œ h ; R Rip; h @t D C j j C 4 h `q i`kqi C h kpi 1 ik 2 j` j` g i H g i h g j h h C 4 r rk j j C r r` jk C r rk i` ik 2 1 1 R 2 Ric hij ; Rij Rip; hip D C j j C 4 h i C 4 h i 1 2 1 ik j` 1 ik j` 1 2 H g g i h g g j h H : 4 j j C 4 r r` jk C 4 r rk i` 4 j j Simplifying the terms, we obtain the required result.

4. Derivative estimates

In this section we are going to prove BBS estimates. At ﬁrst we review several basic identities of commutators Œ; and Œ@=@t; . If A A.t/ is a t-dependency tensor, and @gij =@t vij , then r r D D applying the well-known formulas stated in [Chow and Knopf 2004] on GRF we have @ @ Rm Rm Rm .Rm H H / @t r D r @t C r C . Rm Rm Rm H H Rm 2H H H H / Rm Rm H H Rm D r C C Cr Cr r C r C r X X X . Rm/ i Rm j Rm iH jH k Rm iH jH: D r C r r C r r r C r r i j 0 i j k 0 i j 0 2 (4-1) C D C C D C D C More generally:

Proposition 4.1. For GRF and any nonnegative integer ` we have

@ X X X ` Rm . ` Rm/ i Rm j Rm iH jH k Rm iH jH: @t r D r C r r C r r r C r r i j ` i j k ` i j ` 2 C D C C D C D C (4-2) GENERALIZED RICCI FLOW, I 757

Proof. For ` 1, this is (4-1). Suppose (4-2) holds for 1; : : : ; `. By induction on `, for ` 1 we have D C @ ` 1 Rm @t r C @ . ` Rm/ D @t r r @ . ` Rm/ ` Rm .Rm H H / D r @t r C r r C Ä X X X . ` Rm/ i Rm j Rm iH jH k Rm iH jH D r r C r r C r r r C r r i j ` i j k ` i j ` 2 C D C C D C D C ` Rm Rm H H ` Rm C r r C r r ` 1 ` ` 1 . C Rm/ Rm Rm Rm C Rm D r XC r r C r i 1 Rm j Rm i Rm j 1 Rm C r C r Cr r C i j ` CXD i 1H jH k Rm iH j 1H k Rm iH jH k 1 Rm C r C r r Cr r C r Cr r r C i j k ` C XC D . i 1H jH iH j 1H / H H l Rm : C r C r C r r C C r r i j ` 2 C D C Simplifying these terms, we obtain the required result.

As an immediate consequence, we have an evolution inequality for l Rm 2. jr j Corollary 4.2. For GRF and any nonnegative integer ` we have

@ X ` Rm 2 l Rm 2 2 ` 1 Rm 2 C i Rm j Rm ` Rm @t jr j Ä jr j jr C j C jr j jr j jr j i j ` X C D X C iH jH k Rm ` Rm C iH jH ` Rm ; (4-3) C jr j jr j jr j jr j C jr j jr j jr j i j k ` i j ` 2 C C D C D C where C represents universal constants depending only on the dimension of M .

Next we derive the evolution equations for the covariant derivatives of H.

Proposition 4.3. For GRF and any positive integer ` we have

@ X X `H . `H / iH j Rm iH jH kH: (4-4) @t r D r C r r C r r r i j ` i j k ` C D C C D Proof. From the Bochner formula, the evolution equation for H can be rewritten as @ H H Rm H: (4-5) @t D C 758 YI LI

For ` 1, we have D @ @ H H H .Rm H H/ @t r D r @t C r C .H Rm H / H Rm H H H D r C C r C r .H / H Rm H Rm H H H D r C r Cr C r . H / Rm H H Rm H H H: D r C r C r C r Using (4-2) and the same argument, we can prove the evolution equation for higher covariant derivatives. Similarly, we have an evolution inequality for `H 2. jr j Corollary 4.4. For GRF and for any positive integer l we have @ lH 2 `H 2 2 ` 1H 2 @t jr j Ä jr j jr C j X X C iH j Rm `H C iH j H kH lH ; (4-6) C jr j jr j jr j C jr j jr j jr j jr j i j ` i j k ` C D C C D while @ H 2 H 2 2 H 2 C Rm H 2: (4-7) @t j j Ä j j jr j C j j j j Theorem 4.5. Suppose that .g.x; t/; H.x; t// is a solution to GRF on a closed manifold M n for a short time 0 t T and K1; K2 are arbitrary given nonnegative constants. Then there exists a constant Cn Ä Ä depending only on n such that if

Rm.x; t/ K1; H.x/ K2 j jg.x;t/ Ä j jg.x/ Ä for all x M and t Œ0; T , then 2 2 CnK1t H.x; t/ K2e (4-8) j jg.x;t/ Ä for all x M and t Œ0; T . 2 2 Proof. Since @ 2 2 2 2 2 H H Cn Rm H H CnK1 H ; @t j j Ä j j C j j j j Ä j j C j j using the maximum principle, we obtain u.t/ u.0/eCnK1t , where u.t/ H 2. Ä D j j The main result in this section is the following estimates for higher derivatives of Riemann curvature tensors and three-forms. Some special cases were proved in [Streets 2007; 2008; Young 2008]. Theorem 4.6. Suppose that .g.x; t/; H.x; t// is a solution to GRF on a compact manifold M n and K is an arbitrary given positive constant. Then for each ˛ > 0 and each integer m 1 there exists a constant Cm depending on m; n; max ˛; 1 , and K such that if f g Rm.x; t/ K; H.x/ K j jg.x;t/ Ä j jg.x/ Ä GENERALIZED RICCI FLOW, I 759 for all x M and t Œ0; ˛=K, then 2 2 m 1 m Cm Rm.x; t/ g.x;t/ H.x; t/ g.x;t/ (4-9) jr j C jr j Ä t m=2 for all x M and t .0; ˛=K. 2 2 Proof. In the following computations we always let C be any constants depending on n; m; max ˛; 1 ,and f g K, which may take different values at different places. From the evolution equations and Theorem 4.5, we have @ Rm 2 Rm 2 2 Rm 2 C C 2H C H 2; @t j j Ä j j jr j C C jr j C jr j @ H 2 H 2 2 H 2 C; @t j j Ä j j jr j C @ H 2 H 2 2 2H 2 C Rm H C H 2: @t jr j Ä jr j jr j C jr j jr j C jr j Consider the function u t H 2 H 2 t Rm 2. Directly computing, we obtain D jr j C j j C j j @ u u 2t 2H 2 C t 2H .C 2 / H 2 C C 2t Rm 2 C t Rm H @t Ä jr j C jr j C jr j C C jr j C jr j jr j u 2.C / H 2 C.1 /: Ä C jr j C C If we choose C , then @ u u C which implies that u CeC t since u.0/ C . With this estimate D @t Ä C Ä Ä we are able to bound the ﬁrst covariant derivative of Rm and the second covariant derivative of H. In order to control the term Rm 2, we should use the evolution equations of H 2, H 2 and 2H 2 to jr j j j jr j jr j cancel with the bad terms, i.e., 2 Rm 2, 2H 2, and 3H 2, in the evolution equation of Rm 2: jr j jr j jr j jr j @ Rm 2 @t jr j C C Rm 2 2 2 Rm 2 C Rm 2 Rm C Rm 3H 2H Rm ; Ä jr j jr j C jr j C t 1=2 jr jC jr jjr jC t 1=2 jr jjr j @ C C 2H 2 2H 2 2 3H 2 C 2 Rm 2H Rm 2H C 2H 2 2H : @t jr j Ä jr j jr j C jr jjr jC t 1=2 jr jjr jC jr j C t jr j As above, we deﬁne

u t 2. 2H 2 Rm 2/ tˇ. H 2 Rm 2/ H 2; WD jr j C jr j C jr j C j j C j j @u and therefore, u C . Motivated by cases for m 1 and m 2, for general m, we can deﬁne a @t Ä C D D function

m 1 m m 2 m 1 2 X i i 2 i 1 2 2 u t . H Rm / ˇit . H Rm / H ; WD jr j C jr j C jr j C jr j C j j i 1 D where ˇi and are positive constants determined later. In the following, we always assume m 3. 760 YI LI C Suppose that i 1 Rm iH i , for i 1; 2;:::; m 1. For such i, from Corollary 4.4, we jr j C jr j Ä t i=2 D have

i @ X iH 2 iH 2 2 i 1H 2 C jH i j Rm iH @t jr j Ä jr j jr C j C jr j jr j jr j j 0 i i j D X X C jH i j `H `H iH C jr j jr j jr j jr j j 0 ` 0 D D i i i j i 2 i 1 2 i X Cj Ci j 1 i X X Cj Ci j ` C` H 2 C H C H j i j C1 C H j i j 1 l Ä jr j jr j C jr j C C jr j 2 j 0 t 2 t 2 j 0 ` 0 t 2 t 2 t D D D i 2 i 1 2 Ci i Ci i H 2 C H i 1 H i H : Ä jr j jr j C t C2 jr j C t 2 jr j Similarly, from Corollary 4.2 we also have

i 1 @ X i 1 Rm 2 i 1 Rm 2 2 i Rm 2 C j Rm i 1 j Rm i 1 Rm @t jr j Ä jr j jr j C jr jjr jjr j j 0 i 1 i 1 j D X X C jH i 1 j `H ` Rm i 1 Rm C jr j jr j jr j jr j j 0 ` 0 D D i 1 XC C jH i 1 j H i 1 Rm C jr j jr C j jr j j 0 D i 1 i 1 2 i 2 i 1 X Cj 1 Ci j Rm 2 Rm C Rm j C1 i j Ä jr j jr j C jr j C j 0 t 2 t 2 D i 1 i 1 j i 1 X X Cj Ci 1 j ` C` 1 C Rm j i1 j ` ` C1 C jr j C j 0 ` 0 t 2 t 2 t 2 D D i i 1 X Cj Ci 1 j i 1 Ci C Rm j i C1 j C C H i C jr j C C jr j j 1 t 2 t 2 t 2 D i 1 2 i 2 Ci i 1 Ci i 1 Ci i 1 Rm 2 Rm i 1 Rm i C H i Rm : Ä jr j jr j C t C2 jr j C t 2 jr j C t 2 jr j The evolution inequality for u is now given by

m 1 @u m 1 m 2 m 1 2 X i 1 i 2 i 1 2 mt . H Rm / iˇit . H Rm / @t Ä jr j C jr j C jr j C jr j i 1 D Â Ã m 1 Â Ã m @ m 2 @ m 1 2 X i @ i 2 @ i 1 2 @ 2 t H Rm ˇit H Rm H : C @t jr j C @t jr j C @t jr j C @t jr j C @t j j i 1 D GENERALIZED RICCI FLOW, I 761

It’s easy to see that the second term is bounded by

m 1 m 1 X i 1 Ci X 1 iˇit iˇiCit ; t i D i 1 i 1 D D but this bound depends on t and approaches to inﬁnity when t goes to zero. Hence we use the last second term to control this bad term. The evolution inequality for the third term is the combination of the inequalities @ mH 2 @t jr j m X mH 2 2 m 1H 2 C iH m i Rm mH Ä jr j jr C j C jr j jr j jr j i 0 m m i D X X C jH m i j H iH mH C jr j jr j jr j jr j i 0 j 0 D D m 2 m 1 2 m 2 m m Cm m Cm m H 2 C H C H C Rm H m 1 H m H Ä jr j jr j C jr j C jr j jr j C t C2 jr j C t 2 jr j and m 1 @ X m 1 Rm 2 m 1 Rm 2 2 m Rm 2 C i Rm m 1 i Rm m 1 Rm @t jr j Ä jr j jr j C jr j jr j jr j i 0 m 1 m 1 i D X X C jH m 1 i j H i Rm m 1 Rm C jr j jr j jr j jr j i 0 j 0 D D m 1 XC C iH m 1 iH m 1 Rm C jr j jr C j jr j i 0 D m 1 2 m 2 m 1 2 C m m 1 Rm 2 Rm C Rm 1 H Rm Ä jr j jr j C jr j C t 2 jr j jr j m 1 m 1 Cm m 1 Cm m 1 C C H Rm m 1 Rm m Rm : C jr jjr j C t C2 jr j C t 2 jr j Therefore we have m 1 @u m 1 m 2 m 1 2 X i 1 i 2 i 1 2 mt . H Rm / iˇit . H Rm / @t Ä jr j C jr j C jr j C jr j i 1 Â D m m 2 m 1 2 C m m 2 t H 2 C H m 1 H C H C jr j jr j C t C2 jr j C jr j C m Rm mH m 1 Rm 2 C jr j jr j C jr j m 2 C m 1 m 1 2 2 Rm m 1 Rm C Rm jr j C C jr j C jr j t 2 Ã C m m 1 m 1 m 1 H Rm C C H Rm C t 1=2 jr j jr j C jr j jr j 762 YI LI

m 1 Â X i Ci i 1 i 2 i 1 2 ˇit i 1 Rm H 2 C H C C jr j C jr j jr j i 1 t 2 Ã D i 1 2 Ci i Ci i 1 i 2 Rm i 1 H i C H 2 Rm C jr j C t C2 jr j C t 2 jr j jr j . H 2 2 H 2 C / C j j jr j C u 2t m m 1H 2 C t m m 1H m 1 Rm Ä jr C j C jr C j jr j m 2 m m 2 m m m X i i 1 2 i 2 2t Rm C t Rm H .i 1/ˇi 1t . C H Rm / jr j C jr j jr j C C C jr j C jr j i 0 m 1 D X i i 1 2 i 2 2 2 ˇit . H Rm / 2 H C jr C j C jr j jr j C i 1 D m 1 m 2 m 1 m 1 2 m 1 m m 1 m 1 C t H C t Rm j t 2 H C t 2 Rm C jr j C jr j C jr j C jr j m 1 m m 1 m m 1 m 1 C t 2 H Rm C t H Rm C jr j jr j C jr C j jr j m 1 m 1 X i i 1 X i 1 i 2 i 1 ˇiCit 2 H ˇiCit 2 . H Rm /: C jr C j C jr j C jr j i 1 i 1 D D Choosing A .i 1/ˇi 1 ˇi; ˇi ; i 0; C C D D i! where A is constant which is determined later, and noting that

m 1 m 1 m 1 X i=2 i 1 1 X i i 1 2 1 X 2 ˇiCit H ˇit H ˇiC jr C j Ä 2 jr C j C 2 i i 1 i 1 i 1 D D D and m 1 X i 1 i i 1 ˇiCit 2 . H Rm / jr j C jr j i 1 D m 2 X i i 1 i ˇ1C1. H Rm / ˇi 1Ci 1t 2 . C H Rm / Ä jr j C j j C C C jr j C jr j i 1 D m 2 Â i i 1 2 i i 2 Ã X t C H t Rm ˇi 1Ci 1 ˇ1C1. H Rm / ˇi 1Ci 1 jr j C jr j C C Ä jr j C j j C C C 2ˇi 1Ci 1=ˇi C ˇi i 1 C C D m 2 m 2 2 2 1 X i i 1 2 i 2 X ˇi 1Ci 1 ˇ1C1. H Rm / ˇit . C H Rm / C C ; Ä jr j C j j C 2 jr j C jr j C ˇi i 1 i 1 D D GENERALIZED RICCI FLOW, I 763 we obtain @ u u 2t m m 1H 2 C t m m 1H m 1 Rm @t Ä jr C j C jr C j jr j 2t m m Rm 2 C t m mH m Rm C t m 1 mH 2 C t m 1 m 1 Rm 2 jr j C jr j jr j C jr j C jr j m 1 m m 1 2 2 C t 2 H Rm ˇ0. H Rm / C jr j jr j C jr j C j j m 1 m 2 X i i 1 2 i 2 X i i 1 2 i 2 ˇit . H Rm / ˇit . H Rm / jr C j C jr j C jr C j C jr j i 1 i 1 D D 1 m 1 m 2 2 2 ˇm 1t H ˇ1C1 H 2 H C C C jr j C jr j jr j C C m 1 m 1 2 m 1 m 2 m 1 m 2 m 1 2 2 u C t Rm C t H C t 2 . H Rm / ˇ0 H Ä C jr j C jr j jr j C jr j C jr j 2 1 m 1 m 2 m 1 m 2 ˇ1C1 H 2 H C C 2 ˇm 1t H ˇm 1t Rm C jr j jr j C C jr j jr j 1 p m 1 m 1 2 m 2 u 2 .C t C ˇm 1/t . Rm H / Ä C C jr j C jr j 2 .ˇ0 ˇ1C1 2 / H C C ˇ1C1: C C jr j C C C @u When we chose A and sufﬁciently large, we obtain u C , which implies that u.t/ C since @t Ä C Ä u.0/ is bounded. Finally we give an estimate that plays a crucial role in the next section. Corollary 4.7. Let .g.x; t/; H.x; t// be a solution of the generalized Ricci ﬂow on a closed manifold M . If there are ˇ > 0 and K > 0 such that

Rm.x; t/ K; H.x/ K j jg.x;t/ Ä j jg.x/ Ä for all x M and t Œ0; T , where T > ˇ=K, then there exists for each m ގ a constant Cm depending 2 2 2 on m; n; min ˇ; 1 , and K such that f g m 1 m m=2 Rm.x; t/ H.x; t/ CmK jr jg.x;t/ C jr jg.x;t/ Ä for all x M and t Œmin ˇ; 1 =K; T . 2 2 f g Proof. The proof is the same as in [Chow et al. 2007]; we just copy it here. Let ˇ1 min ˇ; 1 . For any WD f g ﬁxed point t0 Œˇ1=K; T we set T0 t0 ˇ1=K. For t t T0 we let .g.t/; H .t// be the solution 2 WD N WD N N N N of the system @g @H 1 N N 2Ric 2 h; HL;gH ; g.0/ g.T0/; H .0/ H.T0/: @t D C N @t D N N D D N N The uniqueness of solution implies that g.t/ g.t T0/ g.t/ for t Œ0; ˇ1=K. By the assumption N N D NC D N 2 we have

Rm.x; t/ g.x;t/ K; H .x/ g.x/ K j N j N N Ä j j N Ä for all x M and t Œ0; ˇ1=K. Applying Theorem 4.5 with ˛ ˇ1, we have 2 N 2 D m 1 m C m Rm.x; t/ g.x;t/ H.x; t/ g.x;t/ jr N j N C jr N j N Ä t m=2 N N N 764 YI LI

m=2 m=2 m=2 m=2 for all x M and t .0; ˇ1=K. We have t ˇ 2 K if t Œˇ1=2K; ˇ1=K. Taking 2 N 2 N 1 N 2 t ˇ1=K, we obtain N D m=2 m=2 m 1 m 2 C mK Rm.x; t0/ H.x; t0/ jr jg.x;t0/ C jr jg.x;t0/ Ä m=2 ˇ1 for all x M . Since t0 Œˇ=K; T was arbitrary, the result follows. 2 2 5. Compactness theorem

In this section we prove the compactness theorem for our generalized Ricci flow. We follow [Hamilton 1995] on the compactness theorem for the usual Ricci flow. We review several definitions from [Chow et al. 2007]. Throughout this section, all Riemannian manifolds are smooth manifolds of dimensions n. The covariant derivative with respect to a metric g will be denoted by g . r Definition 5.1. Let K M be a compact set and let gk k ގ; g , and g be Riemannian metrics on M . p f g 2 1 For p 0 ގ we say that gk converges in C to g uniformly on K with respect to g if for every 2 f g [ 1 > 0 there exists k0 k0./ > 0 such that for k k0, D g ˛ gk g C p K;g sup sup .gk g /.x/ g < : (5-1) k 1k I WD 0 ˛ p x K j r 1 j Ä Ä 2 Since we consider a compact set, the choice of background metric g does not change the convergence. Hence we may choose g g . D 1 2 Definition 5.2. Suppose Uk k ގ is an exhaustion of a smooth manifold M by open sets and gk are f g 2 Riemannian metrics on Uk . We say that .Uk ; gk / converges in C 1 to .M; g / uniformly on compact 1 sets in M if for any compact set K M and any p > 0 there exists k0 k0.K; p/ such that gk k k0 D f g converges in C p to g uniformly on K. 1 A pointed Riemannian manifold is a 3-tuple .M; g; O/, where .M; g/ is a Riemannian manifold and O M is a basepoint. If the metric g is complete, the 3-tuple is called a complete pointed Riemannian 2 manifold. We say .M; g.t/; H.t/; O/; t .˛; !/, is a pointed solution to the generalized Ricci flow if 2 .M; g.t/; H.t// is a solution to the generalized Ricci flow.

The so-called Cheeger–Gromov convergence in C 1 is deﬁned as follows:

Deﬁnition 5.3. A given sequence .Mk ; gk ; Ok / k ގ of complete pointed Riemannian manifolds con- f g 2 verges to a complete pointed Riemannian manifold .M ; g ; O / if there exist 1 1 1 (i) an exhaustion Uk k ގ of M by open sets with O Uk , and f g 2 1 1 2 (ii) a sequence of diffeomorphisms ˆk M Uk Vk ˆk .Uk / Mk with ˆk .O / Ok W 1 3 ! WD 1 D such that .Uk ; ˆk.gk Vk // converges in C 1 to .M ; g / uniformly on compact sets in M . j 1 1 1 2If for any compact set K M there exists k ގ such that U K for all k k 0 2 k 0 GENERALIZED RICCI FLOW, I 765

The corresponding convergence for the generalized Ricci ﬂow is similar to the convergence for the usual Ricci ﬂow introduced by Hamilton [1995].

Deﬁnition 5.4. A given sequence .Mk ; gk .t/; Hk .t/; Ok / k ގ of complete pointed solutions to the f g 2 GRF converges to a complete pointed solution to the GRF

.M ; g .t/; H .t/; O /; t .˛; !/; 1 1 1 1 2 if there exist

(i) an exhaustion Uk k ގ of M by open sets with O Uk , f g 2 1 1 2 (ii) a sequence of diffeomorphisms ˆk M Uk Vk ˆk .Uk / Mk with ˆk .O / Ok W 1 3 ! WD 1 D 2 such that U .˛; !/; ˆ .g .t/ V / dt ; ˆ .H .t/ V / converges in C to k k k j k C k k j k 1 M .˛; !/; g .t/ dt 2; H .t/ 1 1 C 1 uniformly on compact sets in M .˛; !/. Here we denote by dt 2 the standard metric on .˛; !/. 1 Let injg.O/ be the injectivity radius of the metric g at the point O. The following compactness theorem is due to Cheeger and Gromov.

Theorem 5.5 (compactness for metrics). Let .Mk ; gk ; Ok / k ގ be a sequence of complete pointed f g 2 Riemannian manifolds satisfying these conditions:

(i) For all p 0 and k ގ, there is a sequence of constants Cp < independent of k such that 2 1 gk p Rm.g / g Cp j r k j k Ä on Mk .

(ii) There exists some constant 0 > 0 such that

inj .O / 0 gk k for all k ގ. 2

Then there exists a subsequence jk k ގ such that .Mjk ; gjk ; Ojk / k ގ converges to a complete pointed f g 2 f g 2 Riemannaian manifold .M n ; g ; O / as k . 1 1 1 ! 1 As a consequence of Theorem 5.5, we state our compactness theorem for GRF.

Theorem 5.6 (compactness for GRF). Let .Mk ; gk .t/; Hk .t/; Ok / k ގ be a sequence of complete f g 2 pointed solutions to GRF for t Œ˛; !/ 0 satisfying these conditions: 2 3 (i) There is a constant C0 < independent of k such that 1

sup Rm.gk .x; t// gk .x;t/ C0; sup Hk .x; ˛/ gk .x;˛/ C0: .x;t/ Mk .˛;!/ j j Ä x Mk j j Ä 2 2 (ii) There exists a constant 0 > 0 satisfying

inj .O / 0: gk .0/ k 766 YI LI

Then there exists a subsequence jk k ގ such that f g 2

.Mjk ; gjk .t/; Hjk .t/; Ojk / .M ; g .t/; H .t/; O /; ! 1 1 1 1 converges to a complete pointed solution .M ; g .t/; H .t/; O /; t Œ˛; !/, to GRF as k . 1 1 1 1 2 ! 1 To prove Theorem 5.6 we extend a lemma for Ricci ﬂow to GRF. After establishing this lemma, the proof of Theorem 5.6 is similar to that of Theorem 3.10 in [Chow et al. 2007].

Lemma 5.7. Let .M; g/ be a Riemannian manifold with a background metric g, let K be a compact subset of M , and let .gk .x; t/; Hk .x; t// be a collection of solutions to the generalized Ricci flow defined on neighborhoods of K Œˇ; , where t0 Œˇ; is a fixed time. Suppose that: 2 (i) The metrics g .x; t0/ are all uniformly equivalent to g.x/ on K, i.e., for all V TxM; k, and x K, k 2 2 1 C g.x/.V; V / g .x; t0/.V; V / Cg.x/.V; V /; Ä k Ä where C < is a constant independent of V; k, and x. 1 (ii) The covariant derivatives of the metrics gk .x; t0/ with respect to the metric g.x/ are all uniformly bounded on K, i.e., for all k and p 1, g p g p 1 g .x; t0/ H .x; t0/ Cp j r k jg.x/ C j r k jg.x/ Ä

where Cp < is a sequence of constants independent of k. 1 (iii) The covariant derivatives of the curvature tensors Rm.gk .x; t// and of the forms Hk .x; t/ are uniformly bounded with respect to the metric g .x; t/ on K Œˇ; , i.e., for all k and p 0, k gk p Rm.g .x; t// gk pH .x; t/ C j r k jgk .x;t/ C j r k jgk .x;t/ Ä p0

where Cp0 is a sequence of constants independent of k. Then the metrics g .x; t/ are uniformly equivalent to g.x/ on K Œˇ; , i.e., k 1 B.t; t0/ g.x/.V; V / g .x; t/.V; V / B.t; t0/g.x/.V; V /; Ä k Ä

C t t0 where B.t; t0/ Ce 00 (here the constant C may not be equal to the previous one), and the time- D j j 00 derivatives and covariant derivatives of the metrics gk .x; t/ with respect to the metric g.x/ are uniformly bounded on K Œˇ; , i.e., for each .p; q/ there is a constant Cp;q independent of k such that z ˇ @q ˇ ˇ @q ˇ ˇ g pg .x; t/ˇ ˇ g p 1H .x; t/ˇ C ˇ q k ˇ ˇ q k ˇ zp;q ˇ@t r ˇg.x/ C ˇ@t r ˇg.x/ Ä for all k.

Proof. We use [Chow et al. 2007, Lemma 3.13]: Suppose that the metrics g1 and g2 are equivalent, i.e., 1 .p q/=2 C g1 g2 Cg1. Then for any .p; q/-tensor T we have T g C T g . We denote by h Ä Ä j j 2 Ä C j j 1 GENERALIZED RICCI FLOW, I 767

kp lq the tensor hij g g H Hjpq. In the following we denote by C a constant depending only on n; ˇ, WD ikl and , which may take different values at different places. For any tangent vector V TxM we have 2 @ g .x; t/.V; V / 2 Ric.g .x; t//.V; V / 1 h .x; t/.V; V /; @t k D k C 2 k and therefore ˇ ˇ ˇ @ ˇ ˇ 2 Ric.g .x; t//.V; V / 1 h .x; t/.V; V /ˇ ˇ ˇ ˇ k 2 k ˇ ˇ log gk .x; t/.V; V /ˇ ˇ C ˇ ˇ@t ˇ D ˇ gk .x; t/.V; V / ˇ 2 C 0 C Hk .x; t/ Ä 0 C j jgk .x;t/ C CC 2 C ; Ä 00 C 00 DW since

2 Ric.gk .x; t//.V; V / C 0 gk .x; t/.V; V /; hk .x; t/.V; V / C Hk .x; t/ gk .x; t/.V; V /: j j Ä 0 j j Ä j jgk .x;t/ Integrating on both sides, we have

Z t1 ˇ ˇ ˇZ t1 ˇ ˇ ˇ ˇ @ ˇ ˇ @ ˇ ˇ gk .x; t1/.V; V /ˇ C t1 t0 ˇ log gk .x; t/.V; V /ˇ dt ˇ log gk .t/.V; V / dtˇ ˇlog ˇ ; j j t0 ˇ@t ˇ ˇ t0 @t ˇ D ˇ gk .x; t0/.V; V /ˇ and hence we conclude that

C t1 t0 C t1 t0 e g .x; t0/.V; V / g .x; t1/.V; V / e g .x; t0/.V; V /: j j k Ä k Ä j j k From the assumption (i), it immediately deduces from above that

1 C t1 t0 C t1 t0 C e g.x/.V; V / g .x; t1/.V; V / Ce g.x/.V; V /: j j Ä k Ä j j

Since t1 was arbitrary, the ﬁrst part is proved. From the deﬁnition (or see [Chow et al. 2007, p. 134, (37)]), we have

ec g g g gk e g e .g / . a.g / .g /ac c.g / / 2. / 2. / : k r k bc C rb k r k ab D ab ab Thus gk .x; t/ g.x/ C g g .x; t/ . On the other hand, j jg.x/ Ä j r k jgk .x/ g gk e g e gk e g e a.g / .g / Œ. / . / .g /ecŒ. / . / ; r k bc D k eb ac ac C k ab ab it follows that g g .x; t/ C gk .x; t/ g.x/ and therefore j r k jgk .x;t/ Ä j jgk .x;t/ g g is equivalent to gk g gk g : (5-2) r k D r r The evolution equation for g is @ gk c cd gk gk . / .g / Œ. /a.Ric.g // . / .Ric.g // @t ab D k r k bd C r b k ad gk 1 cd gk gk gk . / .Ric.g // .g / . /a.h / . / .h / . / .h / : r d k ab C 4 k r k bd C r b k ad r d k ab 768 YI LI

Since g does not depend on t, it follows from the assumptions that ˇ @ ˇ ˇ gk g ˇ ˇgk ˇ gk . / C .Ric.g // C .h / g ˇ ˇ ˇ k ˇgk k k ˇ@t ˇgk Ä r C j r j gk CC C H g H g C : Ä 10 C j r k j k j k j k Ä 10 Integrating on both sides, ˇZ t1 @ ˇ ˇ k g ˇ gk g gk g C 0 t1 t0 ˇ .g .t/ / dtˇ .t1/ g .t0/ g : 1j j ˇ @t ˇ j j k j j k t0 gk Hence we obtain

gk g gk g .t/ g C t1 t0 .t0/ g j j k Ä 10 j j C j j k g C t1 t0 C g .t0/ g Ä 10 j j C j r k j k g C t t0 C g .t0/ g Ä 10 j j C j r k j C t t0 C1: Ä 10 j j C The equivalency of metrics tells us that

g 3=2 g 3=2 gk g g .t/ g B.t; t0/ g .t/ g B.t; t0/ C .t/ g j r k j Ä j r k j k Ä j j k 3=2 B.t; t0/ .C t t0 C /: Ä 10 j j C 0 g Since t t0 ˇ, it follows that g .t/ g C1;0 for some constant C1;0. But g and g are j j Ä j r k j Ä z z k equivalent, we have

H .t/ g C H .t/ g CC C1;0: j k j Ä j k j k Ä 10 D z From the assumptions, we also have

g g gk gk H g . /H H g j r k j Ä j r r k C r k j g gk C g g H g C H g Ä j r k j j k j C j r k j k CC C C1;0C1;0 C2;0: Ä 10 C z z WD z Moreover,

@ g g H .g H Rm.g / H / @t r k D r k k C k k g gk gk g g . /g H g H Rm.g / H Rm.g / H D r r k k C r k k C r k k C k r k where gk is the Laplace operator associated to gk . Hence ˇ ˇ ˇ @ g ˇ ˇ Hk ˇ ˇ@t r ˇg

g gk g g C g g g H g C g H g C Rm.g / g H g C Rm.g / g H g Ä j r k j j k k j k C j r k k j C j r k j j k j C j k j j r k j C2;1: Ä z GENERALIZED RICCI FLOW, I 769

For higher derivatives we claim that

g p g p g p g p 1 Ric.g / g C g g C ; g g H g Cp;0; (5-3) j r k j Ä p00j r k j C p000 j r k j C j r k j Ä z for all p 1, where C ; C , and Cp;0 are constants independent of k. For p 1, we have proved the p00 p000 z D second inequality, so we sufﬁce to prove the ﬁrst one with p 1. Indeed, D

g g gk gk Ric.g / g C . / Ric.g / Ric.g / g j r k j Ä j r r k C r k j k g gk gk C g Ric.g / g C Ric.g / g Ä j j j k j k C j r k j k g C g g C : Ä 100j r k j C 1000

Suppose the claim holds for all p < N (N 2), we shall show that it also holds for p N . From D ˇ ˇ ˇ N ˇ g N X g N i g gk gk i 1 gk N Ric.g / g ˇ . / Ric.g / Ric.g /ˇ j r k j D ˇ r r r r k C r k ˇ ˇi 1 ˇg D N ˇ ˇ X g N i g gk gk i 1 gk N ˇ . / Ric.gk /ˇ Ric.gk / g Ä ˇ r r r r ˇg C j r j i 1 D we estimate each term. For i 1, by induction and the assumptions we have D g N 1 g gk . / Ric.g / g j r r r k j g N 1 g C . g Ric.g // g Ä j r r k k j ˇN 1 ˇ ˇ X N 1Ág N 1 j g g j ˇ C ˇ . gk / .Ric.gk //ˇ Ä ˇ j r r r ˇ j 0 g D N 1 X N 1Á g N j g j C gk g Ric.gk / g Ä j j r j j r j j 0 D N 1 X N 1Á g j g N j C .C 00 gk g C 000/ gk g Ä j j j r j C j j r j i 0 D N 1 X N 1Á g N j C .C 00Cj;0 C 000/ gk g Ä j j z C j j r j j 0 D N 1 g N X N 1Á C.N 1/.C000Cj;0 C0000/ gk g C .Cj00Cj;0 Cj000/CN j;0 D z C j r j C j z C z j 1 D g N C g g C : Ä N00 j r k j C N000 770 YI LI

For i 2, we have ˇg N i g gk gk i 1 ˇ ˇg N i g gk i 1 ˇ ˇ . / Ric.gk /ˇ C ˇ . gk Ric.gk //ˇ r r r r g Ä r r r g N i Á X N i g N i j 1 g j gk i 1 C C gk g Ric.gk / g: Ä j j r j j r r j j 0 D If j 0, then D gk i 1 g i 1 Ric.gk / g Ci00 1 gk g Ci0001 Ci00 1Ci 1;0 Ci0001: j r j Ä j r j C Ä z C Suppose in the following that j 1. Hence ˇg j gk i 1 ˇ ˇ g gk gk j gk i 1 ˇ ˇ Ric.gk /ˇ ˇ.. / / Ric.gk /ˇ r r g D r r C r r g j Á X j g l gk j l i 1 C g g Ric.g / g Ä l j r k j j r C k j l 0 D j Xj Á C Cl;0.Cj00 l i 1Cj l i 1;0 Cj000 l i 1/; Ä l z z C C l 0 C C D where we make use of (5-2) from ﬁrst line to second line. Combining these inequalities, we get

g N g N Ric.g / g C g g C : j r k j Ä N00 j r k j C N000 Similarly, we have g N g N h g C g g C : j r k j Ä N00 j r k j C N000 @ Since g 2 Ric.g / 1 h , it follows that @t k D k C 2 k @ g N g g N . 2 Ric.g / 1 h /; @t r k D r k C 2 k ˇ ˇ2 @ g N 2 ˇ @ g N ˇ g N 2 gk g ˇ gk ˇ gk g @t j r j Ä ˇ@t r ˇg C j r j 1 8 g N Ric.g / 2 g N h 2 g N g 2 Ä j r k jg C 2j r k jg C j r k jg .1 18.C /2/ g N g 2 18.C /2: Ä C N00 j r k jg C N00 g g N Integrating the above inequality, we get gk g CN;0 and therefore hk g CN 1;0. We have j r j Ä z j r j Ä z C proved lemma for q 0. When g 1, then D @q @q 1 g pg .t/ g p 2 Ric.g .t// 1 h .t/: q k q 1 k 2 k @t r D r @t C Using the evolution equations for Rm.gk .t// and hk .t/, combining the induction to q and using the above method, we have q q ˇ @ g p ˇ ˇ @ g p 1 ˇ ˇ gk .t/ˇ ˇ hk .t/ˇ Cp;q: ˇ@t q r ˇg C ˇ@t q r ˇg Ä z GENERALIZED RICCI FLOW, I 771

6. Generalization

In this section, we generalize the main results in Sections 4 and 5 to a kind of generalized Ricci ﬂow for which local existence has been established [He et al. 2008]. Let .M; gij .x// be an n-dimensional closed Riemannian manifold and let A Ai and B Bij D f g D f g denote a one-form and a two-form respectively. Set F dA and H dB. The authors in [He et al. D D 2008] proved that there exists a constant T > 0 such that the evolution equations

