Defocusing Nonlinear Schrodinger Equations 9781108472081

Table of contents :
Contents......Page 8
Preface......Page 10
Acknowledgments......Page 13
1.1 Linear Schrodinger Equation and Preliminaries......Page 14
1.2 Strichartz Estimates......Page 23
1.3 Small Data Mass-Critical Problem......Page 37
1.4 A Large Data Global Well-Posedness Result......Page 46
2.1 The Cubic NLS in Three and Four Dimensions with Small Data......Page 54
2.2 Scattering for the Radial Cubic NLS in Three Dimen- sions with Large Data......Page 60
2.3 The Radially Symmetric, Cubic Problem in Four Dimensions......Page 65
3.1 Small Data Energy-Critical Problem......Page 77
3.2 Profile Decomposition for the Energy-Critical Problem......Page 85
3.3 Global Well-Posedness and Scattering When d ≥5......Page 98
3.4 Interaction Morawetz Estimate......Page 109
4.1 Bilinear Estimates......Page 115
4.2 Mass-Critical Profile Decomposition......Page 122
4.3 Radial Mass-Critical Problem in Dimensions d ≥2......Page 141
4.4 Nonradial Mass-Critical Problem in Dimensions d ≥3......Page 159
5.1 The Energy-Critical Problem in Dimensions Three and Four......Page 177
5.2 Three-Dimensional Energy-Critical Problem......Page 191
5.3 The Mass-Critical Problem When d = 1......Page 204
5.4 The Two-Dimensional Mass-Critical Problem......Page 227
References......Page 249
Index......Page 254

Citation preview

C A M B R I D G E T R AC T S I N M AT H E M AT I C S General Editors ´ B. BOLLOB AS, W. FULTON, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO 217 Defocusing Nonlinear Schr¨odinger Equations

CAMBRIDGE TRACTS IN MATHEMATICS GENERAL EDITORS ´ B. BOLLOB AS, W. FULTON, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO A complete list of books in the series can be found at www.cambridge.org/mathematics. Recent titles include the following: 182. Nonlinear Markov Processes and Kinetic Equations. By V. N. Kolokoltsov 183. Period Domains over Finite and p-adic Fields. By J.-F. Dat, S. Orlik, and M. Rapoport ´ ´ and E.M. Vitale 184. Algebraic Theories. By J. Ad amek, J. Rosick y, 185. Rigidity in Higher Rank Abelian Group Actions I: Introduction and Cocycle Problem. By A. Katok and V. Nit¸ ic a˘ 186. Dimensions, Embeddings, and Attractors. By J. C. Robinson 187. Convexity: An Analytic Viewpoint. By B. Simon 188. Modern Approaches to the Invariant Subspace Problem. By I. Chalendar and J. R. Partington 189. Nonlinear Perron–Frobenius Theory. By B. Lemmens and R. Nussbaum 190. Jordan Structures in Geometry and Analysis. By C.-H. Chu 191. Malliavin Calculus for L´evy Processes and Infinite-Dimensional Brownian Motion. By H. Osswald 192. Normal Approximations with Malliavin Calculus. By I. Nourdin and G. Peccati 193. Distribution Modulo One and Diophantine Approximation. By Y. Bugeaud 194. Mathematics of Two-Dimensional Turbulence. By S. Kuksin and A. Shirikyan 195. A Universal Construction for Groups Acting Freely on Real Trees. By I. Chiswell ¨ and T. Muller 196. The Theory of Hardy’s Z-Function. By A. Ivi c´ 197. Induced Representations of Locally Compact Groups. By E. Kaniuth and K. F. Taylor 198. Topics in Critical Point Theory. By K. Perera and M. Schechter 199. Combinatorics of Minuscule Representations. By R. M. Green ´ 200. Singularities of the Minimal Model Program. By J. Koll ar 201. Coherence in Three-Dimensional Category Theory. By N. Gurski 202. Canonical Ramsey Theory on Polish Spaces. By V. Kanovei, M. Sabok, and J. Zapletal 203. A Primer on the Dirichlet Space. By O. El-Fallah, K. Kellay, J. Mashreghi, and T. Ransford 204. Group Cohomology and Algebraic Cycles. By B. Totaro 205. Ridge Functions. By A. Pinkus 206. Probability on Real Lie Algebras. By U. Franz and N. Privault 207. Auxiliary Polynomials in Number Theory. By D. Masser 208. Representations of Elementary Abelian p-Groups and Vector Bundles. By D. J. Benson 209. Non-homogeneous Random Walks. By M. Menshikov, S. Popov and A. Wade 210. Fourier Integrals in Classical Analysis (Second Edition). By C. D. Sogge 211. Eigenvalues, Multiplicities and Graphs. By C. R. Johnson and C. M. Saiago 212. Applications of Diophantine Approximation to Integral Points and Transcendence. By P. Corvaja and U. Zannier 213. Variations on a Theme of Borel. By S. Weinberger 214. The Mathieu Groups. By A. A. Ivanov 215. Slenderness I: Foundations. By R. Dimitric 216. Justification Logic. By S. Artemov and M. Fitting

Defocusing Nonlinear Schr¨odinger Equations BENJAMIN DODSON The Johns Hopkins University

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108472081 DOI: 10.1017/9781108590518 © Benjamin Dodson 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Dodson, Benjamin, 1983– author. Title: Defocusing nonlinear Schr¨odinger equations / Benjamin Dodson (The Johns Hopkins University). Description: Cambridge ; New York, NY : Cambridge University Press, 2019. | Series: Cambridge tracts in mathematics | Includes bibliographical references and index. Identifiers: LCCN 2018047994 | ISBN 9781108472081 (hardback) Subjects: LCSH: Gross-Pitaevskii equations. | Schrodinger equation. | Differential equations, Nonlinear. Classification: LCC QC174.26.W28 D63 2019 | DDC 530.12/4–dc23 LC record available at https://lccn.loc.gov/2018047994 ISBN 978-1-108-47208-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

I dedicate this book to my family, especially my wife Priscilla and my daughter Ella. I also dedicate this book to God, without whom this would not have been possible.

Contents

Preface Acknowledgments

page ix xii

1

A First Look at the Mass-Critical Problem 1.1 Linear Schr¨odinger Equation and Preliminaries 1.2 Strichartz Estimates 1.3 Small Data Mass-Critical Problem 1.4 A Large Data Global Well-Posedness Result

1 1 10 24 33

2

The Cubic NLS in Dimensions Three and Four 2.1 The Cubic NLS in Three and Four Dimensions with Small Data 2.2 Scattering for the Radial Cubic NLS in Three Dimensions with Large Data 2.3 The Radially Symmetric, Cubic Problem in Four Dimensions

41

52

3

The Energy-Critical Problem in Higher Dimensions 3.1 Small Data Energy-Critical Problem 3.2 Profile Decomposition for the Energy-Critical Problem 3.3 Global Well-Posedness and Scattering When d ≥ 5 3.4 Interaction Morawetz Estimate

64 64 72 85 96

4

Mass-Critical NLS Problem in Higher Dimensions 4.1 Bilinear Estimates 4.2 Mass-Critical Profile Decomposition 4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2 4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3

102 102 109 128 146

vii

41 47

viii 5

Contents Low-Dimensional Well-Posedness Results 5.1 The Energy-Critical Problem in Dimensions Three and Four 5.2 Three-Dimensional Energy-Critical Problem 5.3 The Mass-Critical Problem When d = 1 5.4 The Two-Dimensional Mass-Critical Problem

164 164 178 191 214

References Index

236 241

Preface

In this book we study the semilinear Schr¨odinger equation with power-type nonlinearity. This equation has been an active area of research in dispersive partial differential equations. The study of semilinear Schr¨odinger equations is useful in its own right, since it has many applications to physics. It also provides a great deal of insight into other dispersive partial differential equations and geometric partial differential equations. The study of this equation combines tools from harmonic analysis, microlocal analysis, functional analysis, and topology. It is a truly fascinating topic. This book is mainly focused on the mass-critical (or L2 -critical) problem with initial data in L2 , and the energy-critical (or H˙ 1 -critical) problem with initial data in H˙ 1 . These problems have been shown to be globally well posed and scattering in the defocusing case for critical initial data, and moreover, these results are sharp. Presentation of the proofs of these results is the goal of this book. Chapter 1 commences the study of the mass-critical problem. A natural approach to a nonlinear problem is to treat it as a perturbation of a linear problem. Thus, Chapter 1 begins with an examination of the dispersive properties of solutions to the linear equation. In the section that follows, the dispersive estimates are utilized to prove Strichartz estimates for the linear equation. These estimates are very important since they are invariant under time translation of the linear solution operator. In Section 1.3 these estimates are used to prove that the mass-critical problem is scattering when the mass is small. The argument utilizes a standard fixed point argument. Chapter 1 then concludes with a proof of scattering forsolutions to the mass-critical problem for data lying in a subspace of L2 Rd , but with no size restriction on the initial data. The proof combines conserved quantities with perturbative arguments. Chapter 2 addresses the cubic nonlinear Schr¨odinger equation in dimensions ix

x

Preface

three and four, where it is mass supercritical, and either energy subcritical (d = 3) or energy critical (d = 4). There the small data problem is complicated by the need to differentiate the nonlinearity, especially in three dimensions, where the fractional product rule is used. However, the fact that the nonlinearity is cubic makes the analysis far more tractable than the analysis of mass-supercritical problems in higher dimensions. In Section 2.2 the threedimensional cubic problem is discussed. This problem is H˙ 1/2 -critical, but study of this problem has provided a great deal of insight into both the masscritical and energy-critical problems (see for example Kenig and Merle (2010) and Lin and Strauss (1978)). In this section a Morawetz estimate of Lin and Strauss (1978) is proved, directly yielding a scattering result. In the final section of Chapter 2, the scattering result of Bourgain (1999) is presented. This work is widely considered to be the seminal paper for the techniques discussed in this book, although Grillakis (2000) contemporaneously proved global wellposedness in three dimensions for radial initial data. Bourgain (1999) actually proved scattering for the energy-critical, radially symmetric problem in dimensions d = 3 and 4. In keeping with the theme of cubic problems for this chapter, only the d = 4 result is discussed. The three-dimensional proof may be proved using the same argument as the four-dimensional proof, with only minor modifications. The energy-critical problem in higher dimensions is addressed in Chapter 3. The first order of business is to prove small data scattering in dimensions d ≥ 5. Thus, the perturbative energy-critical results of Tao and Visan (2005) are presented in the first section. Section 3.2 discusses Keraani’s’s (2001) profile decomposition for the energy-critical problem, commencing the discussion of the concentration compactness method with regard to the Schr¨odinger equation. In Section 3.3 the profile decomposition is used to prove that the defocusing, energy-critical nonlinear Schr¨odinger equation is scattering in dimensions d ≥ 5. This result was originally proved in Visan (2007) (see also Visan (2006)). However, in this book the result is proved using a slight modification of the argument in Killip and Visan (2010). The argument utilizes the double Duhamel argument and the interaction Morawetz estimate. The interaction Morawetz estimates of Colliander et al. (2004) and Tao et al. (2007a) are proved in Section 3.4. Higher-dimensional mass-critical results are presented in Chapter 4. Bilinear estimates are essential to the study of the mass-critical problem. The interaction Morawetz estimates lend themselves quite well to bilinear estimates, a fact that was well exploited by Planchon and Vega (2009). Section 4.1 discusses both the Fourier analytic approach to bilinear estimates found in

Preface

xi

Bourgain (1998), as well as the interaction Morawetz approach of Planchon and Vega (2009). Both approaches are used to study the mass-critical problem in Chapters 4 and 5. The subsequent section presents the mass-critical profile decomposition of Tao et al. (2008). This profile decomposition crucially relies on the bilinear estimates of Tao (2003), which will not be proved in this book. Then Section 4.3 presents the scattering results of Killip et al. (2009), Killip et al. (2008), and Tao et al. (2007b) for radial data. Section 4.4 then presents the scattering result of Dodson (2012) for nonradial data. Chapter 5 addresses the defocusing, energy-critical and mass-critical problems in low dimensions. The scattering results of Colliander et al. (2008) (d = 3) and Ryckman and Visan (2007) (d = 4) for the defocusing, energy-critical problem are presented in the first two sections. The d = 4 case is completely worked out in the first section, while some of the more technically difficult parts of the d = 3 case are presented in Section 5.2. Sections 5.3 and 5.4 follow a similar pattern for the mass-critical problem. There the scattering results of Dodson (2016a) (d = 1) and Dodson (2016b) (d = 2) are presented for the defocusing, mass-critical problem. The one-dimensional result of Dodson (2016a) is completely worked out in Section 5.3, while some of the technically difficult aspects of the two-dimensional result are resolved in Section 5.4. In the author’s opinion the material would be sufficient for a one-semester course for graduate students who have taken a course in real analysis at the graduate level. The author assumes that the reader is familiar with basic measure theory and integration, such as can be found in Lieb and Loss (2001) or Taylor (2006). The author also assumes that the reader is familiar with basic functional analysis concepts such as Banach spaces, H¨older spaces, and Fr´echet spaces. Conway (1990) and Yosida (1980) provide good overall introductions to the field of functional analysis. Finally, the book uses many techniques from the field of harmonic analysis such as interpolations and stationary phase analysis. Grafakos (2004), Muscalu and Schlag (2013), Sogge (1993), and Stein (1993) provide good introductions to harmonic analysis. The term “nonlinear Schrodinger equation” will often by abbreviated NLS.

Acknowledgments

During the writing of this book, the author was supported by a von Neumann Fellowship, NSF grants 1500424 and 1764358, as well as the NSF postdoctoral fellowship 1103914. Some of this book was written while the author was at the University of California–Berkeley, and some of this book was written while the author was at Johns Hopkins University. The author also wrote parts of this book while a guest at MSRI, Cergy-Pontoise University, the IHES in Paris, the HIM in Bonn, and the IAPCM in Beijing. The author also wishes to thank the following mathematicians who provided many helpful comments on preliminary versions of this book: Aynur Bulut, Jim Colliander, Magdalena Czubak, Chenjie Fan, Luiz Farah, Justin Holmer, Herbert Koch, Anudeep Kumar, Andrew Lawrie, Jonas L¨uhrmann, Jeremy Marzuola, Dana Mendelson, Changxing Miao, Jason Murphy, Camil Muscalu, Nathan Pennington, Svetlana Roudenko, Paul Smith, Chris Sogge, Daniel Tataru, Michael Taylor, Guixiang Xu, Jianwei Yang, Xueying Yu, and Zehua Zhao. The author would also like to thank the students in his classes for their helpful comments and suggestions on all or the parts of this book that were presented in lectures, as well as some of Camil Muscalu’s students at Cornell University for their helpful comments. The author also sincerely apologizes to anyone who should have been listed here but was omitted, it was certainly unintentional.

xii

1 A First Look at the Mass-Critical Problem

1.1 Linear Schr¨odinger Equation and Preliminaries Formally, the solution to the linear Schr¨odinger equation, iut + Δu = 0,

u (0, x) = u0 ,

(1.1)

may be given by eitΔ u0 . Here Δ is a self-adjoint operator on any L2 -based Sobolev space, so iΔ is skew-adjoint on any L2 -based Sobolev space, and thus the operator eitΔ is a perfectly well-defined unitary group. (See Section A.9 of Taylor (2011) for a proper introduction to unitary groups.) When viewed from a Fourier-analytic perspective, the spectral theory of (1.1), and thus the operator eitΔ , is readily apparent.   Definition 1.1 (Fourier transform) For f ∈ L1 Rd let (F f ) (ξ ) = fˆ (ξ ) = (2π )−d/2



e−ix·ξ f (x) dx.

The Fourier transform intertwines translation and multiplication. Let Tx0 be the translation operator Tx0 f = f (x − x0 ) .   Then by a change of variables, for f ∈ L1 Rd ,    F Tx0 f (ξ ) = (2π )−d/2 e−ix·ξ f (x − x0 ) dx = e−ix0 ·ξ fˆ (ξ )

and Tξ0 F ( f ) (ξ ) = F ( f ) (ξ − ξ0 ) = (2π )−d/2   = F eix·ξ0 f (ξ ) . 1



(1.2)

e−ix·(ξ −ξ0 ) f (x) dx (1.3)

2

A First Look at the Mass-Critical Problem

Equations (1.2) and (1.3) are formally equivalent to F (∂xα f ) (ξ ) = (2π )−d/2 = (2π )−d/2

 

e−ix·ξ (∂xα f (x)) dx e−ix·ξ (iξ )α fˆ (ξ ) d ξ = (iξ )α fˆ (ξ )

and F (xα f ) (ξ ) = (2π )−d/2



(1.4)

xα e−ix·ξ f (x) dx

   = (2π )−d/2 iα ∂ξα e−ix·ξ f (x) dx  α = i∂ξ fˆ (ξ ) ,

(1.5)

respectively, where α = (α1 , . . . , αd ) is a multi-index, αi ≥ 0 for all 1 ≤ i ≤ d. Thus if u (t, x) solves (1.1), then the Fourier transform of u (formally) solves i∂t (uˆ (t, ξ )) − |ξ |2 uˆ (t, ξ ) = 0,

uˆ (0, ξ ) = uˆ0 (ξ ) ,

(1.6)

so for each ξ ∈ Rd , (1.6) gives an ordinary differential equation in time whose solution is uˆ (t, ξ ) = e−it|ξ | uˆ0 (ξ ) . 2

(1.7)

It is possible to formally compute u (t, x) from uˆ (t, ξ ) using the inverse Fourier transform.   Definition 1.2 (Inverse Fourier transform) If g ∈ L1 Rd , let   −1  F g (x) = gˇ (x) = (2π )−d/2 g (ξ ) eix·ξ d ξ . −1 The  d quantity F F is the identity on a set of functions that is dense in R for any 1 ≤ p < ∞.   Definition 1.3 (Schwartz space) Let S Rd be the set of functions such that       S Rd = f ∈ C∞ Rd : supx∈Rd |x|β |∂xα f | ≤ C (α , β ) < ∞ ∀α , β ∈ (Z≥0 )d . (1.8)

Lp

Remark The Schwartz space of smooth functions is actually a Fr´echet space (but not a Banach space).  d    d ∞ 2 −d d Clearly  d  S1 R d  ⊂ L R , and since 1 + |x| d  is integrable on R , S R ⊂ L R , so F is well defined on S R . Furthermore, (1.4) and (1.5) imply     F , F −1 : S Rd → S Rd ,   so F −1 F is well defined on S Rd .