@ 1 gij .x; t/ 2Rij .x; t/ hij .x; t/ 2f .x; t/; gij .x; 0/ gij .x/; @t D C 2 C jk D @ k Ai.x; t/ 2 Fi .x; t/; Ai.x; 0/ Ai.x/; @t D rk D @ k Bij .x; t/ 3 H ij .x; t/; Bij .x; 0/ Bij .x/ @t D rk D kl k has a unique smooth solution on m Œ0; T /, where hij H Hj and fij Fi F . We call it D ikl D jk RF.A; B/. According to the deﬁnition of the adjoint operator d , we have

k k .d F/i 2 Fi ;.d H /ij 3 H ij ; (6-1) D rk D rk and hence @ F.x; t/ dd F F F Rm F; (6-2) @t D g.x;t/ D HL;g.x;t/ D C @ H.x; t/ dd H H H Rm H: (6-3) @t D g.x;t/ D HL;g.x;t/ D C They also derived the evolution equations of curvatures: @ R R 2.B B B B / @t ijk` D ijk` C ijk` ij`k i`jk C ikj` pq g .R Rqi R Rqj R R R R / pjk` C ipk` C ijp` qk C ijkp q` 1 pq pq pq pq i .H Hj / i .HjpqH / j .H Hi / j .HipqH / C 4 r r` kpq r rk ` r r` kpq C r rk ` 1 rs pq pq g .H H R HrpqH R / C 4 kpq r ijs` C ` ijks p p p p i .F Fjp/ i .Fj F / j .F Fip/ j .Fi F / C r r` k r rk `p r r` k C r rk `p rs p p g .F FrpR Fr F R /: C k ijs` C `p ijks Under our notation, it can be rewritten as @ X X X Rm Rm i Rm j Rm iH jH iF jF @t D C r r C r r C r r i j 0 i j 0 2 i j 0 2 C D X C D C XC D C iH jH k Rm !! iF jF k Rm : (6-4) C r r r C r r r i j k 0 i j k 0 C C D C C D As before, we have: 772 YI LI

Proposition 6.1. For RF.A; B/ and any nonnegative integer ` we have @ X X X ` Rm . l Rm/ i Rm j Rm iH jH iF jF @t r D r C r r C r r C r r i j ` i j ` 2 i j ` 2 C D X C D C X C D C iH jH k Rm iF jF k Rm : (6-5) C r r r C r r r i j k ` i j k ` C C D C C D In particular, @ X l Rm 2 l Rm 2 2 ` 1 Rm 2 C i Rm j Rm ` Rm @t jr j Ä jr j jr C j C jr j jr j jr j i j ` X C XD C iH jH ` Rm C iF jF ` Rm C jr j jr j jr j C jr j jr j jr j i j ` 2 i j ` 2 C D C C D C X X C iH jH k Rm ` Rm C iF jF k Rm ` Rm : C jr j jr j jr j jr j C jr j jr j jr j jr j i j k ` i j k ` C C D C C D @ Since F F Rm F it follows that @t D C @ @ F F F .Rm H H F F/ @t r D r @t C r C C .F Rm F/ F Rm F H H F F F D r C C r C r C r . F/ Rm F Rm F F H H F F F: D r C r C r C r C r It can be expressed as @ X F . F/ iF j Rm @t r D r C r r i j 1 1 1 1 i C D X X X iF jF kF iF jH 1 i j H: C r r r C r r r i j k 1 i 0 j 0 C C D D D More generally, we can show: Proposition 6.2. For RF.A; B/ and any positive integer ` we have @ X `F . `F/ iF j Rm @t r D r C r r i j ` ` 1 ` i X X X C D iF jF kF iF jH ` i j H: C r r r C r r r i j k ` i 0 j 0 C C D D D In particular, @ X `F 2 `F 2 2 ` 1F 2 C iF j Rm `F @t jr j Ä jr j jr C j C jr j jr j jr j i j ` C D ` 1 ` i X X X C iF jF kF lF C iF jH ` i j H `F : C jr j jr j jr j jr j C jr j jr j jr j jr j i j k ` i 0j 0 C C D D D GENERALIZED RICCI FLOW, I 773

Similarly, we obtain: Proposition 6.3. For RF.A; B/ and any positive integer l we have @ X `H . `H / iH j Rm @t r D r C r r i j ` ` 1 ` i X X X C D iH jH kH iH j F ` i j F: C r r r C r r r i j k ` i 0 j 0 C C D D D In particular, @ X `H 2 `H 2 2 ` 1H 2 C iH j Rm `H @t jr j Ä jr j jr C j C jr j jr j jr j i j ` C D ` 1 ` i X X X C iH jH kH `H C iH jF ` i j F `H : C jr j jr j jr j jr j C jr j jr j jr j jr j i j k ` i 0 j 0 C C D D D From the evolution inequalities @ H 2 H 2 2 H 2 C Rm H 2; @t j j Ä j j jr j C j j j j @ F 2 F 2 2 F 2 C Rm F 2; @t j j Ä j j jr j C j j j j the following theorem is obvious. Theorem 6.4. Suppose that .g.x; t/; H.x; t/; F.x; t// is a solution to RF.A; B/ on a compact manifold n M for a short time 0 t T and K1; K2; K3 are arbitrary given nonnegative constants. Then there Ä Ä exists a constant Cn depending only on n such that if

Rm.x; t/ K1; H.x/ K2; F.x/ K3 j jg.x;t/ Ä j jg.x/ Ä j jg.x/ Ä for all x M and t Œ0; T , then 2 2 CnK1t CnK1t H.x; t/ K2e ; F.x; t/ K3e ; (6-6) j jg.x;t/ Ä j jg.x;t/ Ä for all x M and t Œ0; T . 2 2 Parallel to Theorem 4.6, we can prove: Theorem 6.5. Suppose that .g.x; t/; H.x; t/; F.x; t// is a solution to RF.A; B/ on a compact manifold M n and K is an arbitrary given positive constant. Then for each ˛ > 0 and each integer m 1 there exists a constant Cm depending on m; n; max ˛; 1 , and K such that if f g Rm.x; t/ K; H.x/ K; F.x/ K j jg.x;t/ Ä j jg.x/ Ä j jg.x/ Ä for all x M and t Œ0; ˛=K, then 2 2 m 1 m m Cm Rm.x; t/ g.x;t/ H.x; t/ g.x;t/ F.x; t/ g.x;t/ m ; (6-7) jr j C jr j C jr j Ä t 2 for all x M and t .0; ˛=K. 2 2 774 YI LI

We can also establish the corresponding compactness theorem for RF.A; B/. We omit the detail since the proof is close to the proof in Section 5. In the forthcoming paper, we will consider the BBS estimates for complete noncompact Riemannian manifolds.

Acknowledgment

The author thanks his advisor, Professor Shing-Tung Yau, for helpful discussions. The author expresses his gratitude to Professor Kefeng Liu for his interest in this work and for his numerous help in mathematics. He also thanks Valentino Tosatti and Jeff Streets for several useful conversations.

References

[Bakas 2007] I. Bakas, “Renormalization group equations and geometric flows”, preprint, 2007. arXiv hep-th/0702034v1 [Cao and Zhu 2006] H.-D. Cao and X.-P. Zhu, “A complete proof of the Poincaré and geometrization conjectures: application of the Hamilton–Perelman theory of the Ricci flow”, Asian J. Math. 10:2 (2006), 165–492. MR 2008d:53090 Zbl 1200.53057 [Chow and Knopf 2004] B. Chow and D. Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs 110, American Mathematical Society, Providence, RI, 2004. MR 2005e:53101 Zbl 1086.53085 [Chow et al. 2006] B. Chow, P. Lu, and L. Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics 77, American Mathematical Society, Providence, RI, 2006. MR 2008a:53068 Zbl 1118.53001 [Chow et al. 2007] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci flow: techniques and applications, I: Geometric aspects, Mathematical Surveys and Monographs 135, American Mathematical Society, Providence, RI, 2007. MR 2008f:53088 Zbl 1157.53034 [Chow et al. 2008] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci flow: techniques and applications, II: Analytic aspects, Mathematical Surveys and Monographs 144, American Mathematical Society, Providence, RI, 2008. MR 2008j:53114 Zbl 1157.53035 [Chow et al. 2010] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci flow: techniques and applications, III: Geometric-analytic aspects, Mathematical Surveys and Monographs 163, American Mathematical Society, Providence, RI, 2010. MR 2011g:53142 Zbl 1216.53057 [Gegenberg and Kunstatter 2004] J. Gegenberg and G. Kunstatter, “Using 3D string-inspired gravity to understand the Thurston conjecture”, Classical Quantum Gravity 21:4 (2004), 1197–1207. MR 2004j:53083 Zbl 1046.83024 [Hamilton 1982] R. S. Hamilton, “Three-manifolds with positive Ricci curvature”, J. Differential Geom. 17:2 (1982), 255–306. MR 84a:53050 Zbl 0504.53034 [Hamilton 1995] R. S. Hamilton, “A compactness property for solutions of the Ricci flow”, Amer. J. Math. 117:3 (1995), 545–572. MR 96c:53056 Zbl 0840.53029 [He et al. 2008] C.-L. He, S. Hu, D.-X. Kong, and K. Liu, “Generalized Ricci flow, I: Local existence and uniqueness”, pp. 151–171 in Topology and physics, edited by K. Lin et al., Nankai Tracts Math. 12, World Scientific, Hackensack, NJ, 2008. MR 2010k:53098 Zbl 1182.35145 [Kleiner and Lott 2008] B. Kleiner and J. Lott, “Notes on Perelman’s papers”, Geom. Topol. 12:5 (2008), 2587–2855. MR 2010h:53098 Zbl 1204.53033 [Morgan and Tian 2007] J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3, American Mathematical Society, Providence, RI, 2007. MR 2008d:57020 Zbl 1179.57045 [Oliynyk et al. 2006] T. Oliynyk, V. Suneeta, and E. Woolgar, “A gradient flow for worldsheet nonlinear sigma models”, Nuclear Phys. B 739:3 (2006), 441–458. MR 2006m:81185 Zbl 1109.81058 [Perelman 2002] G. Perelman, “The entropy formula for the Ricci flow and its geometric applications”, preprint, 2002. Zbl 1130.53001 arXiv math.DG/0211159 [Shi 1989] W.-X. Shi, “Deforming the metric on complete Riemannian manifolds”, J. Differential Geom. 30:1 (1989), 223–301. MR 90i:58202 Zbl 0676.53044 GENERALIZED RICCI FLOW, I 775

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Received 22 Sep 2010. Revised 4 Aug 2011. Accepted 27 Sep 2011.

YI LI: [email protected] Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, United States

mathematical sciences publishers msp

ANALYSIS AND PDE Vol. 5, No. 4, 2012 dx.doi.org/10.2140/apde.2012.5.777 msp

SMOOTH TYPE II BLOW-UP SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION

MATTHIEU HILLAIRET AND PIERRE RAPHAËL

We exhibit Ꮿ∞ type II blow-up solutions to the focusing energy-critical wave equation in dimension N = 4. These solutions admit near blow-up time a decomposition 1 x u(t, x) = (Q + ε(t)) , with kε(t), ∂ ε(t)k ˙ 1 2 1, λ(N−2)/2(t) λ(t) t H ×L ∗ where Q is the extremizing proﬁle of the Sobolev embedding H˙ 1 → L2 , and a blow-up speed √ λ(t) = (T − t)e− |log(T −t)|(1+o(1)) as t → T.

1. Introduction

Setting of the problem. We deal in this paper with the energy-critical focusing wave equation (N+2)/(N−2) ∂tt u − 1u − f (u) = 0 with f (t) = t , N (1-1) (u, ∂t u)|t=0 = (u0, u1), (t, x) ∈ ޒ × ޒ . in dimension N = 4. This is a special case of the nonlinear wave equation

∂tt u − 1u − f (u) = 0, (1-2) which, since the pioneering [Jörgens 1961], has been the subject of a considerable amount of work. For the energy-critical nonlinearity f (u) = ±t(N+2)/(N−2), the Cauchy problem is locally well posed in the energy space H˙ 1 × L2 and the solution propagates regularity; see [Sogge 1995] and references therein. Recall that in this case, (1-2) admits a conserved energy Z Z Z 1 1 N −2 − E(u(t)) = E(u , u ) = (∂ u)2 + |∇u|2 ∓ u2N/(N 2) 0 1 2 t 2 2N that is left invariant by the scaling symmetry of the ﬂow, 1 t x u (t, x) = u , . λ λ(N−2)/2 λ λ Global existence in the defocusing case was proved by Struwe [1988] for radial data and Grillakis [1990] for general data. For focusing nonlinearities, a sharp threshold criterion of global existence and scattering

MSC2010: 35Q51. Keywords: wave equation, blow-up.

777 778 MATTHIEU HILLAIRET AND PIERRERAPHAËL or finite time blow-up is obtained by Kenig and Merle [2008] based on the soliton solution to (1-1), 1 (N−2)/2 Q(r) = , (1-3) 1 + r 2/(N(N − 2)) ˙ 1 2∗ which is the extremizing profile of the Sobolev embedding H → L . Indeed, for initial data (u0, u1) such that E(u0, u1) < E(Q, 0), those with k∇u0kL2 < k∇ QkL2 have global solutions and scatter, while those with k∇u0kL2 > k∇ QkL2 lead to finite time blow-up. Note that like in the works of Levine [1974] (see also [Strauss 1989]) and as is standard in a nonlinear dispersive setting, blow-up is derived through obstructive convexity arguments; see also [Karageorgis and Strauss 2007] for refined statements near the soliton Q. However, this approach gives very little insight into the description of the blow-up mechanism and the description of the flow even just near the ground state soliton Q is still only at its beginning.

On the energy-critical wave map problem. There is an important literature devoted to the construction of blow-up solutions for nonlinear wave equations; see [Alinhac 1995; Merle and Zaag 2003; 2008] for the study of the ODE-type of blow-up for subcritical nonlinearities. For energy-critical problems like (1-1), recent important progress has been made through the study of the two-dimensional energy-critical corotational wave map to the 2-sphere, ∂ u k2 sin 2u ∂ u − ∂ u − r − = 0, (1-4) tt rr r 2r 2 where k ∈ ގ∗ is the homotopy number. The ground state is given there by

Q(r) = 2 tan−1(r k).

After the pioneering works of Christodoulou and Tahvildar-Zadeh [1993], Shatah and Tahvildar-Zadeh [1994] and Struwe [2003] and their detailed study of the concentration of energy scenario, the first explicit description of singularity formation for the k = 1 case was derived by Krieger, Schlag and Tataru [2008] who constructed finite energy finite time blow-up solutions of the form x u(t, x) = (Q + ε) t, , with kε(t), ∂ ε(t)k ˙ 1, (1-5) λ(t) t H 1×L2 with a blow-up speed given by λ(t) = (T − t)ν,

3 for any ν > 2 ; see also [Krieger et al. 2009a]. The spectacular feature of this result is that it exhibits arbitrarily slow blow-up regimes further and further from self-similarity which would correspond to the (forbidden; see [Struwe 2003]) self-similar law

λ(t) ∼ T − t. (1-6)

Numerics suggest that this blow-up scenario is nongeneric and corresponds to ﬁnite-codimensional manifolds [Bizon´ et al. 2001]. After the pioneering work [Rodnianski and Sterbenz 2010] for large homotopy number k ≥ 4, Raphaël and Rodnianski [2012] gave a complete description of stable blow-up SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 779 dynamics that originate from smooth data for all homotopy numbers k ≥ 1. The blow-up speed obeys in this regime a universal law that depends in an essential way on the rate of convergence of the ground state Q to its asymptotic value, 1 π − Q ∼ as r → ∞, r k and indeed the stable blow-up regime corresponds to a decomposition (1-5) with blow-up speed  T − t c for k ≥ 2,  k | − |1/(2k−2) λ(t) ∼ log(T √t) (1-7) (T − t)e− |log(T −t)| for k = 1. Note that this work draws an important analogy with another critical problem, the L2 critical nonlinear Schrödinger equation, where a similar universality of the stable singularity formation near the ground state was proved in [Merle and Raphael 2003; 2004; 2005a; 2005b; 2006; Raphael 2005].

Statement of the result. For the power nonlinearity energy-critical problem (1-1), there has been recent progress towards the understanding of the flow near the solitary wave Q. Krieger and Schlag [2007] constructed in dimension N = 3 a codimension one manifold of initial data near Q that yield global solutions asymptotically converging to the soliton manifold. The strategy developed by Krieger et al. [2008] for the wave map problem has been adapted in [Krieger et al. 2009b] to show in dimension N = 3 the existence of finite energy finite time blow-up solutions of the form 1 x u(t, x) = (Q + ε) t, , with kε(t), ∂ ε(t)k ˙ 1, λ(N−2)/2(t) λ(t) t H 1×L2 and with a blow-up speed given by λ(t) = (T − t)ν, (1-8)

3 for any ν > 2 . The quantization of the energy at blow-up for small type II blow-up solutions in dimension N ∈ {3, 5} is proved in [Duyckaerts et al. 2011; 2012] in the radial and nonradial cases. In particular, for radial data, if T < +∞ and

|∇ |2 + |2 ≤ |∇ |2 + ∗ ∗ sup u(t) L2 ∂t u L2 Q L2 α , α 1, t∈[0,T ] then there exists a dilation parameter λ(t) → 0 as t → T and asymptotic proﬁles (u∗, v∗) ∈ H 1 × L2 such that 1 x u(t, x) − Q , ∂ u(t) → (u∗, v∗) in H˙ 1 × L2 as t → T ; λ(N−2)/2(t) λ(t) t see [Merle and Raphael 2005b] for related classiﬁcation results for the L2 critical (NLS). These works however leave open the question of the existence of smooth type II blow-up solutions. We claim that such smooth type II blow-up solutions can be constructed in dimension N = 4 as the formal analogue of the singular dynamics exhibited by Raphaël and Rodnianski [2012] for the wave map problem in the least homotopy number class k = 1. The following theorem is the main result of this paper: 780 MATTHIEU HILLAIRET AND PIERRERAPHAËL

Theorem 1.1 (existence of smooth type II blow-up solutions in dimension N = 4). Let N = 4. Then for ∗ ∞ all α > 0, there exist Ꮿ initial data (u0, u1) with ∗ E(u0, u1) < E(Q, 0) + α such that the corresponding solution to the energy-critical focusing wave equation (1-1) blows up in

ﬁnite time T = T (u0, u1) < +∞ in a type II regime according to the following dynamics: there exist (u∗, v∗) ∈ H˙ 1 × L2 such that 1 x ∗ ∗ u(t, x) − Q , ∂ u(t) → (u , v ) in H˙ 1 × L2 as t → T, (1-9) λ(N−2)/2(t) λ(t) t with a blow-up speed given by √ λ(t) = (T − t)e− |log(T −t)|(1+o(1)) as t → T. (1-10)

Comments on the result. 1. On the smoothness of the initial data. An important feature of Theorem 1.1 is to exhibit a new blow-up speed which is valid for Ꮿ∞ solutions. Indeed, while the Krieger et al. [2009b] approach provides a continuum of blow-up speeds, the exact regularity of the obtained solutions is not known, which is an unpleasant consequence of their construction scheme. In fact, it is expected that Ꮿ∞ initial data should lead to quantized blow-up rates hence breaking the continuum of blow-up speeds (1-8), we refer to [van den Berg et al. 2003] for a related discussion in the context of the energy-critical harmonic heat flow. Hence we expect the blow-up rate (1-10) to correspond to the minimal type II blow-up speed of smooth solutions with small supercritical energy. Such a general lower bound on the blow-up rate in the spirit of the one obtained by Merle and Raphael [2006; Raphael 2005] for the L2 critical NLS is an open problem. The construction of excited blow-up solutions with other speeds and Ꮿ∞ regularity also remains to be done. This problematic is related to the understanding of the structure of the flow near Q, which is still in its infancy. 2. On the codimension one manifold. The proof of Theorem 1.1 involves a detailed description of the set of initial data leading to the type II blow-up with speed (1-10). Indeed, given a small enough parameter b0 > 0 and a suitable deformation Qb0 of the soliton with → → Qb0 Q as b0 0 in some strong sense, we show that for any smooth and radially symmetric excess of energy 2 b0 kη0, η1kH 2×H 1 . , |log(b0)| we can find d+(b0, η0, η1) ∈ ޒ such that the solution to (1-1) with initial data N −2 u = Q + η + d+ψ, u = b Q + y · ∇ Q + η , 0 b0 0 1 0 2 b0 b0 1 blows up in finite time in the regime described by Theorem 1.1. Here ψ is the bound state of the linearized operator close to Q and generates the unstable mode, we refer to Definition 3.4 and Proposition 3.5 for precise statements. Hence the set of blow-up solutions we construct lives on a codimension one SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 781 manifold in the radial class in some weak sense. Following [Krieger and Schlag 2007; Krieger and Schlag 2009], the proof that this set is indeed a codimension one manifold relies on proving some Lipschitz regularity of the map (b0, η0, η1) → d+(b0, η0, η1), and in particular some local uniqueness to begin with. The analysis in [Krieger and Schlag 2009] shows that this may be a delicate step in some cases. Our solution is constructed using a soft continuous topological argument of Brouwer-type coupled with suitable monotonicity properties in the spirit of Cote, Marte and Merle [2009]. In other related settings (see [Martel 2005; Raphaël and Szeftel 2011]) this strategy has proved to be quite powerful for eventually achieving strong uniqueness results. This interesting question in our setting will require additional efforts and needs to be addressed separately in detail. 3. Extension to higher dimensions. We focus on the case of dimension N = 4 for the sake of simplicity. Our main objective is to provide a robust framework to construct Ꮿ∞ type II blow-up solutions. However, following the heuristic developed in [Raphaël and Rodnianski 2012], the blow-up speed (1-10) corresponds to the k = 1 case in (1-7), and we similarly conjecture in dimension N ≥ 5 the existence of type II finite time blow-up solutions close to Q with blow-up speed T − t λ(t) ∼ cN . |log(T − t)|1/(N−4) Note from (1-3) that the higher the dimension, the fastest the decay of the ground state Q, and that this should help avoid some difficulties that occur only in low dimension like in [Raphaël and Rodnianski 2012] for large homotopy number k ≥ 4. We expect the strategy developed in this paper to carry over to the cases N = 5 and 6, but the extension to large dimension will be confronted in particular with the difficulty of the lack of smoothness of the nonlinearity. Let us also insist on the fact that the case N = 4 is in many ways the more delicate one in terms of the strong coupling of the main part of the solution and the outgoing tail due to the slow decay of Q, which results in the somewhat pathological blow-up speed (1-10). This comment becomes even more dramatic in dimension N = 3, where we expect our analysis to be applicable to the construction of Ꮿ∞ type II blow-up solutions, but this seems to require a slightly different approach.

Strategy of the proof of Theorem 1.1.

Step 1: Approximate self-similar solution. Let D, 3 denote the differential operators in (1-18). Exact self-similar solutions to (1-1) of the form 1 x u(t, x) = Q , with b = −λ , λ(N−2)/2(t) b λ(t) t where Qb satisﬁes the self-similar equation

2 3 1Qb − b D3Qb + Qb = 0, (1-11) are known to develop a singularity on the light cone y = (T − t)/λ(t) = 1/b leading to an unbounded

Dirichlet energy k∇ QbkL2 = +∞; see [Kavian and Weissler 1990]. We therefore assume 0 < b 1 and 782 MATTHIEU HILLAIRET AND PIERRERAPHAËL consider a one term expansion approximation

2 Qb = Q + b T1, which injected into (1-11) yields, at the order b2,

HT1 = −D3Q. (1-12) Here H is the linearized operator close to Q given by

N +2 − H = −1 − Q4/(N 2). (1-13) N −2 The spectral structure of H is well known in connection to the fact that Q is an extremizer of the Sobolev ∗ embedding H˙ 1 → L2 , and in the radial sector H admits one nonpositive eigenvalue with well localized eigenvector ψ, Hψ = −ζ ψ, ζ > 0, (1-14) and a resonance at the boundary of the continuum spectrum generated by the scaling invariance of (1-1), C H(3Q) = 0, 3Q(r) ∼ as r → +∞. (1-15) r N−2 = 1 | |2 In order to solve (1-12), we ﬁrst remove the leading-order growth in the exact solution T1 4 y Q which is consequence of the ﬂux computation

(D3Q, 3Q) = 1 lim y4|3Q|2 > 0 (1-16) 2 y→+∞ due to the slow decay of Q in dimension N = 4 from (1-3). For this, we solve (D3Q, 3Q) 1 HT = −D3Q + c 3Q1 ≤ , with c = ∼ as b → 0. 1 b y 1/b b R | |2 | b| y≤1/b 3Q 2 log The purpose of this construction is to yield after a suitable localization process an o(b2) approximate solution to the self-similar equation (1-11) whose dominant term near and past the light cone is still given by Q itself in the sense that 2 b |T1| Q for y ≥ 1/b. This identiﬁes Q as the leading-order radiation term.1 Step 2: Bootstrap estimates. We now roughly consider initial data of the form = + + = + | | + k k 2 u0 Qb0 d+ψ η0, u1 b03Qb0 η1, with d+ η0, η1 H 2×H 1 b0, (1-17) and introduce a modulated decomposition of the ﬂow 1 x u(t, x) = (Q + ε) t, , b(t) = −λ . λ(N−2)/2(t) b(t) λ(t) t

1See [Raphaël and Rodnianski 2012] for a further discussion on this issue and the role played by the nonvanishing Pohozaev integration (1-16). SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 783

Here we face the major difference between the power nonlinearity wave equation (1-1) and the critical wave map problem (1-4), which is the presence of a negative eigenvalue in the first case (1-14) for the linearized operator H close to Q. This induces an instability in the modulation equations for b, λ that is absent in the wave map case, leading to stable blow-up dynamics. However, we claim that the ODE-type instability generated by (1-14) is the only instability mechanism. The situation is conceptually similar to the one studied in [Cote et al. 2009] where multisolitary wave solutions are constructed in the supercritical regime despite the presence of exponentially growing modes for the linearized operator which are absent in the subcritical regime. We adapt a similar scheme of proof that does not rely on a fixed point argument to solve the problem from infinity in time,2 but by directly following the flow for any initial data of the form (1-17). This reduces the full problem to a one-dimensional dynamical system for which a clever classical continuity argument yields the existence of d+(b0, η0, η1) such that the unstable mode is extinct, see Section 5. The key is hence to control the flow under the a priori control of the unstable mode, and here we adapt the technology developed in [Raphaël and Rodnianski 2012] which relies on monotonicity properties of the linearized Hamiltonian at the H 2 level of regularity. However, the analysis in [Raphaël and Rodnianski 2012] heavily relies on the existence of a decomposition of the Hamiltonian,

∗ H = A A, A = −∂y + V (y), which is central to the proof of the main monotonicity property and is lost in our setting. This forces us to revisit the approach in several ways, and to rely in particular on ﬁne algebraic properties of the ﬂow3 near Q and coercivity properties of suitable quadratic forms in the spirit of [Martel and Merle 2002; Merle and Raphael 2005a] (see Lemma 4.7) which remarkably turn out to be almost explicit thanks to the formula

(1-3). We are eventually able to ﬁnd d+(b0, η0, η1) for which, to leading order,

2 2 b ds 1 2 b ∼ −c b ∼ − , b = −λ , = , |d+| + k∂ εk 2 b , s b 2 |log b| t dt λ yy L and whose reintegration in time yields ﬁnite time blow-up in the regime described by Theorem 1.1.

Notation. We deﬁne differential operators N −2 N 3f = f + y · ∇ f (H˙ 1 scaling), D f = f + y · ∇ f (L2 scaling). (1-18) 2 2 Denoting by Z Z +∞ ( f, g) = f g = f (r)g(r)r N−1dr 0 the L2(ޒN ) radial inner product, we observe the integration by parts formulas

(D f, g) = −( f, Dg) and (3f, g) + (3g, f ) = −2( f, g). (1-19)

2After renormalization of the time. 3See in particular (4-23), (4-38). 784 MATTHIEU HILLAIRET AND PIERRERAPHAËL

Given f and λ > 0, we shall write 1 r f (t, r) = f t, , λ λ(N−2)/2 λ and the rescaled space variable will always be denoted by r y = . λ We let χ be a smooth positive radial cut off function, χ(r) = 1 for r ≤ 1 and χ(r) = 0 for r ≥ 2. For a given parameter B > 0, we let r χ (r) = χ . (1-20) B B Given b > 0, we set 2 |log b| B = , B = . (1-21) 0 b 1 b To clarify the exposition we use the notation a . b for when there exists a constant C with no relevant dependency on (a, b) such that a ≤ Cb. In particular, we do not allow constants C to depend on the parameter M except in Appendix A.

2. Computation of the modiﬁed self-similar proﬁle

This section is devoted to the construction of an approximate self-similar solution Qb which describes the dominant part of the blow-up profile inside the backward light cone from the singular point (0, T ) and displays a slow decay at infinity which is eventually responsible for the modifications to the blow-up speed with respect to the self-similar law. The key to this construction is the fact that the structure of the linearized operator H close to Q is completely explicit in the radial sector thanks to the explicit formulas at hand for the elements of the kernel. We introduce the direction 8 = D3Q, (2-1) which displays the cancellation 1 |8(y)| (2-2) . 1+y4 and the crucial nondegeneracy which follows from the Pohozaev integration by parts formula,

(8, 3Q) = lim 1 y4|3Q|2 = 32 > 0. (2-3) y→+∞ 2 Proposition 2.1 (approximate self-similar solution). Let M denote a large enough constant. Then there exists b∗(M) > 0 small enough such that for all 0 < b < b∗(M), there exists a smooth radially symmetric proﬁle T1 satisfying the orthogonality condition

N − 2 y2 = − 3Q(y) N 1 , (2-11) 21 + y2/(N(N − 2)) /2 N(N − 2) which admits the asymptotics dk(3Q) O(1) as y → 0, ∀ k ≥ 0, (y) = (2-12) dyk O(y−(N−2+k)) as y → ∞. Now let Z y ds 0(y) = −3Q(y) N−1 2 1 s (3Q) (s) be another (singular at the origin4) element of the kernel of H, which can be found from the Wronskian relation −1 003Q − 0(3Q)0 = . y N−1 From this we easily ﬁnd the asymptotics of 0(k) for any integer k: dk0 O(y−(N−2+k)) as y → 0, (y) = (2-13) dyk O(y−k) as y → ∞. √ 4Note that 0 must be smooth at y = N(N − 2), where 3Q vanishes, because of the radial ODE H0 = 0. 786 MATTHIEU HILLAIRET AND PIERRERAPHAËL

A smooth solution to Hw = F is Z y Z y w(y) = 0(y) F(s)3Q(s)s N−1ds − 3Q(y) F(s)0(s)s N−1ds. (2-14) 0 0

2 We now look for a solution to the self-similar equation in the form Q + b T1. This yields

2 9b = −1Qb + b D3Qb − f (Qb) 2 4 2 2 0 = b (HT1 + D3Q) + b D3T1 − f (Q + b T1) − f (Q) − b f (Q)T1 . (2-15)

Step 2: Computation of T1. Thanks to the anomalous decay (2-2), we choose T1 to be a solution of

HT = F = −D3Q + c χ 3Q, 1 b B0/4 (2-16) (T1, χM 8) = 0, with cb chosen such that (F, 3Q) = 0. (2-17)

That is, from the Pohozaev integration by parts formula — see (1-21) and (2-3) —

This yields (2-10). Following (2-14), we ﬁrst consider

Z y Z y ˜ 3 3 T1(y) = 0(y) F(s)3Q(s)s ds − 3Q(y) F(s)0(s)s ds. (2-18) 0 0 ˜ The smoothness of T1 at the origin follows from (2-18) together with elliptic regularity from (2-16). We ˜ now examine the behavior of T1 at large y. We ﬁrst observe that, from the orthogonality (2-17),

Z +∞ Z y ˜ 3 3 T1(y) = − 0(y) F(s)3Q(s)s ds + 3Q(y) F(s)0(s)s ds . y 0

−4 2 Hence, from the degeneracy |D3Q| = O(y ), this yields that for B0/2 ≤ y ≤ 1/b ,

Similarly, for 1 ≤ y ≤ B0/2, Z +∞ Z y ˜ 3 3 |T1(y)| = 0(y) F(s)3Q(s)s ds + 3Q(y) F(s)0(s)s ds y 0 Z +∞ s3 Z B0 s3 1 Z y s3 Z y s3 ds + |c | ds + ds + |c | ds . 4 2 b 2 2 2 4 b 2 y (1 + s )(1 + s ) y (1 + s ) 1 + y 0 1 + s 0 1 + s 1 + |log(by)| |log(1 + y)| + . (2-20) . |log b| 1 + y2 We now choose, thanks to (2-3), ˜ ˜ (T1, χM 8) T1(y) = T1(y) − c3Q with c = , (χM 8, 3Q) so that the orthogonality condition (2-4) is fulﬁlled. We note that the bounds (2-19) and (2-20) ensure that c remains bounded by log(M) uniformly in M and b, provided b is chosen sufﬁciently small with respect to M. This yields (2-7) for k = 0; the other cases follow similarly.

Step 3: Estimate on 9B1 and ∂b9B1 . We now cut off the slow decaying tail T1 according to (2-5) and estimate the corresponding error to self-similarity 9B1 given by (2-6). We compute

9 = b2χ (HT + D3Q) + b2−2χ 0 T 0 − T 1χ + (1 − χ )D3Q + b2 D3(χ T ) B1 B1 1 B1 1 1 B1 B1 B1 1 − + 2 − − 0 f (Q b χB1 T1) f (Q) χB1 f (Q)T1 .

= 2 = Outside the support of χB1 we have thus 9B1 b D3Q. On the other hand, in dimension N 4, we have the Taylor expansion

Z 1 f (Q + b2χ T ) − f (Q) − χ f 0(Q)T = b4χ 2 T 2(y) (1 − τ)(Q(y) + τb2χ T (y)) dτ. B1 1 B1 1 B1 1 B1 1 0

We thus estimate from (2-7), (2-15), (2-16) and the degeneracy (2-2) for y ≤ 2B1 that 0 2 2 T1 T1 1 9B − b cbχB /43Q b 1y≥B /2 + + 1 0 . 1 1 + y 1 + y2 1 + y4 Z 1 + 4| | + 4| 2 | − | + 2 | b D3(χB1 T1) b T1 (y) (1 τ) Q(y) τb T1(y) dτ. 0 (2-7) now yields (2-9) for k = 0. Further derivatives are estimated similarly thanks to the smoothness of the nonlinearity. We emphasize here that, given B > 0 large, we have 1/(1 + y) . 1/B . 1/(1 + y) on 0 the support of χB, so that differentiating χB acts as multiplication by 1/(1 + y). Furthermore, we have 1/B = o(b) so that we can always dominate 1/(1 + y) by b on the support of χ 0 . 1 B1

Finally, we compute ∂b PB1 from (2-5). 788 MATTHIEU HILLAIRET AND PIERRERAPHAËL 1 To this end, we note that ∂bcb = O when b → 0, so that the source term for T in (2-16) b|log(b)|2 1 satisﬁes 1 1 ∂b F = O χB + O ρB 3Q, b |log b| 0/4 b |log b| 0/4 0 ∞ where ρ(z) = zχ (z) ∈ Ꮿc (0, ∞) and we keep the convention for function dilation. Hence, the same ˜ arguments as for T1 enable us to show ﬁrst that ∂bT1, and then ∂bT1, satisfy the estimates k d ∂bT1 1 1 + |log(by)| 1 1 + |log(1 + y)| (y) 12≤y≤B /2 + 1y≥B /2 + . (2-21) dyk . b(1 + yk) |log b| 0 b2 y2|log b| 0 1 + y2 Finally, we compute from (2-5) that = + 2 + 2 ∂b PB1 2bχB1 T1 b ∂b log(B1)ρB1 T1 b χB1 ∂bT1. (2-22) This decomposition, together with (2-7) and the previous computation, yield (2-8), which concludes the proof of Proposition 2.1.

3. Description of the trapped regime

We display in this section the regime which leads to the blow-up dynamics described by Theorem 1.1.