1.1 Linear Schr¨odinger Equation and Preliminaries

  f (x) = F −1 fˆ (x) = (2π )−d/2 Proof

 d

If f ∈ S R , then

Lemma 1.4 (Fourier inversion for Schwartz functions) 

3 

eix·ξ fˆ (ξ ) d ξ .

  For any fˆ (ξ ) ∈ S Rd , a direct calculation shows that

(2π )−d/2



eix·ξ fˆ (ξ ) d ξ = (2π )−d/2 lim



ε 0

2 e−ε |ξ | eix·ξ fˆ (ξ ) d ξ ,

(1.9)

uniformly in x. Since f is integrable, Fubini’s theorem implies that for any ε > 0, (1.9) = (2π )−d = (2π )−d

 

ei(x−y)·ξ e−ε |ξ | f (y) dy d ξ 2





f (y)

= (4πε )−d/2



ei(x−y)·ξ e−ε |ξ | d ξ dy 2

f (y) e−

|x−y|2 4ε

dy.

The last equality follows from the computation

2    2 2 −x2 e dx = e−x −y dx dy = 2π

∞

0

(1.10)

e−r rdr = π , 2

and completing the square in the exponent. A change of variables implies that for any ε > 0, x ∈ Rd , (4πε )−d/2



e−

|x−y|2 4ε

dy = 1,

so lim (4πε )−d/2 e−

ε 0

|x−y|2 4ε

= δ (y − x) .

For any x ∈ Rd , δ (y − x) is a tempered distribution, and therefore if f ∈ S Rd , for any x ∈ Rd , lim (4πε )−d/2

ε 0



f (y) e−

|x−y|2 4ε

dy = f (x) .

Further computations also show that this convergence is uniform in Rd , and  d is on S R . uniform in all the seminorms in (1.8). Thus, F −1 F is the identity  d −1 A similar argument shows that F F is the identity on S R . Therefore, the solution to (1.1), formally given by   2 F −1 e−it|ξ | uˆ0 (ξ ) , is well defined for any t ∈ R when u0 is a Schwartz function.

(1.11)

4

A First Look at the Mass-Critical Problem Let ·, · denote the L2 inner product of functions, f , g = Re



f (x) g (x)dx.

  For any f , g ∈ S Rd , F f , g = (2π )−d/2 Re





e−ix·ξ f (x) dx d ξ 

 −d/2 ix· ξ = (2π ) Re f (x) e g (ξ ) d ξ dx g (ξ )

= f , F −1 g .

(1.12)

  Taking g = F f , f ∈ S Rd implies     F f  2 d =  f  2 d . L (R ) L (R )

(1.13)

    Since S Rd is dense in L2 Rd , (1.12) may be extended to the Parseval identity,   F f , g = f , F −1 g for any f , g ∈ L2 Rd , and (1.13) may be extended to the Plancherel identity,   f

L2 (Rd )

  = F f L2 (Rd )

  for any f ∈ L2 Rd .

(1.14)

Since |eit|ξ | | = 1, (1.7) and (1.14) imply that if u solves (1.1), then         2  u (t, x)  2 d =  eit|ξ | uˆ0 (ξ )  2 d = uˆ0 (ξ ) L2 (Rd ) = u0 L2 (Rd ) . L (R ) L (R ) (1.15)     Moreover, since S Rd is dense in L2 Rd , (1.15) implies that (1.11) is well defined for any u0 ∈ L2 Rd .   Next, using the Fourier and inverse Fourier transforms, if u0 ∈ S Rd , 2

u (t, x) = (2π )−d lim





e−ε |ξ | e−it|ξ | eix·ξ e−iy·ξ u0 (y) dy d ξ ε 0

  −d −ε |ξ |2 −it|ξ |2 i(x−y)·ξ = (2π ) lim e e e d ξ u0 (y) dy 

= lim

ε 0

2

2

ε 0

K (t, x − y, ε ) u0 (y) dy.

(1.16)

1.1 Linear Schr¨odinger Equation and Preliminaries

5

Completing the square in the exponent, K (t, x − y, ε ) = (2π )−d = (2π )−d



e−ε |ξ | e−it|ξ | ei(x−y)·ξ d ξ 2



2

e−ε |ξ | e−it|ξ − 2

x−y 2 2t |

ei

|x−y|2 4t

dξ ,

(1.17)

so by stationary phase analysis (see Chapter 8 of Stein (1993)), letting ε  0, u (t, x) = (4π t)−d/2 e−i

dπ 4



ei

|x−y|2 4t

u0 (y) dy.

(1.18)

    Again, since S Rd is dense in L p Rd , for any 1 ≤ p < ∞, (1.18) gives the dispersive estimate  itΔ   1  e u0  ∞ d ≤ u0  1 d . (1.19) L (R ) L (R ) |4π t|d/2 Then by (1.15), (1.19), and the Riesz–Thorin interpolation theorem (see for example Bergh and L¨ofstrom (1976)), for 2 ≤ p ≤ ∞, if p is the Lebesgue dual exponent satisfying 1p + p1 = 1,  itΔ  e u0  p d ≤ L (R )   Therefore, for any u0 ∈ S Rd ,

1 |4π t|d(1/2−1/p)



Rd

  u0 



L p (Rd )

.

(1.20)

|u (t, x) |2 dx

remains constant, while at the same time, sup |u (t, x) |2 → 0 x∈Rd

as t → ±∞. Thus |u (t, x) |2 spreads out as t → ±∞. For this reason (1.19) is called a dispersive estimate and (1.1) is called a dispersive equation. The reader who is familiar with the wave equation should observe that the dispersion in (1.19) is faster by a factor of t −1/2 than for the wave equation in the same dimension. For example, consider the wave equation in one dimension, and choose some f ∈ S (R). The solution to utt − Δu = 0,

u (0, x) = f (x) ,

ut (0, x) = 0

(1.21)

is given by 1 1 f (x + t) + f (x − t) . 2 2 Such a solution consists of two traveling waves, and thus does not disperse at all. u (t, x) =

6

A First Look at the Mass-Critical Problem

On the other hand, the solution to the linear Schr¨odinger equation with u0 = f will have L∞ -norm that decays at the rate of t −1/2 . The reason for this difference is that the wave equation obeys finite propagation speed and the Huygens principle. Thus, a solution to the one-dimensional problem (1.21) cannot disperse at all, since it travels at either speed 1 to the right or speed 1 to the left. For the linear Schr¨odinger equation, following (1.16), and formally taking ε  0, (2π )−d



2 e−it|ξ | eix·ξ fˆ (ξ ) d ξ = (2π )−d



e−it|ξ − 2t | ei x 2

|x|2 4t

fˆ (ξ ) d ξ ,

and thus formally fˆ (ξ ) travels with velocity 2ξ . This computation can be made rigorous using the Littlewood–Paley decomposition. Definition 1.5 (Littlewood–Paley decomposition) Let ψ (ξ ) be a smooth, decreasing, radial function supported on |ξ | ≤ 2, ψ (ξ ) = 1 on |ξ | ≤ 1, and let     (1.22) φk (ξ ) = ψ 2−k−1 ξ − ψ 2−k ξ . Then φk (ξ ) is a radial, smooth function supported on the annulus 2k ≤ |ξ | ≤ 2k+2 . Also, for ξ = 0, ∞

∑

φk (ξ ) = 1.

(1.23)

k=−∞

Let Pk be the Fourier multiplier given by φk (ξ ); that is,   Pk f = F −1 φk (ξ ) fˆ (ξ ) .

(1.24)

Also let P k = Pk−2 + Pk−1 + Pk + Pk+1 + Pk+2 and φ˜k = φk−2 + φk−1 + φk + φk+1 + φk+2 . Since φk (ξ ) is supported on 2k ≤ |ξ | ≤ 2k+2 , (1.23) implies P k Pk = Pk . Define P≤k =

∑ Pj ,

Pk

Also, for any N > 0, define



ξ ˆ f (ξ ) , (PN f ) (x) = F −1 φ0 N



ξ ˆ f (ξ ) , (P 2k+4 |t|, then for any M, |Kk (t, x) | d,M

2dk (1 + 2k |x|)

and if |x| < 2k−4 |t|, then for any M, |Kk (t, x) | d,M

2dk (1 + 22k |t|)

Remark The kernel for the Littlewood–Paley projection operator is given by K j (0, x), where 

Pj f (x) =

K j (0, x − y) f (y) dy.

(1.31)

To simplify notation, let 

P0 f (x) =

K (x − y) f (y) dy,

(1.32)

8

A First Look at the Mass-Critical Problem

and call K (x) the Littlewood–Paley kernel. By (1.29), for any M, |K (x) | M

1 (1 + |x|)M

,

(1.33)

and for any j ∈ Z, (1.22) implies   |K j (x) | = 2 jd |K 2 j x | M

2 jd (1 + 2 j |x|)M

.

(1.34)

For |t| ≤ 2−2k , direct integration and the support of φk (ξ ) implies

Proof

(2π )−d



eix·ξ e−it|ξ | φk (ξ ) d ξ d 2dk . 2

For |t| > 2−2k and |x| ≤ 2k+4 |t|, the same stationary phase argument that gives (1.19) implies 1 |Kk (t, x) |  d/2 . |t| Next suppose |x| > 2k+4 |t|. Then 

eix·ξ e−it|ξ | φk (ξ ) d ξ = 2



φk (ξ )

(−ix + 2it ξ ) · ∇ξ  ix·ξ −it|ξ |2  e , |x − 2t ξ |2

so for any M, 

(1.27) =



(−ix + 2it ξ ) · ∇ξ φk (ξ ) |x − 2t ξ |2

M 

eix·ξ −it|ξ |

2



dξ .

(1.35)

Integrating by parts, when |x| > 2k+4 |t|, |x − 2t ξ | ∼ |x|, so (1.35) M

2kd |t|M 2kd 2kd +  . |x|2M |x|M 2kM |x|M 2kM

This proves (1.29). Equation (1.30) also follows from (1.35) since |x − 2t ξ | ∼ |t| |ξ | when |x| < 2k−4 |t|. Now choose χ (x) ∈ C0∞ (R) such that χ is supported on |x| ≤ 2 and

∑ χ (x − m) = 1

(1.36)

m∈Z

for all x ∈ R. If u0 ∈ L1 (R), then for any N, |x|N χ (x) u0 (x) ∈ L1 (R) . Therefore, by (1.4) and (1.5), fˆ (ξ ) = χ (ξ ) F (χ (x) u0 (x))

(1.37)

1.1 Linear Schr¨odinger Equation and Preliminaries

9

is a Schwartz function. By the Sobolev embedding theorem, if f is given by (1.37),    itΔ  e f  ∞  χ (x) u0  1 . L (R) L (R) Following the computations in the proof of Theorem 1.7,

   −1 itΔ ixξ −it ξ 2 −iyξ e e χ (ξ ) e χ (y) u0 (y) dy d ξ e f = (2π )

   −1 i(x−y)ξ −it ξ 2 = (2π ) χ (ξ ) e e d ξ χ (y) u0 (y) dy

   2 (x−y)2 −1 −it (ξ − x−y ) 2t e χ (ξ ) d ξ ei 4t χ (y) u0 (y) dy. = (2π ) Plugging in t = 1 and making stationary phase arguments, eitΔ f |t=1 =



(x − y) χ (y) u0 (y) dy, K

where for any M, (x − y) | M |K

1 (1 + |x − y|)M

.

(1.38)

Now utilize the Galilean transformation. Lemma 1.8 (Galilean transformation) equation (1.1), then for any ξ0 ∈ Rd ,

If u solves the linear Schr¨odinger

e−it|ξ0 | eix·ξ0 u (t, x − 2t ξ0 ) 2

(1.39)

solves (1.1) with initial data eix·ξ0 u0 (x). For any m ∈ Z, set fˆm (ξ ) = χ (ξ − m) F (χ (x) u0 (x)) , and then by (1.38) and (1.39), eitΔ fm |t=1 =



m (x − y) χ (y) u0 (y) dy, K

where for any M, m (x − y) | M |K Therefore,

1 (1 + |x − 2tm − y|)M

.

    itΔ e (χ u0 ) |t=1  ∞  χ (x) u0 (x)  1 . L (R) L (R)

(1.40)

10

A First Look at the Mass-Critical Problem

Then by (1.36) and (1.40),  itΔ    e u0 |t=1  ∞  u0  1 . L (R) L (R)

(1.41)

Taking a d-dimensional partition of unity and decomposing

∑

uˆ0 (ξ ) =

χ (ξ − m) F (χ (x) u0 (x))

m∈Zd

proves the same in any dimension when t = 1. Time reversal symmetry and the scaling symmetry generalize (1.41) to any t. If u solves (1.1), then for any λ > 0,   λ d/2 u λ 2t, λ x (1.42)

Lemma 1.9 (Scaling symmetry)

solves (1.1) with initial data λ d/2 u0 (λ x). Remark

The invariance in (1.42) is called the mass-critical scaling.

Taking λ = t 1/2 , (1.41) implies        λ d/2 u λ 2 , λ x L∞ (Rd )  λ d/2 u0 (λ x) L1 (Rd ) = λ −d/2 u0 L1 (Rd ) , and therefore,

  u (t, x) 

L∞ (Rd )

 d  t − 2 u0 L1 (Rd ) .

(1.43)

1.2 Strichartz Estimates The inhomogeneous version of (1.1) is given by iut + Δu = F,

u (0, x) = 0.

(1.44)

By unitary group theory, the solution to (1.44) is formally given by u (t) = −i

 t 0

ei(t−τ )Δ F (τ ) d τ .

The dispersive estimate (1.20) implies a space-time integrability estimate for a solution to (1.44).   ˜ p < ∞, Lemma 1.10 (Inhomogeneous estimate) Suppose d 12 − 1q < 1; p, and

1 1 1 1 − d = + . 2 q p˜ p Then

  

t

−∞

  ei(t−τ )Δ F (τ ) d τ 

p q Lt Lx

(R×Rd )

  F  d,p, p,q ˜

p˜ q

Lt Lx (R×Rd )

.

(1.45)

1.2 Strichartz Estimates

11

Proof The inequality (1.45) is a two-dimensional integral in time, and (1.20) implies that the operator ei(t−τ )Δ F (τ ) is singular when t = τ . Restricting τ to τ < 0 and t to t > 0 reduces the singularity from a one-dimensional singularity to a singularity only at the origin. By (1.20) and H¨older’s inequality, for any j, k ∈ Z,    −2k   i(t−τ )Δ  F (τ ) d τ   p q j j+1 d  −2k+1 e Lt Lx ([2 ,2 ]×R )   2 j/p 2k/ p˜  F  p˜ q  ,  1 1 Lt Lx ([−2k+1 ,−2k ]×Rd ) d − (2 j + 2k ) 2 q so for any j ∈ Z,  0    i(t−τ )Δ  e F (τ ) d τ    −∞

 Since

1 p

p q

Lt Lx ([2 j ,2 j+1 ]×Rd )

(1.46)

 

2 j/p 2k/ p˜

 F  p˜ q . 1 1 Lt Lx ([−2k+1 ,−2k ]×Rd ) k∈Z (2 j + 2k )d 2 − q

+ 1p˜ < 1,

∑ 1 p



< 1 − 1p˜ =

  0   i(t−τ )Δ   e F ( τ ) d τ   −∞

1 p˜ ,

so by Young’s inequality and (1.46),

p q

Lt Lx ([0,∞)×Rd )

   p,q,d F 

p˜ q

Lt Lx ((−∞,0]×Rd )

.

(1.47)

Now for almost every (t, τ ) ∈ {[0, 1] × [0, 1] : τ < t}, there exists a unique j, k ∈ Z, j ≤ 0, 0 ≤ k ≤ 2− j − 2 such that τ ∈ Ikj = [k2 j , (k + 1) 2 j ] and t ∈ j = [(k + 1) 2 j , (k + 2) 2 j ]. This decomposition trades off closeness to the Ik+1 singularity,   −j dist

2 −2

j Ikj × Ik+1 , {(t, τ ) : t = τ }

 2 j,

(1.48)

k=0

with decreasing measure, 

μ

j −1 2−

 j Ikj × Ik+1

 2 j.

(1.49)

k=0

Remark This observation was pointed out to the author by Manoussos Grillakis and Matei Machedon.   = 1. Following the Suppose without loss of generality that F  p q Lt Lx (R×Rd ) argument proving the Christ–Kiselev lemma in Christ and Kiselev (2001) and

12

A First Look at the Mass-Critical Problem

Smith and Sogge (2000), define the map G : R → [0, 1],  t    F (τ )  pq G (t) =

Lx (Rd )

−∞

dτ ,

and for any j ≤ 0, 0 ≤ k ≤ 2− j − 1, let   Ikj = G−1 [k2 j , (k + 1) 2 j ) . Inequality (1.47) implies      ei(t−τ )Δ F (τ ) d τ   j 

p q Lt Lx

Ik



j Ik+1 ×Rd



   F 

Since the Ikj intervals are disjoint, (1.47) and d   

t −∞

  ei(t−τ )Δ F (τ ) d τ 

p q Lt Lx

(

R×Rd

)

q,d

  j p˜ q Ik ×Rd

Lt Lx



1 2



= 2 j/ p˜ .

 − 1q < 1 imply 

∑ 2 j/ p˜ 2− j/p

j≤0

=

˜ 1. ∑ 2 j 2− j/ p˜ 2− j/p  p,p

(1.50)

j≤0

This proves the inhomogeneous estimate. Taking p = p, ˜  t    ei(t−τ )Δ F (τ ) d τ   −∞

when p > 2 and

2 p

=d

1 2

p q

Lt Lx (R×Rd )

   p,q,d F 

p q

Lt Lx (R×Rd )

(1.51)

 − 1q .

A pair (p, q) is called admissible if

1 1 2 =d − , (1.52) p 2 q

Definition 1.11 (Admissible pair)

and 4 ≤ p ≤ ∞ when d = 1; 2 < p ≤ ∞ when d = 2; or 2 ≤ p ≤ ∞ when d ≥ 3. Let Ad = {(p, q) : (p, q) is admissible }. If p > 2, (p, q) is called a non-endpoint admissible pair, and when p = 2, (p, q) is called an endpoint admissible pair. Inequality (1.51) directly yields a non-endpoint estimate for the space-time norm of a solution to (1.1). Theorem 1.12 (Non-endpoint Strichartz estimate) If (p, q) is an admissible pair with p > 2,    itΔ  u0  2 d e u0  p q  (1.53) Lt Lx (R×Rd ) p,q,d L (R )

1.2 Strichartz Estimates and

     e−iτ Δ F (τ ) d τ 

L2

R

( ) Rd

13

   p,q,d F 

p q

Lt Lx (R×Rd )

.

(1.54)

The seminal result in this direction was proved in Strichartz (1977) for p = q. The non-endpoint estimate (p > 2) was subsequently proved by Ginibre and Velo (1992) and Yajima (1987). See also Tao (2006). Proof The proof uses a T –T ∗ argument on the L2 inner product and (1.51).   Since L2 Rd is a Hilbert space and Δ is self-adjoint, by (1.51),     2    e−iτ Δ F (τ ) , e−itΔ F (t) L2 dt d τ  e−iτ Δ F (τ ) d τ  2 d = L (R ) R R R     ei(t−τ )Δ F (τ ) , F (t) L2 dt d τ = R R  2   p,q,d F L p Lq R×Rd . (1.55) ) t x ( This proves (1.54). Relation (1.53) follows by duality. This result is actually pretty close to sharp. It is natural to place an L2 -norm in the right-hand side of (1.53). This is because a space-time norm over R × Rd is invariant under the action of the operator eit0 Δ for a fixed t0 ∈ R, so it is appropriate to seek a norm for the initial data that is also invariant under the action of eit0 Δ , such as L2 . Moreover, any p, q, and d such that    itΔ  e f  p q ≤ C (p, q, d)  f L2 (Rd ) , (1.56) Lt Lx (R×Rd ) must satisfy certain algebraic relations. For example, (1.15) shows that    itΔ  e f  ∞ 2 =  f L2 (Rd ) , (1.57) Lt Lx (R×Rd ) but that (1.56) cannot hold for any 1 ≤ p < ∞ and q = 2. Next, under (1.42),     u0 (x)  2 d = λ d/2 u0 (λ x)  2 d , L (R ) L (R ) while  d/2  2  λ u λ t, λ x 

p q

Lt Lx (R×Rd )

  = λ d/2−2/p−d/q u (t, x) L p Lq (R×Rd ) . t

x

Taking λ → 0 or λ → ∞ shows that (1.56) fails unless (p, q) satisfies (1.52). Remark A counterexample to (1.56) for p < 2 could be constructed via N evenly spaced, nonintersecting, identical bump functions that are Galilean translated, so that the Ltp Lxq -norm of the solution is of the order N 1/p , while the L2 -norm of the initial data is of the order N 1/2 .

14

A First Look at the Mass-Critical Problem

When d = 1, 2 ≤ q ≤ ∞ restricts p to p ≥ 4. Therefore, it only remains to consider the case when p = 2, (p, q) satisfies (1.52), and d ≥ 2. When d = 2, there are known counterexamples to the endpoint result of Theorem 1.13. These counterexamples are all nonradial and involve Brownian motion. In fact, these counterexamples also exclude the weaker result of replacing Lt2 Lx∞ with Lt2 (I, BMO). See Montgomery-Smith (1998) for more information and Tao (2000) for a positive result for radially symmetric functions. When d ≥ 3, Keel and Tao (1998) proved the endpoint case of Theorem 1.12, yielding the sharp result for Strichartz estimates. Theorem 1.13 (Strichartz estimates) Suppose (p, q) and ( p, ˜ q) ˜ are admissible pairs, and I ⊂ R is a possibly infinite time interval. Then    itΔ  u0  2 d , e u0  p˜ q˜ (1.58)  ˜ q,d ˜ L (R ) Lt Lx (I×Rd ) p,       (1.59)  e−itΔ F (t) dt  2 d  p,q,d F L p Lq I×Rd , ) L (R ) t x ( R and

  

τ F(t) 4−  Lq

 p  F (t) − 2q . q

(1.66)

Lx (R)

1−θ 4−θ ,



q

Lx (R)

1+θ θ k:2k >F(t) 4−  Lq



p so 1 − 2q =

3 4−θ

= 34 p . Therefore, by

4/3

Ltp Lxq ⊂ Lt Lx1 + Lt1 Lx2 , and furthermore,  (1)    F  1 2 + F (2)  L L (R×R) t

x

4/3 1 Lx (R×R)

Lt

   F 

(1.67)

p q

Lt Lx (R×R)

,

(1.68)

with constant independent of (p , q ). Therefore, (1.51), (1.53), and (1.54) imply   t    ei(t−τ )Δ F (τ ) d τ  ∞ 2 4 ∞ Lt Lx ∩Lt Lx (R×R) 0  (1)         F + F (2)  4/3 1  F  p˜ q˜ . L1 L2 (R×R) t

Lt

x

Lx (R×R)

Lt Lx (R×R)

This directly implies (1.59), and then by duality (1.58). By interpolation, (1.60) also holds for d = 1. If d = 2 set p¯ = min (p, p) ˜ and choose q¯ satisfying ( p, ¯ q) ¯ ∈ A2 . Making an argument almost identical to (1.61)–(1.68), 







Ltp˜ Lxq˜ ⊂ Ltp¯ Lxq¯ + Lt1 Lx2 ,

(1.69)

and again by (1.51), (1.53), and (1.54),   t    ei(t−τ )Δ F (τ ) d τ  ∞ 2 p¯ q¯ Lt Lx ∩Lt Lx (R×Rd ) 0        p,¯ q¯ F1 L1 L2 (R×Rd ) + F2  p¯ q¯  F  p˜ q˜ . d Lt Lx (R×R ) Lt Lx (R×Rd ) t x Therefore, to complete the proof of Theorem 1.13, it only remains to prove that when d ≥ 3,   t     . (1.70)  F   ei(t−τ )Δ F (τ ) d τ  2d 2d 0 Lt2 Lxd+2 (R×Rd ) Lt2 Lxd−2 (R×Rd )

1.2 Strichartz Estimates

17

  Indeed, assuming (1.70) is true, if F ∈ Lt2 Lx R × Rd , then as in (1.55),       e−itΔ F (t) dt, e−iτ Δ F (τ ) d τ 2 = ei(t−τ )Δ F (τ ) d τ , F (t) 2 dt L L R R R R  2    F , 2d Lt2 Lxd+2 (R×Rd ) 2d d+2

which proves   t    ei(t−τ )Δ F (τ ) 

Lt∞ Lx2

0

(

R×Rd

   F 

)

2d

Lt2 Lxd+2 (R×Rd )

.

(1.71)

The estimate   t    ei(t−τ )Δ F (τ ) 

2d Lt2 Lxd−2

0

is the dual to (1.71), and  t     ei(t−τ )Δ F (τ )  0

Lt∞ Lx2

(R×Rd )

(

R×Rd

)

   F L1 L2 (R×Rd ) x

t

   F L1 L2 (R×Rd ) t

(1.72)

x

follows from (1.15). 

2d



As in (1.67) and (1.69), for any (p, q) ∈ Ad , d ≥ 3, Ltp Lxq ⊂ Lt1 Lx2 + Lt2 Lxd+2 , with      (1)  F (t)  1 2 , + F (2) (t)  d F  p q 2d Lt Lx (R×Rd ) Lt Lx (R×Rd ) Lt2 Lxd+2 (R×Rd ) and combining (1.70)–(1.72),   t    ei(t−τ )Δ F (τ )  2d

Lt2 Lxd−2 ∩Lt∞ Lx2

0

(R×Rd )

   F 

2d

Lt1 Lx2 (R×Rd )+Lt2 Lxd+2 (R×Rd )

   F 

p q

Lt Lx (R×Rd )

.

Following Keel and Tao (1998), observe that to prove (1.70), it suffices to 2d

prove that for any F, G ∈ Lt2 Lxd+2 ,   t R 0

   ei(t−τ )Δ F (τ ) , G (t) dt d τ  F 

2d Lt2 Lxd+2

  G (R×Rd )

2d

Lt2 Lxd+2 (R×Rd )

,

2d

and without loss of generality it suffices to prove that for any F ∈ Lt2 Lxd+2 ,   t R 0

 2  ei(t−τ )Δ F (τ ) , F (t) dt d τ  F 

2d

Lt2 Lxd+2 (R×Rd )

.

(1.73)

18

A First Look at the Mass-Critical Problem

To prove (1.73), Keel and Tao (1998) combined the atomic decomposition of L p spaces with the Whitney decomposition in time in a very powerful way. Recall that in the proof of Lemma 1.10, it was enough to use (1.47) combined with the decomposition of {[0, 1] × [0, 1] : τ < t} into unions of rectangles j . Examining such rectangles more closely, however, it is apparent that Ikj × Ik+1 j is occupied by (τ ,t) pairs for which |τ −t| ∼ 2 j . most of the volume of Ikj ×Ik+1 This is potentially quite useful, via the dispersive estimate (1.20). The Whitney decomposition enables this fact to be exploited quite effectively. Definition 1.14 (Dyadic interval) A dyadic interval is an interval of length λ , where λ is some dyadic number, and the coordinates of the endpoints are integer multiples of λ . A general Whitney covering result may be found in Grafakos (2004). In this book it will only be necessary to use a dyadic Whitney decomposition in the special case Ω = {(t, τ ) ∈ R2 : t < τ }. Lemma 1.15 (Dyadic Whitney decomposition) There exists a partition of Ω into a family Q of essentially disjoint squares Q = I × J, where I and J are dyadic intervals, with the property that dist (Q, ∂ Ω) ∼ diam (Q) for any Q ∈ Q. Proof Take the set of all dyadic subintervals of R. If I1 , I2 , and I0 are dyadic intervals, |I1 | = |I2 | = 12 |I0 |, and I1 ∪ I2 = I0 , then I1 and I2 are said to be descendants of I0 and I0 is said to be the parent of I1 and I2 . Say I ∼ J if 1. I and J have the same length, 2. I is to the left of J, 3. I and J are nonadjacent but have adjacent parents, that is, I ⊂ I0 , J ⊂ J0 , I0 and J0 are adjacent, dyadic subintervals of [0, 1], and |I0 | = |J0 | = 2|I|. Then let Q be the set of all squares I × J, where I, J ⊂ R are dyadic intervals and I ∼ J. Also let Qλ be the set of all I × J ∈ Q satisfying |I| = |J| = λ . For almost every τ ,t ∈ R2 , τ < t, there exists I ∼ J such that τ ∈ I and t ∈ J. For each t, τ , decompose F according to the atomic decomposition (1.61) and the time intervals according to the Whitney decomposition,   R τ − j d+2 or l > − j d+2 For k + l ≤ − j d+2 2 4 4 , use the non-endpoint Strichartz of Theorem 1.12. Suppose without loss of generality that   estimates . For any ε > 0, by (1.54) and interpolation, k > − j d+2 4     ei(t−τ )Δ Fk (τ ) , Fl (t) dt d τ I J

ε 2

2l d+2

2

2kε d+2

 I

×

 I

 d  Fk (τ )  d+22d

Lxd+2 (Rd )

  Fk (τ ) 2

 2  F (τ )  d+22d

Lxd+2 (Rd )

1−ε  2

dτ

2d

Lxd+2 (Rd )

J

ε dτ

 2  d+2  F(t)  2d dt . Lxd+2 (Rd ) Lxd+2 (Rd ) (1.75)

 d  Fl (t)  d+22d

By Young’s inequality, the Cauchy–Schwarz inequality, H¨older’s inequality in time, and (1.62),

∑ ∑ j

∑

2

2l d+2

2

2kε d+2

I

I∼J, k+l≤− j( d+2 2 ), |I|=|J|=2 j k>− j( d+2 4 )

×



I

∑ j

×

  Fk (τ ) 2

 d  Fk (τ )  d+22d

Lxd+2 (Rd )

1−ε  2

2d

Lxd+2 (Rd )

∑



∑

dτ

J 2l

 d  Fl (t)  d+22d

2kε

2 d+2 2 d+2 2 j(

Lxd+2 (Rd )

1+ε 2

)

I∼J, k+l≤− j( d+2 2 ), |I|=|J|=2 j k>− j( d+2 4 )



I

 2d  Fk (τ )  d+22d

Lxd+2 (Rd )

 4  F (τ )  d+22d

 2  F(τ )  d+22d

Lxd+2 (Rd )

ε /2 dτ

ε dτ

Lxd+2 (Rd )

 2  F (t)  d+22d

 dt

Lxd+2 (Rd )

1.2 Strichartz Estimates ×

 I

  Fk (τ ) 2

2

2d

Lxd+2 (Rd )

∑

∑

1−ε 

dτ

J

2kε

2l

2 d+2 2 d+2 2 j(

1+ε 2

 2d  Fl (t)  d+22d

Lxd+2 (Rd )

×

×

R

 R

 2d  Fk (τ )  d+22d

Lxd+2 (Rd )

  Fk (τ ) 2

2d

 2  F 

2d

Lt2 Lxd+2

Lxd+2 (Rd )

dτ

Case 2, k + l ≥ − j

1−ε 

R

dτ

 2d  Fl (t)  d+22d

Lxd+2 (Rd )

 4  F (t)  d+22d

 dt

Lxd+2 (Rd )

 d+2  2

and l > − j

 d+2  4

can be treated in identical

 d+2  2

. Combining (1.20) with (1.63),

    k(2−d) l(2−d) ei(t−τ )Δ Fk (τ ) , Fl (t) dt d τ  2 d+2 2 d+2 2−d j/2

×

dt

Lxd+2 (Rd )

.

The case when k + l ≤ − j fashion.

I J



ε /2

 4  F (τ )  d+22d 2

Lxd+2 (Rd )

 4  F (t)  d+22d

)

j k+l≤− j( d+2 ), 2 k>− j( d+2 4 )



21



I

  Fk (τ ) 

2d

Lxd+2 (Rd )

dτ

 

J

  Fl (t) 

2d

Lxd+2 (Rd )



dt ,

so by H¨older’s inequality, if |I| = |J| = 2 j ,     k(2−d) l(2−d) 2−d ei(t−τ )Δ Fk (τ ) , Fl (t) dt d τ  2 d+2 2 d+2 2 j( 2 ) I J

×

 I

  Fk (τ ) 2

2d

Lxd+2 (Rd )

dτ

1/2  J

  Fl (t) 2

1/2 2d

Lxd+2 (Rd )

. (1.76)

dt

By Young’s  inequality, the Cauchy–Schwarz inequality, and (1.62), when k, l ≥ d+2 −j 4 ,

∑ ∑

∑

2

k(2−d) d+2

2

l(2−d) d+2

2

j( 2−d 2 )

I

j |I|=|J| k,l≥ =2 j − j( d+2 4 )

×



J



  Fl (t) 2

1/2 2d

Lxd+2 (Rd )

dt

  Fk (τ ) 2

1/2 2d

Lxd+2 (Rd )

dτ

22 ∑ j

×

A First Look at the Mass-Critical Problem    k(2−d) l(2−d) 2−d j ( ) Fk (τ ) 2 2d ∑ 2 d+2 2 d+2 2 2

 R

∑

  Fl (t) 2

2

dτ

1/2 2d

Lxd+2 (Rd )

∑

j

Lxd+2 (Rd )

R

k,l≥ − j( d+2 4 )

1/2

2k(2−d) d+2

2

dt

j( 2−d 2 )

 R

k≥

  Fk (τ ) 2

 2d

Lxd+2 (Rd )

dτ

− j( d+2 4 )

 2  F 

2d

Lt2 Lxd+2 (R×Rd )

When l < − j

 d+2  4

.

and k + l ≥ − j

 d+2 

, combining (1.75) and (1.76),

2

    ei(t−τ )Δ Fk (τ ) , Fl (t) dt d τ I J

ε A

× ×

 I

 I

 J

 2d  Fk (τ )  d+22d

Lxd+2 (Rd )

  Fk (τ ) 2

ε /4

 4  F (τ )  d+22d  2−ε

Lxd+2 (Rd )

dτ

4

2d Lxd+2

 2d  Fl (t)  d+22d

(Rd )

Lxd+2 (Rd )

dτ

 4  F (t)  d+22d

(1−ε )

1/2 ,

dt

(1.77)

Lxd+2 (Rd )

1−ε

l

(2+2ε −d)

k 2+2ε −d

2+2ε −d

where for clarity we have put A = 2l d+2 2 j 4 2 2(d+2) 2 2(d+2) 2 j 4 . Then by Young’s inequality, the Cauchy–Schwarz inequality, (1.77), and (1.62),

∑ ∑

∑

    2  ei(t−τ )Δ Fk (τ ) , Fl (t) dt d τ  F 

2d

Lt2 Lxd+2 (R×Rd )

j |I|=|J| k+l≥− j( d+2 ), I J 2 =2 j l 0,   λ d/2 u λ 2t, λ x (1.81)  T T solves (1.80) on − λ 2 , λ 2 with initial data

λ d/2 u0 (λ x) , which preserves the L2 -norm, or mass,     u0 (x)  2 d = λ d/2 u0 (λ x)  2 d . Lx (R ) Lx (R ) Theorem 1.17

For any d ≥ 1, there exists ε0 (d) > 0, such that if   u0  2 d ≤ ε0 (d) , L (R )

(1.82)

then (1.80) is globally well posed and scattering. Roughly speaking, global well-posedness and scattering means that (1.80) has a solution, and the solution is close to a solution to the linear Schr¨odinger equation as t → ±∞. Definition 1.18 (Well-posedness) An initial value problem is said to be well posed on an interval I, 0 ∈ I ⊂ R, if 1. there exists a unique solution to the initial value problem, 2. the solution is continuous in time, 3. the solution depends continuously on the initial data.