Modulation of solutions to (1-1). Let us start by describing the set of solutions among which the ﬁnite time blow-up scenario described by Theorem 1.1 is likely to arise. We recall from (1-14) that ψ denotes the bound state of H with eigenvalue −ζ < 0. The following lemma is a standard consequence of the implicit function theorem and the smoothness of the ﬂow; see Appendix A. ∗ Lemma 3.1 (modulation theory). Let M be a large constant to be chosen later and 0 < b0 < b0(M) small enough. Let (η0, η1, d+) satisfy the smallness condition b2 | | + ∇ + − ∇ 0 d+ η0, η0, η1 b0(1 χB1(b0))3Q, η1 H˙ 1×H˙1×L2×L2 . , (3-1) |log b0| 2 2 N 2 N then there exists a time T0 such that the unique solution u ∈ Ꮿ ([0, T0]; L (ޒ )) ∩ Ꮿ([0, T0]; H (ޒ )) to (1-1) with initial data = + + = + u0 PB1(b0) η0 d+ψ, u1 b03PB1(b0) η1, (3-2) admits on [0, T0] a unique decomposition = + u(t) (PB1(b(t)) ε(t))λ(t) (3-3) 2 ∗ with λ ∈ Ꮿ ([0, T0], ޒ+) such that

(ε(t), χM 8) = 0 and b(t) = −λt for all t ∈ [0, T0], (3-4) and the following smallness condition is satisﬁed: 2 2 b0 k∇ε(t)kL2 . b0|log b0|, |b(t) − b0| + |λ(t) − 1| + k∇ ε(t)kL2 . for all t ∈ [0, T0]. (3-5) |log b0| SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 789

Remark 3.2. Recall that the slow decay of Q and the choice of PB1 induces an unbounded tail of 3PB1 in the energy norm, and more speciﬁcally k3QkL2 = +∞, hence the need for the compensation in the norm of the time derivative in (3-1).

Decomposition of the ﬂow and modulation equations. Considering initial data satisfying the assumption of the above lemma, we now write the evolution equation induced by (1-1) in terms of the decomposition (3-3). Let 1 r u(t, r) = P + ε t, = P + w(t, r), (3-6) [λ(t)]N/2−1 B1(b(t)) λ(t) B1(b(t)) λ(t) where b = −λt . Let us derive the equations for w and ε. Let

Z t dτ s(t) = (3-7) 0 λ(τ) be the rescaled time. We shall make an intensive use of the rescaling formulas

1 r ds 1 u(t, r) = v(s, y), y = , = , (3-8) λN/2−1 λ dt λ 1 ∂ u = (∂ v + b3v) , (3-9) t λ s λ 1 ∂ u = ∂2v + b(∂ v + 23∂ v) + b2 D3v + b 3v . (3-10) tt λ2 s s s s λ In particular, we derive from (1-1) the equation for ε,

2 + = − − − + − 2 − + − + ∂s ε HB1 ε 9B1 bs3PB1 b(∂s PB1 23∂s PB1 ) ∂s PB1 b(∂sε 23∂sε) bs3ε N(ε), (3-11)

= where, implicitly, B1 B1(b(t)) and HB1 is the linear operator associated to the proﬁle PB1 ,

= − + 2 − 0 HB1 ε 1ε b D3ε f (PB1 )ε, (3-12) and the nonlinearity = + − − 0 N(ε) f (PB1 ε) f (PB1 ) f (PB1 )ε. (3-13)

Alternatively, the equation for w takes the form

2 + ˜ = − 2 − − + ∂t w HB1 w ∂t (PB1 )λ 1(PB1 )λ f ((PB1 )λ) Nλ(w), with ˜ = − − 0 HB1 w 1w f ((PB1 )λ)w, (3-14) = + − − 0 Nλ(w) f ((PB1 )λ w) f ((PB1 )λ) f ((PB1 )λ)w. (3-15) 790 MATTHIEU HILLAIRET AND PIERRERAPHAËL

We then expand using (3-9), (3-10), obtaining 1 ∂2(P ) − 1(P ) − f ((P ) ) = ∂ P + b(∂ P + 23∂ P ) + b 3P + 9 t B1 λ B1 λ B1 λ λ2 ss B1 s B1 s B1 s B1 B1 λ 1 1 = b3∂ P + b 3P + 9 + ∂ (∂ P ) , λ2 s B1 s B1 B1 λ t λ s B1 λ and rewrite the equation for w as 1 1 ∂2w + H˜ w = − b3∂ P + b 3P + 9 − ∂ (∂ P ) + N (w). (3-16) t B1 λ2 s B1 s B1 B1 λ t λ s B1 λ λ

For most of our arguments we prefer to view the linear operator HB1 acting on w in (3-16) as a perturbation of the linear operator Hλ associated to Qλ. Then 2 + = ∂t w Hλw FB1 (3-17) 1 1 = − b3∂ P +b 3P +9 −∂ (∂ P ) − f 0(Q )− f 0((P ) )w + N (w), λ2 s B1 s B1 B1 λ t λ s B1 λ λ B1 λ λ with 0 Hλw = −1w + f (Qλ)w. (3-18)

The set of bootstrap estimates. First we ﬁx some notations. We introduce the energy Ᏹ(t) associated to the Hamiltonian H , λ Z 2 2 Ᏹ(t) = λ (Hλ∂t w, ∂t w) + (Hλw) . (3-19)

Given the unstable eigenvalue ζ ∈ (0, ∞), we set

1 1 V+ = √ , V− = √ , (3-20) ζ − ζ and introduce the decomposition of the unstable direction,

(ε, ψ) =a ˜+(s)V+ +a ˜−(s)V−. (3-21) (∂sε, ψ) Let us write

bs bs κ+(s) =a ˜+(s) + √ (∂b PB , ψ), κ−(s) =a ˜−(s) − √ (∂b PB , ψ). (3-22) 2 ζ 1 2 ζ 1

We note that the vectors V+, V− given by (3-20) yield an eigenbasis of 0 1 ζ 0 and hence correspond respectively to the unstable and stable mode of the two dimensional dynamical system dY 0 1 = Y, ds ζ 0 SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 791 which, to ﬁrst order in b, is veriﬁed by the projection onto the unstable mode (ε, ψ); see (4-57). The deformation term bs(∂b PB1 , ψ) in (3-22) is present to handle some possible time oscillations induced by 2 the ∂s PB1 term in the right-hand side of (3-11), which cannot be estimated in absolute value but will be proved to be of lower order. With these conventions, we may now parametrize the set of initial data described by Lemma 3.1 by a+ = κ+(0), and then reformulate the initial smallness properties in terms of suitable initial bounds for ε; see Appendix A for the proof, which is standard.

Lemma 3.3 (initial parametrization of the unstable mode and initial bounds). Let M and b0 be given as in Lemma 3.1 and denote by C(M) a sufﬁciently large constant. Then, given (η0, η1, a+) satisfying b2 | | + k ∇ + − ∇ k ≤ 0 a+ η0, η0, η1 b0(1 χB1(b0))3Q, η1 H˙ 1×H˙1×L2×L2 , (3-23) |log b0| | | 2 | | there exists a unique d+ with d+ . b0/ log(b0) and T0 > 0 such that the unique decomposition = + = + u(t) (PB1(b(t)) ε)λ(t) (PB1(b(t)))λ(t) w(t) of the unique smooth solution u to (1-1) on [0, T0] with initial data (3-2) satisﬁes the initialization

κ+(0) = a+ (3-24) and the following smallness conditions on [0, T0]: • Smallness and positivity of b:

0 < b(t) < 5b0. (3-25)

• Pointwise bound on bs: 4 2 [b(t)] |bs(t)| ≤ C(M) . (3-26) |log b(t)|2 • Smallness of the energy norm: b(t) ∇ + − ≤ p w(t), ∂t w(t) (1 χB1(b(t)))3Q λ(t) b0. (3-27) λ(t) L2×L2 • Global H˙ 2 bound: [b(t)]4 |Ᏹ(t)| ≤ C(M) . (3-28) |log b(t)|2 • A priori bound on the stable mode: 2 1/8 [b(t)] |κ−(t)| ≤ (C(M)) . (3-29) |log b(t)| • A priori bound of the unstable mode: [b(t)]2 |κ+(t)| ≤ 2 . (3-30) |log b(t)| We can now describe the bootstrap regime. 792 MATTHIEU HILLAIRET AND PIERRERAPHAËL

∈ [− 2 | | 2 | |] Deﬁnition 3.4 (exit time). Let K (M) be a large constant. Given a+ b0/ log b0 , b0/ log b0 , we let T (a+) be the life time of the solution to (1-1) with initial data (3-2), and T1(a+) > 0 be the supremum of T ∈ (0, T (a+)) such that for all t ∈ [0, T ], the following estimates hold:

• Smallness and positivity of b:

0 < b(t) < 5b0. (3-31)

• Pointwise bound on bs: 4 2 [b(t)] |bs| ≤ K (M) . (3-32) |log b(t)|2 • Smallness of the energy norm: b(t) ∇ + − ≤ p w(t), ∂t w(t) (1 χB1(b(t)))3Q λ(t) b0. (3-33) λ(t) L2×L2 • Global H˙ 2 bound: [b(t)]4 |Ᏹ(t)| ≤ K (M) . (3-34) |log b(t)|2 • A priori bound on the stable and unstable modes:

2 2 [b(t)] 1/8 [b(t)] |κ+(t)| ≤ 2 , |κ−(t)| ≤ (K (M)) . (3-35) |log b(t)| |log b(t)| The existence of blow-up solutions in the regime described by Theorem 1.1 now follows from the following proposition: ∈ [− 2 | | 2 | |] Proposition 3.5. There exists a+ b0/ log b0 , b0/ log b0 such that

T1(a+) = T (a+).

Then the corresponding solution to (1-1) blows up in ﬁnite time in the regime described by Theorem 1.1.

The proof of Proposition 3.5 relies on a monotonicity argument applied to the energy Ᏹ, which is the core of the analysis (see Proposition 4.6), and the strictly outgoing behavior of the unstable mode induced by the nontrivial eigenvalue −ζ < 0 of H (see Lemma 4.10). The fact that the regime described by the bootstrap bounds (3-31)–(3-35) corresponds to a ﬁnite blow-up solution with a speciﬁc blow-up speed will then follow from the modulation equations and the sharp derivation of the blow speed as in [Raphaël and Rodnianski 2012].

4. Improved bounds

This section is devoted to the derivation of the main dynamical properties of the flow in the bootstrap regime described by Definition 3.4. The three main steps are first the derivation of a monotonicity property on Ᏹ, which allows us to improve the bounds (3-31)–(3-34) in [0, T1(a+)], second the derivation of the dynamics of the eigenmode and the outgoing behavior of the unstable direction, and third the derivation SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 793 of the sharp law for the parameter b, which allows to bootstrap its smallness (3-31) and will eventually allow us to derive the sharp blow-up speed.

Remark 4.1. Throughout the proof, we will introduce various constants C(M), δ(M) > 0 that do not depend on the bootstrap constant K (M). An important feature of all these constants is that, up to a smaller choice of b∗(M) or a larger choice of K (M), we assume that any product of the form C(M) f (b), where limb→0 f (b) = 0, or that any ratio δ(M)/K (M) is small in the trapped regime. This will be used implicitly in this section.

Coercivity of Ᏹ. Let us start by showing that the linearized energy Ᏹ yields a control of suitable weighted norms of (w, ε) in the regime t ∈ [0, T1(a+)].

5 Lemma 4.2 (coercivity of Ᏹ). There exists M0 ≥ 1 such that for all M ≥ M0, there exists δ(M) > 0 and C(M) < ∞ such that in the interval [0, T1(a+)), Z Z Z 2 4 1 2 2 2 2 (∂r w) 1/4 b Ᏹ ≥ λ (H w) + δ(M)λ (∇∂t w) + − C(M)[K (M)] . (4-1) 2 λ r 2 |log b|2 Proof of Lemma 4.2. This is a consequence of the explicit distribution of the negative eigenvalues of H and the a priori bound on the unstable mode (3-35). Indeed, let t ∈ [0, T1(a+)), then ﬁrst observe from (3-21), (3-22), (3-35) that | |2 + | |2 | |2 + | |2 + | |2 2 (ε, ψ) (∂sε, ψ) . κ+ κ− bs (∂b PB1 , ψ) 4 1/4 b 2 2 1/4 4 2 [K (M)] + C(M)b |bs| [K (M)] b /|log b| , (4-2) . |log b|2 . where we used the estimates of Proposition 2.1 and the fact that ψ is well localized. This yields 1 1 (w, ψ )2 + (∂ w, ψ )2 = (ε, ψ)2 + (∂ ε + b3ε, ψ)2 λ4 λ λ2 t λ s b4 Z ε2 Z |∇ε|2 [K (M)]1/4 + b2 + , (4-3) . |log b|2 y4(1 + |log(y)|)2 y2 and similarly, using the orthogonality condition (3-4), 1 1 (w, (χ 8) )2 + (∂ w, (χ 8) )2 = (b3ε, χ 8)2 λ4 M λ λ2 t M λ M Z ε2 Z |∇ε|2 b2 MC + . (4-4) . y4(1 + |log(y)|)2 y2 Applying Lemma C.3 yields

Z Z Z |∇ε|2 ε2 λ2 |H w|2 = |Hε|2 ≥ δ(M) + . λ y2 y4(1 + |log(y)|)2

5Recall Remark 4.1. 794 MATTHIEU HILLAIRET AND PIERRERAPHAËL

Introducing the rescaled version (C-13) of Lemma C.3, we then conclude that Z Z Z 2 Z 2 1 2 2 2 2 |∇ε| ε Ᏹ ≥ λ (H w) + δ (M) λ (∇∂t w) + + 2 λ 1 y2 y4(1 + |log(y)|)2 Z ε2 Z |∇ε|2 b4 − b2 MC + − C(M)[K (M)]1/4 y4(1 + |log(y)|)2 y2 |log b|2 Z Z Z 2 4 1 2 2 2 2 (∂r w) 1/4 b ≥ λ (H w) + δ(M)λ (∇∂t w) + − C(M)[K (M)] , 2 λ r 2 |log b|2 where we used the Hardy bound (C-3), and (4-1) is proved. Remark 4.3. Note that (4-1) together with the Hardy estimate (C-1), the coercivity estimate (C-9) and (4-4) yield the following weighted bound on ε which will be extensively used in the paper: Let

(N−2)/2+1 η(s, y) = λ ∂t w(t, λy) = ∂sε(s, y) + b3ε(s, y), (4-5) then Z ε2 Z η2 Z |∇ε|2 Z b4 + + + |∇η|2 c(M) |Ᏹ| + [K (M)]1/4 , (4-6) y4(1 + |log y|2) y2 y2 . |log b|2 p b4 c(M)|Ᏹ| + K (M) . (4-7) . |log b|2

First bound on bs. We now derive a crude bound on bs which appears as an order-one forcing term in the right-hand side of the equation (3-11) for ε. This bound is a simple consequence of the construction of the proﬁle Qb and the choice of the orthogonality condition (3-4). 6 Lemma 4.4 (rough pointwise bound on bs). We have the bound 2 (ε, H8) 1 p b4 bs + |Ᏹ| + K (M) . (4-8) (3Q, 8) . M |log b|2

Remark 4.5. This is in contrast with [Raphaël and Rodnianski 2012], where the bs term could be treated as degenerate with respect to ε thanks to a specific choice of orthogonality conditions and the factorization of the operator H in the wave map case. This difficulty in our case will be treated using a specific algebra generated by our choice of orthogonality condition (3-4) which gives the right sign to the leading-order terms involving bs in the energy identity of Proposition 4.6; see (4-24), (4-38). Proof of Lemma 4.4. Let us recall that the equation for ε in rescaled variables is given by (3-11)–(3-13). 2 N Observe also that from (1-19), the adjoint of HB with respect to the L (ޒ ) inner product is H ∗ = H + 2b2 D. (4-9) B1 B1

To compute bs we take the scalar product of (3-11) with χM 8. Using the orthogonality relations m = m − = ≥ (∂s ε, χM 8) (∂s (PB1 Q), χM 8) 0 for all m 0,

6Recall Remark 4.1. SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 795 we integrate by parts to get the algebraic identity + + bs (3PB1 , χM 8) 2b(3∂b PB1 , χM 8) (3ε, χM 8) = −(9 , χ 8) − (ε, H ∗ (χ 8)) + 2b(∂ ε, 3(χ 8)) + (N(ε), χ 8). (4-10) B1 M B1 M s M M We ﬁrst derive from the estimates of Proposition 2.1 that

4 2 b (9B , χM 8) . (4-11) 1 . |log b|2 Similarly, using (4-6) yields

4 2 p b (∂sε, 3(χM 8)) C(M) c(M)|Ᏹ| + K (M) (4-12) . |log b|2 and q ∗ p (ε, H (χ 8)) = (ε, H8) − (Hε, (1 − χ )8) + O MC b2 c(M)|Ᏹ| + K (M)b4/|log b|2 . B1 M M

We then use the improved decay (2-2) and (4-7) to estimate

2 Z |Hε| |Ᏹ| p b4 (Hε, ( − χ )8)2 + K (M) . 1 M . N . 2 y≥M 1 + y M |log b| Thus 4 ∗ 2 1 p b (ε, H (χM 8)) − (ε, H8) |Ᏹ| + K (M) . (4-13) B1 . M |log b|2 Similarly, + + (3PB1 , χM 8) 2b(3∂b PB1 , χM 8) (3ε, χM 8) b q p = (3Q, 8) + O + MC |Ᏹ| + K (M)b4/|log b|2 |log b| b = (3Q, 8) + O , (4-14) |log b| where we have used that in the trapped regime we have Ᏹ ≤ K (M)b4/[log(b)]2. Finally, on the support of ∗ = + 2 χM and for b < b0(M) small enough, the term Q dominates in Qb Q b T1. Hence, for the nonlinear term, we have from the Sobolev inequality and (4-7) that

Z ε2 ε3 Z |ε|2 p b4 |(N(ε), χM 8)| + [1 + kyεkL∞ ] C(M) Ᏹ + K (M) . . 1 + y6 1 + y4 . (1 + y5) . |log b|2

7 Injecting this, together with (4-11)–(4-14), into (4-10) yields (4-8).

7Recall Remark 4.1. 796 MATTHIEU HILLAIRET AND PIERRERAPHAËL

Global H˙ 2 bound. We derive in this section a monotonicity statement for the energy Ᏹ that provides a global H˙ 2 estimate for the solution. The monotonicity statement involves suitable repulsive properties of the rescaled Hamiltonian Hλ in the focusing regime under the orthogonality condition (3-11) and the a priori control of the unstable mode (3-35), which themselves rely on the positivity of an explicit quadratic form; see Lemma 4.7.

Proposition 4.6 (H 2 control of the radiation). In the trapped regime, there exists a function Ᏺ satisfying

Ᏹ p b4 Ᏺ + K (M) (4-15) . M |log b|2 such that, for some 0 < α < 1 close enough to 1, we have

d Ᏹ + Ᏺ b p b4 ≤ K (M) . (4-16) dt λ2(1−α) λ3−2α |log b|2 Proof.

Step 1: Energy identity. Let N + 2 4/(N−2) 1 r N + 2 4/(N−2) Ve(t, r) = Q (r) = V , V (y) = Q (y). N − 2 λ λ2 λ N − 2 We ﬁrst have an algebraic energy identity that follows by integrating by parts from (3-17),

1 d Z Z Z Z (∂ w)2 Z 2 − 2 + 2 = − t + + (∂tr w) Ve(∂t w) (Hλw) ∂t Ve wHλw ∂t wHλ FB1 . (4-17) 2 dt 2 We now use the w equation and integration by parts to compute Z Z − = − − ∂t VewHλw ∂t Vew(FB1 ∂tt w) (4-18) d Z Z Z Z = − − 2 − ∂t Vew∂t w ∂t VewFB1 ∂t Ve(∂t w) ∂tt Vew∂t w. (4-19) dt We next pick 0 < α < 1 close enough to 1 and combine the above identities to get Z Z Z Z 1 d 2α 2 2 2 λ (∂tr w) − Ve(∂t w) + (Hλw) − 2 ∂t Vew∂t w 2λ2α dt 2αb Z Z = −R1 + R2 + ∂t Vew∂t w − ∂tt Vew∂t w, (4-20) λ where R1 collects the quadratic terms Z Z Z Z Z αb 2 2 2 3 2 bs R1 = (∂tr w) − Ve(∂t w) + (Hλw) + ∂t Ve(∂t w) − ∂t Ve(3Q)λw λ 2 λ2 Z Z Z Z Z b 2 2 2 3 2 = α (∂yη) − α V η +α (Hε) + (2V + y · ∇V )η −bs ε(2V + y · ∇V )3Q (4-21) λ3 2 SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 797 and R2 collects the nonlinear higher-order terms Z Z b = − + s R2 ∂t wHλ FB1 ∂t Vew FB1 (3Q)λ . (4-22) λ2

Step 2: Derivation of the quadratic terms and treatment of the bs term. Let us now obtain a suitable lower bound for the quadratic term R1. The main enemy is the bs term which is of order one in ε and will be treated by using a speciﬁc algebra generated by the choice of the orthogonality condition (3-4). N/2 Observe from H(3Q) = 0 that (3Q/λ)λ(y) = (1/λ) (3Q)(y/λ) satisﬁes 2 −1(3Q/λ)λ(y) − (1/λ) V (y/λ)(3Q/λ)λ(y) = 0. Differentiating this relation at λ = 1 yields H8 = H(D3Q) = (2V + y · ∇V )3Q. We inject this into the modulation equation (4-8) to get Z 4 1/2 2 |Ᏹ| p b −bs ε(2V + y · ∇V )3Q = b (8, 3Q) + |bs|O + K (M) . (4-23) s M |log b|2 We thus conclude using the sign (8, 3Q) > 0 and (4-8), (4-21) that b Z Z Z R ≥ α (∂ η)2 + [(3 − α)V + 3 y · ∇V ]η2 + α (Hε)2 + c (b )2 1 λ3 y 2 1 s |Ᏹ| p b4 + O + K (M) (4-24) M |log b|2 for some universal constant c1 > 0 independent of M.

Step 3: Coercivity of the quadratic form. We now claim the following coercivity property of the quadratic form in η appearing on the right-hand side of (4-24) in the limit case α = 1. The proof is given in Appendix B. ∈ ˙ 1 Lemma 4.7. There exists a universal constant c0 > 0 such that for all η Hrad we have Z Z Z 1 2 + + 3 · ∇ 2 ≥ 2 − 2 + 2 (∂yη) 2V 2 y V η c0 (∂yη) (η, ψ) (η, 8) . c0 From a simple continuity argument, there exists 0 < α∗ < 1 such that given 0 < α∗ < α ≤ 1, for all ∈ ˙ 1 η Hrad, we have Z Z c Z 2 2 + − + 3 · ∇ 2 ≥ 0 2 − 2 + 2 α (∂yη) (3 α)V 2 y V η (∂yη) (η, ψ) (η, 8) . 2 c0 We now pick once and for all such an α < 1 and control the negative directions. Using (4-3) and (4-7) yields p b4 (η, ψ)2 b|Ᏹ| + K (M) . . |log b|2 798 MATTHIEU HILLAIRET AND PIERRERAPHAËL

Similarly, we compute (η, 8) = (η, χM 8) + (η, (1 − χM )8) for which (4-4) and (4-7) yield 4 2 p b (η, χM 8) b|Ᏹ| + K (M) , . |log b|2 and applying (C-1) we have Z 2 Z 2 2 |8| 1 2 (η, (1 − χM )8) ≤ kyηkL∞ . |∂yη| . y≥M/2 y M This, together with (4-24), yields the lower bound on quadratic terms, 4 b 2 p b R ≥ c ((bs) + |Ᏹ|) + O K (M) (4-25) 1 λ3 1 |log b|2 for some universal constant c1 > 0. Indeed, a straightforward integration by parts in (3-19) yields Z Z 2 2 Ᏹ . |∂yη| + |Hε| .

Step 4: Control of lower-order quadratic terms. The lower-order quadratic terms in (4-20) are controlled similarly, Z Z 2 Z 2 4 b ε η 1 p b ∂t Vew∂t w + bC(M)|Ᏹ| + K (M) . λ2 1 + y6 y2 . λ2 |log b|2 1 |Ᏹ| p b4 + K (M) , (4-26) . λ2 M |log b|2 and, with the help of (3-32), Z 2 Z 2 Z 2 4 b |bs| ε η b p b ∂tt Vew∂t w + + bC(M)|Ᏹ| + K (M) . . λ3 λ3 1 + y6 y2 . λ3 |log b|2 Remark 4.8. We note here that (4-26) is sufﬁcient for the proof of our theorem. Indeed, the estimated term R ∂t Vew∂t w has been integrated by parts with respect to time, so that it becomes a part of Ᏺ. Furthermore, 2α R we note that to compute (4-16), we multiply Ᏺ by λ . Consequently, the commutator bα/λ ∂t Vew∂t w appears on the right-hand side.√ However, (4-26) yields that, in the trapped regime, this supplementary term is controlled by b/λ3 K (M)b4/|log b|2. Similar arguments will be repeated implicitly below for the terms that require an integration by parts with respect to time.

Step 5: Rewriting the nonlinear R2 terms. It remains to control the nonlinear R2 terms in (4-20) given by (4-22). According to (3-17), this term contains bss-types of terms which cannot be estimated in absolute value and require a further integration by parts in time. Let 1 F = F − ∂ F , with F = (∂ P ) , (4-27) B1 1 t 2 2 λ s B1 λ and write Z Z bs Z Z R2 = ∂t wHλ F1 − ∂t Vew F1 + (3Q)λ − ∂t wHλ∂t F2 + ∂t Vew∂t F2. λ2 SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 799

We now integrate by parts in time to treat the F2 term, Z Z − ∂t wHλ∂t F2 + ∂t Vew∂t F2 d Z Z Z Z = − ∂t wHλ F2 − ∂t VewF2 − (∂tt Vew + 2∂t Ve∂t w)F2 + ∂tt wHλ F2. dt The last term is rewritten using (3-17) and integration by parts, Z Z ∂tt wHλ F2 = [F1 − ∂t F2 − Hλw]Hλ F2 Z Z Z Z 1 d 2 2 1 2 = − |∇ F2| − VFe − ∂t VFe + [F1 − Hλw]Hλ F2. 2 dt 2 2 2

Eventually we arrive at a manageable expression for R2, Z Z Z Z d 1 2 1 2 R2 = − ∂t wHλ F2 − ∂t VewF2 + |∇ F2| − VFe dt 2 2 2 Z bs Z Z − ∂t Vew F1 + (3Q)λ + ∂t wHλ F1 − (∂tt Vew + 2∂t Ve∂t w)F2 λ2 Z Z 1 2 − ∂t VFe + [F1 − Hλw]Hλ F2. (4-28) 2 2

We now aim at estimating all the terms in the right-hand side of (4-28). According to (3-17), we split F1 into four terms b 1 F + s (3Q) = − 9 + F + F + N(ε) , (4-29) 1 λ2 λ λ2 B1 1,1 1,2 λ with = + − = 0 − 0 F1,1 b3∂s PB1 bs(3PB1 3Q), F1,2 f (Q) f (PB1 ) ε. (4-30)

Step 6: F1 terms. These are the leading-order terms. • 9B1 terms. We ﬁrst extract from (2-9) the rough bound 2 b 4 |9B | + C(M)b 1y≤ B , (4-31) 1 . |log b|(1 + y2) 2 1 which yields Z 2 4 1 + |log y| 2 b |9B | , 1 + y4 1 . |log b|2 and thus, from (4-7), Z Z | || | 1 b ε 9B1 ∂t Vew (9B1 )λ λ2 . λ3 (1 + y4) b b2 q p C(M) |Ᏹ| + K (M)b4/|log b|2 . λ3 |log b| b p b4 K (M) . . λ3 |log b|2 800 MATTHIEU HILLAIRET AND PIERRERAPHAËL

• F1,1 terms. From (2-7) and (2-8) we obtain 2 1 + |log(by)| 1 log(M) + |log y| |F | |bs|b 1 ≤y≤B + 1B ≤y≤ B + , 1,1 . |log b| 2 0/2 b2 y2|log b| 0/2 2 1 1 + y2 and, recalling that differentiation with respect to y acts as a multiplication by 1/(1 + y), 2 |bs|b 1 + |log(by)| 1 log(M) + |log y| HF1,1 C(M) 12≤y≤B /2 + 1B /2≤y≤2B + , . 1 + y2 |log b| 0 b2 y2|log b| 0 1 1 + y2 from which Z 2 Z 2 2 2 2 b (1 + |log y| ) 2 2 2 (1 + y )|H(F )| |bs| , |F | |bs| b . (4-33) 1,1 . |log b|2 (1 + y4) 1,1 .

Hence similar arguments as with the 9B1 terms yield Z b q b b4 p 4 2 p ∂t VewF1,1 b|bs|C(M) |Ᏹ| + K (M)b /|log b| K (M) , . λ3 . λ3 |log b|2 and Z C M b |b | q ( ) s p 4 2 ∂t wHλ F1,1 |Ᏹ| + K (M)b /|log b| . λ3 |log b| 2 4 4 b |bs| Ᏹ p b b p b + + K (M) K (M) . . λ3 |log b| |log b| |log b|2 . λ3 |log b|2

• F1,2 terms. The explicit expansion of the cubic nonlinearity and the bound (2-7) yield C(M)b2 C(M)b2 C(M)b2 |F | |ε| and |∇ F | |ε| + |∇ε|, (4-34) 1,2 . 1 + y2 1,2 . 1 + y3 1 + y2 from which Z 3 Z 2 4 1 C(M)b ε b p b ∂t Vew(F1,2)λ b|Ᏹ| + K (M) , λ2 . λ3 1 + y6 . λ3 |log b|2 SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 801 and, after integration by parts of the Laplacian term,

Z Z 2 Z 2 2 1 C(M) |η| b b b ∂t wHλ(F1,2)λ |ε| + |∇η| |ε| + |∇ε| λ2 . λ3 1 + y4 1 + y2 1 + y3 1 + y2 b |Ᏹ| p b4 + K (M) . . λ3 M |log b|2

• Nonlinear term N(ε). We expand the nonlinearity as

= 2 + 3 N(ε) 3PB1 ε ε .

This yields, using (3-27) and (C-1), the rough bound

ε2 |N(ε)| . . 1 + y

In what follows, we will use the following bound on η, which follows from (4-6), (C-1):

1/2 p b4 kyηkL∞ k∇ηk 2 c(M)|Ᏹ| + K (M) . . L . |log b|2

We then estimate

Z Z 3 4 4 1 b |ε| b p b b p b ∂t Vew(N(ε))λ k∇εkL2 c(M)|Ᏹ| + K (M) K (M) λ2 . λ3 1+y5 . λ3 |log b|2 . λ3 |log b|2

∗ for b0 < b (M) small enough. We split the second term into Z Z Z (N(ε))λ (N(ε))λ (N(ε))λ ∂t wHλ = ∇∂t w · ∇ − Ve∂t w . (4-35) λ2 λ2 λ2

The second of these terms is estimated by brute force:

Z Z 2 Z 2 (N(ε))λ 1 |η||ε| 1 |ε| Ve∂t w kyηkL∞ λ2 . λ3 1 + y5 . λ3 1 + y6 3/2 1 p b4 b b4 c(M)|Ᏹ| + K (M) . . λ3 |log b|2 . λ3 |log b|2

The ﬁrst term in (4-35) is split into two parts:

Z (N(ε)) Z 3 Z ∇∂ w · ∇ λ = ∇∂ w · ∇(w3) + 3(P ) ∇(w2) + ε2∇η · ∇ P . t λ2 t B1 λ λ3 B1

The second term is integrated by parts in space and then estimated by brute force: 802 MATTHIEU HILLAIRET AND PIERRERAPHAËL Z Z 3 2 3 2 ε ∇η · ∇ PB = η ε 1PB + 2ε∇ PB · ∇ε λ3 1 λ3 1 1 1 Z ε2 |ε||∇ε| |η| + . λ3 1 + y4 1 + y3 1 Z ε2 Z |∇ε|2 kyηkL∞ + . λ3 1 + y5 y2 3/2 1 p b4 b b4 c(M)|Ᏹ| + K (M) . . λ3 |log b|2 . λ3 |log b|2 The ﬁrst term is more delicate and requires ﬁrst a time integration by parts, Z Z 3 2 d 2 3 2 ∇∂t w · ∇(w ) + 3(PB ) ∇(w ) = |∇w| w + 3(PB ) w 1 λ dt 2 1 λ Z Z − |∇ |2 − |∇ |2 + 3 w∂t w w 3 w w∂t (PB1 )λ (PB1 )λ∂t w .

We may now estimate all terms by brute force. First,

Z Z 2 4 2 3 2 1 |∇ε| 1 b |∇w| w + 3(PB )λw [kyεkL∞ + ky PB kL∞ ]kyεkL∞ , 2 1 . λ2 1 y2 . λ2 |log b|2 second,

Z Z 2 4 3/2 4 2 1 |∇ε| p b b b w∂t w|∇w| kyεkL∞ kyηkL∞ c(M)|Ᏹ| + K (M) , . λ2 y2 . |log b|2 . λ3 |log b|2 and third,

Z ∞ Z 2 2 kywkL |∇w| b |∇w| w∂t (PB )λ + C(M)b|bs|1y≤B 1 . λ3 y 1 + y2 1 b |log b| Z |∇ε|2 b b4 |∇ε| 2 1 + C(M)|bs| , . λ3 L b y2 . λ3 |log b|2 | | where we used the rough bound extracted from (2-8), ∂b PB1 . C(M)b1y≤B1 . Finally,

Z Z 2 4 3/2 4 2 1 |∇ε| p b b b |∇w| (PB )λ∂t w kyηkL∞ C(M)Ᏹ + K (M) , 1 . λ3 1 + y3 . |log b|2 . λ3 |log b|2

∗ for b0 < b (M) small enough. The above chain of estimates together with Remark 4.8, achieves the control of the nonlinear term N(ε).

Step 7: F2 terms. We estimate from (2-8),

Z 2 Z Z ∂b PB1 2 1 1 2 b + ∇∂b PB and ∂b PB . (4-36) (1 + y) 1 . |log b|2 1 + y3 1 . |log b|2 SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 803

Hence, ﬁrst, Z | | Z | || | Z bs η ∂b PB1 ∂t wHλ F2 + |∇η||∇∂b PB | . λ2 (1 + y4) 1 q 1 |bs| p c(M)|Ᏹ| + K (M)b4/|log b|2 . λ2 |log b| 1 |Ᏹ| p b4 + K (M) , . λ2 M |log b|2 second,

1 Z | | Z | || | | | 4 2 bs b ∂b PB1 ε bs b p b ∂t VewF2 c(M)|Ᏹ| + K (M) . λ2 (1 + y4) . λ2|log b| |log b|2 1 |Ᏹ| p b4 + K (M) , . λ2 M |log b|2 and third, Z Z | |2 Z | |2 Z 2 4 2 2 bs ∂b PB1 2 1 (bs) 1 b |∇ F2| + VF + |∇∂b PB | . 2 . λ2 (1 + y4) 1 . λ2 |log b|2 . λ2 |log b|2 Similarly, Z Z 2 (∂tt Vew + 2∂t Ve∂t w)F2 + ∂t VFe 2 | | Z | | + 2 | | + | | | | Z | |2 bs (( bs b ) ε b η ) ∂b PB1 ∂b PB1 + |bs|b . λ3 (1 + y4) 1 + y4 |b | |b | + b q b2 s ( s ) p 4 2 c(M)|Ᏹ| + K (M)b /|log b| + |bs| . λ3 |log b| |log b|2 b |Ᏹ| p b4 + K (M) . . λ3 M |log b|2 Eventually, (4-32) and (4-33) ensure that Z 6 2 2 6 2 2 b b |bs| b (1 + y )|H(9B + F )| + , 1 1,1 . |log b|2 |log b|2 . |log b|2 which together with (4-36) yields Z 3 4 1 1 b |bs| b b (9B + F1,1)λ Hλ F2 . λ2 1 . λ3 |log b|2 . λ3 |log b|2 We similarly estimate from (4-34), after integration by parts, that Z | | Z 2| || | Z 2| | 2|∇ | 1 bs b ε ∂b PB1 b ε b ε (F1,2)λ Hλ F2 + |∇∂b PB | + λ2 . λ3 1 + y6 1 1 + y3 1 + y2 / b4 Z ε2 Z |∇ε|2 1 2 b b4 C(M) + . . λ3|log b| 1 + y6 1 + y4 . λ3 |log b|2 804 MATTHIEU HILLAIRET AND PIERRERAPHAËL

For the nonlinear term, we extract from (2-8) the rough bound b |H(∂b PB )| [C(M) + log(b)] 1y≤B , 1 . 1 + y2 1 which together with (C-1) ensures that Z Z 2 1 [C(M) + log(b)] b ε (N(ε))λ Hλ F2 |bs| 1y≤B λ2 . λ3 1 + y2 1 + y 1 |b ||log b|4 Z ε2 C(M) s . λ3 (1 + y4)|log y|2 b √ p b4 b b4 b c(M)|Ᏹ| + K (M) . . λ3 |log b|2 . λ3 |log b|2

Step 10: The remaining F2 term has the right sign. It remains to estimate the term Z − HλwHλ F2 on the right-hand side of (4-28). Let us stress the fact that this term is a priori no better O(Ᏹ/λ3) due to the bs contribution and the bound (4-8); recall Remark 4.5. We now claim that the main contribution has the right sign again. Indeed, we ﬁrst compute from the

|Ᏹ| p b4 ≤ O + K (M) . (4-39) M |log b|2

Collecting all the above estimates yields (4-16) and concludes the proof of Proposition 4.6.