1.3 Small Data Mass-Critical Problem

25

Definition 1.19 (Scattering) A solution to an initial value problem is said to scatter forward in time if the solution exists on [0, ∞) and there exists u+ such that u (t) − eitΔ u+ → 0,

(1.83)

as t → +∞. A solution to an initial value problem is said to scatter backward in time if the solution exists on (−∞, 0] and there exists u− such that u (t) − eitΔ u− → 0,

(1.84)

as t → −∞. An initial value problem with initial data in some set is said to be scattering if the problem is globally well posed, scatters both forward and backward in time, and u+ and u− depend continuously on the initial data. In this book, the Banach space in which (1.83) and (1.84) are proved to hold will be the critical L2 -based Sobolev space. This same space will also be used in Definition 1.18 to define continuityin time. As was already mentioned, the  critical Sobolev space for (1.80) is L2 Rd . Proof of Theorem 1.17 For any d ≥ 1, p = q = 2(d+2) lies in Ad , so let X be d the set of functions     ≤ C ε0 , X = u : R × Rd → C : u 2(d+2) Lt,x d (R×Rd ) for some constant C. By Theorem 1.13 and (1.82), there exists some constant C (d) such that  itΔ  C ε0 e u0  2(d+2) . (1.85) ≤ 2 Lt,x d (R×Rd ) Now define the map Φ (u) (t) = eitΔ u0 − i

 t 0

ei(t−τ )Δ F (u (τ )) d τ .

(1.86)

By semigroup theory, if u ∈ X satisfies Φ (u) (t) = u (t) ,

(1.87)

then u solves (1.80). By the contraction mapping principle, to prove the existence of a unique solution to (1.80) in X, it suffices to prove 1. Φ : X → X, 2. the map Φ is a contraction.

26

A First Look at the Mass-Critical Problem

By Theorem 1.13 and H¨older’s inequality, if u ∈ X,  t      d F (u)  2(d+2)  ei(t−τ )Δ F (u (τ )) d τ  2(d+2) 0 Lt,x d (R×Rd ) Lt,xd+4 (R×Rd ) 4  1+ d  u 2(d+2) Lt,x d (R×Rd ) 4

 (Cε0 )1+ d .

(1.88)

Hence, choosing ε0 (d) sufficiently small, (1.85) and (1.88) imply Φ : X → X. To prove Φ is a contraction, for u, v ∈ X,     Φ (u) − Φ (v)  2(d+2)  F (u) − F (v)  2(d+2) Lt,x d (R×Rd ) Lt,xd+4 (R×Rd )  4    4   u − v 2(d+2)  |u| d + |v| d  d+2 Lt,x2 (R×Rd ) Lt,x d (R×Rd )  4  .  (Cε0 ) d u − v 2(d+2) Lt,x d (R×Rd ) For ε0 > 0 sufficiently small,   Φ (u) − Φ (v) 

2(d+2) Lt,x d

 1 ≤ u − v 2(d+2) , 2 L d (R×Rd ) (R×Rd ) t,x

proving Φ is a contraction. Therefore, there exists a unique u ∈ X satisfying (1.87). That u is continuous in time follows directly from Theorem 1.13, the dominated convergence theorem, and the fact that the Schwartz functions are  d 2 dense in Lx R . Continuous dependence on initial data follows from a perturbation lemma. Lemma 1.20 (Perturbation lemma) Let I be a compact time interval and let w be an approximate solution to (1.80) for some d ≥ 1. That is, for some e small, (i∂t + Δ) w = F (w) + e.

(1.89)

Suppose, for some ε0 > 0 small, with 0 ≤ ε ≤ ε0 , that u solves (1.80),     e 2(d+2) u 2(d+2) ≤ ε0 , ≤ ε, Lt,x d (R×Rd ) Lt,xd+4 (R×Rd ) and for some t0 ∈ R,   i(t−t )Δ e 0 (u (t0 ) − w (t0 )) 

≤ ε.

2(d+2)

Lt,x d

(R×Rd )

(1.90)

(1.91)

1.3 Small Data Mass-Critical Problem Then

  u − w

and

ε

2(d+2)

Lt,x d

27

(1.92)

(R×Rd )

   (i∂t + Δ) (u − w) + e

 ε.

2(d+2)

Lt,xd+4

(1.93)

(R×Rd )

If u solves (1.80) and w solves (1.89), then v = w − u solves

Proof

(i∂t + Δ) v = F (w) − F (u) + e. By Theorem 1.13,   v 2(d+2) Lt,x d

   ei(t−t0 )Δ (w (t0 ) − u (t0 ))  (R×Rd )

  + F (w) − F (u) + e

2(d+2)

Lt,x d

.

2(d+2)

Lt,xd+4

Calculating,   F (w) − F (u) 

(R×Rd )

(I×Rd )

2(d+2)

Lt,xd+4

(R×Rd )

4 d       u 2(d+2) + v 2(d+2) . (1.94)  v 2(d+2) Lt,x d (R×Rd ) Lt,x d (R×Rd ) Lt,x d (R×Rd ) Therefore, by Theorem 1.13, (1.90), (1.91), and (1.94),   v

   ε + v

2(d+2)

Lt,x d

(R×Rd )

 2(d+2)

Lt,x d

  ε0 + v

(R×Rd )

4 d

,

2(d+2)

Lt,x d

(R×Rd ) (1.95)

which proves (1.92) and (1.93). Setting e = 0, (1.93) and Theorem 1.13 imply that if u and w solve (1.80) with initial data     w (t0 )  2 d , u (t0 )  2 d ≤ ε0 , L (R ) L (R ) x

x

  then taking ε = w (t0 ) − u (t0 ) L2 implies       u − w 2(d+2) + u − wL∞ L2 (R×Rd )  u (t0 ) − w (t0 ) L2 (Rd ) . x x t Lt,x d (R×Rd )

28

A First Look at the Mass-Critical Problem

This proves continuous dependence on initial data, which completes the proof of global well-posedness. To prove scattering, set u+ = u0 − i and

 ∞

 0

u− = u0 + i

e−itΔ F (u (t)) dt

(1.96)

e−itΔ F (u (t)) dt.

(1.97)

0

−∞

  By (1.54) and (1.88), (1.96) and (1.97) are well defined, and u+ , u− ∈ Lx2 Rd . Additionally, by the dominated convergence theorem,   lim F (u)  2(d+2) = 0. T →∞ Lt,xd+4 ([T,∞)×Rd ) Therefore,     iT Δ e u+ − u (T ) 

Lx2 (Rd )

  =

∞

T

  ei(t−τ )Δ F (u (τ )) d τ 

Lx2 (Rd )

→ 0.

  This proves that each initial data function u0 L2 ≤ ε0 has a solution that is global in time and scatters forward in time. The proof that the solution also scatters backward in time is identical. Expressions (1.94), (1.95), (1.96), and (1.97) prove that u+ and u− depend continuously on the initial data. Theorem 1.17 also holds if (1.82) is replaced by the condition  itΔ  e u0  2(d+2) ≤ ε0 (d) , Lt,x d

(1.98)

(R×Rd )

for some ε0 (d) > 0 sufficiently small. By Strichartz estimates, (1.98) is weaker than (1.82), although it does give a Banach norm on u0 . By a similar argument, to show that (1.80) is globally well posed on [0, ∞), and that the solution scatters forward in time, it is enough to show that  itΔ  e u0  2(d+2) ≤ ε0 (d) , (1.99) Lt,x d ([0,∞)×Rd ) and similarly for (−∞, 0]. To prove local well-posedness of (1.80) on the interval [−T, T ] for initial data near u0 , it is enough to show  itΔ  e u0  2(d+2) ≤ ε0 (d) . (1.100) Lt,x d ([−T,T ]×Rd )

1.3 Small Data Mass-Critical Problem

29

Theorem 1.21 (Local well-posedness)   1. For any u0 ∈ Lx2 Rd , there exists T (u0 ) > 0 such that (1.80) is locally well posed on [−T, T ]. The term T (u0 ) depends on the profile of the initial data as well as its size. Moreover, (1.80) is well posed on an open interval I ⊂ R, 0 ∈ I. 2. If sup (I) < +∞, where I is the maximal interval of existence of (1.80),   = ∞. (1.101) lim u 2(d+2) T sup(I) Lt,x d ([0,T ]×Rd ) The corresponding result also holds for inf (T ) > −∞. Proof Inequality and the dominated convergence theorem imply that  (1.53)  for any u0 ∈ Lx2 Rd ,   = 0. lim eitΔ u0  2(d+2) T 0 Lt,x d ([−T,T ]×Rd ) Therefore, there exists T (u0 ) > 0 sufficiently small such that (1.100) holds, which gives the first part of the theorem. To prove the second part notice that if J ⊂ I is a compact subinterval then   u 2(d+2) < ∞. (1.102) Lt,x d (J×Rd ) Indeed, by Definition 1.18, the first part of Theorem 1.21, (1.53), and the dominated convergence theorem, for any t0 ∈ I, there exists some T (t0 ) such that   u 2(d+2) ≤ ε0 . Lt,x d ([t0 −T (t0 ),t0 +T (t0 )]×Rd ) 

Since t0 ∈J (t0 − T (t0 ) ,t0 + T (t0 )) is an open cover of J, J compact implies the existence of a finite subcover, which in turn implies (1.102). If   < ∞, (1.103) lim u 2(d+2) T sup(I)

Lt,x d

([0,T ]×Rd )

and sup (I) < ∞, then (1.54), (1.86), (1.87), and the dominated   convergence theorem imply that u (t) converges in L2 to some f ∈ Lx2 Rd as t  sup (I). Then by the first result of this theorem and Lemma 1.20, (1.103) implies that (1.80) is locally well posed on [0, sup (I) + δ ) for some δ > 0, which contradicts the maximality of I. Therefore, (1.101) also holds. 2(d+2)

If u is a global solution to (1.80), a uniform bound on the Lt,x d -norm implies scattering.

30

A First Look at the Mass-Critical Problem

Theorem A solution to (1.80) scatters forward in time to some eitΔ u+ ,  1.22  2 d u+ ∈ Lx R , if and only if for some t0 ∈ R,   u 2(d+2) < ∞. (1.104) Lt,x d ([t0 ,∞)×Rd ) Proof

If (1.104) holds, set u+ = u (t0 ) − i

 ∞ t0

e−iτ Δ F (u (τ )) d τ .

  Inequality (1.54) implies u+ ∈ Lx2 Rd , and by the dominated convergence theorem,   lim eitΔ u+ − u (t) L2 (Rd ) = 0. x t∞ Conversely, if u scatters, then (1.53) and the dominated convergence theorem imply that for some T (ε0 , u+ ) sufficiently large,   itΔ   ε0 ε0 e u+  2(d+2) and u (T ) − eiT Δ u+ L2 (Rd ) ≤ . ≤ x 2 2 d Lt,x ([T,∞)×Rd ) (1.105) Then, by Theorem 1.13, (1.105) implies  i(t−T )Δ  e  ε0 , u (T )  2(d+2) Lt,x d ([T,∞)×Rd ) and Theorem 1.21 with (1.99) implies   u 2(d+2)  ε0 . Lt,x d ([T,∞)×Rd )   Since [t0 , T ] is compact, (1.102) implies that u 2(d+2) Lt,x d

< ∞. ([t0 ,T ]×Rd )

The Sobolev norm of u0 in any space more regular than L2 gives sufficient information on the profile of u0 .    Theorem 1.23 For any ε > 0, there exists T u0 H ε (Rd ) > 0 such that (1.80) is locally well posed on [−T, T ]. Also, if (1.80) has a solution on some maximal interval I ⊂ R, with sup (I) < ∞, then   lim uL∞ H ε ([0,T ]×Rd ) = +∞. t T sup(I) An identical result holds when inf (I) > −∞. Proof

By Bernstein’s inequality,     P>N u0  2 d  1 u0  ε d . Lx (R ) H (R ) ε N

1.3 Small Data Mass-Critical Problem

31

Also, by (1.7), the Sobolev embedding theorem, and H¨older’s inequality, d  itΔ    d e P≤N u0  2(d+2)  T 2(d+2) N d+2 −ε u0 H ε (Rd ) . Lt,x d ([−T,T ]×Rd )     Choosing N ε0 , u0 H ε (Rd ) , ε sufficiently large and T = N12 yields (1.100), which proves the theorem.    Upgrading Theorem 1.23 to the existence of T u0 L2 (Rd ) > 0 for which x (1.80) is locally well posed is the goal of the rest of this work, since such a result implies global well-posedness   and scattering. To see this, suppose without loss of generality that T u0 L2 (Rd ) = 1, that is, suppose that (1.80) x

is locally well posed on [−1, 1] for all initial data with L2 -norm given by   u0  2 . Then fix some 1 ≤ T0 < ∞, and set L v0 = λ d/2 u0 (λ x) ,        1/2 with λ = T0 . Since v0 L2 = u0 L2 , T v0 L2 = 1, so (1.80) is locally well posed on [−1, 1] with initial data v0 . Rescaling, 1  t x = u (t, x) v , d/2 λ2 λ λ gives local well-posedness of (1.80) with initial data u0 on [−T0 , T0 ]. Since T 0 is arbitrary, global well-posedness follows. Similar computations also give u 2(d+2) < ∞, with bound independent of T , which by Theorem Lt,x d ([−T,T ]×Rd ) 1.22 proves scattering. The same picture appears when viewing (1.80) through the lens of scaling symmetry. For general s ∈ R,   d/2   λ u0 (λ x)  ˙ s d = λ s u0  ˙ s d . H (R ) H (R ) Thus, for s > 0, proving local well-posedness on a short time interval when u0 H s is large is equivalent to proving local well-posedness on a unit interval when u0 H˙ s is small. This is why in Theorem 1.23, the time interval T has a lower bound that depends on u0 H s . When s < 0, proving local well-posedness on a unit interval for u0 H˙ s of size 1 is equivalent to proving local well-posedness on a short time interval when u0 H˙ s is small. This fact was exploited by Christ et al. (2003, 2008) to prove ill-posedness of (1.80) when u0 ∈ H˙ s , s < 0. In particular, they constructed a solution that is discontinuous in time at t = 0. Thus,it is natural to study the scattering behavior of (1.80) with initial data u0 ∈ Lx2 Rd ; scattering results in L2 are sharp.

32

A First Look at the Mass-Critical Problem   The set of initial data for which scattering occurs is an open set in L2 Rd .

Theorem 1.24 (Long-time perturbations) Let I be a compact time interval and let w be an approximate solution to (1.80) on I × Rd in the sense that (i∂t + Δ) w = F (w) + e. Assume u solves (1.80) and   u

≤ M.

2(d+2)

Lt,x d

(1.106)

(I×Rd )

Let t0 ∈ I, and suppose that u (t0 ) is close to w (t0 ) in the sense that  i(t−t )Δ  e 0 (u (t0 ) − w (t0 ))  2(d+2) ≤ ε. Lt,x d (I×Rd ) Also assume

  e

≤ ε.

2(d+2)

Lt,xd+4

(I×Rd )

Then there exists ε1 (M, d) > 0 such that if 0 < ε ≤ ε1 (M, d),   u − w 2(d+2) ≤ C (M, d) ε Lt,x d (I×Rd ) and

  w

≤ C (M, d) .

2(d+2)

Lt,x d

(1.107)

(I×Rd )

Proof This argument first appeared in Bourgain (1999) for the energy-critical problem.  2(d+2)  d subintervals Jk such that on each Jk , Divide I into ∼ 1 + εM0   u

2(d+2)

Lt,x d

(Jk ×Rd )

≤ ε0 ,

where ε0 is small. For each Jk = [ak , bk ], by Theorem 1.13 and (1.94),     v 2(d+2)  ei(t−ak )Δ v (ak )  2(d+2) Lt,x d (Jk ×Rd ) Lt,x d (Jk ×Rd )  4 d       v 2(d+2) + u 2(d+2) + ε; + v 2(d+2) Lt,x d (Jk ×Rd ) Lt,x d (Jk ×Rd ) Lt,x d (Jk ×Rd ) (1.108)

1.4 A Large Data Global Well-Posedness Result  i(t−a )Δ  k v (a )  so, if e ≤ ε0 , we have 2(d+2) k Lt,x d (R×Rd )     v 2(d+2)  ei(t−ak )Δ v (ak )  2(d+2) + ε. Lt,x d (Jk ×Rd ) Lt,x d (R×Rd )

33

Also, from Theorem 1.13 and (1.108), it follows that     i(t−b )Δ k v (b )  e  ei(t−ak )Δ v (ak )  2(d+2) + ε . (1.109) 2(d+2) k Lt,x d (R×Rd ) Lt,x d (R×Rd ) Iterating (1.109) over each Jk , there exists ε1 (M, d) sufficiently small such that if 0 < ε ≤ ε1 ,   u − w 2(d+2) ≤ C (M, d) ε . Lt,x d (I×Rd ) This gives (1.107). Then by (1.106),   w 2(d+2) ≤ C (M, d) . Lt,x d (I×Rd )

1.4 A Large Data Global Well-Posedness Result The proof of global well-posedness and scattering for generic large data cannot rely solely on perturbative arguments. Instead, some knowledge of the solution to the nonlinear problem is required, beyond what can be gleaned from the behavior of a solution to the linear equation and perturbation theory. Suppose u solves the nonlinear Schr¨odinger equation iut + Δu = N = μ |u| p u,

(1.110)

on I × Rd , where I is a compact interval, μ = ±1, and p > 0. If Y is a scalar, vector, tensor, or linear operator, then Y (iut + Δu − N ) = 0,

(1.111)

and a well-chosen Y can yield important local conservation results. Conservation laws typically     arise via integrating by parts, so it is convenient to assume that u (t)∈ S Rd . This is a safe assumption to make since S Rd is dense in H s Rd for any s ∈ R. The local well-posedness result in Theorem 1.21 then implies that the solution u (t) is sufficiently regular, at least for short times. Remark See Sulem and Sulem (1999) for derivation of conservation laws using Noether’s theorem.