Improved bound. We now claim that the a priori bound on the unstable direction (3-35), coupled with the monotonicity property of Proposition 4.6, implies the following:

Lemma 4.9 (improved bounds under the a priori control (3-35)). In [0, T1(a+)] we have b(t) ∇ + − | | w(t), ∂t w(t) 1 χB1(b(t)) 3Q λ(t) . b0 log b0 , (4-40) λ(t) L2×L2 b4(t) b4(0) ≥ , (4-41) |log b(t)|2λ2(1−α)(t) |log b(0)|2λ2(1−α)(0) 4 2 K (M) b |bs| ≤ , (4-42) 2 |log b|2 K (M) b4 |Ᏹ(t)| ≤ . (4-43) 2 (log b)2 Proof.

Step 1: Energy bound. The energy bound (4-40) is a consequence of the conservation of energy. Indeed, conservation of energy and the initial bounds of Lemma 3.1 ensure that p E(u, ∂t u) = E(u0, u1) = E(Q) + O(b0 |log b0|),

(see Appendix A) and thus give Z Z Z 1 2 1 2 1 4 E(Q) + O(b0|log b0|) = ∂t (PB )λ + ∂t w + ∇(PB )λ + ∇w − (PB )λ + w . (4-44) 2 1 2 1 4 1 We lower bound the ﬁrst term by expanding

b b b3 b ∂ (P ) + ∂ w = ∂ w + ((1 − χ )3Q) + (χ 3Q) + (3[χ T ]) + s (∂ P ) t B1 λ t t λ B1 λ λ B1 λ λ B1 1 λ λ b B1 λ b = ∂ w + ((1 − χ )3Q) + 6, t λ B1 λ with Z 2 2 6 . b0|log b0|, where we used the bootstrap bounds (3-31) and (3-32). Finally,

Z Z 2 2 1 b 2 ∂t (PB ) + ∂t w ≥ ((1 − χB )3Q) + ∂t w − O(b |log b |). (4-45) 1 λ 2 λ 1 λ 0 0 806 MATTHIEU HILLAIRET AND PIERRERAPHAËL

We then expand the second term as Z Z 1 2 1 4 ∇(P ) + ∇w − (P ) + w 2 B1 λ 4 B1 λ Z Z 1 2 1 4 = ∇ P + ∇ε − P + ε 2 B1 4 B1 Z Z Z Z 1 2 1 4 3 1 2 2 2 1 3 4 = ∇ PB − PB − (ε, 1PB + P ) + |∇ε| − 3 P ε − 4PB ε + ε . 2 1 4 1 1 B1 2 B1 4 1

From the construction of PB1 , Z Z 1 2 1 4 2 ∇ PB − PB = E(Q) + O(b |log b|). (4-46) 2 1 4 1 The linear term is treated using (2-9), the improved decay (2-2) and (4-31). We get

3 2 2 (ε, 1P + P ) = (ε, b D3P − 9 ) kε/yk 2 ky(b D3P − 9 )k 2 b|∇ε| 2 . (4-47) B1 B1 B1 B1 . L B1 B1 L . L We now rewrite the quadratic term as a small deformation of H and use the coercivity bound (C-8) to ensure that Z Z Z |∇ε|2 − 3 P2 ε2 ≥ c |∇ε|2 + Def , (4-48) B1 0 with Z (ε, ψ)2 Def := 3 (Q2 − P2 )ε2 − . B1 c0 Collecting (2-7) and (C-1), on the one hand, and (4-2) on the other hand, we compute Z 2 2 2 2 2 2 ∞ 2 2 2 2 (Q − P )ε ≤ ky (Q − P )kL k∇εk 2 bk∇εk 2 and |(ε, ψ)| b |log b|. (4-49) B1 B1 L . L . The nonlinear term is easily estimated by the Sobolev inequality: Z 3 2 p 2 + ≤ k k ∞ k k ∞ k∇ k k∇ k (3PB1 ε)ε y PB1 L yε L ε L2 . b0 ε L2 . (4-50)

Injecting (4-45), (4-47), (4-46), (4-49), (4-48), (4-50) into (4-44) now yields (4-40).

Step 2: Lower bound on b. We now turn to the proof of (4-41). First observe from the bootstrap estimate (3-32) that 2 p b 1 − α 2 |bs| ≤ K (M) ≤ b . (4-51) |log b| 10 This implies 4 3 d b 4b 1 1 − α 2 = bs 1 − + b > 0 ds (log b)2λ2(1−α) λ2(1−α)(log b)2 2 log b 2 and (4-41) follows. SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 807

(ε, ψ) Y (t) = =a ˜+(t)V+ +a ˜−(t)V−, (4-54) (∂sε, ψ) and the variables given by (3-22),

bs bs κ+(s) =a ˜+(s) + √ (∂b PB , ψ), κ−(s) =a ˜−(s) − √ (∂b PB , ψ). 2 ζ 1 2 ζ 1

Proof. We compute the equation satisﬁed by the unstable direction (ε, ψ) by taking the inner product of (3-11) with the well localized direction ψ and get d2 (ε, ψ) − ζ(ε, ψ) = E(ε) − (∂2 P , ψ), (4-57) ds2 s B1 with = − − − + − + E(ε) (9B1 , ψ) bs(3PB1 , ψ) b(∂s PB1 23∂s PB1 , ψ) b(∂sε 23∂sε, ψ) − + + 2 + 0 − 0 bs(3ε, ψ) (N(ε), ψ) b (3ε, Dψ) (( f (PB1 ) f (Q))ε, ψ). (4-58) Simple algebraic manipulations using (4-54) and (3-22) and the initial condition yield the equivalent system d p E+(s) d p E−(s) κ+ = ζ κ+(s) + √ , κ− = − ζ κ−(s) − √ κ−(0), (4-59) ds 2 ζ ds 2 ζ with bs bs E+(s) = E(s) − (∂b PB , ψ), E−(s) = E(s) + (∂b PB , ψ). (4-60) 2 1 2 1 We now have from the explicit formula (4-58) and (4-60), the exponential localization of ψ, the orthogonality (ψ, 3Q) = 0, the estimates of Proposition 2.1 and the bootstrap estimate (3-32) the bound 2 2 1 p p b √ b √ |E±| |b| |bs| + |Ᏹ| + K (M) ≤ b , (4-61) ζ . |log b| |log b| which together with (4-59) yields (4-56). Let then |log b|2 Ᏻ = κ2 , − b4 then from (4-59), (4-61), (3-32), we estimate that 2 2 dᏳ dκ− |log b| 2 4|log b| 2 log b = 2κ− + κ b − + ds ds b4 − s b5 b5 2 2 |log b| p E− 2 |log b| |bs| = 2 κ− − ζ κ− − √ + κ O b4 ζ − b4 b √ √ 2 2 √ 2 ζ |log b| 2 |log b| b ζ ≤ − κ + κ− b − Ᏻ + 1. 2 b4 − b4 |log b| . 2 We integrate this in time and get √ ζ Z s s− √ − s − ( σ ) ζ Ᏻ(s) ≤ Ᏻ(0)e 2 + e 2 dσ . 1, 0 where we used the initial inequality (A-18) yielding that Ᏻ(0) . 1. This concludes the proof of (4-55) and of Lemma 4.10. SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 809

Derivation of the sharp law for b. We now turn to the derivation of the sharp law for b, which will yield the monotonicity statement on b needed to obtain the smallness bootstrap estimate (3-31), and will eventually lead to the derivation of the sharp blow-up speed (1-10). Lemma 4.11 (sharp law for b). Let

˜ = PB0 χB0/4 Q, (4-62) Z b = | ˜ |2 + ˜ ˜ ˜ ˜ G(b) b 3PB0 L2 b(∂b PB0 , 3PB0 ) db, (4-63) 0 = ˜ + + ˜ + ˜ ˜ − − ˜ ˜ Ᏽ(s) (∂sε, 3PB0 ) b(ε 23ε, 3PB0 ) bs(∂b PB0 , 3PB0 ) bs ∂b(PB1 PB0 ), 3PB0 . (4-64) Then

G(b) = 64b |log b| + O(b), |Ᏽ| . K (M)b, (4-65) 2 d 2 b {G(b) + Ᏽ(s)} + 32b . K (M) . (4-66) ds p|log b| Remark 4.12. Observe that (4-65) and (4-66) essentially yield a pointwise differential equation b2 bs ∼ − , 2|log b| which will allow us to derive the sharp scaling law via the relationship −λs/λ = b. Proof of Lemma 4.11. The proof is inspired by the one in [Raphaël and Rodnianski 2012]. We multiply ˜ (3-11) by 3PB0 and compute + + + 2 ˜ (bs3PB1 b(∂s PB1 23∂s PB1 ) ∂s PB1 , 3PB0 ) = − ˜ − ˜ − 2 + + + ˜ + ˜ (9B1 , 3PB0 ) (HB1 ε, 3PB0 ) ∂s ε b(∂sε 23∂sε) bs3ε, 3PB0 (N(ε), 3PB0 ). We further rewrite this as ˜ + ˜ + ˜ + 2 ˜ ˜ bs3PB0 b(∂s PB0 23∂s PB0 ) ∂s PB0 , 3PB0 = − ˜ − − ˜ + − ˜ + − ˜ + 2 − ˜ ˜ (9B1 , 3PB0 ) bs3(PB1 PB0 ) b(∂s(PB1 PB0 ) 23∂s(PB1 PB0 )) ∂s (PB1 PB0 ), 3PB0 − ˜ − 2 + + + ˜ + ˜ (HB1 ε, 3PB0 ) ∂s ε b(∂sε 23∂sε) bs3ε, 3PB0 (N(ε), 3PB0 ). (4-67) We now estimate all terms in this identity.

We now turn to the key step in the derivation of the sharp b law which corresponds to the following outgoing ﬂux computation:8 ˜ 2 1 (9B , 3PB ) = 32b 1 + O as b → 0. (4-69) 1 0 |log b|

Indeed, we ﬁrst estimate from (2-9) that

Z 1 + |log(by)| 1 + |log(1 + y)| b2 (9 − c b2χ 3Q, 3P˜ ) b4 + . B1 b B0/4 B0 . | | + 2 + 2 2 . | | y≤B0/2 log b (1 y ) (1 y ) log b

The remainder term is computed from (2-10) and the explicit formula for Q (1-3),

d −(∂2(P − P˜ ), 3P˜ ) = − b (∂ (P − P˜ ), 3P˜ ) + b2∂ (P − P˜ ), ∂ 3P˜ . s B1 B0 B0 ds s b B1 B0 B0 s b B1 B0 b B0 The remaining terms are estimated in brute force using (2-8) and (3-32), which yield

− ˜ + − ˜ + − ˜ ˜ + 2 − ˜ ˜ bs3(PB1 PB0 ) b(∂s(PB1 PB0 ) 23∂s(PB1 PB0 )), 3PB0 bs (∂b(PB1 PB0 ), ∂b3PB0 ) 2 2 |bs| b |bs| + K (M) . . b2 . |log b|

Step 2: ε terms. We are left with estimating the third line on the right-hand side of (4-67). We ﬁrst treat the linear term from (4-1), (4-7) and (3-34) and get Z |(H ε, 3P˜ )| |(Hε, 3P˜ )| + |ε||P2 − Q2||3P˜ | + b2|(D3ε, 3P˜ )|. (4-70) B1 B0 . B0 B1 B0 B0

On the one hand, (4-7) together with bootstrap estimates yields

Z Z | | Z | |2 1/2 2 2 2 ˜ 2 ε 3/2 ε b |ε||P − Q ||3PB | b ≤ b . B1 0 . + 2 2 + 5 . | | y≤B0 (1 y ) (1 y) log(b)

On the other hand, after integration by parts, we repeat the same arguments as before and apply (C-4).

8See again [Raphaël and Rodnianski 2012] for more details about the ﬂux computation statement and its connection to the Pohozaev integration by parts formula. SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 811

This yields Z |ε| Z |ε| Z y b2 (D3ε, 3P˜ ) ≤ b2 + b2 + b2 |∇ε| B0 + 4 + 2 + 2 y≤B0 (1 y ) B0/4≤y≤B0/2 (1 y ) y≤B0 1 y / / / Z |ε|2 1 2 Z |ε|2 1 2 Z |∇ε|2 1 2 b3/2 + + . + 5 + 4 + 2 (1 y ) B0/4≤y≤B0/2 (1 y ) y≤B0 1 y 4 1/2 p p b |log(b)| c(M)|Ᏹ| + K (M) . |log b|2 p b2 . K (M) . p| log(b)| Finally, b2 | ˜ | k k p| | + p (Hε, 3PB0 ) . Hε L2 log b K (M) plog(b) p q p p b2 . |log b| |Ᏹ| + K (M) b4/|log b|2 . K (M) . p|log b| We further integrate by parts in time to obtain 2 + + + ˜ ∂s ε b(∂sε 23∂sε) bs3ε, 3PB0 d = (∂ ε, 3P˜ ) + b(ε + 23ε, 3P˜ ) − b (∂ ε + b3ε, 3∂ P˜ ) + (ε, 8 ), ds s B0 B0 s s b B0 b with = − ˜ − 2 ˜ − ˜ − 2 ˜ 8b 3PB0 3 PB0 b3∂b PB0 b3 ∂b PB0 . We thus estimate from (4-1), (4-5), (4-7), (3-32) and (3-34) that Z |η| Z |ε| |b ||(∂ ε + b3ε, 3∂ P˜ ) + (ε, 8 )| |b | + s s b B0 b . s + 2 B0/4≤y≤B0 y y≤B0 1 y q 2 |bs||log b| p b . C(M) |Ᏹ| + K (M) b4/|log b|2 . K (M) . b2 p|log b| The nonlinear term is estimated as before. Indeed, we have Z | ˜ | | | + | | 2| ˜ | (N(ε), 3PB0 ) . ( PB1 ε )ε 3PB0

Z B0 2 1 2 |ε| ky(|P | + |ε|)k ∞ k( + y )3P˜ k ∞ . 2 B1 L 1 B0 L 4 b 0 y(1 + y ) C(M) b4 b2 . Ᏹ + K (M) . K (M) . b2 |log b|2 p|log b| Step 3: Control of G(b) and Ᏽ. Injecting the estimates of Steps 1 and 2 into (4-67) yields (4-66). It remains to prove (4-65). The estimate for G(b) is a straightforward consequence of the choice (4-62) and 812 MATTHIEU HILLAIRET AND PIERRERAPHAËL the explicit formula (1-3). It remains to control Ᏽ. We integrate by parts in space in (4-64) and get = + ˜ − ˜ + 2 ˜ + ˜ ˜ − − ˜ ˜ Ᏽ(s) (∂sε b3ε, 3PB0 ) b(ε, 3PB0 3 PB0 ) bs(∂b PB0 , 3PB0 ) bs ∂b(PB1 PB0 ), 3PB0 . The b terms are estimated as in Step 1, ˜ ˜ ˜ ˜ |bs| |bs| (∂b PB , 3PB ) − (∂b(PB − PB ), 3PB ) b. 0 0 1 0 0 . b . The linear term is estimated using (4-1), (4-5), (4-7), (3-32) and (3-34), + ˜ − ˜ + 2 ˜ (∂sε b3ε, 3PB0 ) b(ε, 3PB0 3 PB0 ) 1/2 1/2 Z |η| Z |ε| 1 Z |η|2 |log b| Z |ε|2 + b + K (M)b, . 2 2 . 2 2 4 + | |2 . y≤B0 y y≤B0 y b y b y≤B0 y (1 log y ) and (4-65) is proved. This concludes the proof of Lemma 4.11.

5. Sharp description of the singularity formation

We are now in position to conclude the proofs of Proposition 3.5 and Theorem 1.1 as simple consequences of the a priori bounds obtained in the previous section. The proofs rely on a topological argument that ﬁnishes the bootstrap argument, and then the sharp description of the blow-up dynamic is a consequence of the a priori bounds obtained on the solution and in particular the modulation equation (4-66). Proof of Proposition 3.5. We argue by contradiction and assume that for all 2 2 b0 b0 a+ ∈ − , , T1(a+) < T (a+). |log b0| |log b0| In view of what Lemma 4.9 says about the bootstrap regime and the improved bounds of Lemmas 4.9 and 4.10, a simple continuity argument ensures that T1(a+) is attained at the ﬁrst time t where |b(t)|2 |κ+(t)| = . (5-1) 2|log(b(t))| The fundamental fact used now is the outgoing behavior (4-56), which together with (5-1), ensures that

dκ+ (T1(a+)) > 0. dt Thus from a standard argument,9 the map

9See [Cote et al. 2009, Lemma 6] for a complete exposition. SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 813

On the one hand, (5-1) implies b2 b2 8 − 0 , 0 ⊂ {−1, 1}. |log b0| |log b0|

On the other hand, the outgoing behavior (4-56) together with the initialization κ+(0) = a+ ensure that b2 b2 8 − 0 = −1 and 8 0 = 1, |log b0| |log b0| 10 and a contradiction follows. This concludes the proof of Proposition 3.5. Proof of Theorem 1.1. Step 1: Finite time blow-up and derivation of the blow-up speed. Choose from Proposition 3.5 initial data with T1(a+) = T (a+). We ﬁrst claim that u blows up in ﬁnite time,

T = T (a+) < +∞. (5-2)

Indeed, from (4-41),

2(1−α) 3 2/3 2(1−α)/3 λ . b and thus λ . λ . b = −λt . Integrating this differential inequality yields

1/3 1/3 t . λ (0) − λ (t) . 1 ˙ 1 ˙ 2 2 ˙1 and (5-2) follows. The (H ∩ H )×(L ∩ H ) bounds (3-33) and (3-34) on (ε, ∂t ε), and hence on (u, ∂t u) in the bootstrap regime, and standard H 2 local well posedness theory ensures that blow-up corresponds to

λ(t) → 0 as t → T (a+).

We now derive the blow-up speed by reintegrating the ODE (4-66) and brieﬂy sketch the proof which follows as in [Raphaël and Rodnianski 2012]. First recall the standard scaling lower bound

λ(t) ≤ C(u0)(T − t), which implies that the rescaled time is global, Z t dτ s(t) = → +∞ as t → T. 0 λ(τ) Let ᏶ = G + Ᏽ so that from (4-65) we get 1 ᏶ 1 ᏶ = 64b |log b| 1 + O and b = 1 + O , (5-3) |log b| 64|log ᏶| p|log ᏶|

10This topological argument is the one-dimensional version of Brouwer’s ﬁxed-point argument used in [Cote et al. 2009]. 814 MATTHIEU HILLAIRET AND PIERRERAPHAËL and ᏶ satisﬁes from (4-66) the ODE ᏶2 1 ᏶s + 1 + O = 0. 128|log ᏶|2 p|log ᏶|

Appendix A: Modulation theory

This appendix is devoted to the proof of Lemmas 3.1 and 3.3. The arguments are standard in the framework of modulation theory and we brieﬂy sketch the main computations. Proof of Lemma 3.1. First note that the bounds k∇ − k + k − − k | | (PB1 Q) L2 b 3PB1 b(1 χB1 )3Q L2 . b log b ensure that our initial data are of the form

u0 = Q +η ˜0, u1 =η ˜1, for a small excess of energy in the sense that

2 k∇η ˜0, η˜1kL2×L2 . b0|log b0|, k∇ η˜0, ∇η ˜1kL2×L2 . b0. (A-1)

Hence the continuity of the ﬂow associated to (1-1) ensures the existence of a time T0 > 0 (uniform in η˜0, η˜1) for which the solution u to (1-1) with initial data (u0, u1) satisﬁes on [0, T0] that

sup k∇(u − Q), ∂t ukL2×L2 . b0|log b0|. (A-2) [0,T0] SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 815

11 Step 1: Modulation near Q. The nondegeneracy (3Q, 8) 6= 0 ensures that u admits on [0, T0] a decomposition

u(t) = (Q +ε( ˜ t))λ(t), (A-3) with

(ε(˜ t), χM 8) = 0. (A-4)

2 ∗ Moreover, λ ∈ Ꮿ ([0, T0]; ޒ+), and noting that η˜0 satisﬁes 2 b0 |(η˜0, χM 8)| . , |log b0| we obtain the bound 2 b0 |λ(0) − 1| . . (A-5) |log b0|

We then let b(t) = −λt (t) on [0, T0]. Step 2: Positivity of b. Straightforward computations yield b(t) ∂t ε(˜ t) = ∂t u − 3u . λ(t) 1/λ(t)

Taking the scalar product with χM 8, we obtain at the initial time (u1)1/λ(0), χM 8 b(0) = λ(0) , (A-6) (3u0)1/λ(0), χM 8 where (2-5) together with (A-5) imply 2 b0 (u1)1/λ(0), χM 8 = b0(3Q, χM 8) + O , (A-7) |log(b0)| 2 (3u0)1/λ(0), χM 8 = (3Q, χM 8) + O b0|log(b0)| . (A-8) This yields the positivity of b(0) and the positivity of b(t) for small time, together with 2 b0 b(t) = b0 + O . (A-9) |log(b0)| As b > 0, we may introduce the decomposition = + ˜ = + = ˜ − − u(t) (Q ε)λ(t) (PB1(b(t)) ε)λ(t), where ε(t) ε(t) (PB1(b(t)) Q). (A-10) Observe from (2-4) and (A-4) that

∀ t ∈ [0, T0], (ε(t), χM 8) = 0. (A-11) The uniqueness of such a decomposition is guaranteed by the (local) uniqueness of (λ, ε)˜ .

11This is a direct consequence of the implicit function theorem and the smoothness of the ﬂow (1-1). 816 MATTHIEU HILLAIRET AND PIERRERAPHAËL

Step 3: Smallness of ε. To complete the proof, we obtain smallness of ε in H˙ 1 and H˙ 2. To this end, we note that = − = − + + ε(0) (u0)1/λ(0) PB1(b(0)) PB1(b0) 1/λ(0) PB1(b(0)) (η0 d+ψ)1/λ(0). Simple computations based on the estimates of Proposition 2.1 yield the expected result, 2 ε(0) 2 b0 k∇ε( )k 2 b | (b )| + k∇ ε( )k 2 . 0 L . 0 log 0 and 4 0 L . (A-12) 1 + y L2 |log(b0)| This concludes the proof.

Proof of Lemma 3.3. The proof of this lemma is divided into two steps. First, given (η0, η1, d+) satisfying the smallness condition (3-1) for small b0, we prove that b, bs and w satisfy (3-31)–(3-34). Then, we show that given (b0, η0, η1), we can apply the inverse mapping theorem to d+ 7→ κ+(0) close to 0. The arguments used are standard and we refer to [Cote et al. 2009] for a detailed proof in a similar setting.

Step 1: Smallness of initial modulation. Given (η0, η1, d+) satisfying the smallness condition (3-1), we can apply Lemma 3.1. This yields T0 and b, ε, w such that (3-31) holds and 2 2 b0 k∇w(t)k 2 b | (b )|, k∇ w(t)k 2 . L . 0 log 0 L . 2 (A-13) |log(b0)|

We emphasize that Lemma 3.1 implies in particular that b0/2 < b(0) < 2b0 for sufficiently small b0. As before, we focus now on bounds satisfied initially. We first compute bs(0) using (1-1) and the k = k−1 = orthogonality condition (A-11). Recalling that (∂b PB1 , χM 8) (∂s ε, χM 8) 0 for any integer k, we get, like for (4-10),

[ + + ] bs (3PB1 , χM 8) 2b(3∂b PB1 , χM 8) (3ε, χM 8) = −(9 , χ 8) − (ε, H ∗ (χ 8)) + b(∂ ε, 3(χ 8)) + (N(ε), χ 8), B1 M B1 M s M M where, denoting by LHS and RHS the two sides at initial time, we compute, for b0 small enough with respect to M that 2 b0 |bs(0)| |RHS| ≤ C(M) + k∂sεkL2(y

1 1 bs κ− = (ε, ψ) − (∂sε, ψ) − (∂b PB , ψ). 2 ζ 2ζ 1

Consequently, we apply (3-28), noting that w(t) = (ε(t))λ(t), and (A-15) because of the exponential decay of ψ to get 2 b0 |κ−(0)| . . (A-18) |log b0|

Step 2: Computation of d+. We now claim from an explicit computation that given a+, the initialization (3-24) can be reformulated in the form ∂ F = = 1 k k2 + F(d+) a+, with 2 ψ L2 O(b0), (A-19) ∂d+ d+=0 from which the implicit function theorem concludes the proof of Lemma 3.3. Let us brieﬂy justify (A-19). We want to study the mapping

4 ᐂ → ޒ , d+ 7→ [b(t), bs(t), (ε(0), ψ), (∂sε(t), ψ)], where ᐂ is a neighborhood of 0. To this end, it is necessary to study the dependencies of all initial parameters on d+. For conciseness, we denote by d differentiation with respect to d+ in what follows.

Computation of (λ(0), ε(˜ 0)). As a first step in modulation theory, we proved that (λ(0), ε(˜ 0)) = 8(u0), where 8 is a smooth mapping H˙ 1(ޒN ) → ޒ × H˙ 1(ޒN ) defined on a neighborhood of Q. Due to the ∞ N exponential decay of ψ ∈ Ꮿ (ޒ ) we thus have that λ(0) is a smooth function of d+ with differential dλ(0) = dλ ∈ ޒ. We have the same result for ε with differential dε(˜ 0) = dε˜ ∈ H˙ 1(ޒN ). By definition, we have

ε(˜ 0) = u0 − Q1/λ, 818 MATTHIEU HILLAIRET AND PIERRERAPHAËL so that dλ dε˜ = ψ + (3Q) . λ(0) 1/λ(0)

Computation of b(0). From (A-6), b(0) is a Ꮿ1 mapping with

" 2 # (u1)1/λ(0), χM 8 (3 u0)1/λ(0), χM 8 − (3u1)1/λ(0), χM 8 db(0) = dλ + 2 (3u0)1/λ(0), χM 8 (3u0)1/λ(0), χM 8 (u1)1/λ(0), χM 8 (3ψ)1/λ(0), χM 8 −λ(0) , 2 (3u0)1/λ(0), χM 8 where (A-6) and (A-7) ensure that, for some db ∈ ޒ, we have

db(0) = db + O(b0).

Computation of ε(0). Next, = ˜ − − ε(0) ε(0) (PB1(b(0)) Q).

Consequently, (ε(0), ψ) is also a smooth function of d+ with derivative dps1(0) satisfying

= ˜ − dps1(0) (dε, ψ) db(0)(∂b PB1(b(0)), ψ).

˜ = | − | 2 | | Replacing dε by its values, and applying that (3Q, ψ) 0 together with λ(0) 1 . b0/ log(b0) , we get ˜ = k k2 + (dε, ψ) ψ L2 O(b0), so that = k k2 + dps1(0) ψ L2 O(b0).

Computation of ∂sε(0) + bs(0)∂b PB1(b(0)). From (A-15),

= − − + ∂sε(0) bs(0)∂b PB1(b(0)) b(0)3u0 λ(0) b03PB1(b0) 1/λ(0) , + so that (∂sε(0) bs(0)∂b PB1(b(0)), ψ) is a smooth function of d+ with derivative

= − + + 2 − dps2(0) db(0)(3u0, ψ) dλ b03PB1(b0) 1/λ(0) b03 PB1(b0) 1/λ(0) , ψ b(0)(3ψ, ψ), where, for the same orthogonality reason (3Q, ψ) = 0, we have

(3u0, ψ) = (3Q, ψ) + O(b0) = O(b0).

Consequently dps2(0) = O(b0). SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 819

Conclusion. Finally, we have

1 1 κ+(0) = (ε(0), ψ) + √ (∂sε(0) + bs(0)∂b PB (b(0)), ψ) , 2 ζ 1 and κ+(0) = a+ reduces to a simple one-dimensional equation F(d+) = a+, with F computed as combination of the above functions so that it is smooth in a neighborhood of 0. Moreover, 1 1 1 2 dF = dps1(0) + √ dps2(0) = kψk 2 + O(b0), 2 ζ 2 L and (A-19) is proved. This concludes the proof of Lemma 3.3.

Appendix B: Coercivity estimates

The aim of this section is to prove the coercivity properties of the quadratic form Z Z 2 2 B(η, η) = (Ꮾv, v) = |∂r η| + Wη , ޒ4 ޒ4 where 6 9 r 2 W(r) = 2V + 3 rV 0 = − . 2 (1 + r 2/8)2 4 (1 + r 2/8)3

We use the elementary method developed in [Fibich et al. 2006]. The coercivity property of Lemma 4.7 is a consequence of the two following facts. First the index of B on

Z Z 2 1 2 u H˙ = u radial |∇u| + < +∞ r r 2 is at most 2. From standard Sturm–Liouville oscillation theorems, see Theorem XIII.8 [Reed and Simon 1978], this is equivalent to counting the number of zeroes of the solution to

ᏮU = 0, (B-1) U(0) = 1, U 0(0) = 0, on (0, ∞), and this can be analytically reduced to counting the number of zeroes of a Bessel function. Then we need to show that the orthogonality conditions (η, ψ) = (η, 8) = 0 are enough to treat the two negative directions. Arguing exactly as in [Fibich et al. 2006] — see also [Marzuola and Simpson ˙ 1 2011] — this is equivalent to first inverting the operator Ꮾ on Hrad, and then showing that B restricted to Span{Ꮾ−1ψ, Ꮾ−18} is negative definite, which is an elementary numerical check. We shall check these two facts below and refer to [Fibich et al. 2006] for the proofs that this implies the claimed coercivity property. The proofs there are given for exponentially decaying functions and potentials, but one checks easily that the decay of the potential |W(r)| ∼ 1/r 4 at infinity and |8(r)| ∼ 1/r 4 are more than enough to have all proofs go through. 820 MATTHIEU HILLAIRET AND PIERRERAPHAËL

˙ 1 Computation of the index of B. We ﬁrst show that the index of Ꮾ on Hr is at most 2. We start by noting that W(r) ≥ Wb(r), where 3 r 2 Wˆ (r) = − . 2 (1 + r 2/8)3

Hence, classical Sturm–Liouville theory ensures that U has less zeros than Uˆ , the unique solution to 1 d 3 d 0 − r Ub + WbUb = 0, Ub(0) = 1, Ub (0) = 0, (B-2) r 3 dr dr on (0, ∞). Second, we look for Ub of the form Ub(r) = (2/r 2) U(r 2/2), with U a sufﬁciently smooth function. Denoting by s the new variable r 2/2, straightforward calculations yield that U is a solution to d2 − U + W U = 0, U(0) = 0, U 0(0) = 1, (B-3) ds2 on (0, ∞), where 3 1 W(s) = − . 2 (1 + s/4)3 √ √ Setting then U(s) = 1 + s/4 Ue(1/ 1 + s/4), we obtain that U is a solution to (B-3) if and only if Ue is a solution to 2 2 d d 2 0 τ Ue + τ Ue + (96τ − 1)Ue = 0, Ue(1) = 0, Ue (1) = −8, dτ 2 dτ √ √ on (0, 1). Hence, Ue is a combination of Bessel functions: Ue(τ) = C1 J(1, 4 6τ) + C2Y (1, 4 6τ). We compute (C1, C2) and draw the explicit combination with Maple (Figure 1). The computed solution Ue has two zeros on (0, 1). Moreover, it diverges at 0 so that Ue(τ) ∼ K/τ close to 0 with K 6= 0, As a

Figure 1. Solution to (B-3) computed by Maple. SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 821

Figure 2. Solution to (B-1) computed by MAPLE. consequence. ˆ ∼ 1 6= → ∞ U(r) 4 K 0 when r , − + ˆ ˙ 1 and thus the index of 1 W on Hrad is exactly two. Hence the index of Ꮾ is at most 2.

Choice of the orthogonality conditions. We now invert Ꮾ. We ﬁrst check numerically that the solution U does not vanish at inﬁnity, that is, lim U(r) > 0; r→+∞ see Figure 2. Hence U is not a resonance — note that if U had been a resonance, we could have removed the resonance by diminishing a bit the potential and getting a potential with index 2 and no resonance — and ˙1 thus from standard ODE arguments [Fibich et al. 2006] there exists unique smooth solution in H rad of 1 d d ᏮU = − r 3 U + WU = ψ, U 0(0) = 0, (B-4) r 3 dr dr on (0, ∞), with (1 + r 2)U ∈ L∞, and 1 d d ᏮU = − r 3 U + WU = 8, U 0(0) = 0, (B-5) r 3 dr dr on (0, ∞), with (1 + r 2/log r)U ∈ L∞. We denote by Ꮾ−1ψ and Ꮾ−18 the respective solutions to these systems. We recall the explicit formula

2 − 3r 2/4 8(r) = D3Q(r) = . (1 + r 2/8)3 822 MATTHIEU HILLAIRET AND PIERRERAPHAËL

In the remainder of this section we check numerically that the restriction of B to Span(Ꮾ−1ψ, Ꮾ−18) is negative deﬁnite, or equivalently: (Ꮾ−1ψ, ψ) (Ꮾ−18, ψ) Lemma. The symmetric matrix ނ = satisﬁes (Ꮾ−18, ψ) (Ꮾ−18, 8)

(Ꮾ−1ψ, ψ) < 0 and det ނ > 0, (B-6) and is thus negative deﬁnite.

Numerical proof. We use standard MATLAB routines for the computation of solutions to (B-4) and (B-5). We note that we only fixed the initial value for U 0(0). The value U(0) is left open in order to achieve the expected decay at infinity that characterizes the inverse. To obtain Ꮾ−1ψ, we first compute ψ. We obtain that the corresponding eigenvalue is approximately l = −0.5860808922. Because ψ decays exponentially, we only need to obtain an approximation on a short time-range. We computed our solutions until Tψ,max = 30. We emphasize here that we use an explicit scheme. As a drawback, the accumulation of errors tends to make the numerical solution become negative when the exact solution is exponentially small. Hence, our scheme becomes unstable after time Teψ,max = 18. Nevertheless, we extend our numerical solution by 0 after this time. This induces an exponentially small error. The pictures in Figure 3 illustrate this computation. On√ the left-hand side we draw the obtained solution. On the right-hand side, we draw ψtest(r) = ψ(r) exp( −lr). We observe here that our solution enters the exponential asymptotic regime before the instability comes into play. The solution Ꮾ−1ψ is computed with the extension of ψ. Straightforward ODE analysis shows that the unique solution decaying fast at infinity behaves like 1/r 2 asymptotically. The choice of U(0) is made with respect to this criterion. Figure 4 illustrates that we obtained a solution with the suitable decay. As previously, on the left-hand side is a picture of the numerical solution. On the right-hand side we −1 2 −1 plot Ꮾ ψtest(r) = r Ꮾ ψ(r). In the latter computations, this solution is involved in scalar products

Figure 3. Numerical simulations for ψ. SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 823

Figure 4. Numerical simulations for Ꮾ−1ψ. with ψ. Hence even if drawn until Tmax = 300, we only need a precise computation of this solution until

TᏮ−1ψ,max = 18. The last solution Ꮾ−18 is computed with the same method. In this second case, the expected decay of the solution is log(r)/r 2. Figure 5 illustrates that we obtained a solution with the suitable decay. The picture on the right-hand side restricts to the time-interval r = 0 ... 100 because this is the significant region. In the latter computations, this solution is involved in integrals which converge slowly. Hence, we compute this solution until TᏮ−18,max = 1000. We now compute numerically the entries of the matrix ނ. We first compute (Ꮾ−18, ψ) = (Ꮾ−1ψ, 8). The exponential decay of ψ implies that we need to compute the first integral (Ꮾ−18, ψ) on a shorter time-interval. Hence, we prefer this computation to the second one. We compute the L2 scalar products with a standard trapezoidal method. Changing the time-interval and the time-step, the computations are

Figure 5. Numerical simulations for Ꮾ−18. 824 MATTHIEU HILLAIRET AND PIERRERAPHAËL

Figure 6. Computations for (Ꮾ−18, 8). stable up to an error of 10−2. We get the following approximations for the integrals involving ψ:

(Ꮾ−1ψ, ψ) = −4.63 ± 10−2 and (Ꮾ−18, ψ) = 32.65 ± 10−2.

The last integral is a more involved computation. Indeed, standard real analysis implies that Z M I (M) := Ꮾ−18Q(r)8(r)r 3dr = (Ꮾ−18, 8) + err(M), 0 with a remainder satisfying err(M) = (K + o(1)) ln(M)/M2 for some constant K . This remainder goes to 0 slowly; we see numerically that our computations have not converged even after integrating until TᏮ−18,max = 1000 (see Figure 6, red crosses). To improve the rate of convergence we compute an approximation of coefﬁcient K and subtract the estimated error term of our computations. This yields the blue circles in Figure 6. In this second computation we obtain a very good rate of convergence. Hence, we get the approximation (Ꮾ−18, 8) = −574.25 ± 10−2, which leads to

det(ނ) = 1591 ± 10, concluding the numerical proof of the lemma.

Appendix C: Some linear estimates

We start by recalling some obvious integration-by-part results: ≥ ∈ 1 N Lemma C.1. For any N 3, there exists a constant C for which there holds, for any v Hrad(ޒ ), / Z |v(y)|2 1 2 Z 1/2 + |y|(N−2)/2|v(y)| ≤ C |∇v(y)|2 . 2 sup (C-1) ޒN |y| y∈ޒN ޒn Looking for control on further derivatives, we prove a lemma. SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 825

= ∈ 2 N Lemma C.2 (Hardy inequalities). Let N 4. Then for all R > 2 and v Hrad(ޒ ), we have

Z |∂ v|2 Z y (1v)2, (C-2) y2 . Z |v|2 Z |∂ v|2 Z y + |v|2, 4 2 . 2 (C-3) y≤R y (1 + |log y|) y≤R y y≤2 Z |v|2 Z |∂ v|2 Z R y + |v|2. 4 . log 2 (C-4) R≤y≤2R y y≤R y y≤2

Proof. Let v be smooth. (C-2) follows from the explicit formula after integration by parts,

Z Z N − 1 2 Z Z |∂ v|2 (1v)2 = ∂ v + ∂ v = (∂ v)2 + (N − 1) y . yy y y yy y2

To prove (C-3), let a ∈ [1, 2] be such that

Z |v(a)|2 ≤ |v|2. (C-5) 1≤y≤2 3 4 2 Let f (y) = −(1/y (1 + log(y))) ey so that ∇ · f = 1/(y (1 + |log y|) ), and integrate by parts to get

Z |v|2 Z = |v|2∇ · f 4 2 a≤y≤R y (1 + log y) a≤y≤R R |v|2 Z v∂ v = − + y 2 3 1 + log(y) a y≤R y (1 + log y) / 1/2 Z |v|2 1 2Z |∂ v|2 |v(a)|2 + y . . 4 2 2 (C-6) y≤R y (1 + |log y|) y≤R y

˜ 3 similarly, using f (y) = (1/y (1 − log(y))) ey, we get

Z |v|2 Z = |v|2∇ · f˜ 4 2 ε≤y≤a y (1 − log y) a≤y≤R a |v|2 Z 1 = + v∂ v 2 y 3 1 − log(y) ε y≤a y (1 − log y) / 1/2 Z |v|2 1 2Z |∂ v|2 |v(a)|2 + y . . 4 2 2 (C-7) y≤R y (1 + |log y|) y≤R y

(C-5)–(C-7) now yield (C-3). The last inequality (C-4) is a straightforward variant of [Raphaël and Rodnianski 2012, Lemma B.1, (B.4)] and is left to the reader. 826 MATTHIEU HILLAIRET AND PIERRERAPHAËL

Lemma C.3 (coercivity estimates with H). Let ψ be the ﬁrst eigenvector of H. Then there exist c > 0 ≥ ≥ ∈ 1 N and M0 1 such that for M M0, there exists δ(M) > 0 such that given u Hrad(ޒ ), we have Z 1 (Hu, u) ≥ c (∂ u)2 − (u, ψ)2 + (u, χ 8)2, (C-8) y c M Z Z 2 Z 2 2 (∂yu) u 1 2 (Hu) ≥ δ(M) + − (u, χM 8) . (C-9) y2 y4(1 + |log y|)2 δ(M) Proof. (C-8) is a standard consequence of the coercivity of the linearized energy which admits exactly ψ as bound state and 3Q as resonance at the origin, the good enough localization of 8 from (2-1) and the nondegeneracy from (2-2). The detailed proof is left to the reader. To prove (C-9), we ﬁrst observe the key subcoercivity property Z Z Z Z Z 2 2 2 2 2 2 (Hu) = (1u + V u) = (1u) − 2 V (∂yu) + (1V + V )u

Z Z u2 1 Z (∂ u)2 Z u2 ≥ c (1u)2 + − y + , (C-10) 1 + y6 c 1 + y4 1 + y8 where we used the asymptotic value N(N + 2)(N − 2) 1 V (y) = 1 + O as y → +∞. y4 y2

(C-9) now follows by contradiction. Let M > 0 ﬁxed and consider a sequence un such that Z (∂ u )2 Z u2 y n + n = 1 (C-11) y2 y4(1 + |log y|)2 and Z 1 (Hu )2 ≤ ,(u , χ 8) = 0. (C-12) n n n M ∈ 1 Then by semicontinuity of the norm, a subsequence of un weakly converges to a solution u∞ Hloc of Hu∞ = 0. The solution u∞ is smooth away from the origin and hence the explicit integration of the ODE ∈ 1 and the regularity assumption at the origin u∞ Hloc imply that

u∞ = α3Q.

On one hand, the uniform bound (C-11) together with the local compactness of Sobolev embeddings ensure that, up to a subsequence,

2 2 2 2 Z (∂yun) Z |un| Z (∂yu∞) Z |u∞| + → + and (un, χM 8) → (u∞, χM 8), 1 + y4 1 + y8 1 + y4 1 + y8 thanks to the χM localization. We thus conclude that

α(3Q, χM 8) = (u∞, χM 8) = 0 and thus α = 0. On the other hand, the subcoercivity property (C-10), the Hardy control (C-2), (C-3) and SMOOTH SOLUTIONS TO THE FOUR-DIMENSIONAL ENERGY-CRITICAL WAVE EQUATION 827

(C-11), (C-12) ensure that Z (∂ u )2 Z u2 y n + n ≥ C > 0, 1 + y4 1 + y8 from which 2 2 2 2 Z (∂ 3Q) Z |3Q| Z (∂ u∞) Z |u∞| α2 y + = y + ≥ C > 0, 1 + y4 1 + y8 1 + y4 1 + y8 and thus α 6= 0. A contradiction follows. This concludes the proof of (C-9) and of Lemma C.3. Straightforward computations show that the coercivity estimates with H can be adapted to any of the ∈ 1 N operators Hλ yielding, for any λ > 0 and u Hrad(ޒ ), Z 1 (H u, u) ≥ c (∂ u)2 − (u, (ψ) )2 + (u, (χ 8) )2 (C-13) λ y cλ4 λ M λ for the same c and δ(M) as in Lemma C.3.

Acknowledgements

The authors thank Igor Rodnianski for stimulating discussions about this work. P. Raphaël is supported by the French ANR Jeune Chercheur SWAP.

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Received 8 Oct 2010. Accepted 13 May 2011.

MATTHIEU HILLAIRET: [email protected] Ceremade, Université Paris Dauphine, 75775 Paris Cedex 16, France http://www.ceremade.dauphine.fr/~mhillair/

PIERRE RAPHAËL: [email protected] Université Paul Sabatier, Institut de Mathématiques de Toulouse, 31062 Toulouse Cedex 9, France http://www.math.univ-toulouse.fr/~raphael/

mathematical sciences publishers msp

ANALYSIS AND PDE Vol. 5, No. 4, 2012 dx.doi.org/10.2140/apde.2012.5.831 msp

NONCONCENTRATION IN PARTIALLY RECTANGULAR BILLIARDS

LUC HILLAIRET AND JEREMY L.MARZUOLA

In speciﬁc types of partially rectangular billiards we estimate the mass of an eigenfunction of energy E in the region outside the rectangular set in the high-energy limit. We use the adiabatic ansatz to compare the Dirichlet energy form with a second quadratic form for which separation of variables applies. This allows us to use sharp one-dimensional control estimates and to derive the bound assuming that E is not resonating with the Dirichlet spectrum of the rectangular part.

1. Introduction

We study concentration and nonconcentration of eigenfunctions of the Laplace operator in stadium-like billiards. As predicted by the quantum/classical correspondence, such concentration is deeply linked with the classical underlying dynamics. In particular, the celebrated quantum ergodicity theorem roughly states that when the corresponding classical dynamics is ergodic then almost every sequence of eigenfunctions equidistributes in the high energy limit (see [Schnirelman 1974; Colin de Verdière 1985; Zelditch 1987] and [Gérard and Leichtnam 1993; Zelditch and Zworski 1996] in the billiard setting for a more precise statement). In strongly chaotic systems such as negatively curved manifolds, it is expected that every sequence of eigenfunctions equidistributes. This statement is the quantum unique ergodicity conjecture (Q.U.E.) and remains open in most cases despite several recent striking results (see for instance [Faure et al. 2003; Lindenstrauss 2006; Anantharaman 2008; Anantharaman and Nonnenmacher 2007]). On the other extreme, the Bunimovich stadium, although ergodic, is expected to violate Q.U.E. Indeed, it is expected that there exist bouncing ball modes, i.e., exceptional sequences of eigenfunctions concentrating on the cylinder of bouncing ball periodic orbits that sweep out the rectangular region (see [Bäcker et al. 1997] for instance). The existence of such bouncing ball modes is still open and only recently did Hassell prove that the generic Bunimovich stadium billiard indeed fails to be Q.U.E. (see [Hassell 2010]). Our work is closely related to the search for bouncing ball modes but proceeds loosely speaking in the other direction. We actually aim at understanding how strong concentration of eigenfunctions in the rectangular part cannot be. We thus follow [Burq and Zworski 2005], where it is proved that even bouncing ball modes couldn’t concentrate strictly inside the rectangular region. This was made precise by

Marzuola was supported in part by a Hausdorff Center Postdoctoral Fellowship at the University of Bonn, in part by an NSF Postdoctoral Fellowship at Columbia University. He also thanks the MATPYL program, which supported his coming to the University of Nantes, where this research began. Hillairet was partly supported by the ANR programs NONaa and Methchaos. MSC2010: primary 35P20; secondary 35Q40, 58J51. Keywords: eigenfunctions, billiards, nonconcentration.

831 832 LUC HILLAIRET AND JEREMYL.MARZUOLA

Burq, Hassell and Wunsch in [Burq et al. 2007], where the following estimate was proved:

1 u 2 E u 2 ; k kL .W / k kL ./ 2 in which u 2 and u 2 denote the L norm of the eigenfunction u in the wings and in the k kL .W / k kL ./ billiard, respectively. Our main result for the Bunimovich stadium is the following:

Theorem 1. Let be a Bunimovich stadium with rectangular part R Œ B0; 0 Œ0; L0. We set WD W R and denote by † the Dirichlet spectrum of R, i.e., D n k22 l22 † ; k; l ގ : 2 2 D L0 C B0 2

For any " 0 there exists E0 and C such that if u is an eigenfunction of energy E such that E > E0 and " dist.E; †/ > E then the following estimates holds:

5 8" C u 2 CE 6 u 2 ; k kL ./ Ä k kL .W / 1 This bound improves on the Burq–Hassell–Wunsch bound provided that " < 8 . It is natural that the smaller " is the better the bound is. Indeed, the condition on the distance between E and † is comparable to a nonresonance condition and should imply heuristically that u must have some mass in the wing region. It is quite interesting to have a quantitative statement conﬁrming this heuristics. We will actually give a more general statement concerning more general billiards (see Theorem 2). In particular we will consider billiards with smoother boundaries (see Section 2) disregarding the fact that these may not be ergodic. Here again we expect the bound to be better when the billiard becomes smoother and this statement is made quantitative in Theorem 2. The method we propose relies on comparing the Dirichlet energy quadratic form with another quadratic form arising from the adiabatic ansatz presented in the numerical study of eigenfunctions by Bäcker, Schubert and Stifter [1997]. This adiabatic quadratic form has also appeared recently in [Hillairet and Judge 2009] in the study of the spectrum of the Laplacian on triangles. These two quadratic forms are close provided we do not enter too deeply into the wing region so that the nonconcentration estimate really takes place in a neighborhood of the rectangle that becomes smaller and smaller when the energy goes to inﬁnity (see Sections 4.3.3 and 4.6.1). Since the new quadratic form may be addressed using separation of variables, we will show precise one-dimensional control estimates and then use them to prove our results. We have separated these one dimensional estimates in an appendix since they may be of independent interest. Finally, we remark that the method can be applied to quasimodes with some caution (see Remark 5.2) but there are no reasons to think that the bound we obtain is optimal.

2. The setting

Let L be a function deﬁned on Œ B0; B1 with the following properties: - For nonpositive x, L.x/ L0 > 0. D NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 833 L.x/

. B0; L0/

R W

. B0; 0/ .b ; 0/ .B1; 0/ 0

Figure 1. An example of a billiard .

- On .0; B1/, L is smooth, nonnegative and nonincreasing.

- When x goes to B1, L has a negative limit (either ﬁnite or ). 0 1 - For small positive x, we have the asymptotic expansions

1 1 L.x/ L0 cLx o.x /; L .x/ cL x o.x / (2-1) D C 0 D C 3 for some positive cL and . 2 The billiard is then deﬁned by

.x; y/ B0 x B1; 0 y L.x/ : D f j Ä Ä Ä Ä g See Figure 1 for an example of an applicable billiard. For any b < B1, we will denote by b WD x b and by W 0 x b . \ f Ä g b WD \ f Ä Ä g We study eigenfunctions of the positive Dirichlet Laplacian, , on . Namely, we study solutions uE such that 2 2Á uE @ @ uE EuE and uE 0; D x C y D j@ D where E > 0. We may formulate this equation using quadratic forms. We thus introduce q deﬁned on H 1./ by Z q.u/ u 2dx dy: D jr j The Euclidean Laplacian with Dirichlet boundary condition in is the unique self-adjoint operator 1 1 associated with q deﬁned on H0 ./. We denote by qb the restriction of q to H .b/ and by b the Dirichlet Laplace operator on b. We will also denote by Ᏸb the set of smooth functions with compact support in b.

3. Adiabatic approximation

Motivated by the well-known eigenvalue problem on a rectangular billiard and computational results in

[Bäcker et al. 1997], we introduce a second family of quadratic forms ab and compare it to qb. 834 LUC HILLAIRET AND JEREMYL.MARZUOLA

For any b < B1 and any u Ᏸ , Fourier decomposition in y implies that 2 b X Â k Ã u.x; y/ u .x/ sin y : (3-1) D k L.x/ k Since Z L.x/ ˇ Â Ãˇ2 ˇ y ˇ L.x/ ˇsin k ˇ dy 0 ˇ L.x/ ˇ D 2 each Fourier coefﬁcient uk is given by 2 Z L.x/ Â k Ã uk .x/ u.x; y/ sin y dy: D L.x/ 0 L.x/ For such u, we deﬁne

X Z b Â k22 Ã L.x/ a .u/ u .x/ 2 u .x/ 2 dx; b k0 2 k D B0 j j C L .x/j j 2 k ގ 2 Z b X 2 L.x/ Nb.u/ uk .x/ dx: D B0 j j 2 k ގ 2 Observe that for each ﬁxed x, Plancherel’s formula reads Z L.x/ X 2 L.x/ 2 uk .x/ u.x; y/ dy; j j 2 D 0 j j k ގ 2 2 so that we get Nb.u/ u 2 by integration with respect to x. D k kL .b/ Fixing some 0 < b0 < B1, and using that L is uniformly bounded above and below on Œ B0; b0 we 2 ﬁnd a constant C such that for any b b0 and u L . /: Ä 2 b

1 2 X1 2 2 C u uk 2 C u : (3-2) k kb Ä k kL . B0;b/ Ä k kb k 1 D The quadratic form ab appears as the direct sum of the following quadratic forms ab;k (that can be 1 deﬁned on the whole function space H . B0; b/): Z b Â k22 Ã L.x/ a .u/ u 2 u 2 dx: (3-3) b;k 0 2 WD B0 j j C L .x/j j 2 Recall that, on an interval I, the standard H 1 norm is deﬁned by 1 2 2 2 u 1 u u ; (3-4) k kH WD k 0kL2.I/ C k kL2.I/ so that, for any k and b < B1 and any u Ꮿ . B0; b/ we have 2 01 Â 2 2 Ã Â 2 2 Ã k 2 k 2 min L.b/; u H 1 ab;k .u/ max L0 ; u H 1 : (3-5) L0 k k Ä Ä L.b/ k k NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 835

1 1 1 The norm a 2 thus deﬁnes on H . B0; b/ a norm that is equivalent to the standard H norm. b;k

3.1. Comparing ab and qb. To compare ab and qb, we introduce the following operators D and R deﬁned on Ᏸb by yL0.x/ Ru @yu; D L.x/

Du @xu Ru: D C Using the Plancherel formula for each ﬁxed x and then integrating, we obtain Z 2 2 ab.u/ Du @yu dx dy: D b j j C j j from which the following holds for any u; v Ᏸ : 2 b a .u; v/ q .u; v/ Du; Dv @xu;@xv b b D h i h i @xu; Rv Ru; Dv ; (3-6) D h i C h i @xu; Rv Ru;@xv Ru; Rv : (3-7) D h i C h i C h i We thus obtain the following lemma. Lemma 3.1. Let ı be the function deﬁned by

2 ı.b/ sup L0.x/ sup L0.x/ : D .0;b j j C .0;b j j Then for all u; v Ᏸ 2 b 1 1 a .u; v/ q .u; v/ ı.b/ q 2 .u/ q 2 .v/: j b b j Ä b b 1 Remark 3.1. The function ı is continuous on .0; B1/ and ı.b/ O.b / when b goes to 0. D 1 Proof. In (3-7), we use the Cauchy–Schwarz inequality, max. Du ; @yu / a 2 .u/, and the fact that k k k k Ä b y=L.x/ is uniformly bounded by 1 on . The following corollary is then straightforward. 1 Corollary 3.2. For any 0 < b < B1 and any u H ./, the linear functional ƒ deﬁned by ƒ.v/ 2 WD a .u; v/ q .u; v/ belongs to H 1. /. Moreover b b b

ƒ H 1. / ı.b/ u H 1. /: k k b Ä k k b 4. Nonconcentration

4.1. Preliminary reduction. Let u be an eigenfunction of q with eigenvalue E. And deﬁne the associated linear functional ƒ using Corollary 3.2. Integration by parts shows that for any v H 1. / we have 2 0 b

q .u; v/ E u; v 2 ; b D h iL ./ 836 LUC HILLAIRET AND JEREMYL.MARZUOLA so that a .u; v/ E N .u; v/ ƒ.v/: (4-1) b b D

We now deal with this equation using the adiabatic decomposition. We thus deﬁne ƒk as the distribution over Ᏸ such that, for any v Ᏸ , b 2 b Â Â y ÃÃ ƒ .v/ ƒ v.x/ sin k : (4-2) k WD L.x/ Remark 4.1. From now on, u will always denote the eigenfunction that we are dealing with. We will denote by uk the functions entering in the adiabatic decomposition of u, by ƒ the linear functional associated with u and by ƒk the one-dimensional linear functionals that are associated with ƒ.

A straightforward computation yields, that for any v Ᏸ we have 2 b Z b L.x/ ab;k .uk ; v/ E uk .x/v.x/ dx ƒk .v/; B0 2 D where ab;k is the quadratic form defined in (3-3). An integration by parts then shows that, in the distributional sense in . B0; b/, we have 1 d Âk22 Ã .Lu / E u ƒ ; (4-3) L dx k0 C L2 k D Q k where the linear functional ƒ is defined by Q k 2 Á ƒ .v/ ƒ v : (4-4) Q k WD k L Remark 4.2. Since L is not smooth, this definition of ƒ doesn’t make sense as a distribution. However, Q k 1 in the next section, we will prove that ƒk actually is in H and, since multiplication by 2=L is a bounded operator from H 1. B ; b/ into itself, we thus get that ƒ is a perfectly legitimate element of 0 Q k 1 H . Moreover, for any b0 there exists C.b0/ such that for any b b0, and v Ᏸ , we have Ä 2 b 2 v C.b0/ v 1 : 1 H . B0;b/ L H . B0;b/ Ä k k

We denote by Pk the operator that is deﬁned by

1 d Âk22 Ã P .u/ .Lu / E u; k D L dx 0 C L2 and we try to analyze the way a solution to equation (4-3) on . B0; b/ may be controlled by its behavior on .0; b/.

The strategy will depend upon whether k is large or not, but ﬁrst we have to get a bound on ƒk in some reasonable functional space of distributions. NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 837

1 4.2. Bounding ƒk. In this section, we prove that each ƒk is actually in H . B0; b/ and provide a 1 bound for its H norm. 1 We ﬁrst note that, using (3-4), for any F H . B0; b/: 2 F./ F./ 1 F H . B0;b/ sup j j sup j j : (4-5) 1 2 k k WD Ᏸb H Ä Ᏸb 0 L 2 k k 2 k k Using (3-6) in the deﬁnition of ƒk — see (4-2) — we obtain Â Â y ÃÃ Â Â y ÃÃ ƒ .v/ @xu; R v.x/ sin k Ru; D v.x/ sin k : k D L.x/ C L.x/

Denote by Ak .v/ the first term on the right and Bk .v/ the second term. By inspection, we have Z b Z b k L0.x/ 1 Ak .v/ v.x/ Fk .x/ dx and Bk .v/ v0.x/L0.x/Gk .x/ dx; WD 2 0 L.x/ WD 2 0 where we have set 2 Z L.x/ Â y Ã Fk .x/ 1W [email protected]; y/ cos k dy; (4-6) WD L.x/ 0 L.x/ 2 Z L.x/ Â y Ã Gk .x/ 1W [email protected]; y/ sin k dy (4-7) WD L.x/ 0 L.x/ Since u H 1./, F and G are L2.0; b/ and we can estimate the H 1 norm of ƒ using them. 2 k k k Lemma 4.1. For any b0 < B1, and given ƒ and F ; G defined as above, there exists C C. / k k k D b0 such that 1 ƒk H 1 C.kb Fk L2.0;b/ b Gk L2.0;b//: (4-8) k k Ä k k C k k Proof. We estimate Ak .v/, using first an integration by parts Z b ÂZ x Ã k L0./ Ak .v/ v0.x/ Fk ./ d dx: WD 2 B0 0 L./ Using the Cauchy–Schwarz inequality and the fact that L ./ O. 1/ we have 0 D ˇZ x ˇ 1 ˇ L0./ ˇ 2 ˇ Fk ./ dˇ Cx Fk L2.0;b/: ˇ 0 L./ ˇ Ä C k k

Inserting into Ak .v/ and using the Cauchy–Schwarz inequality again we get

2 2 Ak .v/ C .kb / Fk L .0;b/ v0 L . B0;b/; j j Ä k k k k which gives the claimed bound using (4-5). Next, the second term is estimated using directly the Cauchy–Schwarz estimate and the fact that sup L .x/ Cb 1. We get Œ0;b j 0 j Ä 1 2 2 Bk .v/ C b Gk L .0;b/ v0 L . B0;b/: j j Ä k k k k That gives the claimed bound using again (4-5). 838 LUC HILLAIRET AND JEREMYL.MARZUOLA

Define F 1W @xu and G 1W @yu. By definition, F .x/ is the Fourier coefficient of the function WD WD k F.x; / with respect to the Fourier basis Â Â y ÃÃ y cos k : 7! L.x/ k ގ 0 2 [f g Using the Plancherel formula we get Z L.x/ X 2 L.x/ 2 Fk .x/ F.x; y/ dy: 2 Ä 0 j j k 1 For the same reason, but using this time the sin basis, we have Z L.x/ X 2 L.x/ 2 Gk .x/ G.x; y/ dy: 2 D 0 j j k 1 Integrating with respect to x and bounding y from above and L.x/ from below uniformly we get:

Lemma 4.2. For any b0 there exists C depending only on the billiard and b0 such that, for any b < b0, X 2 2 Fk 2 C @xu 2 ; (4-9) k kL .0;b/ Ä k kL .Wb/ k 1 X 2 2 Gk 2 C @yu 2 : (4-10) k kL .0;b/ Ä k kL .Wb/ k 1 k22 We now switch to the control estimate. We begin by dealing with the modes for which E E. 2 L0 4.3. Large modes. 4.3.1. A control estimate. Equation (4-3) may be rewritten as Â k22 Ã u E u h ; (4-11) k00 C L2.x/ k D k 1 where hk is the element of H deﬁned by L h ƒ 0 u (4-12) k WD Q k C L k0 1 The H norm of hk is now estimated as follows: k22 Lemma 4.3. There exists a constant C C.b0/ such that for any b b0 and any k with E E WD Ä L2 the following estimate holds: 0 1 1 1 2 2 2 hk H . B0;b/ C.b0/ kb Fk L .0;b/ b Gk L .0;b/ b uk L .0;b/ : (4-13) k k Ä k k C k k C k k Proof. Using Remark 4.2, the norm of ƒ is uniformly controlled by the norm of ƒ and the latter is Q k k estimated using Lemma 4.1. To estimate the H 1 norm of .L =L/u , we ﬁrst set v .L =L/u and 0 k0 D 0 k0 remark that Â Ã Â 2 Ã L 0 L .L / v 0 u 00 0 u : D L k L L2 k NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 839

We choose a test function and estimate Z b Â Ã L0 0 I1 uk dx: D B0 L 1 We perform an integration by parts, use that L .x/=L.x/ Cb 1x>0, then apply the Cauchy–Schwarz 0 Ä inequality to get ˇ Z b Â Ã ˇ ˇ Z b ˇ ˇ L0 0 ˇ ˇ L0.x/ ˇ 1 2 2 ˇ uk dxˇ ˇ uk .x/0.x/ dxˇ Cb uk L .0;b/ 0 L . B0;b/: ˇ B0 L ˇ D ˇ B0 L.x/ ˇ Ä k k k k We then estimate ˇZ b ÂL .x/ .L .x//2 Ã ˇ I ˇ 00 0 u.x/.x/ dxˇ: 2 ˇ 2 ˇ D ˇ B0 L.x/ L .x/ ˇ We perform an integration by parts, use that ˇL .x/ .L .x//2 ˇ ˇ 00 0 ˇ Cx 2; ˇ 2 ˇ ˇ L.x/ L .x/ ˇ Ä C then twice apply the Cauchy–Schwarz inequality to get Z b ÂZ x Ã 2 1 I2 C u./ d 0.x/ dx Cb u L2.0;b/ 0 L2.0;b/: Ä B0 0 C j j j j Ä k k k k The claim follows using (4-5). The variational formulation of equation (4-11) is given by Z b Z b Â k22 Ã u v dx E u v dx h .v/: (4-14) k0 0 2 k k B0 C B0 L .x/ D 2 2 2 1 Since k =L0 E E, the left-hand side is a continuous quadratic form on H0 . B0; b/, so that, 1 by Lax–Milgram theory, there is a unique vk in H0 . B0; b/ satisfying (4-11) in the distributional sense. 2 The following lemma allows us to estimate the L norm of this vk .

Lemma 4.4. There exists a constant C depending only on b0 but not on b < b0, k, or E such that, if 2 2 2 1 E 1 and k =L E E, the variational solution v in H . B0; b/ to equation (4-11) satisﬁes 0 k 0 1 1 1 1 2 2 2 2 2 2 vk L . B0;b/ C.b0/ b Fk L .0;b/ E b Gk L .0;b/ E b uk L .0;b/ : (4-15) k k Ä k k C k k C k k Proof. Since v is a variational solution, putting v v in (4-14) we get k D k Z b Z b Â k22 Ã v .x/ 2dx E v .x/ 2dx h .v /: (4-16) k0 2 k k k B0 j j C B0 L .x/ j j D In the regime we are considering the second integral on the left is positive, so that we obtain Z b 2 v0 .x/ dx hk .vk / hk 1 vk 1 : k H H B0 j j Ä j j Ä k k k k 840 LUC HILLAIRET AND JEREMYL.MARZUOLA

1 Since v is in H . B0; b/, Poincaré’s inequality gives c.b/, a positive continuous function of b deﬁned k 0 for b > B0 and satisfying Z b 2 2 vk0 .x/ dx c.b/ vk H 1 : B0 j j k k This gives a constant C depending only on b0 such that, for any 0 < b < b0, we have

vk H 1 C hk H 1 : k k Ä k k We now use (4-16) again to obtain

Â 2 2 Ã Z b k 2 2 E vk .x/ dx hk H 1 vk H 1 C hk 1 2 H L B0 j j Ä k k k k Ä k k 0 with the preceding bound. Using the estimate (4-13) we obtain

1 Â 2 2 Ã2 k 1 1 E v 2 C kb F 2 b G 2 b u 2 : 2 k L . B0;b/ k L .0;b/ k L .0;b/ k L .0;b/ L0 k k Ä k k C k k C k k

2 2 2 1 We divide both sides by .k =L E/ 2 . The coefﬁcient in front of b F 2 is bounded by a 0 k k kL .0;b/ constant that is uniform in k, using the fact that

k2 L2 Â E Ã L2 Â E Ã sup sup 0 1 0 1 : 2 2 2 2 2 2 Z E k22=L E E k =L0 E D Z E C D C 0 For the two other terms, we use simply that k22=L2 E E. This gives the lemma. 0 We can now let w u v . By construction, w is a solution to the homogeneous equation k D k k k Â k22 Ã w E w 0: (4-17) 00 C L2.x/ D

Moreover, since both u and v satisfy Dirichlet boundary condition at B0 we have that w . B0/ 0. k k k D Since the “potential” part in equation (4-17) is bounded below by E, concentration properties of solutions may be obtained using convexity estimates.

Lemma 4.5. For any b b0, any solution w to (4-17) such that w. B0/ 0 satisﬁes Ä D Z b Z b 2 2 b w .x/ dx .B0 b0/ w .x/ dx: B0 j j Ä C 0 j j Proof. Multiplying the equation by w we ﬁnd

Â k22 Ã w w E w2 0: 00 C L2.x/ D

It follows that .w2/ ˇ2w2; for some positive ˇ (here ˇ2 2E). 00 D NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 841

Since w. B0/ 0, using the maximum principle on Œ B0; , we obtain for all B0 x b0 D Ä Ä Ä sinh.ˇ.x B0// w2.x/ w2./ C : Ä sinh.ˇ. B0// C For any t Œ0; 1, deﬁne x.t/ B0 t.B0 b/ and .t/ tb. Since for any t we have B0 x.t/ 2 D C C D Ä Ä .t/ b0, we may integrate the preceding relation: Ä Z 1 Z 1 sinh.ˇ.x.t/ B0// w2.x.t// dt w2..t// C dt: Ä sinh.ˇ..t/ B0// 0 0 C Since sinh is increasing the quotient of sinh is bounded above by 1 and we obtain Z b Z b 2 2 b w .x/ dx .B0 b/ w .x/ dx: B0 Ä C 0 Putting these two lemmas together we obtain:

Proposition 4.6. There exists a constant C depending only on b0 such that for any b b0, for any k and Ä E such that k22=L2 E E and E 1, 0 1 1 3 1 2 2 2 2 2 2 2 2 uk L . B0;b/ C b Fk L .0;b/ E b Gk L .0;b/ b uk L .0;b/ (4-18) k k Ä k k C k k C k k for C C.b0/. D Proof. According to Lemma 4.5 we have

1 2 2 2 wk L . B0;b/ Cb wk L .0;b/; k k Ä k k where w u v and v is the variational solution constructed above. Using the reverse triangle k D k k k inequality, we obtain

1 1 1 2 2 2 2 2 2 uk L . B0;b/ Cb uk L .0;b/ .C b0 /b vk L . B0;b/: k k Ä k k C C k k The claim will follow using estimate (4-15) of Lemma 4.4. Observe that the prefactor of u 2 is k k kL .0;b/ at ﬁrst (up to a constant prefactor) 1 1 1 1 b 2 b 2 E 2 b : C 1 1 Since E 2 b is uniformly bounded we obtain the given estimate. 4.3.2. Summing over k. We will now sum the preceding estimates over k. We thus introduce X ky u .x; y/ uk .x/ sin C L.x/ D 2 2 k =L0 E E and prove the following proposition.

Proposition 4.7. There exist b0 and E0 and a constant C depending only on E0 and b0 such that, if u is an eigenfunction with energy E > E0 and b < b0, then u 2 C b2 1 @ u 2 E 1b2 3 @ u 2 b 1 u 2 : L2.R/ x Wb y L2.W / L2.W / k Ck Ä k k C k k b C k k b 842 LUC HILLAIRET AND JEREMYL.MARZUOLA

Proof. We square estimate (4-18), sum over k, and use (3-2) and Lemma 4.2. Observe that the controlling term in the preceding estimate is supported in the wing region. However, compared to the usual bounds (as in [Burq et al. 2007]) there is a loss of derivatives since we need @xu and @yu in the wings.

Corollary 4.8. Let b0 and E0 be ﬁxed. There exists C depending on the billiard b0 and E0 but not on the eigenfunction nor on b < b0 such that 2 2 1 2 3 2 1 2 u L2.R/ C .b E b / u L2./ b u L2.W / : k Ck Ä C k k C k k 2 2 2 Proof. We bound @xu 2 and @yu 2 by E u 2 and use the fact that the norm over Wb k kL .Wb/ k kL .Wb/ k kL ./ is less than the norm over W . It remains to choose b in a clever way to obtain the desired bound.

1 ˛ 4.3.3. Optimizing b. We will choose b to be of the form M E for some constants M and ˛ to be ˛ chosen. As long as ˛ is positive, there is some large E0 such that for any E E0 then b ME < b0 D so that we can use the preceding proposition. We obtain 2 1 2 1 ˛.2 1/ ˛.2 3/ 2 ˛ 2 u L2.R/ C .M E E / u L2./ ME u L2.W / : (4-19) k Ck Ä C k k C k k It remains to make good choices to obtain the following proposition.

Proposition 4.9. There exists E0 and C depending only on the billiard such that for any u eigenfunction with energy E > E0 the following holds:

1 2 1 2 2 1 2 u L2.R/ 4 u L2./ CE u L2.W / (4-20) k Ck Ä k k C k k Proof. We choose ˛ 1=.2 1/ and M such that CM 1 2 1 . For E large enough, E ˛.2 3/ WD D 8 goes to zero. It is thus bounded by 1=.8C / for E large enough. Substituting in (4-19) we get (4-20).