34

A First Look at the Mass-Critical Problem

Lemma 1.25 (First local conservation law) Let T00 (t, x) be the mass density, T00 (t, x) = |u (t, x) |2 , and for j = 1, . . . , d let T0 j (t, x) be the momentum density T0 j (t, x) = 2 Im (u¯∂ j u) . If u solves (1.110) on I × Rd , then

∂t T00 + ∂ j T0 j = 2{N , u}m = 0,

(1.112)

where {·, ·}m is the mass bracket { f , g}m = Im ( f g) ¯ .

(1.113)

Remark This work follows the usual convention of summing repeated indices over j = 1, . . . , d. Since we will always be working on Rd with the Euclidean metric, we will not concern ourselves with whether indices are raised or lowered. See Tao (2006) for more on tensor notation. Proof

Take Y = −iu¯ in (1.111).

Re ((−iu) ¯ (iut + Δu − N )) = Re (u¯ (ut − iΔu + iN )) =

1 ∂ |u (t, x) |2 + ∇ · Im[u∇u] ¯ − {N , u}m . 2 ∂t

  Since {N , u}m = Im |u| p+2 = 0, the proof is complete. Corollary 1.26 (Conservation of mass) I × Rd , then the mass 

Rd

If u solves (1.110) on the interval 

T00 (t, x) dx =

|u (t, x) |2 dx

is conserved for all t ∈ I. Remark

The identity

∂t T00 (t, x) + ∂ j T0 j (t, x) = 0

(1.114)

also implies that the quantity T00 (t, x) changes in time at the rate of T0 j (t, x). This is why T0 j (t, x) is called the momentum. Integrating by parts,

∂t



x j |u (t, x) |2 dx =



T0 j (t, x) dx.

Lemma 1.27 (Second local conservation law) Let { f , g} p be the momentum bracket   { f , g} p = Re f ∇g¯ − g∇ f¯ ,

1.4 A Large Data Global Well-Posedness Result

35

and for j = 1, . . . , d let   { f , g} pj = Re f ∂ j g¯ − g∂ j f¯ . Then

∂t T0 j + ∂k L jk = 2{N , u} pj , where L jk (t, x) is the linear part of the energy current    L jk (t, x) = Lk j (t, x) = −∂ j ∂k |u (t, x) |2 + 4 Re ∂ j u (∂k u) . Proof

(1.115)

(1.116)

By (1.111), {(iut + Δu − N ) , u} pj = 0.

(1.117)

Expanding (1.117),   ∂ 1 Im[u∇u] ¯ + 2∂k Re (∇u∂k u) ¯ − ∇Δ |u|2 − {N , u} p = 0, ∂t 2 which directly gives (1.115). Corollary 1.28 (Conservation of momentum) then the total momentum

If u solves (1.110) on I × Rd



Rd

T0 j (t, x) dx

is a conserved quantity for any t ∈ I. Proof

If N = μ |u| p u then {N , u} p = −

 μp  ∇ |u (t, x) | p+2 . p+2

(1.118)

Therefore,

∂t T0 j + ∂k T jk = 0,

(1.119)

where T jk = L jk + δ jk and



∂t T0 j (t, x) dx =

2μ p |u (t, x) | p+2 p+2



−∂k T jk (t, x) dx = 0,

which yields Corollary 1.28. Plugging Y = u¯t into (1.111) yields conservation of energy.

(1.120)

36

A First Look at the Mass-Critical Problem

Lemma 1.29 (Conservation of energy) E (u (t)) =

1 2



|∇u (t, x) |2 dx +

The total energy

μ p+2



|u (t, x) | p+2 dx

(1.121)

is conserved. Proof 

If Y = u¯t , then integrating by parts,

Re (u¯t (iut + Δu − N )) dx =



Re (u¯t (Δu − N )) 

 1 μ d |∇u (t, x) |2 + |u (t, x) | p+2 dx =− dt 2 p+2

= 0. When μ = +1, the energy and the mass are positive definite quantities that together control the Hx1 -norm. This directly yields a global result for large data. Theorem 1.30

The defocusing, mass-critical initial value problem   4 iut + Δu = |u| d u, u (0) ∈ H 1 Rd ,

(1.122)

is globally well posed. Proof

By Theorem 1.21, (1.122) is locally well posed on an open interval I.    2(d+2)  By the Sobolev embedding theorem, Hx1 Rd → L d Rd , so E (u (0)) < ∞, where E (u (t)) is the energy given by (1.121). Then by conservation of energy,   < ∞. (1.123) sup u (t)  2(d+2) t∈I Lx d (Rd ) Therefore, for any T ∈ I,   u

d

2(d+2) Lt,x d

([0,T ]×Rd )

E(u(0)) T 2(d+2) ,

so by (1.101), sup (I) = ∞. A similar computation also shows inf (I) = −∞. Remark Conservation of energy does not imply (1.123) in the case when μ = −1. This is not merely a technical difficulty, but is indicative of a significant separation between the behavior of (1.110) for large data when μ = +1 and when μ = −1. In fact, there are known counterexamples to global wellposedness and scattering of (1.122) with μ = −1. This distinction does not arise in the case of small data, where the nonlinearity is treated perturbatively, ignoring its sign altogether.

1.4 A Large Data Global Well-Posedness Result

37

The case μ = +1 is called defocusing because in that case a solution to (1.110) will disperse faster than a linear solution disperses. In the case μ = −1, a solution to (1.110) will disperse at a slower rate than a solution to the linear equation, and could, at least in principle, not disperse at all, or even concentrate: see Weinstein (1982, 1989). The pseudoconformal conservation law quantifies this fact. Observe that (1.18) implies that if u solves (1.110) for N = 0, the quantity    (x + 2it∇) u (t)  2 d (1.124) L (R ) is a conserved quantity. Indeed, (x + 2it∇)



1

ei

t d/2

|x−y|2 4t

f (y) dy =



1 t d/2

ei

|x−y|2 4t

y f (y) dy,

and therefore (1.15) implies the conservation of (1.124). Theorem 1.31 (Pseudoconformal conservation law) I × Rd ,    (x + 2it∇) u (t) 2 2 L

8μ t 2 + ( ) p+2

2  = xu (0)  2

Rd

L (Rd )

Proof

 t

+ 0

4μ s



If u solves (1.110) on

|u (t, x) | p+2 dx



4−dp p+2 |u (s, x) | dx ds. p+2

(1.125)

By (1.114),

∂ 2 |x| |u (t, x) |2 = −2|x|2 ∇ · Im[u∇u] ¯ (t, x) ∂t  2  = −2∇ · |x| Im[u∇u] ¯ (t, x) + 4x · Im[u∇u] ¯ (t, x) .

(1.126)

By (1.119) and (1.120), −4

∂ ∂ (tx · Im[u∇u]) ¯ = −4x · Im[u∇u] ¯ − 4t (x · Im[u∇u]) ¯ ∂t ∂t   = −4x · Im[u∇u] ¯ − 2tx j · ∂ j Δ |u (t, x) |2   4μ pt x j · ∂ j |u (t, x) | p+2 . + 8tx j · ∂k Re ((∂ j u) ¯ (∂k u)) + p+2

Therefore, integrating by parts,

∂ −4 t ∂t



x · Im[u∇u] ¯ dx

= −4



x · Im[u∇u] ¯ dx − 8t



|∇u|2 dx −

4μ d pt p+2



|u| p+2 dx.

(1.127)

38

A First Look at the Mass-Critical Problem

Finally, by conservation of energy, 

 2  d 8μ t 2 |u (t, x) | p+2 dx = 16tE (u (t)) . 4t 2 ∇u (t) L2 (Rd ) + dt p+2

(1.128)

Summing (1.126), (1.127), and (1.128) proves (1.125). In the mass-critical case, p = d4 , (1.125) implies    (x + 2it∇) u (t) 2 2 L

(Rd ) +

8μ t 2 p+2



2  |u (t, x) | p+2 dx = xu (0) L2 (Rd ) .

This implies a scattering result for the defocusing, mass-critical problem. Theorem 1.32

The initial value problem

  u (0, x) = u0 ∈ L2 Rd   has a global solution for all u0 ∈ H 1 Rd . Moreover, 4

iut + Δu = |u| d u,

  2(d+2) u d

u0 

2(d+2)

Lt,x d

Proof

(R×Rd )

L2

u0 H˙ 1 (Rd ) xu0 L2 (Rd ) .

(1.129)

(1.130)

By the Sobolev embedding theorem and interpolation,

 2(d+2)  u (0)  d 2(d+2) Lx d

2  2(d+2) 4     u (0) H˙ 1 (Rd ) u (0)  d2 d .  u (0)  dd Lx (R ) H˙ d+2 (Rd ) (Rd )

By conservation of energy and H¨older’s inequality, for T = xu0 L2 (Rd ) / u0 H˙ 1 (Rd ) ,   2(d+2) u d

2(d+2) Lt,x d

([−T,T ]×Rd )

 4/d      u0 L2 Rd u0 H˙ 1 (Rd ) xu0 L2 (Rd ) . ( )

For |t| > T , the pseudoconformal conservation law implies 4dt 2 d +2

 Rd

|u (t, x) |

2(d+2) d

  dx ≤ xu0 L2 (Rd ) .

Integrating over |t| > T completes the proof of (1.130). Remark (1.81).

Both sides of (1.130) are invariant under the L2 -critical scaling

For the focusing case (μ = −1), let τ = − 1t , y = xt , and

1 i |x|2 1 x 4t v (τ , y) = d/2 e u − , . t t t

(1.131)

1.4 A Large Data Global Well-Posedness Result

39

By direct calculation,    (x + 2iτ ∇) u (τ ) 2 2

 8μτ 2  u (τ )  p+2 + p+2 = 8E (v (τ )) . Lx ( ) Lx p+2 Rd

Also by direct calculation, if u solves (1.129) on the interval I = (T1 , T2 ) then  v (τ , y) also solves (1.129) on − T11 , − T12 . Definition 1.33 (Pseudoconformal transformation) Equation (1.131) is called the pseudoconformal transformation of u (t, x). The sharp interpolation estimate of Weinstein (1982),    2  f  2 d  2   2(d+2) f d ∇ f  2 , L ≤  2(d+2) L Q 2 d d L Lx (R )     implies that whenever u0 L2 < QL2 ,  E (u (t))  u0 

L2

 2(d+2)  u (t)  d . 2(d+2) L

(1.132)

d 2(d+2)

Since (1.131) preserves the L2 -norm, as well as the Lt,x d -norm, (1.131)– (1.132) imply that for any 0 < T < ∞,   2(d+2) u d 2(d+2) Lt,x d

(R×Rd )

  u0 

L2

T E (u0 ) +

 1 xu0 2 2 . L T

Taking T ∼ E(u0 )1/2 /xu0 L2 concludes the proof of the corresponding focusing result. Theorem 1.34

The initial value problem

  (1.133) u (0, x) = u0 ∈ L2 Rd     1 d 2   has  a global solution for all u0 ∈ H R , xu0 ∈ L , whenever u0 L2 < Q 2 . Moreover, L 4

iut + Δu = −|u| d u,

  2(d+2) u d

2(d+2) Lt,x d

(R×Rd )

   u0 

( )

L 2 Rd

    u0  ˙ 1 d xu0  2 d . L (R ) H (R )

Remark The interpolation result of Weinstein (1982) is extremely useful for the focusing nonlinear Schr¨odinger problem because Q is the positive H 1 solution to the elliptic partial differential equation 4

ΔQ + |Q| d Q = Q,

(1.134)

and eit Q is certainly not a scattering solution to (1.133). Berestycki and Lions

40

A First Look at the Mass-Critical Problem

(1979) proved the existence of a solution to (1.134). Applying the pseudoconformal transformation to eit Q gives a finite time blowup solution to (1.133), x |x|2 i 1 u (t, x) = d/2 e− t ei 4t Q . (1.135) t t Theorem 1.34  suggests that scattering could hold for initial data u0 satisfying   u0  2 < Q 2 . See Dodson (2015) for a proof of this fact. L L Remark See Holmer and Roudenko (2008) and Holmer et al. (2010) for results for the focusing, cubic nonlinear Schr¨odinger equation when d = 3.

2 The Cubic NLS in Dimensions Three and Four

2.1 The Cubic NLS in Three and Four Dimensions with Small Data In general, the power-type nonlinear Schr¨odinger equation iut + Δu = F (u) = μ |u| p u,

u (0, x) = u0 ,

μ = ±1

(2.1)

is invariant under the scaling symmetry   u (t, x) → λ 2/p u λ 2t, λ x .

(2.2)

The critical Sobolev norm for (2.2) is H˙ sc , where sc =

d 2 − . 2 p

(2.3)

Thus, the cubic nonlinear Schr¨odinger equation iut + Δu = F (u) = μ |u|2 u,

u (0, x) = u0

is mass critical when d = 2, H˙ 1/2 -critical when d = 3, and energy critical when d = 4. Indeed,     u (0, x)  ˙ 1/2 3 = λ u (0, λ x)  ˙ 1/2 3 , H (R ) H (R )     u (0, x)  ˙ 1 4 = λ u (0, λ x)  ˙ 1 4 , H (R ) H (R ) and

    u (0, x)  4 4 = λ u (0, λ x)  4 4 . L (R ) L (R ) x

x

Global well-posedness and scattering hold for the cubic problem with small initial data in the critical Sobolev spaces in dimensions d = 3, 4. 41

42

The Cubic NLS in Dimensions Three and Four

Theorem 2.1

There exists ε0 > 0 such that

  (2.4) u (0, x) = u0 ∈ H˙ sc Rd   is globally well posed and scattering for u0 H˙ sc (Rd ) ≤ ε0 when d = 3, 4. iut + Δu = F(u) = μ |u|2 u,

Proof

Using Strichartz spaces, let     X = u : uS˙sc (R×Rd ) ≤ Cε0 ,

and define the operator Φ (u) (t) = eitΔ u0 − i

 t 0

ei(t−τ )Δ F (u (τ )) d τ .

By Strichartz estimates, when d = 4 and sc = 1,       u0  ˙ 1 4 + ∇F (u)  3/2 Φ (u)  ˙1  . 4 H (R ) S (R×R ) Lt,x (R×R4 )

(2.5)

By   rule and the Sobolev embedding theorem, which implies   the product u 6  u ˙1 , for ε0 sufficiently small, Lt,x

S

     2 (2.5)  u0 H˙ 1 (R4 ) + ∇uL3 (R×R4 ) uL6 (R×R4 )  ε0 +(Cε0 )3  ε0 . (2.6) t,x t,x Thus, Φ : X → X when d = 4. When d = 3, Strichartz estimates imply       Φ (u)  ˙1/2  u0 H˙ 1/2 (R3 ) +  |∇|1/2 F (u)  10/7 . S (R×R3 ) Lt,x (R×R3 ) Use the fractional product rule to estimate |∇|1/2 F (u). Theorem 2.2 (Fractional product rule) For 0 < s < 1, 1 < p < ∞,            f g ˙ s,p d   f  q d g ˙ s,p d + g r d  f  ˙ s,r d , L 1 (R ) L 1 (R ) W (R ) W 2 (R ) W 2 (R ) where 1 1 1 1 1 = + = + , p q1 q2 r1 r2   q2 , r2 ∈ (1, ∞), and (q1 , r1 ) ∈ (1, ∞]. Here W˙ s,p Rd is the L p -based Sobolev space of order s. Proof

See Taylor (2000).

By the fractional product rule, for any interval I ⊂ R,    1/2   2  |∇| F (u)  10/7 u 5 .   |∇|1/2 u 10/3 3 3 Lt,x (I×R3 ) Lt,x (I×R ) Lt,x (I×R )

(2.7)

Therefore, as in (2.6), for ε0 > 0 sufficiently small, Φ : X → X when d = 3.

2.1 3D and 4D Cubic NLS with Small Data

43

To prove that Φ is a contraction, by the fundamental theorem of calculus, F (u) − F (v) =

 1 d 0

dt

F (v + t (u − v)) dt = (u − v) ·

 1 0

F  (v + t (u − v)) dt.

(2.8) By the product rule, when d = 4,   Φ (u) − Φ (v)  ˙1 S (R×R4 )         2 2  u − vS˙1 (R×R4 ) uL6 (R×R4 ) + vL6 (R×R4 ) + u − vL6 (R×R4 ) t,x t,x t,x           × uL6 (R×R4 ) + vL6 (R×R4 ) uS˙1 (R×R4 ) + vS˙1 (R×R4 ) t,x t,x   2     Cε0 u − v S˙1 (R×R4 ) . When d = 3, the fractional product rule implies  1/2   |∇| (F (u) − F (v))  10/7 Lt,x (R×R3 )     1/2  2    u2 5   |∇| (u − v)  10/3 3 ) + v L5 (R×R3 ) 3 R×R L ( Lt,x (R×R ) t,x t,x        + u − vL5 (R×R3 ) uL5 (R×R3 ) + vL5 (R×R3 ) t,x t,x t,x

 1/2   1/2      |∇| u 10/3 + |∇| v 10/3 × Lt,x (R×R3 ) Lt,x (R×R3 )

  1/2   2     u − v L5 (R×R3 ) + |∇| (u − v) 10/3  (Cε0 ) . (2.9) Lt,x (R×R3 ) t,x Therefore,     Φ (u) − Φ (v)  ˙s  (Cε0 )2 u − vS˙sc (R×Rd ) , S c (R×Rd ) proving that Φ is a contraction. Then by the contraction mapping principle, Strichartz  estimates,  and the  product  (or fractional product) rule, the solution u ∈ Ct0 R; H˙ xsc Rd ∩ S˙sc R × Rd exists and is unique. The proof of continuous dependence on initial data again relies on a perturbation lemma. Lemma 2.3 (Perturbation lemma) Let I be a compact time interval and let w be an approximate solution to (2.4), that is, for some e small, (i∂t + Δ) w = F (w) + e. Also suppose

  w

Lt∞ H˙ xsc (I×Rd )

≤ E.