4.4. Small modes. We now consider modes for which k22=L2 E E, and this time we rewrite the 0 Ä equation P .u / ƒ in the form k k D k u z u h ; (4-21) k00 k k D k in which we have set z E k22=L2 and k WD 0 L k2 Â 1 1 Ã h ƒ 0 u u : k Q k L k0 2 2 2 k WD C C L0 L

4.4.1. The control estimate. Since zk E we can use the results of the Appendix to control the term 1 2 uk L . B0;b/. To do so, we need to estimate the norm of hk in H . B0; b/. k k Lemma 4.10. There exists some constant C depending only on b0 such that, for any b b0 and any k Ä such that k22=L2 E E, the following holds: 0 Ä 1 1 2 1 1 2 2 2 hk H . B0;b/ C kb Fk L .0;b/ b Gk L .0;b/ .b k b C / uk L .0;b/ : (4-22) k k Ä k k C k k C C k k NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 843

Proof. From the definition, L k2 Â 1 1 Ã h ƒ 0 u u k Q k L k0 2 2 2 k D C C L0 L and estimate each term separately. The first term is estimated using Lemma 4.1 and Remark 4.2. The second is estimated as in the proof of Lemma 4.3. The same method applies to estimate the third term. We introduce ˇZ b Â 1 1 Ã ˇ I ˇ u .x/.x/ dxˇ; 3 ˇ 2 2 k ˇ D ˇ B0 L L .x/ ˇ 0 and observe that the quantity in parentheses is O.x /. Integrating by parts and using the Cauchy–Schwarz inequality twice gives C 1 I3 Cb 2 u 2 : Ä C k 0kL .0;b/k k kL .0;b/ 1 Using the definition of the H norm (see (4-5)) and putting these estimates together yields the lemma. For any E ޒ, define 2 ˇ k22 l22 ˇ .E/ min ˇE ˇ;.k; l/ ގ ގ : ˇ 2 2 ˇ WD ˇ L0 B0 ˇ 2 Remark 4.3. Taking l 1 in the definition shows that, for E large, we have D .E/ < cpE (4-23) for some constant c. Lemma 4.11. For any ˇ > 0, there exists some c such that the following holds. For any k such that 2 2 2 2 zk E k =L0 ˇ , D .E/ sin.B0pzk / c : j j pzk Proof. First we use that there exists some c such that

sin x c dist.x; ޚ/ for all x ޒ: j j 2 We denote by lk the integer such that Â Ã ˇ ˇ ˇ lk ˇ dist pzk ; ޚ ˇpzk ˇ; B0 D ˇ B0 ˇ so that we have k22 l22 E k ˇ ˇ 2 2 2 2 2 ˇ ˇ ˇ lk ˇ zk lk =B0 L0 B0 ˇsin.B0pzk /ˇ c ˇpzk ˇ c c ; ˇ B0 ˇ pzk lk =B0 C pzk where, for the last bound, we have used the Lemma 4.12 below. The claim follows by the deﬁnition of .E/. 844 LUC HILLAIRET AND JEREMYL.MARZUOLA

Lemma 4.12. Fix ˛ > 0 and denote by l the (step-like) function on Œ0; / deﬁned by 1 l./˛ dist.; ˛ޚ/ j j D Then there exists some C such that

l./˛ C : for all Œ0; /: C Ä 2 1 Proof. Deﬁne f by l./˛ f ./ C : D Since l vanishes on Œ0; ˛=2, we have f ./ 1 on this interval. Next, f tends to the limit 2 when goes D to inﬁnity. Finally, on Œ˛=2; M we have l./ 2M 1 f ./ 1 ˛ 1 C : D C Ä C ˛ Putting these estimates together, we get:

Proposition 4.13. There exists b0 and E0 and a constant C C.b0; E0/ such that the following holds. 2 2 2 WD For any E > E0, for any k such that k =L E E and for any b < b0, we have the estimate 0 Ä 1 E 2 1 1 1 2 1 2 2 2 2 2 2 2 2 uk L . B0;b/ C E b C Fk L .0;b/ b Gk L .0;b/ .1 Eb C /b uk L .0;b/ : k k Ä .E/ k k C k k C C k k (4-24) Proof. For any k we let z E k22=L2 and use the estimates of the appendix combined with the k D 0 bound on hk given by Lemma 4.10. For k such that zk corresponds to estimates (A-10) and (A-12) of Theorem 3 we obtain 1 1 2 2 1 2 2 uk L . B0;b/ C b hk H . B0;b/ b uk L .0;b/ k k Ä k k C k k 1 1 2 2 1 C kb 2 F 2 b 2 G 2 .b k b 1/b 2 u 2 : Ä C k k kL .0;b/ C k k kL .0;b/ C C C C k k kL .0;b/ 1 2 2 We now use that k O.E 2 / in the regime we are considering. We also remark that b k b 1 D C C C D O.1 Eb 2/. C C In the opposite case (for k such that zk corresponds to estimate (A-11)), we have to add a global 1 sin.B0pz / prefactor. Using Lemma 4.11, we have j k j 1 1 pzk E 2 sin.B0pz / C C : j k j Ä .E/ Ä .E/ We thus obtain that, for any k,

2 uk L . B0;b/ k k Â 1 Ã E 2 1 1 1 2 1 C max 1; E 2 b 2 F 2 b 2 G 2 .1 Eb /b 2 u 2 : Ä .E/ C k k kL .0;b/ C k k kL .0;b/ C C C k kL .0;b/ Using (4-23), for large E we have E1=2=.E/ bounded from below, so that the claim follows. NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 845

2 4.5. Summing over k. We use the estimates of the preceding sections to obtain a control on u 2 k kL .R/ in which we have set X Â ky Ã u .x; y/ uk .x/ sin : L.x/ D 2 2 k =L0 E E Ä Proposition 4.14. There exists b0 and E0 and a constant C depending only on E0 and b0 such that if u is an eigenfunction with energy E > E0 and b < b0, then

2 E 2 1 2 2 1 2 2 2 1 2 u L2.R/ C Eb C @xu L2.W / b @yu L2.W / .1 Eb C / b u L2.W / : k k Ä .E/2 k k C k k C C k k

Proof. We square (4-24) and sum with respect to k. The Lemma 4.2 controls P F 2 and P G 2. k k k k k k Plancherel formula takes care of P u 2. We also use as before that the norm over W is smaller than k k k b the norm over W .

2 2 As for the large mode case, we get a corollary using the fact that @xu and @yu are bounded k k k k above by E u 2 . k kL2./ Corollary 4.15. There exists b0 and E0 and a constant C depending only on E0 and b0 such that if u is an eigenfunction with energy E > E0 and b < b0, then ÄÂ 3 2 Ã 1 2 E 2 1 E 2 1 2 2 2 Eb 2 u L2.R/ C b C b u L2./ .1 Eb C / u L2.W / : k k Ä .E/2 C .E/2 k k C C .E/2 k k 4.6. A nonresonance condition. We now want to make the previous estimates explicit with respect to E and b so that we can use a similar optimization procedure as for the large modes case. We thus impose some condition on .E/. Namely, for any " 0, we introduce the set ˇ ˇ k22 l22 ˇ ᐆ ˚E ޒ .E/ c E "« E ޒ ˇ ˇE ˇ c E " for k; l ގ : " 0 ˇ ˇ 2 2 ˇ 0 WD 2 j D 2 ˇ L0 B0 ˇ 2

In other words, the set ᐆ" consists in energies that are far from the Dirichlet spectrum of the rectangle Œ B0; 0 Œ0; L0. It is natural to say that such energies are not resonating with the rectangle. The coefﬁcient c0 which is irrelevant when " > 0 has been chosen in such a way that Weyl’s law for the rectangle implies that ᐆ0 is not empty. Note however that, although expected, it is not clear that there actually are eigenvalues in ᐆ0, nor for that matter in ᐆ". Once " is ﬁxed, the estimate of the Corollary 4.15 becomes

2 2 1 3 2" 2 1 2 2" 2 2 2 1 1 2" 2 u L2.R/ C .b C E C b E C / u L2./ .1 Eb C / b E C u L2.W / : k k Ä C k k C C k k (4-25)

4.6.1. Optimizing b. As before we let b ME ˛ for some positive ˛ and try to optimize the bound. D Proposition 4.16. Deﬁne ˛ by Â 3 2" 2 2" Ã ˛ max C ; C : D 2 1 2 1 C 846 LUC HILLAIRET AND JEREMYL.MARZUOLA

There exists E0 and C such that for any u eigenfunction with energy E in ᐆ" such that E > E0, the following holds: 2 1 2 1 2" ˛ 2 u L2.R/ 4 u L2./ C E C C u L2.W /: (4-26) k k Ä k k C k k Proof. With the given choice of ˛ it is possible to choose M so that the prefactor of u 2 is 1 for E k kL2./ 4 large enough. The claim follows remarking that the deﬁnition of ˛ implies 3 1 ˛ > ; 2 1 2 C C so that the prefactor .1 Eb 2/2 is uniformly bounded above. C C 5. Nonconcentration estimate

We now put all the estimates together to obtain the following theorem. Theorem 2. Fix ", and deﬁne by Â2 2. 1/" 1 2 4 "Ã max C C C ; C C : WD 2 1 4 2 C There exists E0 and C such that any eigenfunction u of with energy E in ᐆ" such that E > E0 satisﬁes:

u 2 C E u 2 : k kL ./ Ä k kL .W / Proof. We ﬁrst remark that whatever the exponent ˛ is we always have 1 2" ˛ 1 > 1 so that C C 2 1 the exponent for the small modes is always larger than the exponent for the large modes. Thus, adding the estimates from propositions 4.9 and 4.16, we obtain

u 2 1 u 2 CE1 2" ˛ u 2 : k kL2.R/ Ä 2 k kL2./ C C C k kL2.W / Since u 2 u 2 u 2 we get k kL2.R/ D k kL2./ k kL2.W / 1 u 2 .1 CE1 2" ˛/ u 2 2 k kL2./ Ä C C C k kL2.W / When E is large the constant 1 can be absorbed in the term with a power of E. The claim follows by computing 1 2" ˛ for both possible choices of ˛ and taking square roots. C C We state as a corollary the corresponding statement for the Bunimovich billiard (see Theorem 1).

Corollary 5.1. In the Bunimovich stadium, for any " 0 there exists E0 and C such that if u is an eigenfunction of energy E in ᐆ" such that E > E0 then the following estimate holds:

5 8" C u 2 CE 6 u 2 : k kL ./ Ä k kL .W / 4 6" 5 8"Á 4 6" 5 8" Proof. We let 2, so ˛ max C ; C . Since C C for any nonnegative ", the D D 5 6 5 Ä 6 proof is complete. NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 847

Remark 5.1. The bounds in [Burq et al. 2007] gives a similar control with 1 as the exponent of E. Our 1 bound thus gives a better estimate as long as " < 8 . As it has been recalled in the introduction, it is quite natural that the nonresonance condition allows to get better bounds. Remark 5.2. We could deal with quasimodes by adding an error term to ƒ that is controlled by some negative power of E. There will be mainly two differences in the analysis. First the second term ƒ will not have support away from the rectangle anymore and second, in the optimization process, we will have to take care of the new error term (which will possibly change the range of applicable exponents). Remark 5.3. By adding the estimates in propositions 4.7 and 4.14, we get a different control estimate, where the control still is in the wings but now with a loss in derivatives. We haven’t tried to optimize this bound.

Appendix: One-dimensional control estimates

The aim of this appendix is to provide a control estimate for the equation

u z u h 00 D on Œ B0; b of the form

2 1 2 u L . B0;0/ C1 h H . B0;b/ C2 u L .0;b/; k k Ä k k C k k in which we want an explicit dependence of the constants C1 and C2 on z and b. It is now standard (see [Burq and Zworski 2005]) that if b is ﬁxed then we can choose C1 and C2 to be independent of z but what we need is an estimate when b goes to 0. We ﬁrst need a few preparatory lemmas. Lemma A.2. For any " > 0, there exists a constant C C."/ such that for any b, for any h 1 2 2 WD 1 2 H . B0; b/ and any z such that z .1 "/ =b , there exists a solution vp H .0; b/ to Ä 2 0 v zvp h; p00 D in Ᏸ0.0; b/ and

2 1 vp L .0;b/ Cb h H . B0;b/: (A-1) k k Ä k k 1 Proof. First we note that h, when restricted to .0; b/ also belongs to H .0; b/ and that h H 1.0;b/ k k Ä h H 1. B ;b/. The proof follows from a standard resolvent estimate since, on .0; b/, the bottom of the k k 0 spectrum of the self-adjoint operator v v with Dirichlet boundary condition is 2=b2. We include 7! 00 it for the convenience of the reader. We decompose vp in Fourier series: X k Á vp.x/ a sin x : D k b k 1 We have XÂk22 Ã k Á h.x/ z a sin x D b2 k b I k 1 848 LUC HILLAIRET AND JEREMYL.MARZUOLA hence 2 2 2 2 2 X k =b z 2 h H 1.0;b/ ak k k D k22=b2 j j k 1 or zb2 Á2 1 ! 2 2 2 k22 2 2 2 2 h H 1.0;b/ b inf vp H 1.0;b/ c b vp L2.0;b/: k k k 1 2k2 k k k k The claim follows since the inf is bounded away from zero in the regime we are considering. 2 2 1 Lemma A.3. Given z .1 "/ =b , let w H . B0; b/ be a solution to Ä 2 0 w zw 0 00 D in Ᏸ .. B0; b/ 0 /. Then there exists a constant A such that w AG, in which the function G is 0 n f g D defined by 8sin.pz.x B0// sin.pzb/ ˆ C if x < 0; <ˆ pz pz G.x/ (A-2) D ˆsin.pz.b x// sin.pzB0/ :ˆ if x > 0: pz pz Proof. Let w be such a function then necessarily there exist two constants A such that ˙ 8 sin.pz.x B0// Â C if x < 0; < pz w.x/ D ˆ sin.pz.b x// :Â if x > 0: C pz By assumption w H 1 and hence is continuous at 0, so 2 sin.pzB / sin.pzb/ A 0 A : pz D C pz In the regime we are considering sin.pzb/=pz 0, hence we can divide by this expression and express ¤ A in terms of A . The claim follows. C We finish these preparatory lemmas by establishing the control estimate for multiples of G.

Lemma A.4. (1) For ˇ such that 0 < ˇ =B0, there exists B1 B1.ˇ/ and C C.ˇ/ such that, 2 Ä D WD for any z ˇ and any b < B1, the following estimate holds: Ä 1 2 2 2 G L . B0;0/ Cb G L .0;b/: (A-3) k k Ä k k (2) For any ˇ; " > 0 there exists B1 B1.ˇ/ and C C.ˇ; "/ such that, for any b B1 and WD WD Ä ˇ2 z .1 "/2=b2, the following estimate holds: Ä Ä 1 b 2 2 2 G L . B0;0/ C G L .0;b/: (A-4) k k Ä sin.pzB0/k k NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 849

2 2 Proof. (1) We ﬁrst assume that z < Z for some positive Z0. We set z ! and compute 0 D Z 0 sinh2.!b/ Z 0 G.x/ 2 dx sinh2.!.B x// dx; 2 0 B0 j j D ! B0 C Z b sinh2.!B / Z b G.x/ 2 dx 0 sinh2.!.b x// dx: 2 0 j j D ! 0 By a straightforward change of variables we get

Z 0 sinh2.!b/ Z !B0 G.x/ 2 dx sinh2./ d; 3 B0 j j D ! 0 Z b sinh2.!B / Z b G.x/ 2 dx 0 sinh2./ d: 3 0 j j D ! 0

R X sinh2./ d We set F.X / 0 , so that we ﬁnally obtain WD sinh2.X /

Z 0 F.!B / Z b G.x/ 2 dx 0 G.x/ 2 dx: B0 j j D F.!b/ 0 j j 1 It is straightforward that F.X / is positive, tends to 1 at inﬁnity and that F.X /=X tends to 3 at 0. As a 2 consequence, there exists some C.Z0/ such that, for any z < Z , 0 Z 0 Z b 2 1 2 G.x/ dx C max.1; .!b/ / G.x/ dx; B0 j j Ä 0 j j 1 1 1 1 For b < B1 and ! > Z0, we have max.1; .!b/ / max.1;! /b Cb which gives the claim Ä Ä for this range of parameters. We now assume that we have Z2 < z < ˇ2. We have 0 Z 0 ˇ ˇ2 Z 0 ˇ ˇ2 2 ˇsin.pzb/ˇ ˇsin.pz.x B0//ˇ G.x/ dx ˇ ˇ ˇ C ˇ dx B0 j j D ˇ pz ˇ B0 ˇ pz ˇ ˇ ˇ2 Z 0 ˇ ˇ2 2 ˇsin.pzb/ˇ ˇsin.pz.x B0//ˇ 2 2 b ˇ ˇ ˇ C ˇ .x B0/ dx Cb ; D ˇ pzb ˇ B0 ˇ pz.x B0/ ˇ C Ä C where the constant C comes from the fact that the function sin.w/=w is continuous and its argument belongs to a ﬁxed compact set. On the other hand, by a simple change or variables we have

Z b ˇ ˇ2 Z b ˇ ˇ2 3 2 ˇsin.pzB0/ˇ ˇsin.pzx/ˇ 2 b G.x/ dx ˇ ˇ ˇ ˇ dx cB0 ; 0 j j D ˇ pz ˇ 0 ˇ pz ˇ 3 in which c is given by ˇ ˇ2 ˇ ˇ2 ˇsin.pzB0/ˇ ˇsin.pzx/ˇ c ˇ ˇ inf ˇ ˇ : D ˇ pzB0 ˇ 0 x B1 ˇ pzx ˇ Ä Ä 850 LUC HILLAIRET AND JEREMYL.MARZUOLA

Using that sin.w/=w is continuous and does not vanish on . ; / and choosing B1 accordingly we 1 obtain the first bound. (2) We first use homogeneity and prove the bound for G zG. We have Q WD 0 0 Z 2 2 Z 2 ˇG.x/ˇ dx ˇsin.pzb/ˇ ˇsin.pz.x B //ˇ dx B sin.X / 2 ; ˇ Q ˇ ˇ ˇ ˇ 0 ˇ 0 B0 D B0 C Ä j j in which we have set X pzb. On the other hand we have WD Z b Z b Z X ˇ ˇ2 ˇ ˇ2 ˇ ˇ2 ˇ ˇ2 1 2 ˇG.x/ˇ dx ˇsin.pzB0/ˇ ˇsin.pzx/ˇ dx b ˇsin.pzB0/ˇ sin./ d; 0 Q D 0 D X 0 j j with the same X . Under the assumptions, X belongs to a compact subinterval of Œ0; /. Since on this interval the function 1 Z X X sin./ 2d 2 7! X sin.X / 0 j j j j is continuous, the claim follows. 1 Proposition A.5. There exist ˇ and B1 B1.ˇ/, such that if b B1 and v H . B0; b/ satisfies WD Ä 2 0 v zv h; 00 D with h that vanishes on . B0; 0/, then the following estimates hold: (1) If z ˇ2, then Ä 1 1 2 2 1 2 2 v L . B0;0/ C1 b h H . B0;b/ b v L .0;b/ ; (A-5) k k Ä k k C k k 1 (2) If ˇ2 z , Ä Ä b2 Â 1 1 Ã b 2 b 2 2 1 2 v L . B0;0/ C1 h H . B0;b/ v L .0;b/ ; (A-6) k k Ä sin.B0pz/ k k C sin.B0pz/ k k 1 j j j j (3) If z then b2 Ä 1 1 2 2 1 2 2 v L . B0;0/ C3 b h H . B0;b/ b v L .0;b/ : (A-7) k k Ä k k C k k 2 Proof. 1 In the first two cases, we have z b2 < b2 . We may thus consider vp as given by Lemma A.2 and Ä 1 define vp by extending vp by 0 for negative x. Observe that w v vp is in H . B0; b/ and satisfies Q WD Q 0 w zw 0 00 D in Ᏸ.. B0; b/ 0 / so that v vp AG for some A according to Lemma A.3. Using Lemma A.4 we n f g Q D obtain in the first case 1 2 2 2 v vp L . B0;0/ Cb v vp L .0;b/: k Q k Ä k Q k We use the triangle inequality on the right-hand side and the fact that vp is 0 for negative x and coincide Q with vp for positive v. We obtain 1 2 2 2 2 v L . b0;0/ Cb v L .0;b/ vp L .0;b/ : k k Ä k k C k k NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 851

The claim then follows from the estimate on vp 2 in Lemma A.2. We prove the second case by k kL .0;b/ following the same argument, inserting the corresponding bound for G. The third case will follow the same lines but we will introduce a different particular solution vp, following then even more closely the proof of [Burq and Zworski 2005]. We set pz. 2 D Denote by H the unique L function on . B0; b/ that vanishes on . B0; 0/ and such that H 0 h in 2 1 D the distributional sense. The L norm of H is related to the H norm of h by the relation ÂZ b Ã 2 1 H H.y/ dy L . B0;b/ h H . B0;b/: k 0 k D k k The Cauchy–Schwarz inequality then implies that

1 1 2 2 1 H L . B0;b/ .1 b / h H . B0;b/: (A-8) k k C k k Set Z x sin..x y// vp.x/ H 0.y/ dy: D B0 Then vp satisﬁes 2 v vp H p00 D 0 in Ᏸ . B0; b/ and vp. B0/ 0 but the boundary condition need not be satisﬁed at b. We thus have 0 D sin..x B0// v.x/ vp.x/ vp.b/ C : D sin..B0 b// C The function v vp is thus a multiple of sin..x B0//. C We have Z 0 ˇ ˇ2 ˇsin..x B0//ˇ dx B0 B0 C Ä and Z b 2 1 1 Á sin..x B0// dx b : 0 j C j 2 2 Hence, in the regime under consideration we have

1 2 2 2 v vp L . B0;b/ Cb v vp L .0;b/: (A-9) k k Ä k k We perform an integration by parts in vp and observe that the boundary contributions vanish because H vanishes near B0 and sin..y x// vanishes at y x. D Finally, we obtain Z x vp.x/ cos..x y//H.y/ dy: D B0 It follows that vp is identically 0 on . B0; 0/ and that, on .0; b/, it satisﬁes

2 p vp.x/ H L . B0;b/ x: j j Ä k k 852 LUC HILLAIRET AND JEREMYL.MARZUOLA

Squaring and integrating, we get

2 2 vp L .0;b/ b H L . B0;b/: k k Ä k k Using the triangle inequality in (A-9) and inserting this bound, the result follows for b 1 using (A-8). Ä 2 In the paper, we will need to relax the condition that v.b/ 0. This can be done using a standard D construction related to a commutator method. We will get the following

Theorem 3. There exist ˇ and four constants B1; C1; C2; C3 depending only on ˇ such that the following 1 holds. For any b B1, for any function u in H . B0; b/ that satisfies Ä u zu h; 00 D 1 with h H . B0; b/ and such that u. B0/ 0 and h vanishes on . B0; 0/. Then, the following 2 D estimates hold: (1) If z ˇ2, then Ä 1 1 2 2 1 2 2 u L . B0;0/ C1 b h H . B0;b/ b u L .0;b/ : (A-10) k k Ä k k C k k 1 (2) If ˇ2 z , then Ä Ä b2 Â 1 1 Ã b 2 b 2 2 1 2 u L . B0;0/ C1 h H . B0;b/ u L .0;b/ : (A-11) k k Ä sin.B0pz/ k k C sin.B0pz/ k k 1 j j j j (3) If z, then b2 Ä 1 1 2 2 1 2 2 u L . B0;0/ C3 b h H . B0;b/ b u L .0;b/ : (A-12) k k Ä k k C k k 1 Proof. Define a smooth cutoff function 1 such that 1.x/ is identically 1 if x 2 and identically 0 if 1 Ä x 1 and let be the function x 1.x=b/. Define v u then v H . B0; b/ and satisfies b 7! WD b 2 0 v zv h 2. u/ u: 00 D C b0 0 b00

The right-hand side vanishes on . B0; 0/ so that, in order to use Proposition A.5, we have to estimate its 1 H norm. The strategy is the same as in the proofs of Lemmas 4.3 and 4.10. An integration by parts followed by the use of the Cauchy–Schwarz inequality gives

ˇZ B1 ˇ ˇ ˇ C ˇ .b0 u/0ˇ 0u L2.0;b/ 0 L2 u L2.0;b/ 0 L2 : ˇ B0 ˇ Ä k k k k Ä b k k k k Thus, C .0 u/0 H 1 u L2.0;b/: k b k Ä b k k The third term can be estimated using the same method. Indeed, ˇZ ˇ ˇZ b ÂZ x Ã ˇ Z x ˇ ˇ ˇ ˇ ˇ b00uˇ ˇ b00.y/u.y/ dy 0.x/ dxˇ b00.y/u.y/ dy L2.0;b/ 0 L2 : ˇ ˇ D ˇ 0 0 ˇ Ä k 0 k k k NONCONCENTRATIONINPARTIALLYRECTANGULARBILLIARDS 853

Using again Cauchy–Schwarz inequality and the fact that .y/ Cb 2 we get j b00 j Ä ˇZ x ˇ ˇ ˇ 2 ˇ b00.y/u.y/ dyˇ Cb u L2.0;b/px: ˇ 0 ˇ Ä k k We obtain Z x 2 1 b00.y/u.y/ dy Cb u L2.0;b/ px L2.0;b/ Cb u L2.0;b/: 0 L2.0;b/ Ä k k k k Ä k k It follows that

1 1 1 2 h 2.b0 u/0 b00u H . B0;b/ h H . B0;b/ Cb u L .0;b/: k C k Ä k k C k k We obtain the theorem by plugging this bound into the estimates of the Proposition A.5.

Acknowledgement

The authors wish to thank Andrew Hassell and Alex Barnett for very helpful conversations during the preparation of this article.

References

[Anantharaman 2008] N. Anantharaman, “Entropy and the localization of eigenfunctions”, Ann. of Math. .2/ 168:2 (2008), 435–475. MR 2011g:35076 Zbl 1175.35036 [Anantharaman and Nonnenmacher 2007] N. Anantharaman and S. Nonnenmacher, “Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold”, Ann. Inst. Fourier .Grenoble/ 57:7 (2007), 2465–2523. MR 2009m:81076 Zbl 1145.81033 [Bäcker et al. 1997] A. Bäcker, R. Schubert, and P. Stifter, “On the number of bouncing ball modes in billiards”, J. Phys. A 30:19 (1997), 6783–6795. MR 1481345 Zbl 0925.81030 [Burq and Zworski 2005] N. Burq and M. Zworski, “Bouncing ball modes and quantum chaos”, SIAM Rev. 47:1 (2005), 43–49. MR 2006d:81111 Zbl 1072.81022 [Burq et al. 2007] N. Burq, A. Hassell, and J. Wunsch, “Spreading of quasimodes in the Bunimovich stadium”, Proc. Amer. Math. Soc. 135:4 (2007), 1029–1037. MR 2007h:35240 Zbl 1111.35013 [Colin de Verdière 1985] Y. Colin de Verdière, “Ergodicité et fonctions propres du Laplacien”, Comm. Math. Phys. 102:3 (1985), 497–502. MR 87d:58145 Zbl 0592.58050 [Faure et al. 2003] F. Faure, S. Nonnenmacher, and S. De Bièvre, “Scarred eigenstates for quantum cat maps of minimal periods”, Comm. Math. Phys. 239:3 (2003), 449–492. MR 2005a:81076 Zbl 1033.81024 [Gérard and Leichtnam 1993] P. Gérard and É. Leichtnam, “Ergodic properties of eigenfunctions for the Dirichlet problem”, Duke Math. J. 71:2 (1993), 559–607. MR 94i:35146 Zbl 0788.35103 [Hassell 2010] A. Hassell, “Ergodic billiards that are not quantum unique ergodic”, Ann. of Math. .2/ 171:1 (2010), 605–619. MR 2011k:58050 Zbl 1196.58014 [Hillairet and Judge 2009] L. Hillairet and C. Judge, “Generic spectral simplicity of polygons”, Proc. Amer. Math. Soc. 137:6 (2009), 2139–2145. MR 2009k:58065 Zbl 1169.58007 [Lindenstrauss 2006] E. Lindenstrauss, “Invariant measures and arithmetic quantum unique ergodicity”, Ann. of Math. .2/ 163:1 (2006), 165–219. MR 2007b:11072 Zbl 1104.22015 [Schnirelman 1974] A. I. Schnirelman, “Ergodic properties of eigenfunctions”, Uspehi Mat. Nauk 29:6(180) (1974), 181–182. In Russian. MR 53 #6648 Zbl 0324.58020 854 LUC HILLAIRET AND JEREMYL.MARZUOLA

[Zelditch 1987] S. Zelditch, “Uniform distribution of eigenfunctions on compact hyperbolic surfaces”, Duke Math. J. 55:4 (1987), 919–941. MR 89d:58129 Zbl 0643.58029 [Zelditch and Zworski 1996] S. Zelditch and M. Zworski, “Ergodicity of eigenfunctions for ergodic billiards”, Comm. Math. Phys. 175:3 (1996), 673–682. MR 97a:58193 Zbl 0840.58048

Received 30 Nov 2010. Revised 14 Jul 2011. Accepted 24 Oct 2011.

LUC HILLAIRET: [email protected] Laboratoire de Mathématiques Jean Leray, UMR 6629 CNRS-Université de Nantes, 2 rue de la Houssinière, BP 92208, F-44 322 Nantes cedex 3, France

JEREMY L.MARZUOLA: [email protected] Mathematics Department, University of North Carolina, Chapel Hill, Phillips Hall, Chapel Hill, NC 27599, United States

mathematical sciences publishers msp ANALYSIS AND PDE Vol. 5, No. 4, 2012 dx.doi.org/10.2140/apde.2012.5.855 msp

GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE DEFOCUSING QUINTIC NLS IN THREE DIMENSIONS

ROWAN KILLIPAND MONICA VIS, AN

We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Collian- der, Keel, Stafﬁlani, Takaoka, and Tao.

1. Introduction

The defocusing quintic nonlinear Schrödinger equation,

4 iut + 1u = |u| u, (1-1)

3 describes the evolution of a complex-valued function u(t, x) of spacetime ޒt × ޒx . This evolution conserves energy: Z := 1 |∇ |2 + 1 | |6 E(u(t)) 2 u(t, x) 6 u(t, x) dx. (1-2) ޒ3 ˙ 1 3 By Sobolev embedding, u(0) has ﬁnite energy if and only if u(0) ∈ Hx (ޒ ), which is the space of initial data that we consider. This is also a scale-invariant space; both the class of solutions to (1-1) and the energy are invariant under the scaling symmetry

u(t, x) 7→ uλ(t, x) := λ1/2u(λ2t, λx). (1-3)

For this reason, the equation is termed energy-critical. A function u : I × ޒ3 → ރ on a nonempty time interval I 3 0 is called a strong solution to (1-1) if it 0 ˙ 1 3 10 3 lies in the class Ct Hx (K × ޒ )∩ Lt,x (K × ޒ ) for all compact K ⊂ I , and obeys the Duhamel formula Z t u(t) = eit1u(0) − i ei(t−s)1|u(s)|4u(s) ds, (1-4) 0 for all t ∈ I . We say that u is a maximal-lifespan solution if the solution cannot be extended (in this class) to any strictly larger interval. Our main result is a new proof of the following:

The ﬁrst author was partially supported by NSF grant DMS-1001531. The second author was partially supported by the Sloan Foundation and NSF grant DMS-0901166. This work was completed while the second author was a Harrington Faculty Fellow at the University of Texas at Austin. MSC2010: 35Q55. Keywords: energy critical, nonlinear Schrödinger.

855 856 ROWAN KILLIP AND MONICAVIS, AN

˙ 1 3 Theorem 1.1 (global well-posedness and scattering). Let u0 ∈ Hx (ޒ ). Then there exists a unique 0 ˙ 1 3 global strong solution u ∈ Ct Hx (ޒ × ޒ ) to (1-1) with initial data u(0) = u0. Moreover, this solution satisfies Z Z 10 |u(t, x)| dx dt ≤ C ku k ˙ 1 . (1-5) 0 Hx ޒ ޒ3 ˙ 1 Further, scattering occurs:(i) there exist asymptotic states u± ∈ Hx such that it1 u(t) − e u± ˙ 1 → 0 as t → ±∞ (1-6) Hx ˙ 1 ˙ 1 and (ii) for any u+ ∈ Hx (or u− ∈ Hx ) there exists a unique global solution u to (1-1) such that (1-6) holds. Theorem 1.1 was proved by Colliander, Keel, Staffilani, Takaoka, and Tao in the ground-breaking paper [Colliander et al. 2008]. The key point is to prove the spacetime bound (1-5); scattering is an easy consequence of this. Note also that the solution described in Theorem 1.1 is in fact unique in the larger 0 ˙ 1 class of Ct Hx functions obeying (1-4); this unconditional uniqueness statement is proved in [Colliander et al. 2008, §16] by adapting earlier work. The paper [Colliander et al. 2008] advanced the induction on energy technique, introduced by Bourgain in [1999], and presaged many recent developments in dispersive PDE at critical regularity. The argument may be outlined as follows: (i) If a bound of the form (1-5) does not hold, then there must be a minimal almost-counterexample, that is, a minimal-energy solution with (prespecified) enormous spacetime norm. (ii) By virtue of its minimality, such a solution must have good tightness and equi-continuity properties. (iii) To be consistent with the interaction Morawetz identity such a solution must undergo a dramatic change of (spatial) scale in a short span of time. (iv) Such a rapid change is inconsistent with simultaneous conservation of mass and energy. As just described, the argument appears to be by contradiction, but this is not the case. In fact, it is entirely quantitative, showing that in order to achieve such a large spacetime norm, the solution must have at least a certain amount of energy. The energy requirement diverges as the spacetime norm diverges and so yields an effective bound for the function C appearing in (1-5). This style of argument adapts also to other equations and dimensions; see, for example, [Nakanishi 1999; Ryckman and Vis,an 2007; Tao 2005; Vis,an 2007]. The downside to the induction on energy argument is its complexity. It is monolithic, as opposed to modular; the value of a small parameter introduced at the very beginning of the proof is not determined until the very end. In recent years, the induction on energy argument has been supplanted by a related contradiction argument that is completely modular and is much easier to understand; it is not quantitative. The genesis of this new method comes from the discovery of Keraani [2006] that the estimates underlying the proof that minimal almost-counterexamples have good tightness/equicontinuity properties can be pushed further to show that failure of Theorem 1.1 guarantees the existence of a minimal counterexample. This insight was first applied to the well-posedness problem in an important paper of Kenig and Merle [2006], which considered the focusing equation with radial data in dimensions three, four, and five. Subsequent papers (by a wide array of authors) have greatly refined and expanded this methodology. GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 857

In this paper, we revisit the proof of Theorem 1.1 using this “minimal criminal” approach, which, we believe, results in significant expository simplification. We will also endeavor to convey that much of the original argument lives on, both in spirit and in the technical details, by explicit reference to [Colliander et al. 2008] as well as by maintaining their notations, as much as possible. In some very striking recent work [Dodson 2012; 2011b; 2011a], Dodson has proved the analogue of Theorem 1.1 for the mass-critical nonlinear Schrödinger equation in arbitrary dimension. The most significant difference between [Colliander et al. 2008] and the argument presented here comes from the adaptation of some of his ideas (present already in the first paper [Dodson 2012]) to the problem (1-1). We postpone a fuller discussion of these matters until we have described some of the key steps in the proof.