(2.10)

(2.11)

44

The Cubic NLS in Dimensions Three and Four

Let u solve (2.4), and suppose that for some t0 ∈ R,   u (t0 ) − w (t0 )  ˙ s d ≤ E  , H c (R )

(2.12)

for some E  > 0. Moreover, assume the smallness condition, 0 ≤ ε ≤ ε0 (E, E  ), along with   u d+2 ≤ ε0 , (2.13) Lt,x (R×Rd )   ∇e ˙ s ≤ ε, (2.14) N c (R×Rd ) and

  i(t−t )Δ e 0 (u (t0 ) − w (t0 )) 

d+2 Lt,x (R×Rd )

Then

  u − w

d+2 Lt,x (R×Rd )

and

   (i∂t + Δ) (u − w) + e ˙ s

≤ ε.

ε

N c (R×Rd )

(2.15)

 ε.

(2.16)

Remark The proofs of Theorem 2.1 and Lemma 2.3 are presented in Holmer and Roudenko (2008). See also Guevara (2014) and Kenig and Merle (2010). If u solves (2.4) and w solves (2.10), then v = w − u solves

Proof

(i∂t + Δ) v = F (w) − F (u) + e. By Strichartz estimates, the Sobolev embedding theorem, and Duhamel’s principle,     v d+2  ei(t−t0 )Δ (w (t0 ) − u (t0 )) Ld+2 (R×Rd ) Lt,x (R×Rd ) t,x   + F (w) − F (u) + eN˙ sc (R×Rd ) and   v ˙s

S c (R×Rd )

     w (t0 ) − u (t0 ) H˙ sc (Rd ) + F (w) − F (u) + eN˙ sc (R×Rd ) .

By (2.8) and (2.9),   F (w) − F (u)  ˙ s N c (R×Rd )         2 2  vS˙sc (R×Rd ) uLd+2 (R×Rd ) + vLd+2 (R×Rd ) + vLd+2 (R×Rd ) t,x t,x t,x                   × u S˙sc (R×Rd ) + v S˙sc (R×Rd ) u Ld+2 (R×Rd ) + v Ld+2 (R×Rd ) . t,x t,x (2.17)

2.1 3D and 4D Cubic NLS with Small Data

45

Therefore, by Strichartz estimates, the product rule, (2.13), (2.14), and (2.17),   v d+2 Lt,x (R×Rd )  2      ε + vS˙sc (R×Rd ) ε0 + vLd+2 (R×Rd ) t,x           + vLd+2 (R×Rd ) uS˙sc (R×Rd ) + vS˙sc (R×Rd ) ε0 + vLd+2 (R×Rd ) t,x t,x and   v ˙s

S c (R×Rd )

 2      E + E  + vS˙sc (R×Rd ) ε0 + vLd+2 (R×Rd ) t,x           + vLd+2 (R×Rd ) uS˙sc (R×Rd ) + vS˙sc (R×Rd ) ε0 + vLd+2 (R×Rd ) . t,x t,x   Strichartz estimates, (2.11), (2.12), and (2.13) imply uS˙sc (R×Rd )  E +E  , so 1 taking ε0 (E, E  ) sufficiently small, say ε0  1+E+E  , proves (2.15) and (2.16). Setting e = 0, then (2.16) and Strichartz estimates imply that if u and w solve (2.4) with initial data     w (t0 )  ˙ sc d , u (t0 )  ˙ sc d ≤ ε0 , Hx (R ) Hx (R ) then

  u − w

d+2 Lt,x (R×Rd )

    + u − wS˙sc (R×Rd )  u (t0 ) − w (t0 ) H˙ sc (Rd ) .

This proves continuous dependence on the initial data, which completes the proof of global well-posedness. Setting u+ = u (t0 ) − i and u− = u (t0 ) + i

 ∞ t0

 t0 −∞

e−iτ Δ F (u (τ )) d τ

e−iτ Δ F (u (τ )) d τ ,

  implies scattering, since u+ , u− ∈ H˙ sc Rd , and (2.6) and (2.7) imply that u (t) scatters forward to eitΔ u+ as t  +∞, and u (t) scatters backward in time to eitΔ u− as t  −∞. Taking w = 0, Lemma 2.3 implies global well-posedness and scattering for (2.4) when  itΔ     e u0  d+2 ≤ ε0 u0 H˙ sc , Lt,x (R×Rd )

46

The Cubic NLS in Dimensions Three and Four   where in this case, ε0  0 as u0 H˙ sc  ∞. Likewise, to prove global wellposedness on [0, ∞) and that a solution scatters forward in time, it would be enough to show  itΔ     e u0  d+2 (2.18) ≤ ε0 u0 H˙ sc . Lt,x ([0,∞)×Rd ) Compare to (1.99), in which ε0 is independent of the size of the initial data. Following the arguments used to prove Theorems 1.21 and 1.22 we obtain the following theorems. Theorem 2.4 (Local well-posedness)   1. For any u0 ∈ H˙ xsc Rd , there exists T (u0 ) > 0 such that (2.1) is locally well posed on [−T, T ]. The quantity T (u0 ) depends on the profile of the initial data as well as its size. Moreover, (2.1) is well posed on an open interval I ⊂ R, 0 ∈ I. 2. If sup (I) < +∞, where I is the maximal interval of existence of (2.1),     lim uL∞ H˙ sc ([0,T ]×Rd ) = ∞. lim uLd+2 ([0,T ]×Rd ) = ∞ or t,x t T sup(I) T sup(I) The corresponding result also holds for inf (T ) > −∞. Theorem 2.5 (Scattering) A solution to (2.1) scatters forward in time to some  d itΔ s c ˙ e u+ , u+ ∈ Hx R if and only if for some t0 ∈ R,     u ∞ ˙ s u d+2 < ∞ and < ∞. d Lt,x ([t0 ,∞)×R ) Lt H c ([t0 ,∞)×Rd ) Arguing as in the proof of Theorem 1.24 implies long-time perturbation results. Theorem 2.6 (Long-time perturbations) Let I be a compact time interval and let w be an approximate solution to (2.4) on I × Rd in the sense that (i∂t + Δ) w = F (w) + e. Assume that u is a solution to (2.4) satisfying   u d+2 ≤M Lt,x (I×Rd ) and

  u

Lt∞ H˙ xsc (I×Rd )

≤ E,

for some M, E > 0. Let t0 ∈ I, and suppose that u (t0 ) is close to w (t0 ) in the sense that   u (t0 ) − w (t0 )  ˙ sc d ≤ E  Hx (R )

2.2 Large Data Scattering for Radial, 3D Cubic NLS and

 i(t−t )Δ  e 0 (u (t0 ) − w (t0 )) 

d+2 Lt,x (I×Rd )

Also assume

  e ˙ s

N c (I×Rd )

47

≤ ε.

≤ ε,

for some 0 < ε ≤ ε1 (E, E  , M). Then      u − w d+2 L (I×Rd ) ≤ C E, E , M, d ε t,x

and

     w d+2 L (I×Rd ) ≤ C E, E , M, d . t,x

Proof As in the proof of Theorem 1.24, Theorem 2.5 may be proved by partitioning I into subintervals and iterating the argument used to prove Lemma 2.3.

2.2 Scattering for the Radial Cubic NLS in Three Dimensions with Large Data Local conservation laws imply a large data scattering result for the defocusing, three-dimensional cubic nonlinear Schr¨odinger equation, iut + Δu = |u|2 u,

u (0, x) = u0 ,

u : R × R3 → C.

(2.19)

Theorem 2.7 The initial value problem (2.19) is globally well posed and scattering   for radial initial data that lies in the inhomogeneous Sobolev space Hx1 R3 . Furthermore,  3/2    3/2  3/2    u0 L2 R3 u0 H˙ 1 R3 1 + u0 L2 (R3 ) u0 H˙ 1 (R3 ) . x x x( ) x( ) (2.20) Both the left- and right-hand sides of (2.20) are invariant under the scaling symmetry (2.2), when p = 2 and d = 3.  5 u 5

Lt,x (R×R3 )

Remark

Compare Theorem 2.7 to Theorem 1.32.    1/2  Proof By Theorem 2.4, since Hx1 R3 → H˙ x R3 , (2.19) is locally well posed on some interval I with [−T, T ] ⊂ I, T (u0 ) > 0. Proof of global well-posedness is a straightforward application of conser vation of mass and energy. By the Sobolev embedding theorem, H˙ x1 R3 →

48

The Cubic NLS in Dimensions Three and Four

 6

 Lx R3 , so

     2   E (u0 )  u0 H˙ 1 (R3 ) 1 + u0 L2 (R3 ) u0 H˙ 1 (R3 ) , x

x

x

(2.21)

and then by interpolation, for any t ∈ I,         u (t) 2 1/2 3  u (t)  2 3 u (t)  ˙ 1 3  u (t)  2 3 E (u (t))1/2 Lx (R ) Lx (R ) Hx (R ) H˙ x (R )  1/2          u0 L2 (R3 ) u0 H˙ 1 (R3 ) 1 + u0 L2 (R3 ) u0 H˙ 1 (R3 ) , x x x x (2.22)   which implies the uniform bound u ∞ ˙ 1/2 < ∞. Moreover, by H¨older’s Lt Hx (I×R3 ) inequality in time, the Sobolev embedding theorem, and conservation of energy, for any t0 ∈ I,   i(t−t )Δ e 0 u (t0 )  5 Lt,x ([t0 −T,t0 +T ]×R3 )  1/10  9/10    T 1/5 u (t0 )  9  T 1/5 u (t0 ) L2 R3 u (t0 ) H˙ 1 R3 ( ) x x( ) H˙ x10 (R3 ) 9/20    1/10  9/10     T 1/5 u0 L2 R3 u0 H˙ 1 R3 1 + u0 L2 (R3 ) u0 H˙ 1 (R3 ) . (2.23) x x x( ) x( ) The estimates (2.22) and (2.23) imply a uniform lower bound on T (u (t0 )), t0 ∈ I. Iterating local well-posedness directly implies I = R. The proof of scattering uses the Morawetz estimate. Recall that the momentum  T0 j (t, x) dx is a conserved quantity. Moreover, when μ = +1, the time derivative of T0 j is the derivative of a negative definite quantity, plus three derivatives of a lowerorder term. Indeed, by (1.115) and (1.116),    ∂t T0 j (t, x) = −4∂k Re ∂ j u (∂k u) − ∂ j |u|4 + ∂ j Δ|u|2 . Integrating by parts,

∂t





x j T0 j (t, x) dx = 4

|∇u (t, x) |2 dx + 3



|u (t, x) |4 dx ∼ E (u (t)) .

This fact, combined with an appropriate choice of Y in (1.111), and the fundamental theorem of calculus, gives estimates of certain space-time integrals of u (t, x). Take (1.111) with Y = h (x) u¯ (t, x), and let 

M (t) =

h (x) |u (t, x) |2 dx.

(2.24)

2.2 Large Data Scattering for Radial, 3D Cubic NLS

49

If N = |u| p u, (1.112) implies

∂t M (t) = −



h (x) ∂ j T0 j (t, x) dx,

and (1.119) and (1.120) imply        ∂tt M (t) = h (x) Δ|u|4 + ∂ j ∂k 4 Re ∂ j u (∂k u) − ∂ j ∂k |u|2 dx. (2.25) Integrating by parts,

∂tt M (t) 

=

Rd

(−ΔΔh (x)) |u (t, x) |2 + 4∂ j ∂k h (x) · Re (∂ j u∂k u) ¯ + (Δh (x)) · |u (t, x) |4 dx. (2.26)

The choice h (x) = |x| gives a Morawetz estimate for the nonlinear Schr¨odinger equation. Theorem 2.8 (Morawetz estimate) If u is a smooth solution to (2.4) on I ⊂ R when d = 3, 4,   I

and

    |u (t, x) |4 dx dt  uL∞ L2 (I×Rd ) uL∞ H˙ 1 (I×Rd ) x t t x |x|   I |x||I|1/2

Remark (2.28). Proof

|u (t, x) |4 dx dt  |I|1/2 E (u0 ) . |x|

(2.27)

(2.28)

Lin and Strauss (1978) proved (2.27) and Bourgain (1999) proved

When h (x) = |x|, Δh (x) =

d −1 , |x|

∂ j ∂k h (x) =

δ jk x j xk − 3, |x| |x|

(2.29)

and when d = 3, ΔΔh (x) = −8πδ (x) ,

(2.30)

where (2.30) holds in a distributional sense. Indeed,





2 2 2 2 − ΔΔ|x| = − ∂rr + ∂r . ∂rr + ∂r r = − ∂rr + ∂r r r r r If f is a Schwartz function then switching to polar coordinates and integrating

50

The Cubic NLS in Dimensions Three and Four

by parts, 



2 2 dr d θ f (r, θ ) r2 ∂rr + ∂r r r S2 0

 ∞ 2 = fr (r, θ ) r2 ∂r dr d θ r S2 0

f (x) (−ΔΔ|x|) dx = −

 ∞

= −2

 ∞ 0

S2

fr (r, θ ) dr d θ = 8π f (0) .

Therefore, by the fundamental theorem of calculus in time, (2.26), (2.29), and (2.30) imply 8π

 I

|u (t, 0) |2 dt +

 

 

 4  |∇u (t, x) |2 − |∂r u (t, x) |2 dx dt |x|

I

2 + |u (t, x) |4 dx dt |x| I   ≤ 2 sup |∂t M (t) | ≤ 2u ∞

Lt Lx2 (I×R3 )

t∈I

  u ∞ ˙ 1 . L H (I×R3 ) t

(2.31)

To make (2.31) completely rigorous, one could take a smooth approximation of the initial data and then show that (2.31) holds in the limit, using, by now standard, perturbative arguments. Since |∂r u| ≤ |∇u|, all three terms on the left-hand side of (2.31) are positive, proving (2.27) when d = 3.     To prove (2.28) choose h (x) = |x|ψ Rx , where ψ ∈ C∞ R3 is a smooth function, ψ (x) = 1 on |x| ≤ 1, and ψ (x) = 0 for |x| > 2. For |x| < R, (2.29) and (2.30) hold. When |x| > R, |∇h (x) |  1,

|∂ j ∂k h (x) | 

1 , R

|Δ2 h (x) | 

1 . R3

(2.32)

By the Sobolev embedding theorem and H¨older’s inequality, |∂t M (t) | 

 |x|≤2R

 2 |u (t, x) | |∇u (t, x) | dx  R∇uL2 (Rd ) .

By (2.29) and (2.32), 

 −1  ∇u2 2 d . L (R ) R   By (2.30), (2.32), Hardy’s inequality, and the support of ψ Rx , 4

Re (∂k u∂ j u) ¯ (∂ j ∂k h (x)) dx 



− |u (t, x) | (ΔΔh (x)) dx = −



2



|u| (ΔΔh (x)) dx −



2

|x|≤R

 −1  ∇u2 2 d . Lx (R ) R

|x|>R

|u|2 (ΔΔh (x)) dx

2.2 Large Data Scattering for Radial, 3D Cubic NLS   Finally, by (2.29), (2.32), and the support of ψ Rx , 

|u (t, x) |4 · Δh (x) dx ≥ ≥

 

|u (t, x) |4 dx −C |x| |x|≤R

51



|u (t, x) |4 dx |x| |x|≥R

C |u (t, x) |4 dx − E (u0 ) . |x| R

Integrating ∂tt M (t) in time, by the fundamental theorem of calculus,   I

|I| |u (t, x) |4 dx dt  E (u0 ) + RE (u0 ) . |x| R |x|≤R

Choosing R ∼ |I|1/2 proves (2.28) when d = 3. When d = 4 the proofs of (2.27) and (2.28) are similar, except 3 . |x|3

−ΔΔ|x| =

  Next, by the fundamental theorem of calculus, if f ∈ H 1 R3 is a radial function, f 2 (r) = − ≤

 ∞

2 r2

r

   ∂s f 2 (s) ds = −2

 ∞ r

∞

r

s2 | f  (s) | | f (s) | ds 

f  (s) f (s) ds

   1  f ˙1 3  f 2 3 . R Lx (R ) H 2 ( ) r

(2.33)

This is the only time the radial symmetry is used in the proof of Theorem 2.7. Combining (2.27) and (2.33),  5 u 5

Lt,x (

R×R3

)

  ≤  |x|u  3/2  u ∞

∞ Lt,x

(

R×R3

Lt H˙ x1 (I×R3 )

)·

  R R3

 3/2 u ∞ L L2 t

x

|u (t, x) |4 dx dt |x|

(I×R3 )

.

(2.34)

Then conservation of energy and (2.21) directly give (2.20). Remark The interaction Morawetz estimate (Lemma 3.23), conservation of mass, and conservation of energy imply nonradial scattering for initial data satisfying the conditions of Theorem 2.7. Remark Kenig and Merle (2010) proved global and scatter  well-posedness   ing for (2.19), assuming an a priori bound on u (t) ˙ 1/2 3 . Hx (R ) Remark See Duyckaerts et al. (2008) and Holmer and Roudenko (2008) for results in the focusing case.

52

The Cubic NLS in Dimensions Three and Four

2.3 The Radially Symmetric, Cubic Problem in Four Dimensions Study of the four-dimensional cubic problem with large data represented an important development in the study of the scattering theory for critical, defocusing nonlinear Schr¨odinger equations. Observe that the small data results of Theorem 1.17 and Theorem 2.1 prove global well-posedness and scattering for the nonlinear Schr¨odinger equation with initial data lying in the critical Sobolev space, while the large data results in Theorem 1.32 and Theorem 2.7 hold only for initial data in a subspace of the critical Sobolev space. In other words, for (1.130) and (2.34), the right-hand sides are norms that bound the critical Sobolev spaces by interpolation, and are invariant under the critical scaling, but are strict subsets of the critical Sobolev space, which naturally raises the question of the long-term behavior of generic initial data in the critical Sobolev space. Bourgain (1999) proved global well-posedness and scattering for the defocusing, energy-critical problem for radially symmetric data lying in H˙ 1 in dimensions three and four. In keeping with the theme of this chapter, only the proof of scattering for the cubic, four-dimensional nonlinear Schr¨odinger equation will be discussed here. Proof of the corresponding result for the threedimensional, quintic problem is similar. This result proved to be quite foundational to the study of other nonlinear Schr¨odinger problems, and many of the same ideas were then incorporated into the concentration compactness method, which will be studied in the next chapter. Therefore, there is great advantage in studying the argument of Bourgain (1999) in detail. The reader is encouraged to pay careful attention to how the Morawetz estimate and conservation laws are used in concert to bound the scattering size of a solution. To be sure, Grillakis (2000) independently proved global well-posedness for the radial, three-dimensional energy-critical problem. These results were subsequently extended to higher dimensions by Tao (2005). The upcoming proof contains many of the arguments found in Grillakis (2000) and Tao (2005) when such arguments simplify the exposition, without obscuring the ideas in Bourgain (1999). Theorem 2.9 The defocusing, cubic, initial value problem iut + Δu = F (u) = |u|2 u,

  u (0, x) = u0 ∈ H˙ 1 R4

(2.35)

is globally well posed and scattering for u0 radial. Moreover, there exists a function C : [0, ∞) → [0, ∞), such that   u 6 ≤ C (E (u0 )) , (2.36) Lt,x (R×R4 )

2.3 The Radially Symmetric, Cubic Problem in Four Dimensions

53

where E (u (t)) is the conserved energy. Proof

By the Sobolev embedding theorem,     u 4 4  u ˙ 1 4 , Lx (R ) H (R )

and therefore

 4  2 E (u0 )  u0 H˙ 1 (R4 ) + u0 H˙ 1 (R4 ) ,

so Theorem 2.1 implies that for E (u0 ) small,       u 6  u0 H˙ 1 (R4 )  E (u0 )1/2 ∼ u0 H˙ 1 (R4 ) . Lt,x (R×Rd )

(2.37)

Starting with (2.37) as a base case, and then arguing in a manner analogous to the induction argument for natural numbers, to prove (2.36) it suffices to show that there exists some decreasing, continuous function ε (E) : [0, ∞) → (0, ∞), such that C (E) C(E−ε (E)) 1.

(2.38)

Naively applying Theorem 2.6 does not suffice to prove (2.38), because Theorem 2.6 alone is not strong enough to disprove the possibility that ε (E)  0 as E  E0 , for some E0 < ∞. The proof of (2.38) is considerably more complicated and involves considering a number of different cases. The first case to consider is the case in which the initial data has small Besov norm.   Lemma 2.10 For u0 ∈ H˙ 1 R4 ,

2/3  itΔ   1/3   e u0  6     u P sup u . (2.39)  j 0 H˙ 1 (R4 ) 0 H˙ 1 R4 Lt,x (R×R4 ) ( ) j Then by (2.18), there exists an ε (E) > 0 such that (2.35) is globally well posed and scattering when      sup Pj u0 H˙ 1 (R4 ) ≤ ε u0 H˙ 1 , j with scattering size bounded by

2/3  1/3       sup Pj u0 H˙ 1 (R4 ) . 6 R×R4  u0 ˙ 1 Lt,x ( ) H (R4 ) j

  u Proof

Make a Littlewood–Paley decomposition,      itΔ 6 e u0   ∑ eitΔ Pj1 u0  eitΔ Pj2 u0  eitΔ Pj3 u0  j1 ≤ j2 ≤ j3 ≤ j4 ≤ j5 ≤ j6

    × eitΔ Pj4 u0  eitΔ Pj5 u0  eitΔ Pj6 u0 .

54

The Cubic NLS in Dimensions Three and Four

Then  itΔ 6 e u0  6 L

t,x



∑

j1 ≤ j2 ≤ j3 ≤ j4 ≤ j5 ≤ j6

 itΔ      e Pj u0  ∞ eitΔ Pj u0  ∞ eitΔ Pj u0  ∞ 4 1 2 3 L L L L t,x

t,x

t

x

      × eitΔ Pj4 u0 L∞ L4 eitΔ Pj5 u0 L2 L4 eitΔ Pj6 u0 L2 L4 . (2.40) t

x

t

x

t

x

By Strichartz estimates, the Sobolev embedding theorem, and the Cauchy– Schwarz inequality,       (2.40)  ∑ 2 j1 + j2 Pj1 u0 H˙ 1 (R4 ) Pj2 u0 H˙ 1 (R4 ) eitΔ Pj3 u0 L∞ L4 j1 ≤ j2 ≤ j3 ≤ j4 ≤ j5 ≤ j6

t

x

      × eitΔ Pj4 u0 L∞ L4 Pj5 u0 L2 (R4 ) Pj6 u0 L2 (R4 ) x x t x

4  2    u0 H˙ 1 (R4 ) sup Pj u0 H˙ 1 (R4 ) , (2.41) j which proves (2.39). The right-hand side of (2.41) can be improved to  itΔ 6 e u0  6 Lt,x (R×R4 )

2

2  2      u0 H˙ 1 (R4 ) sup Pj u0 H˙ 1 (R4 ) sup eitΔ Pj u0 L∞ L4 (R×R4 ) , t x j j (2.42) which will be relevant later. √  Now let ε0 = ε 2E , where E = E (u0 ). Suppose u is a solution to (2.35) on the maximal interval of existence I. For any t ∈ I, let     (2.43) j (t) = inf j ∈ Z : Pj u (t) H˙ 1 ≥ ε0 .   If Pj u (t) H˙ 1 < ε0 for all j ∈ Z then set j (t) = +∞. If there exists t0 ∈ I such that j (t0 ) = ∞ then by conservation of energy,   u (t0 ) 2˙ 1 ≤ 2E, so Theorem 2.1 and (2.41) imply H  6 u 6  E ε04 . Lt,x (R×R4 ) If j (t) < ∞ for all t ∈ I then consider two cases separately: one in which u (t) is large in a Besov sense at frequencies far away from j (t), and the other in which u (t) is small in a Besov sense far away from j (t). In the former, the scattering size may be estimated using induction on energy.

2.3 The Radially Symmetric, Cubic Problem in Four Dimensions 



ε02

55



Theorem 2.11 There exists L E,C E − 8 ∈ Z such that if for some t0 ∈ I     ε 2  ε 2  there exists j1 > j (t0 ) + L E,C E − 80 or j1 < j (t0 ) − L E,C E − 80 satisfying   Pj u (t0 )  ˙ 1 4 ≥ ε0 , 1 H (R ) 2 then I = R and

  u

6 R×R4 Lt,x ( )



ε2 C E− 80



1.

Proof If u (t0 ) satisfies the conditions of the theorem, then u (t0 ) may be split into two pieces with energy strictly less than E. By induction, each piece will yield a global, scattering solution, and the two solutions will interact very weakly, and can be estimated using Theorem 2.6. For any l ∈ Z, j (t0 ) ≤ l ≤ j1 ,   u (t0 ) 2˙ 1 4 H (R )  2 2   = u≤l (t0 ) H˙ 1 (R4 ) + u>l (t0 ) H˙ 1 (R4 ) + 2 u≤l (t0 ) , u>l (t0 ) H˙ 1 (R4 ) , (2.44) and

      u (t0 ) 4 4 4 = u≤l (t0 ) 4 4 4 + u>l (t0 ) 4 4 4 Lx (R ) Lx (R ) Lx (R )    + O  |u>l (t0 ) | |u≤l (t0 ) | |u (t0 ) |2 L1 (R4 ) . x

By the Cauchy–Schwarz and Bernstein inequalities,    1  1 u (t0 ) 2˙ 1 4 2 u>l (t0 ) , u≤l (t0 ) H˙ 1 (R4 )   ∑ H (R ) L − 1 j(t )l (t0 )  2 u≤l (t0 )  ∞ L L L

j(t0 ) 0 sufficiently large, by (2.70) and Bernstein’s inequality,  j

   χ 2 x Pj u (t)    4 4 C Lx (R )  1/2  1/2  2 j/2  ≥ Pj u (t) L4 (R4 ) − 1/2 Pj u (t) L2 R4  |x| Pj u (t) L∞ R4 x x( ) x( ) C E 1. Furthermore,  j      χ 2 x , Pj u   4 C Lx    j 

j

 4j   j  2x 2y  2 = K 2 (x − y) χ χ − u (y) dy   4 C C Lx      4j    j  2 j |x − y| 1   u (y) dy  K 2 (x − y)  4  C u Lx4 . 2 C Lx Therefore, for C (E) sufficiently large,  j    Pj χ 2 x u  E 1,   4 C Lx

2.3 The Radially Symmetric, Cubic Problem in Four Dimensions

61

so by the Littlewood–Paley theorem, 

1 2j |u (t, x) |4 dx  . |x| C

(2.71)

Then (2.62), (2.69), and (2.71) imply 

 Ik,l

1 1 |u (t, x) |4 dx dt  2− j ∼ε1 ,E |Ik,l |1/2 . |x| C

(2.72)

Meanwhile, by the Morawetz estimate in (2.28), 

 (0)

Ik

|u (t, x) |4 (0) dx dt  E|Ik |1/2 , |x|

which by (2.72) implies

∑

(0)

|Ik,l |1/2 ε1 ,E |Ik |1/2 .

(2.73)

(0) Ik,l ⊂Ik

Observe that (2.73) gives an upper bound on the number of consecutive (0) intervals Ik,l ⊂ Ik of the same length. For example, if |Ik,l | = 1 for each l, then the number of Ik,l intervals would be bounded by a constant that depends on E and ε1 (E), which would then imply the desired (2.51) bound. However, (2.73) does not automatically imply a uniform bound on the total (0) number of Ik,l subintervals of Ik under the general constraint of (2.50). For example, one could take |Ik,l | = 2−l , l ≥ 0 and then (2.73) would not give any kind of upper bound on the number of Ik,l intervals. However, in the second case, conservation of energy would give a bound. Indeed, by (2.73), the general fact that l 1 ⊂ l 2 ⊂ l ∞ , and H¨older’s inequality,

∑

(0)

|Ik,l |1/2 ∼ε1 ,E |Ik |1/2

(0) Ik,l ⊂Ik

(0)

sup |Ik,l |1/2 ∼ε1 ,E |Ik |1/2 .

and

(2.74)

(0) Ik,l ⊂Ik

Therefore, (2.74) implies that there are ≤ N (ε1 , E) intervals Ik,l such that (0)

|Ik,l | ∼ |Ik |,

(2.75)

where N (ε1 , E) is some fixed quantity. If    u 6  (0) 4 = M, Lt,x Ik ×R

with M  N, then by the pigeonhole principle, after removing the intervals Ik,l (1) (0) that satisfy (2.75), there exists an interval Ik ⊂ Ik such that   u

  6 I (1) ×R4 Lt,x k

≥

M . N

62

The Cubic NLS in Dimensions Three and Four

Moreover, by (2.74) there exists ε2 (E) > 0, ε2  ε1 such that (1)

(0)

|Ik | ≤ (1 − ε2 ) |Ik |. (1)

n Now if M N  N the same procedure may be applied to Ik . In fact, if M  N , there is a nested sequence of intervals (n)

(n−1)

Ik ⊂ Ik

(0)

⊂ · · · ⊂ Ik ,

for each 1 ≤ m ≤ n, (m)

(m−1)

|Ik | ≤ (1 − ε2 ) |Ik

|,

(2.76)

and there exists (m)

(m)

Ik,l ⊂ Ik

(m)

(m)

such that |Ik,l | ∼ε1 ,E |Ik |.

(2.77)

(n)

(m)

Fix t0 ∈ Ik . Relation (2.77) implies that for all 0 ≤ m ≤ n, there exists Ik,l ⊂ (m)

Ik

such that

  (m) (m) |Ik,l | ε1 ,E dist t0 , Ik,l .

(2.78)

Inequality (2.78) implies that some of the energy at level j (t) (see (2.43)) remains at t0 . By (1.112), for any j ∈ Z,     d P> j u (t) 2 2 = Im P> j uP> j F (u) dx Lx dt    = Im P> j u (P> j F (u) − F (P> j u)) dx     2   P> j uL2 P≤ j uL∞ uL4

(2.79)

    d P≤ j u (t) 2 2 = Im P≤ j uP≤ j F (u) dx L x dt    = Im P≤ j u (P≤ j F (u) − F (P≤ j u)) dx     2   P≤ j uL∞ P> j uL2 uL4 .

(2.80)

x

and

x

x

x

x

x

(m)

Choose tm ∈ Ik,l . Assume without loss of generality that tm > t0 . Now for L1 ≥ 2L, (2.45) and (2.62) imply 1 L1 − L

 tm

∑

      u (t) 2 4 4 P≤ j u (t)  ∞ 4 P> j u (t)  2 4 dt L (R ) L (R ) L (R )

t0 j(t )−L ≤ j≤ j(t )−L m m 1

ε1 ,E

x

1 1 (m) |I j,k | ε1 ,E 2−2 j(tm ) . L1 − L L1 − L

x

x

(2.81)

2.3 The Radially Symmetric, Cubic Problem in Four Dimensions

63

Therefore, the pigeonhole principle implies that there exists some j (tm ) − L1 ≤ j ≤ j (tm ) − L such that the integral  tm  t0

     P> j u (t)  2 P≤ j u (t)  ∞ u (t) 2 4 dt L L L x

x

x

is bounded by the right-hand side of (2.81). Then by definition of j (tm ) (see (2.43)) and Bernstein’s inequality, for L1 (ε1 , E, L) sufficiently large, (2.79) and (2.80) imply   Pj(t )−L ≤·≤ j(t )+L u (t0 )  2 4  ε0 2−2 j(tm )−L , m m 1 1 Lx (R ) and therefore

 Pj(t

m )−L1 ≤·≤ j(tm )+L1

 u (t0 ) H˙ 1 (R4 )  ε0 2−L−L1 . x

(2.82)

Inequalities (2.66) and (2.76) also imply that there exists some constant J (L1 , ε2 ) such that if M  N n , then there exist at least J(L n,ε ) integers 1 2 0 ≤ m ≤ n such that the intervals j (tm ) − L1 ≤ · ≤ j (tm ) + L1 are disjoint. Conservation of energy and (2.82) then give an upper bound on the number of such m , which in turn gives an upper bound on J(L n,ε ) , which finally gives 1 2 an upper bound on M, such that    u 6  (0) 4 = M. Lt,x Ik ×R

This finally concludes the proof of Theorem 2.9.

3 The Energy-Critical Problem in Higher Dimensions

3.1 Small Data Energy-Critical Problem Plugging sc = 1 into (2.3) shows that the general, energy-critical problem 4 . In dimensions d ≥ 5, this means that the is given by (2.1) with p = d−2 energy-critical problem has the added difficulty that the nonlinearity is no 4 longer smooth as a function of u. Rather, the function F (x) = |x| d−2 x has a 4 . This fact complicates derivative that is only of order d−2   H¨older continuous   the computation of F (u) − F (v) N˙ 1 , a necessary part of proving scattering for small data. Remark The small data problem when d = 3 (p = 4) may be analyzed in a manner very similar to the analysis in dimension d = 4. Theorem 3.1 (Global well-posedness  and scattering for small critical data) There exists ε0 (d) > 0 such that if u0 H˙ 1 (Rd ) = ε < ε0 , then 4

iut + Δu = F (u) = μ |u| d−2 u, μ = ±1, (3.1)   u (0) = u0 ∈ H˙ 1 Rd   is globally well posed and scattering in H˙ 1 Rd . In fact, this result holds for     itΔ  e u0  2(d+2) = ε ≤ ε0 u0 H˙ 1 (Rd ) . Lt,xd−2 (R×Rd ) Proof This result was proved in Cazenave and Weissler (1988). The proof presented here is the proof in Tao and Visan (2005), which also contained a perturbative result for the energy-critical problem. When d = 3, 4, 5, and denoting 2(d+2) d−2 by B, define the set of functions         X = u : uS˙1 (R×Rd ) ≤ C (d) u0 H˙ 1 (Rd ) ; uLB (R×Rd ) ≤ 2ε0 . t,x 64

3.1 Small Data Energy-Critical Problem

65

By the chain rule and Strichartz estimates,       Φ (u)  ˙1  u0 H˙ 1 (Rd ) + ∇u S (R×Rd )

2(d+2) Lt,x d

(R×Rd )

4   d−2 u B Lt,x (R×Rd )

and     4     u ˙1 Φ (u)  B ≤ eitΔ u0 LB (R×Rd ) +C (d) u d−2 . B R×Rd Lt,x (R×Rd ) S (R×Rd ) L ) t,x t,x (     4 Therefore, when d = 3, 4, 5, for ε0 d, u0 H˙ 1 > 0 sufficiently small, d−2 > 1, ε < ε0 , and u ∈ V imply that Φ (u) ∈ V . When d ≥ 6, choose q satisfying

4 1 4 2 + − = 1. q d − 2 q d (d + 2) Then let q1 satisfy 2 =d q1



1 1 d −2 − , − 2 q d +2

and define the set of functions      X = u : uS˙1 (R×Rd ) ≤ C (d) u0 H˙ 1 (Rd ) ;

  4  |∇| d+2 u

q q Lt 1 Lx

 ≤ 2 ε 1 . (R×Rd )

By the fractional chain rule, Duhamel’s principle, the Sobolev embedding theorem, 1q > 12 − d1 , and Lemma 1.10,   4  |∇| d+2 Φ (u) 

q

q

Lt 1 Lx (R×Rd )

   d+2  4 4   |∇| d+2 eitΔ u0 Lq1 Lq (R×Rd ) +  |∇| d+2 u d−2 q1 t

q

Lt Lx (R×Rd )

x

.