Outline of the proof. We argue by contradiction. Simple contraction mapping arguments show that Theorem 1.1 holds for solutions with small energy; thus, if the theorem were not to hold there must be a transition energy above which the energy no longer controls the spacetime norm. The first step in the argument is to show that there is a minimal counterexample and that, by virtue of its minimality, this counterexample has good compactness properties. ∞ ˙ 1 3 Definition 1.2 (almost periodicity). A solution u ∈ Lt Hx (I × ޒ ) to (1-1) is said to be almost periodic (modulo symmetries) if there exist functions N : I → ޒ+, x : I → ޒ3, and C : ޒ+ → ޒ+ such that for all t ∈ I and η > 0, Z Z 2 2 2 ∇u(t, x) dx + |ξ| |u ˆ(t, ξ)| dξ ≤ η. (1-7) |x−x(t)|≥C(η)/N(t) |ξ|≥C(η)N(t) We refer to the function N(t) as the frequency scale function for the solution u, to x(t) as the spatial center function, and to C(η) as the modulus of compactness. ˙ 1 Remark 1.3. Together with boundedness in Hx , the tightness plus equicontinuity statement (1-7) illustrates that almost periodicity is equivalent to the (co)compactness of the orbit modulo translation and dilation symmetries. In particular, from compactness we see that for each η > 0 there exists c(η) > 0 so that for all t ∈ I , Z Z 2 2 2 ∇u(t, x) dx + |ξ| |u ˆ(t, ξ)| dξ ≤ η. |x−x(t)|≤c(η)/N(t) |ξ|≤c(η)N(t) Similarly, compactness implies Z Z 2 6 |∇u(t, x)| dx .u |u(t, x)| dx ޒ3 ޒ3 uniformly for t ∈ I . This last observation plays the role of Proposition 4.8 in [Colliander et al. 2008]. With these preliminaries out of the way, we can now describe the first major milestone in the proof of Theorem 1.1: Theorem 1.4 (reduction to almost periodic solutions [Kenig and Merle 2006; Killip and Vis,an 2010]). Suppose Theorem 1.1 fails. Then there exists a maximal-lifespan solution u : I × ޒ3 → ރ to (1-1) which 858 ROWAN KILLIP AND MONICAVIS, AN is almost periodic and blows up both forward and backward in time in the sense that for all t0 ∈ I ,

Z sup I Z Z t0 Z |u(t, x)|10 dx dt = |u(t, x)|10 dx dt = ∞. 3 3 t0 ޒ inf I ޒ The theorem does not explicitly claim that u is a minimal counterexample; nevertheless, this is how it is constructed and, more importantly, how it is shown to be almost periodic. In [Colliander et al. 2008], the role of this theorem is played by Corollary 4.4 (equicontinuity) and Proposition 4.6 (tightness). A précis of the proof of Theorem 1.4 can be found in [Kenig and Merle 2006], building on Keraani’s method [2006]; for complete details see [Killip and Vis,an 2010] or [Killip and Vis,an 2008]. Just as for the results from [Colliander et al. 2008] mentioned above, the key ingredients in the proof are improved Strichartz inequalities, which show that concentration occurs, and perturbation theory, which shows that multiple simultaneous concentrations are inconsistent with minimality. Continuity of the flow prevents rapid changes in the modulation parameters x(t) and N(t). In particular, from [Killip et al. 2009, Corollary 3.6] or [Killip and Vis,an 2008, Lemma 5.18] we have Lemma 1.5 (local constancy property). Let u : I × ޒ3 → ރ be a maximal-lifespan almost periodic solution to (1-1). Then there exists a small number δ, depending only on u, such that if t0 ∈ I then −2 −2 t0 − δN(t0) , t0 + δN(t0) ⊂ I and −2 N(t) ∼u N(t0) whenever |t − t0| ≤ δN(t0) . We recall next a consequence of the local constancy property; see [Killip et al. 2009, Corollary 3.7; Killip and Vis,an 2008, Corollary 5.19]. Corollary 1.6 (N(t) at blowup). Let u : I × ޒ3 → ރ be a maximal-lifespan almost periodic solution to −1/2 (1-1). If T is any finite endpoint of I , then N(t) &u |T − t| ; in particular, limt→T N(t) = ∞. Finally, we will need the following result linking the frequency scale function N(t) of an almost periodic solution u and its Strichartz norms: Lemma 1.7 (spacetime bounds). Let u be an almost periodic solution to (1-1) on a time interval I . Then Z Z 2 q 2 N(t) dt k∇uk q 1 + N(t) dt (1-8) .u L Lr (I ×ޒ3) .u I t x I 2 + 3 = 3 ≤ ∞ for all q r 2 with 2 q < . Proof. We recall that Lemma 5.21 in [Killip and Vis,an 2008] shows that Z Z Z Z 2 10 2 N(t) dt .u |u(t, x)| dx dt .u 1 + N(t) dt. (1-9) I I ޒ3 I The second inequality in (1-8) follows from the second inequality above and an application of the Strichartz inequality. The first inequality follows by the same method used to prove the corresponding − /q 6≡ 2 k∇ k r result in (1-9): The fact that u 0 ensures that N(t) u(t) L x never vanishes. Almost periodicity then implies that it is bounded away from zero and the inequality follows. GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 859

Let u : I ×ޒ3 → ރ be an almost periodic maximal-lifespan solution to (1-1). As a direct consequence of the preceding three results, we can tile the interval I with inﬁnitely many characteristic intervals Jk, which have the following properties:

• N(t) ≡ Nk is constant on each Jk. −2 •| Jk| ∼u Nk , uniformly in k. 2 3 3 • k∇uk q r 3 ∼ 1, for each + = with 2 ≤ q ≤ ∞ and uniformly in k. Lt L x (Jk ×ޒ ) u q r 2 Note that the redeﬁnition of N(t) may necessitate a mild increase in the modulus of compactness. We may further assume that 0 marks a boundary between characteristic intervals, which we do, for expository reasons. Returning to Theorem 1.4, a simple rescaling argument (see, for example, the proof of Theorem 3.3 in [Tao et al. 2007]) allows us to additionally assume that N(t) ≥ 1 at least on half of the interval I , say, on [0, Tmax). Inspired by [Dodson 2012], we further subdivide into two cases dictated by the control given by the interaction Morawetz inequality. Putting everything together, we obtain

Theorem 1.8 (two special scenarios for blowup). Suppose Theorem 1.1 failed. Then there exists an 3 almost periodic solution u :[0, Tmax) × ޒ → ރ, such that

kuk 10 4 = +∞ Lt,x ([0,Tmax)×ޒ ) and [0, Tmax) = ∪k Jk where Jk are characteristic intervals on which N(t) ≡ Nk ≥ 1. Furthermore,

Z Tmax Z Tmax either N(t)−1 dt < ∞ or N(t)−1 dt = ∞. 0 0 Thus, in order to prove Theorem 1.1 we just need to preclude the existence of the two types of almost periodic solution described in Theorem 1.8. By analogy with the trichotomies appearing in [Killip et al. 2009; Killip and Vis,an 2010], we refer to the ﬁrst type of solution as a rapid low-to-high frequency cascade and the second as a quasisoliton. In each case, the key to showing that such solutions do not exist is a fundamentally nonlinear relation obeyed by the equation. In the cascade case, it is the conservation of mass; in the quasisoliton case, it is the interaction Morawetz identity (a monotonicity formula introduced in [Colliander et al. 2004]). Un- fortunately, both of these relations have energy-subcritical scaling and so are not immediately applicable ∞ ˙ 1 to Lt Hx solutions; additional control on the low frequencies is required. It is in how this control is achieved that we deviate most from [Colliander et al. 2008]. The argument in [Colliander et al. 2008] relies heavily on the interaction Morawetz identity. To cope with the noncritical scaling, a frequency localization is introduced. This produces error terms which are then controlled by means of a highly entangled bootstrap argument. Dodson’s paper [2012] also uses a frequency-localized interaction Morawetz identity; however, the error terms are handled via spacetime estimates that are proved independently of this identity. Indeed, the proof of these estimates does not even rely on the defocusing nature of the nonlinearity. 860 ROWAN KILLIP AND MONICAVIS, AN

In this paper, we adopt Dodson’s strategy (see also [Vis,an 2012]). The requisite estimates on the low- frequency part of the solution appear in Theorem 4.1. It seems to us that this theorem represents the limit of what can be achieved without the use of intrinsically nonlinear tools such as monotonicity formulae. The rationale for this assertion comes from consideration of the focusing equation and is discussed in Remark 4.3. Nevertheless, Theorem 4.1 does just suffice to treat the error terms in the frequency-localized interaction Morawetz identity (see Section 6), which is then used to preclude quasisolitons in Section 7. The proof of Theorem 4.1 relies on a type of Strichartz estimate that we have not seen previously. This estimate, Proposition 3.1, has the flavor of a maximal function in that it controls the worst Littlewood– Paley piece at each moment of time. The necessity of considering a supremum over frequency projections (as opposed to a sum) is borne out by an examination of the ground-state solution to the focusing equation; see Remark 4.3. The proof of this proposition is adapted from the double Duhamel trick first introduced in [Colliander et al. 2008, §14]. The original application of this trick also appears here, namely, as Proposition 3.2. The nonexistence of cascade solutions is proved in Section 5. The argument combines the following proposition and Theorem 4.1 to prove first that the mass is finite and then (to reach a contradiction) that it is zero. It is equally valid in the focusing case.

Proposition 1.9 (no-waste Duhamel formula, [Killip and Vis,an 2008; Tao et al. 2008]). Let u :[0, Tmax)× 3 ޒ → ރ be a solution as in Theorem 1.8. Then for all t ∈ [0, Tmax), Z T u(t) = i lim ei(t−s)1|u(s)|4u(s) ds, T → Tmax t ˙ 1 where the limit is to be understood in the weak Hx topology.

2. Notation and useful lemmas

We use the notation X . Y to indicate that there exists some constant C so that X ≤ CY . Similarly, we write X ∼ Y if X . Y . X. We use subscripts to indicate the dependence of C on additional parameters. For example, X .u Y denotes the assertion that X ≤ CuY for some Cu depending on u. We will make frequent use of the fractional differential/integral operators |∇|s together with the corresponding homogeneous Sobolev norms:

s s s ˆ k f k ˙ s := k|∇| f k 2 where |∇|\f (ξ) := |ξ| f (ξ). Hx L x We will also need some Littlewood–Paley theory. Speciﬁcally, let ϕ(ξ) be a smooth bump supported in the ball |ξ| ≤ 2 and equaling one on the ball |ξ| ≤ 1. For each dyadic number N ∈ 2ޚ we deﬁne the Littlewood–Paley operators ˆ ˆ ˆ P\≤N f (ξ) := ϕ(ξ/N) f (ξ), P\>N f (ξ) := (1−ϕ(ξ/N)) f (ξ), [PN f (ξ) := (ϕ(ξ/N)−ϕ(2ξ/N)) f (ξ).

Similarly, we can deﬁne P

The Littlewood–Paley operators commute with derivative operators, the free propagator, and complex p ˙ s conjugation. They are self-adjoint and bounded on every L x and Hx space for 1 ≤ p ≤ ∞ and s ≥ 0. They also obey the following Sobolev and Bernstein estimates:

±s ±s 3 − 3 k|∇| P f k p ∼ N kP f k p kP f k q N p q kP f k p N L x s N L x , N L x .s N L x , whenever s ≥ 0 and 1 ≤ p ≤ q ≤ ∞. We will frequently denote the nonlinearity in (1-1) by F(u), that is, F(u) := |u|4u. We will use the notation Ø(X) to denote a quantity that resembles X, that is, a ﬁnite linear combination of terms that look like those in X, but possibly with some factors replaced by their complex conjugates and/or restricted to various frequencies. For example,

5 X j 5− j 4 F(u + v) = Ø(u v ) and F(u) = F(u>N ) + Ø(u≤N u ) for any N > 0. j=0

q r We use Lt L x to denote the spacetime norm Z Z q/r 1/q r kuk q r := |u(t, x)| dx dt , Lt L x ޒ ޒ3 with the usual modiﬁcations when q or r is inﬁnity, or when the domain ޒ × ޒ3 is replaced by some q r q smaller spacetime region. When q = r we abbreviate Lt L x by Lt,x . Let eit1 be the free Schrödinger propagator. In physical space this is given by the formula

1 Z 2 eit1 f (x) = ei|x−y| /4t f (y)dy. 3/2 (4πit) ޒ3 In particular, the propagator obeys the dispersive inequality

it1 −3/2 ke f k ∞ 3 |t| k f k 1 3 (2-1) L x (ޒ ) . L x (ޒ ) for all times t 6= 0. As a consequence of this dispersive estimate, one obtains the Strichartz estimates; see, for example, [Ginibre and Velo 1992; Keel and Tao 1998; Strichartz 1977]. The particular version we need is from [Colliander et al. 2008]. Lemma 2.1 (Strichartz inequality). Let I be a compact time interval and let u : I ×ޒ3 → ރ be a solution to the forced Schrödinger equation

iut + 1u = G for some function G. Then we have

1/2 n X 2 o k∇ k k k + k∇ k 0 u N q r × 3 . u(t0) H˙ 1(ޒ3) G q˜ r˜0 3 (2-2) Lt L x (I ޒ ) x Lt L x (I ×ޒ ) N∈2ޚ ∈ ˜ ˜ 2 + 3 = 2 + 3 = 3 ≤ ˜ ≤ ∞ for any time t0 I and any exponents (q, r) and (q, r) obeying q r q˜ r˜ 2 and 2 q, q . Here, as usual, p0 denotes the dual exponent to p, that is, 1/p + 1/p0 = 1. 862 ROWAN KILLIP AND MONICAVIS, AN

Elementary Littlewood–Paley theory shows that (2-2) implies

k∇ k q k k + k∇ k 0 u L Lr (I ×ޒ3) . u(t0) H˙ 1(ޒ3) G q˜ r˜0 3 , t x x Lt L x (I ×ޒ ) which corresponds to the usual Strichartz inequality; however, the Besov variant given above allows us ∞ to “Sobolev embed” into L x : Lemma 2.2 (an endpoint estimate). For any u : I × ޒ3 → ޒ we have n o1/4 k k k∇ k1/2 X k∇ k2 u L4 L∞(I ×ޒ3) u ∞ 2 u N 2 6 3 . t x . Lt L x Lt L x (I ×ޒ ) N∈2ޚ In particular, for any frequency N > 0, n o1/4 k k k∇ k1/2 X k∇ k2 u≤N L4 L∞(I ×ޒ3) u≤N ∞ 2 u M 2 6 3 . t x . Lt L x Lt L x (I ×ޒ ) M≤N Proof. Using Bernstein’s inequality we have,

Z 4 4 n X o kuk ku (t)k ∞ dt L4 L∞(I ×ޒ3) . N L x t x I N∈2ޚ X ku N kL∞ L∞ ku N kL∞ L∞ ku N k 2 ∞ ku N k 2 ∞ . 1 t x 2 t x 3 Lt L x 4 Lt L x N1≤N2≤N3≤N4 1 h i 2 X N1 N2 k∇u N k ∞ 2 k∇u N k ∞ 2 k∇u N k 2 6 k∇u N k 2 6 . N3 N4 1 Lt L x 2 Lt L x 3 Lt L x 4 Lt L x N1≤···≤N4 1 X h i 2 k∇ k2 N3 k∇ k k∇ k u ∞ 2 u N3 2 6 u N4 2 6 . . Lt L x N4 Lt L x Lt L x N3≤N4

3 All spacetime norms above are over I × ޒ . The claim now follows from Schur’s test.

3. Maximal Strichartz estimates

Proposition 3.1. Let (i∂t + 1)v = F + G on a compact interval [0, T ]. Then for each 6 < q ≤ ∞,

3 −1 − 1 − 1 q q 2 2 M(t) kPM(t)v(t)kL 2 |∇| v ∞ 2 + |∇| G 2 6/5 + kFk 2 1 x Lt . Lt L x Lt L x Lt L x uniformly for all functions M :[0, T ] → 2ޚ. All spacetime norms are over [0, T ] × ޒ3. It is not difﬁcult to see that the conclusion is weaker than (and has the same scaling as) |∇|−1/2v ∈ 2 6 Lt L x . In fact, if F ≡ 0, this stronger result can be deduced immediately from the Strichartz inequality. 1 However, this argument does not extend to give a proof of the proposition because F ∈ L x does not imply −1/2 6/5 |∇| F ∈ L x . Indeed, the whole theory of the energy-critical NLS in three dimensions is dogged by the absence of endpoint estimates of this type. The freedom of choosing an arbitrary function M(t) makes this a maximal function estimate; at each time one can take the supremum over all choices of the parameter. Writing maximal functions in this GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 863 way yields linear operators and so one may use the method of TT ∗; this is an old idea dating at least to the work of Kolmogorov and Seliverstov in the 1920s (cf. [Zygmund 2002, Chapter XIII]). As we will see, the double Duhamel trick, which underlies the proof of Proposition 3.1, is a variant of the TT ∗ idea. Speciﬁcally, one takes the inner-product between two different representations of v(t). The double Duhamel trick was introduced in [Colliander et al. 2008, §14]. There it was used for a different purpose, namely, to obtain control over the mass on balls. This is then used to estimate error terms in the (localized) interaction Morawetz identity. We will also need this information and for exactly the same reasons; see (6-16). The following proposition captures the main thrust of [Colliander et al. 2008, §14]:

Proposition 3.2. Let (i∂t + 1)v = F + G on a compact interval [0, T ] and let Z 1/2 2 −3/2 2 −|y|2/R2 [᏿Rv](t, x) := (π R ) |v(t, x + y)| e dy . (3-1) ޒ3 Then for each 0 < R < ∞ and 6 < q ≤ ∞,

1 − 3 1 2 q − R ᏿Rv 2 q kvkL∞ L2 + kGk 2 6/5 + R 2 kFk 2 1 , (3-2) Lt L x . t x Lt L x Lt L x where all spacetime norms are over [0, T ] × ޒ3.

We use the letter ᏿ for the operator appearing in (3-1) to signify both “smudging” and “square function”. It is easy to see that the Gaussian smudging used here could be replaced by other methods without affecting the result; indeed, the analogous estimate in [Colliander et al. 2008] averages over balls. That paper also sets q = 100 and sums over a lattice rather than integrating in x. As ᏿v is slowly varying, summation and integration yield comparable norms. To control ᏿Rv we need to estimate some complicated oscillatory (and nonoscillatory) integrals. By choosing a Gaussian weight, some of the integrals can be done both quickly and exactly; see the proof of Lemma 3.4. Before turning to that subject, we ﬁrst show how the two propositions are interconnected. The proof of the next lemma also demonstrates how bounds on ᏿R can be used to deduce analogous results with other weights.

Lemma 3.3. Fix 6 < q ≤ ∞. Then

3 −1 3 −1 q q − sup M fM Lq . sup M ᏿M 1 fM Lq . (3-3) M>0 x M>0 x ˜ Proof. Let PM = PM/2 + PM + P2M denote the fattened Littlewood–Paley projector. The basic relation ˜ PM = PM PM reduces our goal to showing that

3 −1 3 −1 q ˜ q − sup M PM g Lq . sup M ᏿M 1 g Lq (3-4) M>0 x M>0 x

3 for general functions g : ޒ → ރ, say, g = fM . 864 ROWAN KILLIP AND MONICAVIS, AN

˜ 3 Recall that the convolution kernel for PM takes the form M ψ(Mx) for some Schwartz function ψ. By virtue of its rapid decay, we can write ∞ Z 2 2 |ψ(x)| ≤ π −3/2e−|x| /λ dµ(λ) 0 where µ is a positive measure with all moments ﬁnite. Indeed, since ψ is radial one can choose dµ(λ) = 20|ψ0(λ)| dλ. Thus by the Cauchy–Schwarz inequality, Z Z ∞ ˜ 2 2 3 2 3 [PM g](x) ≤ |g(x + y)| M |ψ(My)| dy ≤ [᏿λM−1 g](x) λ dµ(λ). ޒ3 0

q/2 3 Applying Minkowski’s inequality in L x (ޒ ) then easily yields (3-4); indeed, one can take the constant R 1+6/q 1/2 to be [ λ dµ(λ)] . Lemma 3.4. For ﬁxed 6 < q ≤ ∞, the integral kernel

2 −3/2 iτ1 −| · −x|2/R2 is1 K R(τ, z; s, y; x) := (π R ) hδz, e e e δyi obeys Z ∞Z ∞ 2− 6 2 q k ; ; k + − [ ] sup R K R(τ, z s, y x) L∞ Lq/2 f (t τ) f (t s) ds dτ . ᏹ f (t) , (3-5) R>0 0 0 z,y x where ᏹ denotes the Hardy–Littlewood maximal operator and f : ޒ → [0, ∞).

Proof. From the exact formula for the propagator,

Z exp{i|z − x0|2/4τ − |x0 − x|2/R2 + i|x0 − y|2/4s} K (t, z; s, y; x) = dx0. R 3/2 3/2 2 3/2 (3-6) ޒ3 (4πiτ) (4πis) (π R ) Completing the square and doing the Gaussian integral yields 2 + 2| − ∗|2 −3 2 2 4 2−3/4 n R (s τ) x x o |K R(t, z; s, y; x)| = (2π) 16s τ + R (s + τ) exp − 16s2τ 2 + R4(s + τ)2 where x∗ = (sz + ty)/(s + t). One more Gaussian integral then yields

−3 3/q −6/q −6/q 2 2 4 2 −3/4+3/q kK R(. . .)k q/2 = (2π) (2π/q) R |s + τ| [16s τ + R (s + τ) ] . L x Notice that there is no dependence on z or y. This is due to simultaneous translation and Galilean invariance. In this way, we deduce that Z ∞Z ∞ ∗ 2 2 LHS(3-5) . sup Kq (α, β) f (t + R α) f (t − R β) dα dβ, (3-7) R>0 0 0 where we have changed variables to α = R−2τ and β = R−2s and written

∗ −6/q 2 2 2−3/4+3/q Kq (α, β) := [α + β] α β + (α + β) . GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 865

∗ 1 To ﬁnish the proof, we just need to show that Kq can be majorized by a convex combination of (L - normalized) characteristic functions of rectangles of the form [0, `] × [0, w]. In fact, we can write it exactly as a positive linear combination of such rectangles: Z ∞Z ∞ Z ∞Z ∞ ρ(`, w) d` dw ∗ = 1 1 = Kq (α, β) ` χ[0,`](α) w χ[0,w](β) ρ(`, w) d` dw 0 0 α β `w

∗ 1 where ρ(`, w) := `w∂`∂w Kq (`, w) ≥ 0. Thus, we just need to check that ρ ∈ L . With a little patience, ∗ one ﬁnds that ρ(`, w) .q Kq (`, w), which leaves us to integrate the latter over a quadrant. We use polar coordinates, ` + iw = reiθ : Z ∞Z ∞ Z ∞Z π/2 ∗ −6/q 4 2 2 −3/4+3/q Kq (`, w) d` dw . r [r sin (2θ) + r ] r dθ dr 0 0 0 0 Z ∞ −1/2 −3/2+6/q . r (1 + r) dr . 1. 0 Notice that convergence of the r integral relies on q > 6. The estimate for the θ integral given above is only valid in the range 6 < q < 12. When q > 12, the correct form is R r −1/2(1 + r)−1 dr and when R −1/2 −1 q = 12, it is r log(2 + r)(1 + r) dr. Nevertheless, both of these integrals are also ﬁnite. We now have all the necessary ingredients to complete the proofs of Propositions 3.1 and 3.2. We only provide the details for the former because the two arguments are so similar. Indeed, the proof of the latter essentially follows by choosing M(t) ≡ R−1 and throwing away the Littlewood–Paley projector PM(t) in the argument we are about to present.

Proof of Proposition 3.1. In view of Lemma 3.3 we need to show that

3 −1 2 q − ∈ [ ] sup M ᏿M 1 PM v(t) Lq Lt ( 0, T ) M>0 x (with suitable bounds), where the supremum is taken pointwise in time. As noted earlier, we will use the double Duhamel trick, which relies on playing two Duhamel formulae off against one another, one from each endpoint of [0, T ]: Z t Z t v(t) = eit1v(0) − i ei(t−s)1G(s) ds − i ei(t−s)1 F(s) ds (3-8) 0 0 Z T Z T = e−i(T −t)1v(T ) + i e−i(τ−t)1G(τ) dτ + i e−i(τ−t)1 F(τ) dτ. (3-9) t t

2 The idea is to compute the L x norm of PM v(t) with respect to the Gaussian measure that deﬁnes [᏿M−1 PM v](t, x) by taking the inner product between these two representations. Actually, we deviate slightly from this idea because it is not clear how to estimate a pair of cross-terms. Our trick for avoiding this is the following simple fact about vectors in a Hilbert space:

v = a + b = c + d =⇒ kvk2 ≤ 3kak2 + 3kck2 + 2|hb, di|. (3-10) 866 ROWAN KILLIP AND MONICAVIS, AN

(The numbers are neither optimal nor important.) To prove this, write

kvk2 = ha, vi + hv, ci − ha, ci + hb, di and then use the Cauchy–Schwarz inequality. Let us invoke (3-10) with a and c representing (PM applied to) the ﬁrst two summands in (3-8) and (3-9), respectively, while b and d represent the summands which involve F. In this way, we obtain the pointwise statement

Z t 2 2 it1 i(t−s)1 [᏿M−1 (PM v)](t, x) . ᏿M−1 e vM (0) − i e G M (s) ds (x) 0 Z T 2 −i(T −t)1 −i(τ−t)1 + ᏿M−1 e vM (T ) + i e G M (τ) dτ (x) + h M (t, x), t where h M is an abbreviation for Z T Z t −3/2 3 −i(τ−t)1 −M2| · −x|2 i(t−s)1 h M (t, x) := π M e FM (τ) dτ, e e FM (s) ds t 0 The contributions of the ﬁrst two summands are easily estimated: For any function w, Young’s and Bernstein’s inequalities imply

3 −1 − 1 − 1 q 2 2 M [᏿M−1 (PM w)](t, x) q 3 M PM w(t) 6 3 |∇| w(t) 6 3 . L x (ޒ ) . L x (ޒ ) . L x (ޒ ) This can then be combined with Strichartz inequality, which shows Z t − 1 − − 1 − 1 2 it1 i(t s)1 2 2 |∇| e v(0) − i e G(s) ds |∇| v(0) 2 + |∇| G 2 6/5 . L L L x 2 6 x t 0 Lt L x and similarly for the second summand. The third summand, h M , is the crux of the matter. Using the notation from Lemma 3.4 and changing variables, we have Z T −tZ t ZZ ¯ 0 0 0 0 0 0 h M (t, x) = FM (t + τ , z)K M−1 (τ , z; s , y; x)FM (t − s , y) dy dz ds dτ . 0 0

Note also that by Bernstein’s inequality and the maximal inequality, the function f (t) := kF(t)k 1 obeys L x

kFM (t)k 1 f (t) and kᏹ f k 2 kFk 2 1 . L x . Lt . Lt L x Thus using Lemma 3.4 (with f as just deﬁned), we obtain

6 − q 2 2 sup M kh M (t)k q/2 kFk 2 1 . L x 1 . L L M>0 Lt t x

Recalling that h M appears in an upper bound on the square of the size of PM v, the proposition follows. GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 867

4. Long-time Strichartz estimates

The main result of this section is a long-time Strichartz estimate. As will be evident from the proof, the ∞ ˙ 1 3 result is also valid for Lt Hx (ޒ ) solutions to the focusing equation; see also Remark 4.3 at the end of this section.

3 Theorem 4.1 (long-time Strichartz estimate). Let u : (Tmin, Tmax)×ޒ → ރ be a maximal-lifespan almost periodic solution to (1-1) and I ⊂ (Tmin, Tmax) a time interval that is tiled by ﬁnitely many characteristic intervals Jk. Then for any ﬁxed 6 < q < ∞ and any frequency N > 0,

1/2 n X 2 o A(N) := k∇u M k 2 6 3 (4-1) Lt L x (I ×ޒ ) M≤N and 3 − ˜ 3/2 q 1 A (N) := N sup M ku (t)k q 3 (4-2) q M L x (ޒ ) L2(I ) M≥N t obey ˜ 3/2 1/2 A(N) + Aq (N) .u 1 + N K , (4-3) := R −1 where K I N(t) dt. The implicit constant is independent of the interval I . The proof of this theorem will occupy the remainder of this section. Throughout, we consider a single ˜ interval I and so the implicit dependence of A(N), Aq (N), and K on the interval should not cause confusion. Additionally, all spacetime norms will be on I × ޒ3, unless speciﬁed otherwise. ˜ By Bernstein’s inequality, Aq (N) is monotone in q. Thus q = ∞ is also allowed. The analogue of Theorem 4.1 in [Colliander et al. 2008] is Proposition 12.1. Our proof is very different and is inspired by Dodson’s work [2012] on the mass-critical NLS (see also [Vis,an 2012]). In [Colliander 4 et al. 2008], this estimate is derived on the assumption that u>N obeys certain Lt,x spacetime bounds. That the solution does admit these spacetime bounds is derived from the interaction Morawetz estimate, using the analogue of (4-3) to control certain error terms. This results in a tangled bootstrap argument across several sections of the paper. The argument that follows does not use the Morawetz identity, merely Strichartz and maximal Strichartz estimates, and so is equally valid in the focusing case. We also contend that it is simpler. The attentive reader will discover that the implicit constant in (4-3) depends only on u through its ∞ ˙ 1 Lt Hx norm and its modulus of compactness (cf. Deﬁnition 1.2). Indeed, the dependence on the latter can be traced to the following: Let η > 0 be a small parameter to be chosen later. Then, by Remark 1.3 and Sobolev embedding, there exists c = c(η) such that

ku≤ k ∞ 6 + k∇u≤ k ∞ 2 ≤ η. (4-4) cN(t) Lt L x cN(t) Lt L x By elementary manipulations with the square function estimate and Lemma 2.2, respectively, we have

1/2 1/2 1/2 k∇u≤N kL2 L6 A(N), ku≤N kL4 L∞ A(N) k∇u≤N k ∞ 2 u A(N) . (4-5) t x . t x . Lt L x . 868 ROWAN KILLIP AND MONICAVIS, AN

As noted earlier, the only reason for considering the Besov-type norm that appears in (4-1), rather than 2 6 4 ∞ the simpler Lt L x norm, is that it allows us to deduce these Lt L x bounds. By combining the Strichartz inequality (Lemma 2.1) with Lemma 1.7 we have Z Z 2 2 2 A(N) .u 1 + N(t) dt .u N(t) dt. (4-6) I I Note that the second inequality relies on the fact that I contains at least one whole characteristic interval Jk. Similarly, using Proposition 3.1 and then Bernstein’s inequality we ﬁnd

˜ 3/2n −1/2 −1/2 o 4 Aq (N) N |∇| u≥N ∞ 2 + |∇| P≥N F(u) 2 6/5 1 + k∇uk 2 6 kuk ∞ 6 . Lt L x Lt L x . Lt L x Lt L x Z 1/2 2 .u N(t) dt . I Thus R N(t)2 dt 1/3 A(N) + A˜ (N) N 3/2 K 1/2 whenever N ≥ I (4-7) q .u R −1 I N(t) dt and so, in particular, when N ≥ Nmax := supt∈I N(t). This is the base step for the inductive proof of Theorem 4.1. The passage to smaller values of N relies on the following:

˜ Lemma 4.2 (recurrence relations for A(N) and Aq (N)). For η sufﬁciently small,

−3/2 3/2 1/2 2 ˜ A(N) .u 1 + c N K + η Aq (2N) (4-8) ˜ −3/2 3/2 1/2 2 ˜ Aq (N) .u 1 + c N K + ηA(N) + η Aq (2N), (4-9) uniformly in N ∈ 2ޚ. Here c = c(η) as in (4-4).

˜ Proof. The recurrence relations for A(N) and Aq (N) rely on Lemma 2.1 and Proposition 3.1, respectively. To estimate the contribution of the nonlinearity, we decompose u(t) = u≤cN(t)(t) + u>cN(t)(t) and then selectively u = u≤N + u>N . Recalling that the Ø notation incorporates possible additional Littlewood–Paley projections, we may write

2 3 2 3 F(u) = Ø u>cN(t)u + Ø u≤cN(t)u 2 3 2 2 2 2 = Ø u>cN(t)u + Ø u≤cN(t)u>N u + Ø u≤cN(t)u≤N u . (4-10)

Using this decomposition together with Lemma 2.1 and Bernstein’s inequality, we obtain

2 3 A(N) k∇u≤N kL∞ L2 + ∇ P≤N Ø u u 2 6/5 . t x >cN(t) Lt L x 2 2 2 2 + ∇ P≤N Ø u≤ u u 2 6/5 + ∇ P≤N Ø u≤ u≤ u 2 6/5 cN(t) >N Lt L x cN(t) N Lt L x 3/2 2 3 3/2 2 2 2 2 u 1 + N ku u k 2 1 + N ku≤ u uk 2 1 + ∇Ø u≤ u≤ u 2 6/5 . (4-11) . >cN(t) Lt L x cN(t) >N Lt L x cN(t) N Lt L x GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 869

Using instead Proposition 3.1 and Bernstein’s inequality, we ﬁnd

˜ 3/2n −1/2 2 3 2 2 Aq (N) N |∇| u≥N ∞ 2 + ku cN t u k 2 1 + ku≤cN t u N uk 2 1 . Lt L x > ( ) Lt L x ( ) > Lt L x −1/2 2 2 o + |∇| P≥N Ø u≤ u≤ u 2 6/5 cN(t) N Lt L x 3/2 2 3 3/2 2 2 2 2 u 1 + N ku u k 2 1 + N ku≤ u uk 2 1 + ∇Ø u≤ u≤ u 2 6/5 . (4-12) . >cN(t) Lt L x cN(t) >N Lt L x cN(t) N Lt L x Therefore, to obtain the desired recurrence relations it remains to estimate the (identical) last three terms on the right-hand sides of (4-11) and (4-12). We will consider these terms individually, working from left to right. To treat the ﬁrst term, we decompose the time interval I into characteristic subintervals Jk where N(t) ≡ Nk. On each of these subintervals, we apply Hölder’s inequality, Sobolev embedding, Bernstein’s inequality, and Lemma 1.7 to obtain

k 2 3k k k2 k k3 u cN t u L2 L1 J ×ޒ3 u>cNk 4 3 u ∞ 6 > ( ) t x ( k ) . Lt,x (Jk ×ޒ ) Lt L x −3/2 −3/2k∇ k2 −3/2 −3/2 u c N u>cNk 4 3 3 u c N . . k Lt L x (Jk ×ޒ ) . k

Squaring and summing the estimates above over the subintervals Jk, we ﬁnd

3/2 2 3 −3/2 3/2 1/2 N ku u k 2 1 u c N K , (4-13) >cN(t) Lt L x . which is the origin of this term on the right-hand sides of (4-8) and (4-9). To estimate the second term, we begin with a preliminary computation: Using Bernstein’s inequality and Schur’s test (for the last step), we estimate

2 X Ø u u 2 3/2 ku (t)k 2 ku (t)k q ku (t)k 6q >N L L . M1 L x M2 L x M3 t x q−6 2 L x L M1≥M2≥M3 t M2>N 3 −1 X M3 3/q sup kM q u M (t)k q k∇u M (t)k 2 k∇u M (t)k 2 . L x M1 1 L x 3 L x M>N L2 M1≥M3 t −3/2 ˜ .u N Aq (2N). (4-14) Using this, Hölder, and (4-4), we ﬁnd

3/2 2 2 3/2 2 2 2 ˜ N ku≤ u uk 2 1 N ku≤cN(t)k ∞ 6 kØ u u k 2 3/2 u η Aq (2N). (4-15) cN(t) >N Lt L x . Lt L x >N Lt L x . This is the origin of the last term on the right-hand sides of (4-8) and (4-9). Finally, to estimate the contribution coming from the last term in (4-11) and (4-12), we distribute the gradient, use Hölder’s inequality, and then (4-4) and (4-5):

2 2 ∞ 3 ∇Ø u≤ u≤ u 2 6/5 k∇u≤N k 2 6 ku≤cN(t)kL L6 kuk ∞ 6 cN(t) N Lt L x . Lt L x t x Lt L x 2 + k∇ukL∞ L2 ku≤cN(t)kL∞ L6 ku≤N k 4 ∞ kukL∞ L6 t x t x Lt L x t x .u ηA(N). (4-16) 870 ROWAN KILLIP AND MONICAVIS, AN

As A(N) is known to be finite (cf. (4-6)), this can be brought to the other side of (4-8); naturally, this requires η to be sufficiently small depending on u and certain absolute constants, but not on I . Collecting estimates (4-13) through (4-15) and choosing η sufficiently small, this completes the proof of the lemma.

We now have all the ingredients needed to complete the proof of Theorem 4.1.

Proof of Theorem 4.1. With the base step (4-7) and Lemma 4.2 in place, Theorem 4.1 follows from a straightforward induction argument, provided η is chosen sufﬁciently small depending on u.

Remark 4.3. In the introduction it was asserted that the long-time Strichartz estimates in Theorem 4.1 are essentially best possible in the focusing case. We now elaborate that point. For the energy-critical equation, the principal difﬁculty is to obtain control over the low frequencies, because all known conservation laws (with the exception of energy) and monotonicity formulae are energy-subcritical. If (by ∞ 2 some miracle) we knew our putative minimal counterexample u belonged to Lt L x , the whole argument could be brought to a swift conclusion, even in the focusing case (cf. [Killip and Vis,an 2010]). Thus any potential improvement of Theorem 4.1 should be judged by whether it gives better control on the low frequencies. It is well-known that

= + 1 | |2 −1/2 + 5 = W(x) 1 3 x ) obeys 1W W 0 (4-17) and so is a static solution of the focusing energy-critical NLS. In particular, it is almost periodic with ≡ ≡ parameters N(t) 1 and√x(t) 0. As R W(x)5 dx = 4π 3, we can read off from (4-17) that √ Wˆ (ξ) = 4π 3|ξ|−2 + O|ξ|ε as ξ → 0 (4-18)

1−3/q and so deduce kWM kLq ∼ M for M small and 6 ≤ q ≤ ∞. This shows that the supremum is essential in (4-2); we cannot expect the bound (4-3) for the sum of the Littlewood–Paley pieces. It also shows that 2 6 3/2 the Lt L x norm of ∇W≤N on long time intervals decays no faster than the N rate proved for A(N).

5. Impossibility of rapid frequency cascades

In this section, we show that the ﬁrst type of almost periodic solution described in Theorem 1.8 (for which R Tmax −1 ∞ 0 N(t) dt < ) cannot exist. We will show that its existence is inconsistent with the conservation := R | |2 of mass, M(u) ޒ3 u(t, x) dx. The argument does not utilize the defocusing nature of the equation ∞ ˙ 1 beyond the fact that the solution belongs to Lt Hx .

3 Lemma 5.1 (ﬁnite mass). Let u :[0, Tmax) × ޒ → ރ be an almost periodic solution to (1-1) with kuk 10 3 = +∞ and Lt,x ([0,Tmax)×ޒ ) Z Tmax K := N(t)−1 dt < ∞. (5-1) 0 GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 871

∞ 2 (Note Tmax = ∞ is allowed.) Then u ∈ Lt L x ; indeed, for all 0 < N < 1,

1/2 1 n X 2 o k k ∞ 2 3 + k∇ k u N≤ · ≤1 L L ([0,Tmax)×ޒ ) u M 2 6 [ × 3 .u 1. (5-2) t x N Lt L x ( 0,Tmax) ޒ ) M

Proof. The key point is to prove (5-2); ﬁniteness of the mass follows easily from this. Indeed, letting N → ∞ 2 0 in (5-2) to control the low frequencies and using ∇u ∈ Lt L x and Bernstein for the high frequencies, we obtain

kuk ∞ 2 ≤ ku≤ k ∞ 2 + ku k ∞ 2 1. (5-3) Lt L x 1 Lt L x >1 Lt L x .u

3 In the inequality above and for the remainder of the proof all spacetime norms are over [0, Tmax) × ޒ . As K is ﬁnite, the conclusion (4-3) of Theorem 4.1 extends (by exhaustion) to the time interval −1 [0, Tmax). Observe that the second summand in (5-2) is N A(N/2), in the notation of that theorem. We will estimate the left-hand side of (5-2) by a small multiple of itself plus a constant. For this statement to be meaningful, we need the left-hand side of (5-2) to be ﬁnite. This follows easily from Theorem 4.1 and Bernstein’s inequality:

−1 −1 −1 3 1/2 LHS(5-2) N k∇uk ∞ 2 + N A(N/2) N (1 + N K ) < ∞. (5-4) . Lt L x .u

The origin of the small constant lies with the almost periodicity of the solution. Indeed, by Remark 1.3 and Sobolev embedding, for η > 0 (a small parameter to be chosen later) there exists c = c(η) such that

ku≤ k ∞ 6 + k∇u≤ k ∞ 2 ≤ η. (5-5) cN(t) Lt L x cN(t) Lt L x

To continue, ﬁx 0 < N < 1. Using the Duhamel formula from Proposition 1.9 together with the Strichartz inequality we obtain

1 LHS(5-2) k∇ P

To estimate the nonlinearity, we decompose u(t) = u≤cN(t)(t)+u>cN(t)(t) and then u = u1. As the Ø notation incorporates possible additional Littlewood–Paley projections, we may write

2 3 2 2 2 F(u) = Ø u>cN(t)u + Ø u≤cN(t)u1u . (5-7)

Next, we estimate the contributions of each of these terms to (5-6), working from left to right. Using Bernstein’s inequality and (4-13), we bound the contribution of the ﬁrst term as follows:

1 2 3 2 3 ∇ PcN(t) Lt L x >cN(t) Lt L x 1/2 2 3 (N + 1)kØ(u u )k 2 1 . >cN(t) Lt L x −3/2 1/2 .u c K . 872 ROWAN KILLIP AND MONICAVIS, AN

To estimate the contribution of the second term in (5-7) to (5-6), we use Bernstein’s inequality on the second summand and distribute the gradient, followed by Hölder’s inequality, (4-5), and (5-5):

1 2 2 2 2 ∇ P

1 ∞ 2 2 1 ∞ 3 k∇u≤cN(t)kL L2 ku

1 2 2 ∇ P

1 2 2 2 2 ∇ P≤N/2Ø u≤cN(t)u u 2 6/5 + PN≤ · ≤1Ø u≤cN(t)u u 2 6/5 N >1 Lt L x >1 Lt L x 1/2 2 2 (N + 1) Ø u≤cN(t)u u 2 1 . >1 Lt L x 2 ku≤cN(t)kL∞ L6 kØ(u u)k 2 3/2 kukL∞ L6 . t x >1 Lt L x t x 1/2 .u η(1 + K ). Collecting all the estimates above, (5-6) implies

1/2 −3/2 1/2 LHS(5-2) .u η(1 + K ) LHS(5-2) + 1 + c K .

Recalling (5-1) and (5-4) and taking η small enough depending on u and K yields (5-2). We are now ready to prove the main result of this section:

3 Theorem 5.2 (no rapid frequency-cascades). There are no almost periodic solutions u :[0, Tmax)×ޒ →

ރ to (1-1) with kuk 10 3 = +∞ and Lt,x ([0,Tmax)×ޒ )

Z Tmax N(t)−1 dt < ∞. (5-8) 0 Proof. We argue by contradiction. Let u be such a solution. By Corollary 1.6,

lim N(t) = ∞, (5-9) t→Tmax when Tmax is ﬁnite; this is also true when Tmax is inﬁnite by virtue of (5-8). GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 873

We will prove that the existence of such a solution u is inconsistent with the conservation of mass. In Lemma 5.1 we found that the mass is finite; to derive the desired contradiction we will prove that the mass is not only finite, but zero! We first show that the mass at low frequencies is small. To do this, we use the Duhamel formula from Proposition 1.9 together with the Strichartz inequality, followed by Bernstein’s inequality:

1/2 ku≤N kL∞ L2 kP≤N F(u)k 2 6/5 N kF(u)k 2 1 . t x . Lt L x . Lt L x 3 In the display above and for the remainder of the proof all spacetime norms are over [0, Tmax) × ޒ . To estimate the nonlinearity we decompose it as follows:

3 2 3 2 F(u) = Ø(u≤1u ) + Ø(u>1u ). By Theorem 4.1, (4-5), (5-8), Bernstein, and ﬁniteness of the mass,

3 2 2 2 k k 2 k k k k ∞ k k ∞ Ø(u≤1u ) L L1 u≤1 4 ∞ u≤1 Lt x u 2 u 1, t x . Lt L x , Lt L x . while by Theorem 4.1, (4-14), and (5-8),

3 2 2 3 kØ(u u )k 2 1 kuk ∞ 6 ku k 2 3/2 u 1. >1 Lt L x . Lt L x >1 Lt L x . Thus, 1/2 ku≤ k ∞ 2 N . N Lt L x .u By comparison, control over the mass at middle and high frequencies can be obtained with just Bern- stein’s inequality and the fact that for any η > 0 there exists c = c(u, η) > 0 so that

k∇u≤ (t)k 2 ≤ η, cN(t) L x which was noted in Remark 1.3. Altogether, we have that for any t ∈ [0, Tmax),

ku(t)k 2 ku≤ (t)k 2 + kP u≤ (t)k 2 + ku (t)k 2 L x . N L x >N cN(t) L x >cN(t) L x 1/2 −1 −1 −1 N + N k∇u≤ (t)k 2 + c N(t) k∇uk ∞ 2 .u cN(t) L x Lt L x 1/2 −1 −1 −1 .u N + N η + c N(t) . Using (5-9), we can make the right-hand side here as small as we wish. (Choose N small, then η small, and then t close to Tmax.) Because mass is conserved under the ﬂow, this allows us to conclude that kuk ∞ 2 = 0 and thus u ≡ 0 in contradiction to the hypothesis kuk 10 3 = +∞. Lt L x Lt,x ([0,Tmax)×ޒ )

6. The frequency-localized interaction Morawetz inequality

In this section, we prove a spacetime bound on the high-frequency portion of the solution:

3 Theorem 6.1 (a frequency-localized interaction Morawetz estimate). Suppose u :[0, Tmax) × ޒ → ރ is an almost periodic solution to (1-1) such that N(t) ≥ 1 and let I ⊂ [0, Tmax) be a union of contiguous 874 ROWAN KILLIP AND MONICAVIS, AN characteristic intervals Jk. Fix 0 < η0 ≤ 1. For N > 0 sufﬁciently small (depending on η0 but not on I ), Z Z 4 −3 |u>N (t, x)| dx dt .u η0 N + K , (6-1) I ޒ3 := R −1 where K I N(t) dt. Importantly, the implicit constant in the inequality above does not depend on η0 or the interval I . Unlike Theorem 4.1, the argument does not rely solely on estimates for the linear propagator and is not indifferent to the sign of the nonlinearity. Instead, we use a special monotonicity formula associated with (1-1), namely, the interaction Morawetz identity. This is a modiﬁcation of the traditional Morawetz identity (cf. [Lin and Strauss 1978; Morawetz 1975]) introduced in [Colliander et al. 2004]. We begin with a general form of the identity:

4 Proposition 6.2. Suppose i∂t φ = −1φ + |φ| φ + Ᏺ and let ZZ 2 ¯ M(t) := 2 |φ(y)| ak(x − y) Im{φk(x)φ(x)} dx dy, (6-2) ޒ3×ޒ3 for some weight a : ޒd → ޒ. Then Z Z n = 4 − | |6| |2 ∂t M(t) 3 akk(x y) φ(x) φ(y) (6-3) ޒ3 ޒ3 2 ¯ ¯ + 2ak(x − y)|φ(y)| Re φk(x)Ᏺ(x) − Ᏺk(x)φ(x) (6-4) ¯ ¯ + 4ak(x − y)(Im Ᏺ(y)φ(y))(Im φk(x)φ(x)) (6-5) 2 ¯ ¯ ¯ + 4a jk(x − y) |φ(y)| φ j (x)φk(x) − (Im φ(y)φ j (y))(Im φ(x)φk(x)) (6-6) 2 2o − a j jkk(x − y) |φ(y)| |φ(x)| dx dy. (6-7)

Subscripts denote spatial derivatives and repeated indices are summed.

The signiﬁcance of this identity to our problem is best seen by choosing a(x) = |x| and φ to be a solution to (1-1). In this case, Ᏺ = 0 and the fundamental theorem of calculus yields Z Z 4 ∞ 3 8π |φ(t, x)| dx dt ≤ 2kM(t)kL I ≤ 4kφk ∞ 2 3 kφk ∞ ˙ 1 3 . t ( ) L L (I ×ޒ ) Lt Hx (I ×ޒ ) I ޒ3 t x The left-hand side originates from (6-7); the terms (6-6) and (6-3) are both positive. 2 Unfortunately for us, a minimal blowup solution need not have ﬁnite L x norm at any time. Thus it is necessary to localize the identity to high frequencies, that is, choose φ = u>N . Naturally, this produces myriad error terms; nevertheless, in spatial dimensions four and higher they can be controlled (cf. [Ryckman and Vis,an 2007; Vis,an 2007; 2012]). In the three-dimensional case under consideration here, there is one error term (originating from (6-5)) that cannot be satisfactorily controlled. (See also Remark 6.9 at the end of this section.) This was observed already in [Colliander et al. 2008] and as there, our solution is to truncate the function a. This truncation ruins the convexity properties of a that made some of the terms in Proposition 6.2 positive, thus creating more error terms to control. GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 875

For reasons we will explain in due course, it is important to perform the cutoff of a in a very careful fashion. We choose a to be a smooth spherically symmetric function, which we regard interchangeably as a function of x ∈ ޒ3 or r = |x|. We specify it further in terms of its radial derivative:

1 if r ≤ R,  −1 J−J a(0) = 0, ar ≥ 0, arr ≤ 0, ar = 1 − J log(r/R) if eR ≤ r ≤ e 0 R, (6-8)  0 if eJ R ≤ r, where J0 ≥ 1, J ≥ 2J0, and R are parameters that will be determined in due course. It is not difficult to see that one may fill in the regions where ar is not yet defined so that the function obeys

k −1 −k |∂r ar | .k J r for each k ≥ 1, (6-9) uniformly in r and in the choice of parameters. When |x| ≤ R, we see that a(x) = |x|, while a is a constant when |x| ≥ eJ R. The key point about the transition between these two regimes is that 2 2J 1 a ≥ 0 but |a | ≤ (6-10) r r Jr rr Jr

≤ ≤ J−J0 = + 2 when eR r e R. Thus the Laplacian akk arr r ar is dominated by the ﬁrst derivative term and so remains coercive at these radii. (This also appears implicitly in [Colliander et al. 2008, §11] and is the key point behind the “averaging over R” argument there.) As noted above, we will be applying Proposition 6.2 with

φ = uhi := u>N , and so Ᏺ = Phi F(u) − F(uhi). (6-11)

(We will also write ulo := u≤N .) Here N is an additional parameter that will be chosen small (depending on η0 and u). We require that N, R, and J are related via

eJ RN = 1. (6-12)

Actually, it is merely essential that eJ RN ≤ 1, but choosing equality makes the exposition simpler. Our ﬁrst restriction on these parameters is that N is small enough and R is large enough so that given

η = η(η0, u), Z Z Z 2 2 2 2 |∇ulo(t, x)| dx + |Nuhi(t, x)| dx + |∇uhi(t, x)| dx < η (6-13) 3 3 | − | R ޒ ޒ x x(t) > 2 uniformly for 0 ≤ t < Tmax. The possibility of doing this follows immediately from the fact that u is almost periodic modulo symmetries and N(t) ≥ 1. Before moving on to estimating the terms in Proposition 6.2, we pause to review the tools at our disposal. Besides using the norm ku k 4 to estimate itself, we will also make recourse to Theorem 4.1 hi Lt,x and Proposition 3.2. For ease of reference, we record these results in the forms we will use: 876 ROWAN KILLIP AND MONICAVIS, AN

2 + 3 = 3 ≤ ≤ ∞ − 3 Corollary 6.3 (a priori bounds). For all q r 2 with 2 q and any s < 1 q ,

− 1/q q 1 s s 3 k∇ulokL Lr + N |∇| uhi q r u 1 + N K . (6-14) t x Lt L x . Under the hypothesis (6-13), 1/2 3 1/4 ku k 4 ∞ u η (1 + N K ) . (6-15) lo Lt L x . Furthermore, for any ρ ≤ ReJ = N −1, Z Z 2 −3 sup |uhi(t, y)| dy dt .u ρ K + N . (6-16) I x∈ޒ3 |x−y|≤ρ Proof. Recall that Theorem 4.1 implies n X o1/2 0 2 3 1/2 A(M) := k∇u M k 2 6 3 u (1 + M K ) Lt L x (I ×ޒ ) . M0≤M uniformly in M. Setting M = N yields all the estimates on ulo stated in the corollary. More explicitly, the q = 2 case of (6-14) as well as (6-15) follow from this statement and (4-5). The other values of q can then be deduced by interpolation with the (conserved) energy.

Similarly, to estimate uhi we write

− (q−2)/q 1 s s q q 2/q 3 1/q M k|∇| u M kL Lr k∇u M kL Lr A(M) k∇uk ∞ 2 u (1 + M K ) , t x . t x . Lt L x . s−1 ≥ 3 + multiply through by M , and sum over M N. Notice that the condition q s < 1 guarantees the convergence of this sum. Claim (6-16) will follow by combining Proposition 3.2 and Theorem 4.1. First we write (i∂t + = + = 2 3 = 4 1)u>N F G with F P>N Ø(u>N u ) and G P>N Ø(u≤N u) and then estimate these as follows: By Theorem 4.1 and (4-14),

2 2 −3/2 1/2 kFk 2 1 kuk ∞ 6 kØ(u u)k 2 3/2 u N + K , Lt L x . Lt L x >N Lt L x . while by Bernstein, Theorem 4.1, and (4-5),

−1 4 −1 2 2 kGk 2 6/5 N k∇Ø(u≤N u)k 2 6/5 N k∇ukL∞ L2 kuk ∞ 6 ku≤N k 4 ∞ Lt L x . Lt L x . t x Lt L x Lt L x −1 1/2 1/2 .u N + N K . Putting these together with Proposition 3.2 yields

1/2 −1 1/2 −1/2 −31/2 ρ ᏿u>N 2 ∞ 3 u N + N + ρ K + N . Lt L x (I ×ޒ ) .

−3/2 2 Noting from (3-1) that, modulo a factor of ρ , ᏿u>N (t, x) controls the L x norm on the ball around x, and recalling the restriction on ρ, we deduce the claim. We now will analyze the individual terms in Proposition 6.2, beginning with the most important one: GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 877

Lemma 6.4 (mass-mass interactions). Z ZZ η2e2J ku k4 − −a x − y |u y |2|u x |2 dx dy dt K + N −3 8π hi L4 (I ×ޒ3) j jkk( ) hi( ) hi( ) .u ( ). t,x I J Proof. In three dimensions, 1|x| = 2|x|−1 and −(4π|x|)−1 is the fundamental solution of Laplace’s equation. In this way, we are left to estimate the error terms originating from the truncation of a at radii |x − y| ≥ R. Combining (6-9) and (6-16) yields Z ZZ J − | |2| |2 −1k k2 X j −3 j + −3 a j jkk(x y) uhi(y) uhi(x) dx dy dt .u J uhi L∞ L2 (Re ) (Re )(K N ). | − |≥ t x I x y R j=0 To obtain the lemma, we simply invoke (6-13) as well as (6-12). The second most important term originates from (6-3). Its importance stems from the fact that it contains additional coercivity that we will use to estimate other error terms below. Lemma 6.5. We estimate (6-3) in two pieces: Z ZZ := 4 − | |6| |2 ≥ BI 3 akk(x y) uhi(x) uhi(y) dx dy dt 0, (6-17) I |x−y|≤eJ−J0 R as akk ≥ 0 there, and on the complementary region,

Z ZZ J 2 | − || |6| |2 0 + −3 akk(x y) uhi(x) uhi(y) dx dy dt . J K N . (6-18) I |x−y|≥eJ−J0 R −1 Proof. That akk ≥0 and hence BI ≥0 is immediate from (6-10). Further, by construction, |akk|. J0(Jr) when r ≥ eJ−J0 R. In this way, we see that (6-18) relies only on controlling Z ZZ J |u (x)|6|u (y)|2 0 hi hi dx dy dt, I eJ−J0 R≤|x−y|≤eJ R J|x − y| which by (6-16) is

J 2 6 X j −1 j −3 J0 −3 u J0kuhik ∞ 6 (Je R) · (e R) K + N u K + N , . Lt L x . J j=J−J0 as needed. Now we come to the most dangerous-looking term, (6-6). Satisfactory control relies on the full strength of (6-10). Lemma 6.6. Let

2 8 jk(x, y) := |uhi(y)| ∂ j uhi(x)∂kuhi(x) − (Im uhi(y)∂ j uhi(y))(Im uhi(x)∂kuhi(x)). Then Z ZZ − − 2 + J0 + −3 + 1 4a jk(x y)8 jk(x, y)dx dy dt .u η (K N ) J BI . I J 0

(For the BI notation, refer to (6-17).) 878 ROWAN KILLIP AND MONICAVIS, AN

Proof. As a jk(x − y) is invariant under x ↔ y, we may replace 8 by the matrix 1 + 1 2 8 jk(x, y) 2 8 jk(y, x), which is Hermitian-symmetric. Moreover, for each x, y this matrix deﬁnes a positive semideﬁnite quadratic form on ޒ3. To see this, notice that for any vector e ∈ ޒ3 and any function φ, ¯ ¯ eke j (Im φ(y)φ j (y))(Im φ(x)φk(x)) ≤ |φ(y)| |e · ∇φ(y)| |φ(x)| |e · ∇φ(x)| ≤ 1 | |2| · ∇ |2 + 1 | |2| · ∇ |2 2 φ(x) e φ(y) 2 φ(y) e φ(x) .

As a jk is a real symmetric matrix (for any x and y), its eigenvectors are real. Thus, wherever a jk is positive semideﬁnite (i.e., a is convex), the integrand has a favorable sign. In general, the eigenvalues −1 of the Hessian of a spherically symmetric function are arr and r ar with the latter having multiplicity −1 −1 two (ambient dimension minus one). In our case ar ≥ 0 and |arr | . J r . Therefore, we are left to estimate Z ZZ |∇u (x)|2|u (y)|2 hi hi dx dy dt. (6-19) I R<|x−y| R/2 and |x − x(t)| ≤ R/2. In the former case, we use (6-13) and (6-16) to obtain the bound

J 2 X j −1 j −3 2 −3 u k∇uhik ∞ 2 (Je R) · (e R) K + N u η K + N . . Lt L x (|x−x(t)|>R/2) . j=0

When |x − x(t)| ≤ R/2, we further subdivide into two regions. When additionally |x − y| ≥ ReJ−J0 , we estimate in much the same manner as above to obtain the bound J 2 X j −1 j −3 J0 −3 u k∇uk ∞ 2 (Je R) · (e R) K + N u K + N . . Lt L x . J j=J−J0 This leaves us to consider the integral (6-19) over the region where |x − x(t)| ≤ R/2 and |x − y| < ReJ−J0 . Here we use the fact that by the almost periodicity of u (cf. also Remark 1.3 and (6-13)), Z Z Z 2 6 6 |∇uhi(t, x)| dx .u |uhi(t, x)| dx .u |uhi(t, x)| dx, ޒ3 ޒ3 |x−x(t)|≤R/2

−1 uniformly for t ∈ [0, Tmax). We also observe from (6-10) that J0(Jr) ≤ akk; recall J0 ≥ 1. Therefore, 1 the remaining integral is u BI . . J0 The terms appearing in (6-4) are referred to as momentum bracket terms on account of the notation ¯ ¯ {Ᏺ, φ}p := Re(Ᏺ∇φ − φ∇Ᏺ). (6-20)

Note that applying Proposition 6.2 with φ = u>N gives Ᏺ = Phi F(u) − F(uhi). These error terms are comparatively easy to control: GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 879

Lemma 6.7 (Momentum bracket terms). For any ε ∈ (0, 1],

Z Z Z J 2 | |2∇ − ·{ } + k k4 + −1 + 0 −3+ uhi(t, y) a(x y) Ᏺ, φ p dx dy dt .u εBI η uhi L4 ε η ε J (N K ). (6-21) I ޒ3 ޒ3 t,x

Proof. We begin by expanding the momentum bracket into several terms. First, we note that {F(φ), φ}p = − 2 ∇| |6 3 φ and so { } = − 2 ∇| |6 − | |6 − | |6 − { − } − { } Ᏺ, uhi p 3 u ulo uhi F(u) F(ulo), ulo p Plo F(u), uhi p.

Then, using { f, g}p = ∇( f g) + Ø( f ∇g), we obtain

{Ᏺ, uhi}p = 5 ∇ X j 6− j + 2 2 ∇ + 3 2 ∇ + ∇ + ∇ O(uhiulo ) Ø(u uhiulo ulo) Ø(u uhi ulo) Ø(uhi Plo F(u)) Ø(uhi Plo F(u)). (6-22) j=1 We will treat each of these terms in succession. The presence of the gradient in front of a term is a signal that we will integrate by parts in (6-21) before estimating its contribution. We begin with the ﬁrst term in (6-22). Integrating by parts and using

5 X j 6− j 6 −1 2 3 |uhi| |ulo| . ε|uhi| + ε |ulo| |uhi| |uhi| + |ulo| , j=1 we ﬁnd that we need to obtain satisfactory estimates for Z ZZ 2 6 ε |akk(x − y)||uhi(t, y)| |uhi(t, x)| dx dy dt, (6-23) I which follow already from Lemma 6.5, and for

Z ZZ |u (t, y)|2|u (t, x)|2|u (t, x)|[|u (t, x)| + |u (t, x)|]3 hi lo hi hi lo dx dy dt. (6-24) I ε|x − y|

−1 (To obtain this compact form, we use the fact that |akk(x − y)| . |x − y| .) To bound this second integral, we use the Hölder and Hardy–Littlewood–Sobolev inequalities, as well as Corollary 6.3 and (6-13):

−1 −1 2 2 3 (6-24) ε |x| ∗ |uhi| 4 6 kuhikL4 L3 kulok 4 ∞ kuk ∞ 6 . Lt L x t x Lt L x Lt L x − 1 ∞ 2 2 3 ε kuhikL L2 kuhik 4 3 kulok 4 ∞ kulok ∞ 6 . t x Lt L x Lt L x Lt L x −1 −3 .u ε η(N + K ). We now move on to estimating the contribution of the second term in (6-22). This is easily estimated using Corollary 6.3:

2 2 2 2 −1 3 kØ(u uhiulo∇ulo)kL1 kuhikL∞ L2 k∇ulokL2 L6 kulok 4 ∞ kuk ∞ 6 u ηN (1 + N K ). t,x . t x t x Lt L x Lt L x . 880 ROWAN KILLIP AND MONICAVIS, AN

This takes the desired form when multiplied by Z 2 2 −2 |uhi(t, y)| dy .u η N . (6-25) ޒ3 To control the third term in (6-22), we use Bernstein together with Corollary 6.3:

3 2 2 3 kØ(u u ∇ulo)kL1 k∇ulokL2 L∞ kuhik 4 kuk ∞ 6 hi t,x . t x Lt,x Lt L x 1/2 2 u N k∇ulokL2 L6 kuhik 4 . t x Lt,x 2 4 −1 3 u N kuhik 4 + N (1 + N K ). . Lt,x Next, we estimate the contribution from the fourth term in (6-22), which, after integration by parts, this takes the form Z ZZ 2 − |uhi(t, y)| akk(x − y)Ø uhi Plo F(u) (t, x) dx dy dt. I −1 To continue, we write uhi(t, x) = div(∇1 uhi(t, x)) and integrate by parts once more. This breaks the contribution into two parts; after applying Hölder’s inequality and the Mikhlin multiplier theorem, the total contribution is bounded by

−1 2 −1 |x| ∗ |uhi| 4 12 k|∇| uhik 2 6 k∇ Plo F(u)k 4 4/3 (6-26) Lt L x Lt L x Lt L x −2 2 −1 + |x| ∗ |uhi| 4 12/5 k|∇| uhik 2 6 kPlo F(u)k 4 12/5 . (6-27) Lt L x Lt L x Lt L x Applying the Hardy–Littlewood–Sobolev inequality to the ﬁrst factor in each term and using Sobolev embedding on the very last factor, yields

2 −1 (6-26) + (6-27) |uhi| 4 4/3 |∇| uhi 2 6 k∇ Plo F(u)k 4 4/3 . (6-28) . Lt L x Lt L x Lt L x 4 To estimate ∇ Plo F(u), we decompose F(u) = F(ulo) + Ø(uhiu ). Using Hölder, Bernstein, and Corollary 6.3, we obtain

3/4 3/4 4 3/4 3 1/4 k∇ Plo F(ulo)k 4 4/3 N k∇ F(ulo)k 4 1 N k∇ulok 4 3 kulok ∞ 6 u N (1 + N K ) , Lt L x . Lt L x . Lt L x Lt L x . 4 3/2 4 3/2 4 3/2 k∇ PloØ(uhiu )k 4 4/3 N kuhiu k 4 12/11 N kuhik 4 kuk ∞ 6 u N kuhik 4 , Lt L x . Lt L x . Lt,x Lt L x . Lt,x Putting these together with Corollary 6.3 and (6-13) yields

−1 3/4 3 1/4 3/2 (6-28) ku k ∞ 2 ku k 4 k|∇| u k 2 6 N (1 + N K ) + N ku k 4 . hi Lt L x hi Lt,x hi Lt L x hi Lt,x −1 −2 3 1/2 3/4 3 1/4 3/2 ηN ku k 4 N (1 + N K ) N (1 + N K ) + N ku k 4 .u hi Lt,x hi Lt,x 4 −3 u η kuhik 4 + (N + K ) . Lt,x −1 For the ﬁfth (and last) term in (6-22), we again write uhi = div(∇1 uhi). After integrating by parts once, the contribution splits into two pieces, one of which is controlled by (6-26) and another which we bound by 2 −1 (∇a) ∗ |uhi| ∞ ∞ |∇| uhi 2 6 k1Plo F(u)k 2 6/5 . (6-29) Lt L x Lt L x Lt L x GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 881

= + 2 2 + 2 3 We now decompose F(u) F(ulo) Ø(uhiulou ) Ø(uhiu ). Using the Hölder and Bernstein inequalities, we deduce

4 3 1/2 k1Plo F(ulo)k 2 6/5 Nk∇ulok 2 6 kulok ∞ 6 u N(1 + N K ) , Lt L x . Lt L x Lt L x . 2 2 2 2 2 3 1/2 k1PloØ(uhiulou )k 2 6/5 N kuhikL∞ L2 kulok 4 ∞ kuk ∞ 6 u N(1 + N K ) , Lt L x . t x Lt L x Lt L x . and 2 3 5/2 2 3 5/2 2 k1PloØ(u u )k 2 6/5 N kuhik 4 kuk ∞ 6 u N kuhik 4 . hi Lt L x . Lt,x Lt L x . Lt,x Putting it all together we ﬁnd

2 −2 3 1/2 3 1/2 5/2 2 (6-29) kuhik ∞ 2 N (1 + N K ) N(1 + N K ) + N kuhik 4 . Lt L x Lt,x 2 4 −3 u η kuhik 4 + (N + K ) . . Lt,x With the last term estimated satisfactorily, the proof of Lemma 6.7 is now complete.

Looking back to Proposition 6.2, we are left with just one term in ∂t M(t) to estimate, namely, (6-5). As in [Colliander et al. 2008], we call this the mass (Poisson) bracket term and use the notation ¯ {Ᏺ, φ}m := Im(Ᏺφ).

4 Notice that {|φ| φ, φ}m = 0 for any function φ. Lemma 6.8 (mass bracket terms). For any ε > 0, Z Z Z { } ∇ − · ∇ 1/4k k4 + −3 + Im Ᏺ, uhi m(t, y) a(x y) uhi(t, x)uhi(t, x) dx dy dt . η uhi L4 N K . I ޒ3 ޒ3 t,x (6-30) Proof. Exploiting the cancellation noted above and

3 F(u) − F(uhi) − F(ulo) = Ø ulouhiu , we write

{Ᏺ, uhi}m = {Phi F(u) − F(uhi), uhi}m

= {Phi[F(u) − F(uhi) − F(ulo)], uhi}m − {Plo F(uhi), uhi}m + {Phi F(ulo), uhi}m 2 3 = Ø ulouhiu − {Plo F(uhi), uhi}m + {Phi F(ulo), uhi}m. (6-31) We will treat their contributions in reverse order (right to left) since this corresponds to increasing complexity. The contribution of the third term is easily seen to be bounded by 2 −1 kuhi∇uhikL∞ L1 kuhi Phi F(ulo)k 1 k∇uhikL∞ L2 kuhik ∞ 2 N k∇ F(ulo)k 1 2 t x Lt,x . t x Lt L x Lt L x 2 −3 2 2 u η N k∇ulokL2 L6 kulok 4 ∞ kulok ∞ 6 . t x Lt L x Lt L x 2 −3 .u η (N + K ). 882 ROWAN KILLIP AND MONICAVIS, AN

−1 For the second term in (6-31) we write uhi = div(∇1 uhi) and integrate by parts. This yields two contributions to LHS(6-30), which we bound as follows:

− − ∞ 1 ∞ 2 3 1/2 3/2 kuhi∇uhikL L1 |∇| uhi 2 6 ∇ Plo F(uhi) 2 6/5 u kuhikL L2 N (1 + N K ) N kF(uhi)k 2 1 t x Lt L x Lt L x . t x Lt L x −3 1/2 2 3 u η(N + K ) kuhik 4 kuhik ∞ 6 . Lt,x Lt L x 4 −3 u η kuhik 4 + N + K . Lt,x and

−1 −1 |x| ∗ |uhi∇uhi| 4 12 |∇| uhi 2 6 Plo F(uhi)k 4 4/3 Lt L x Lt L x Lt L x −2 3 1/2 3/4 k∇u k ∞ 2 ku k 4 N (1 + N K ) N kF(u )k 4 1 . hi Lt L x hi Lt,x hi Lt L x 1/4 −3 1/2 15/4 1/4 u kuhikL4 N (N + K ) kuhikL4 kuhik ∞ 6 kuhik ∞ 2 . t,x t,x Lt L x Lt L x 1/4 4 −3 u η kuhik 4 + N + K . . Lt,x

5 We now move to the ﬁrst term in (6-31). This term, or more precisely, the term Ø(ulouhi) contained therein, is the reason we needed to introduce the spatial truncation on a. Using ReJ = N −1, we estimate this term via

∞ ∞ 2 3 3 J 3/4 1/2 3 1/4 k∇uhikL L2 kuhikL4 k∇akL L4 kuhik 4 kulokL4 L∞ kuk ∞ 6 u kuhik 4 (e R) η (1 + N K ) t x t,x t x Lt,x t x Lt L x . Lt,x 1/2 4 −3 u η kuhik 4 + N + K . . Lt,x This completes the control of the mass bracket terms.

Proof of Theorem 6.1. From Hölder’s inequality, we see that when φ = uhi and a is as above, the interaction Morawetz quantity deﬁned in (6-2) obeys

3 3 −3 sup |M(t)| ≤ 2ku k ∞ k∇u k ∞ 2 3 η N , hi L L2 (I ×ޒ3) hi Lt L x (I ×ޒ ) .u t∈I t x provided, of course, that N is small enough so that (6-13) holds. Applying the fundamental theorem of calculus to the identity in Proposition 6.2 and putting together all the lemmas in this section, we reach the conclusion that 2 2J 4 1 1 4 1 η J 2 e −3 ku k + B + B + 4 ku k + 4 + + 0 + N + K 8π hi L4 (I ×ޒ3) I .u ε I η hi L4 (I ×ޒ3) η η ( ). t,x J0 t,x ε J J We remind the reader that this estimate is uniform in ε, η ∈ (0, 1], but was derived under several overar- J ching hypotheses: (6-13), N Re = 1, and J ≥ 2J0 ≥ 2. −1 We now choose our parameters as follows: First ε and J0 are made small enough so that the BI term on the RHS can be absorbed by that on the LHS. Next η and J −1 are chosen small enough both to 4 −3 handle the Lt,x on the RHS and to ensure that the prefactor in front of (N + K ) is smaller than η0. We now choose R and N −1 large enough so that (6-13) holds and then further increase N −1 or R so as to ensure N ReJ = 1. GLOBALWELL-POSEDNESSFORTHEQUINTICNLSIN3D 883

To fully justify bringing the two terms across the inequality, we need to verify that they are indeed ﬁnite. This is easily done: Z k k4 k|∇|1/4 k4 −3k∇ k4 −3 + −3 2 uhi L4 . u≥N L4 L3 . N uhi L4 L3 .u N N N(t) dt, t,x t x t x I by Sobolev embedding, Bernstein, and Lemma 1.7. Similarly,

−1 2 5/4 19/4 BI |x| ∗ |uhi| 4 12 kuhik ∞ 2 kuhik 19/3 114/7 . Lt L x Lt L x L L t x Z 9/4 19/4 −3 −3 2 ku k 4 4 ku k ∞ k∇u k u N + N N(t) dt, . hi Lt L x hi L L2 hi 19/3 38/15 . t x Lt L x I by also using the Hardy–Littlewood–Sobolev inequality. Remark 6.9. As noted in the course of the proof, the necessity of truncating a(x) stems from our inability to estimate one term. It would be possible to give a much simpler proof if we could show (a priori) that

5 −2 ku u k 1 N + ηNK, (6-32) hi lo Lt,x .u for N sufﬁciently small. We will now describe what appears to be an intrinsic obstacle to doing this. With current technology, proving (6-32) without using the interaction Morawetz identity seems to require proving that it also holds for almost periodic solutions of the focusing equation; however, the static solution W described in Remark 4.3 shows (6-32) does not hold in that setting. From (4-18) and simple arguments, Z Z −1 5 −1 5 lim N W>N (x) W≤N (x) dx = lim N W(x) W≤N (x) dx N→0 ޒ3 N→0 ޒ3 Z = lim N −1 |ξ|2|Wˆ (ξ)|2ϕ(ξ/N) dξ ∼ 1. N→0 ޒ3 5 As N(t) ≡ 1, it follows that K = |I | and so kW W k 1 NK for N small. hi lo Lt,x &

7. Impossibility of quasisolitons

In this section, we show that the second type of almost periodic solution described in Theorem 1.8, R Tmax −1 = ∞ namely, those with 0 N(t) dt , cannot exist. This is because their existence is inconsistent with the interaction Morawetz estimate obtained in the last section. 3 Theorem 7.1 (no quasisolitons). There are no almost periodic solutions u :[0, Tmax) × ޒ → ރ to (1-1) with N(t) ≡ Nk ≥ 1 on each characteristic interval Jk ⊂ [0, T ) which satisfy kuk 10 3 = +∞ max Lt,x ([0,Tmax)×ޒ ) and Z Tmax N(t)−1 dt = ∞. (7-1) 0 Proof. We argue by contradiction and assume there exists such a solution u. First we observe that there exists C(u) > 0 such that Z N(t) |u(t, x)|4 dx ≥ 1/C(u) (7-2) |x−x(t)|≤C(u)/N(t) 884 ROWAN KILLIP AND MONICAVIS, AN uniformly for t ∈ [0, Tmax). That this is true for a single time t follows from the fact that u(t) is not identically zero. To upgrade this to a statement uniform in time, we use the fact that u is almost periodic. More precisely, we note that the left-hand side of (7-2) is both scale- and translation-invariant and that 6 ˙ 1 the map u(t) 7→ LHS(7-2) is continuous on L x and hence also on Hx . Moreover, by Hölder’s inequality, Z | |4 k k4 N(t) u≤N (t, x) dx .u u≤N (t) L6 for any N > 0, |x−x(t)|≤C(u)/N(t) x uniformly for t ∈ [0, Tmax). Combining this with (7-2) and Theorem 6.1 shows that for each η0 > 0 there exists some N = N(η0) sufﬁciently small so that Z Z −1 −3 −1 N(t) dt .u η0 N + η0 N(t) dt I I uniformly for time intervals I ⊂ [0, Tmax) that are a union of characteristic subintervals Jk. In particular, we may choose η0 small enough to defeat the implicit constant in this inequality and so deduce that

Z Tmax Z T −1 −1 N(t) dt = lim N(t) dt .u 1, 0 T %Tmax 0 which contradicts (7-1).

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Received 5 Mar 2011. Revised 12 May 2011. Accepted 11 Jun 2011.

ROWAN KILLIP: [email protected] Department of Mathematics, University of California, Box 951555, Los Angeles, CA 90095-1555, United States

MONICA VIS, AN: [email protected] Department of Mathematics, University of California, Box 951555, Los Angeles, CA 90095-1555, United States

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