(3.2)

By the product rule, the Sobolev embedding theorem, and Strichartz estimates,       Φ (u)  ˙1  u0 H˙ 1 (Rd ) + ∇u S (R×Rd )

2d Lt2 Lxd−2

(R×Rd )

 4  4  |∇| d+2 u d−2 q

q

Lt 1 Lx (R×Rd )

.

(3.3) By interpolation and Strichartz estimates, there exists 0 < α (d) < 1 such that  1−α   α  4  |∇| d+2 (3.4) eitΔ u0 Lq1 Lq (R×Rd )  u0 H˙ 1 (Rd ) eitΔ u0  2(d+2) x t Lt,xd−2 (R×Rd ) and

  4  |∇| d+2 u

q q Lt 1 Lx

(

R×Rd

 1−α  α u ˙1 u 2(d+2)  d S (R×R ) ) Lt,xd−2

, (R×Rd )

(3.5)

and therefore ε1 d,u0  ˙ 1 ε0α . So, for ε0 (d, u0 H˙ 1 ) small enough, Φ : X → X. H

66

The Energy-Critical Problem in Higher Dimensions To prove a contraction, by Taylor’s theorem,

F (u) − F (v) = (v − u)

 1 0

Fz (u + τ (v − u)) d τ + (v − u)

 1 0

Fz¯ (u + τ (v − u)) d τ , (3.6)

where Fz and Fz¯ are the complex derivatives



∂F ∂F 1 ∂F 1 ∂F −i +i Fz = and Fz¯ = . 2 ∂x ∂y 2 ∂x ∂y By the chain rule, ∇F (u (x)) = Fz (u (x)) ∇u (x) + Fz¯ (u (x)) ∇u¯ (x) . For 3 ≤ d ≤ 5, Fz and Fz¯ are differentiable functions, so by the chain rule and product rule (and again using the notation B for 2(d+2) d−2 ),   Φ (u) − Φ (v)  ˙1 S (R×Rd )    6−d   d u − vS˙1 (R×Rd ) uS˙1 (R×Rd ) + vS˙1 (R×Rd )  |u| + |v|  d−2 B R×Rd Lt,x ( )  6−d   ε0d−2 uS˙1 (R×Rd ) + vS˙1 (R×Rd ) u − vS˙1 (R×Rd ) .     This gives a contraction in S˙1 R × Rd for ε0 d, u0 H˙ 1 (Rd ) sufficiently small and d = 3, 4, 5. For dimensions d ≥ 6 the proof will utilize (3.2). To simplify notation, write     4   u =  |∇| d+2 u q1 q . S (I×Rd ) Lt Lx (I×Rd ) By the fractional product rule, fractional chain rule, (3.2), and (3.6),   Φ (u) − Φ (v)  S (R×Rd )    4     d−2 d u − vS (R×Rd ) uS (R×Rd ) + vS (R×Rd ) 4    ε1d−2 u − vS (R×Rd ) .   d This implies a contraction in S R  × R for ε1 > 0 sufficiently small, which by (3.4) and (3.5) follows from ε0 d, u0 H˙ 1 (Rd ) > 0 sufficiently small. Thus   there exists a unique u ∈ S R × Rd satisfying u (t) = eitΔ u0 − i

 t 0

ei(t−τ )Δ F (u (τ )) d τ .

       By (3.3), F (u) N˙ 1 (R×Rd )  u0 H˙ 1 (Rd ) , proving u (t) ∈ Ct0 R, H˙ x1 Rd .

3.1 Small Data Energy-Critical Problem

67

Furthermore, by the chain rule,   F (u) − F (v)  ˙ 1 N (R×Rd )    4     d−2  u − vS˙1 (R×Rd ) uS (R×Rd ) + vS (R×Rd )     4    v ˙1 u ˙1 + + u − v d−2 d d d R×R R×R S S ( ) ( ) , S (R×R ) which will be useful later for perturbative results, since then, 4   4       u − v ˙1  u0 H˙ 1 (Rd ) u − v d−2 + ε1d−2 u − vS˙1 (R×Rd ) . d S (R×Rd ) S (R×R )

Thus ε1 > 0 is small and u − vS (R×Rd ) → 0 implies Φ (u) − Φ (v) S˙1 (R×Rd ) → 0. Continuous dependence on initial data again follows from a perturbation lemma. Lemma 3.2 (Perturbation lemma) Let I be a compact time interval and let w be an approximate solution to (3.1), (i∂t + Δ) w = F (w) + e, for some e. Suppose also that   w

Lt∞ H˙ 1 (I×Rd )

≤ E.

  Let u solve (3.1) on I × Rd , and for t0 ∈ I let u (t0 ) ∈ H˙ 1 Rd be close to w (t0 ),   u (t0 ) − w (t0 )  ˙ 1 d ≤ E  , H (R ) for some E  > 0. Moreover, assume the smallness conditions     u ˙ = ∇u 2(d+2) ≤ ε0 , W (I×Rd ) Lt,x d (I×Rd )   e ˙ 1 N

and

(I×Rd ) ≤ ε ,

  i(t−t )Δ e 0 (u (t0 ) − w (t0 ))  ˙ ≤ ε, W (R×Rd )

for some 0 < ε ≤ ε0 (E, E  ). Then   d 4 2 u − w ˙  ε + ε d−2 ( d−2 − d ) , W (I×Rd )

(3.7)

(3.8)

68

The Energy-Critical Problem in Higher Dimensions   d 4 2 u − w ˙1  E  + ε + ε d−2 ( d−2 − d ) , S (I×Rd )   w ˙1 S

and

d

4

(3.9)

2

 d−2 ( d−2 − d ) , (I×Rd )  E + E + ε + ε

   (i∂t + Δ) (u − w) + e ˙ 1 N

(I×Rd )  ε + ε

d d−2

(3.10)

4 −2 ( d−2 d ).

(3.11)

Taking e = 0 implies 4 −2     d ( d−2   d) u − w ˙1  u (t0 ) − w (t0 ) H˙ 1 (Rd ) + u (t0 ) − w (t0 )  d−2 , S (R×Rd ) H˙ 1 (Rd ) (3.12) which proves continuous dependence on the initial data.   To prove scattering forward in time, since F (u) ∈ N˙ 1 R × Rd , set

u+ = u (t0 ) − i Then

 ∞

e−itΔ F (u (t)) dt.

t0

  lim F (u) N˙ 1 ([T,∞)×Rd ) = 0,

T →∞

so

  iT Δ  e u+ − u (T )  ˙ 1 d =   H (R )

∞

T

  ei(t−τ )Δ F (u (τ )) d τ 

H˙ 1 (Rd )

→ 0.

The argument proving scattering backward in time is identical. This completes the proof Theorem 3.1, assuming Lemma 3.2 is true. Proof of Lemma 3.2 The proof of Lemma 3.2 is also simpler for 3 ≤ d ≤ 5 than for d ≥ 6. By Strichartz estimates, if v = u − w,   v ˙ W (I×Rd )      ei(t−t0 )Δ (w (t0 ) − u (t0 )) W˙ (I×Rd ) + F (w) − F (u) + eN˙ 1 (I×Rd ) . When 3 ≤ d ≤ 5, applying the product rule to F (w) − F (u),

4 4       d−2   d−2    F (w) − F (u)  ˙ 1     v W˙ (I×Rd ) u ˙ + w ˙ . N (I×Rd ) W (I×Rd ) W (I×Rd ) This implies 4

4     d−2 d−2      ε + v W˙ (I×Rd ) ε0 + v ˙ , W (I×Rd ) W (I×Rd )

  v ˙

3.1 Small Data Energy-Critical Problem

69

which proves (3.8) and (3.11). Next,       u − w ˙1  w (t0 ) − u (t0 ) H˙ 1 (Rd ) + F (w) − F (u) + eN˙ 1 (I×Rd ) S (I×Rd )  E + ε. This takes care of (3.9). Finally, since       u ˙1  u (t0 ) − w (t0 ) H˙ 1 (Rd ) + w (t0 ) H˙ 1 (Rd ) S (I×Rd ) 4     d−2 u + ∇u 2(d+2) , 1 ˙ S (I×Rd ) Lt,x d (I×Rd ) we have

  u ˙1 S

 (I×Rd )  E + E + ε ,

which proves (3.10). For d ≥ 6, since ∂t eitΔ = iΔeitΔ , the Sobolev embedding theorem in time and (3.7) imply   i(t−t )Δ e 0 v (t0 )   ε. S (I×Rd ) This implies       v  ei(t−t0 )Δ v (t0 ) W˙ (I×Rd ) + w − uS (I×Rd ) S (I×Rd )

  4   4   × w d−2 d + u d−2 d + eN˙ 1 (I×Rd ) , S (I×R ) S (I×R ) which implies



4 4   d−2     d−2        ε + v S (I×Rd ) u ˙ + v ˙ S (I×Rd ) W (I×Rd ) W (I×Rd ) 4       4  ε + ε0d−2 vS (I×Rd ) + vS (I×Rd ) v d−2 . W˙ (I×Rd )

  v

(3.13)

By Strichartz estimates and the product rule, 4     d−2   v ˙ u v ˙  ε + d d W (I×R ) W (I×R ) W˙ (I×Rd )    4 −2 2    d−2    d   v d u ˙ v + v + W (I×Rd ) W˙ (I×Rd ) S (I×Rd ) W˙ (I×Rd ) 4   4 2 2   d−2 − d   v  ε + ε0d−2 vW˙ (I×Rd ) + ε0 v d˙ S (I×Rd ) W (I×Rd )   4 − 2  1+ 2 + v d−2 d d v ˙ d d . (3.14) S (I×R ) W (I×R )

70

The Energy-Critical Problem in Higher Dimensions

Therefore,        4    d+2     v ˙ + v  ε + ε0 v + v ˙ d−2 + v + v ˙ d−2 , S S S W W W which implies d−2     v ˙ + v  ε + ε d−6 . 0 S W

Then, since ε < ε0 and ε0 is small,     v + vW˙  ε0 . S (I×Rd )

(3.15)

Plugging (3.15) into (3.13) implies   v

(3.16)

S (I×Rd )

 ε.

Then plugging (3.16) into (3.14) implies   d 4 2 v ˙  ε d−2 ·( d−2 − d ) . W (I×Rd ) This completes the proof of Lemma 3.2. Combining the modifications in higher dimensions used in the above theorem, and arguing as in Theorems 2.4 and 2.5, we obtain the following theorems. Theorem 3.3 (Energy-critical local well-posedness)   1. For u0 ∈ H˙ 1 Rd , there exists T (u0 ) > 0 such that (3.1) is locally well posed in H˙ 1 Rd on [0, T ]. In particular, (3.1) is well posed on an open interval I ⊂ R, 0 ∈ I. 2. If sup (I) < +∞, where I is the maximal interval of existence of (3.1),   = ∞. lim u 2(d+2) T sup(I) Lt,xd−2 ([0,T ]×Rd ) The corresponding result holds for inf (T ) > −∞. Theorem 3.4 (Scattering) A solution to (3.1) scatters forward in time to some  d itΔ 1 ˙ e u+ , u+ ∈ Hx R , if and only if, for some t0 ∈ R,     u 2(d+2) < ∞ and uL∞ H˙ 1 ([t ,∞)×Rd ) < ∞. 0 t Lt,xd−2 ([t0 ,∞)×Rd ) The backward-in-time scattering result is similar. Proof of long-time perturbative results is also more difficult in dimensions d ≥ 5.

3.1 Small Data Energy-Critical Problem

71

Theorem 3.5 (Long-time perturbations) Let I be a compact time interval and let w be an approximate solution to (3.1) on I × Rd in the sense that (i∂t + Δ) w = F (w) + e. Assume that u solves (3.1),

  u

and

≤ M,

2(d+2)

Lt,xd−2

  u

(I×Rd )

Lt∞ H˙ 1 (I×Rd )

≤ E,

for some M, E > 0. Let t0 ∈ I, and suppose that u (t0 ) is close to w (t0 ),   u (t0 ) − w (t0 )  ˙ 1 d ≤ E  . (3.17) H (R ) Also assume that

  i(t−t )Δ e 0 (u (t0 ) − w (t0 ))  ˙ ≤ε W (I×Rd )

and

  e ˙ 1 N

(I×Rd ) ≤ ε ,

for some 0 < ε ≤ ε1 (E, E  , M, d). Then      d 4 2  u − w ˙ d−2 ( d−2 − d ) , ≤ C E, E , M, d ε + ε d W (I×R )       d 4 2  u − w ˙1 d−2 ( d−2 − d ) , ≤ C E, E , M, d + ε + ε E S (I×Rd ) and

  w ˙1 S

(I×Rd )

  ≤ C E, E  , M, d .

Proof This result was also proved in Tao and Visan (2005). Divide I into  2(d+2)  ∼ 1 + εM0 d−2 subintervals Jk such that on each Jk ,   u 2(d+2) ≤ ε0 , Lt,xd−2 (Jk ×Rd ) where ε0 (d) is a small fixed constant. For each Jk = [ak , bk ],         u ˙1  u (ak ) H˙ 1 (Rd ) + F (u) N˙ 1 (J ×Rd ) + eN˙ 1 (J S (J ×Rd ) k

4 d−2

 E + ε0 This implies

  u ˙1 S

  u ˙1 S

k

(Jk ×Rd ) + ε .

(Jk ×Rd ) M,d E,

d k ×R

)

72

The Energy-Critical Problem in Higher Dimensions

and thus

  u ˙1 S

(I×Rd ) ≤ C (M, d) E.

(3.18)

Therefore, it is possible to partition I into C (E, M, d) subintervals I j such that   u ˙ ≤ ε0 . W (I j ×Rd ) Suppose

   i(t−t j )Δ  (u (t j ) − w (t j ))  e

S (I j ×Rd )

≤ ε,

(3.19)

for ε < ε0 sufficiently small. Then      i(t−t j )Δ  u − w  (u (t ) − w (t )) + ε. e  j j d S (I j ×R ) S (I j ×Rd ) Moreover,

       i(t−t j+1 )Δ   u t j+1 − w t j+1  e S (I×Rd )    i(t−t j )Δ  E,E  e (u (t j ) − w (t j ))  + ε. S (I×Rd )

Therefore, there exists ε1 (M, E, E  , d) sufficiently small such that if 0 < ε ≤ ε1 ,     u − w ≤ C M, E, E  , d ε . (3.20) S (I×Rd ) Plugging (3.20) into (3.14) implies      d 4 2 u − w ˙ ≤ C M, E, E  , d ε + ε d−2 ( d−2 − d ) . W (I×Rd )

(3.21)

Next, by (3.17), (3.21), and Strichartz estimates,       d 4 2  u − w ˙1 d−2 ( d−2 − d ) . ≤ C M, E, E , d E + ε + ε d S (I×R ) Then by (3.18) the proof of Theorem 3.5 is complete.

3.2 Profile Decomposition for the Energy-Critical Problem To prove scattering for the defocusing, energy-critical problem 4

iut + Δu = |u| d−2 u,

u (0, x) = u0 ,

u : R × Rd → C,

with large data, define the quantity corresponding to (2.36).

(3.22)

3.2 Profile Decomposition for the Energy-Critical Problem

73

Definition 3.6 (Scattering size function) For any energy 0 ≤ E < ∞, let   : I is the maximal interval C (E) = sup u 2(d+2) Lt,xd−2 (I×Rd )  of existence of u, and E (u (t)) = E . It is clear that C (0) = 0, and by perturbation theory, C (E) is a continuous function of E, so if C (E) < ∞ for all E < ∞ does not hold, there must exist a minimum E0 for which C (E) < ∞ for all E < E0 , and C (E0 ) = ∞.   Consider a sequence of initial data u0,n ∈ H˙ 1 Rd such that un is the corresponding solution to (3.22), with E (u0,n )  E0 , and   un  2(d+2)  ∞. Lt,xd−2 (R×Rd ) Keraani’s (2001) that u0,n must converge weakly to some nonzero  showed  element of H˙ 1 Rd , modulo symmetries of the equation (3.22).   Theorem 3.7 Suppose { fn } ⊂ H˙ 1 Rd is a bounded sequence of functions,    fn  ˙ 1 d ≤ A, H (R ) and

  lim eitΔ fn 

n→∞

= ε > 0.

2(d+2)

Lt,xd−2

(R×Rd )

  Then after passing to a subsequence there exists φ ∈ H˙ 1 Rd , and sequences λn ∈ (0, ∞), (tn , xn ) ∈ R × Rd such that d−2   λn 2 eitn Δ fn (λn x + xn )  φ (x) ,

(3.23)

weakly in H˙ 1 , and taking −( d−2 2 ) −itn Δ

φn (x) = λn

e



φ

x − xn λn



−( d−2 2 )

= λn



−i tn2 Δ λn

e

φ

 x−x

n

λn

,

 ε 2 α   2  2 2 , lim  fn H˙ 1 (Rd ) −  fn − φn H˙ 1 (Rd ) = φ H˙ 1 (Rd )  ε 2 n→∞ A and

 ε β  B  itΔ B   lim sup e ( fn − φn ) LB (R×Rd ) ≤ ε 1 − c , t,x A n→∞

where B =

2(d+2) d−2

(3.24)

and c, α , and β are constants that depend on dimension.

74

The Energy-Critical Problem in Higher Dimensions Both the energy and (3.22) are preserved under spatial translation Tx0 : u (x) → u (x − x0 ) .

(3.25)

Also by (2.2), the energy and (3.22) are preserved under the scaling symmetry Rλ : u (x) → λ

d−2 2

u (λ x) .

(3.26)

The invariances (3.25) and (3.26) also preserve scattering size. The proof is straightforward for (3.25). For (3.26) make a change of variables x = λ x, t  = λ t. Then 

T0 λ2

 Rd

0

  d−2  2   2(d+2) λ 2 u λ t, λ x  d−2 dx dt =

0

T0  Rd

|u (t, x) |

2(d+2) d−2

dx dt.

Also, if (p, q) is an admissible pair, then   ∇u

p q

Lt Lx ([0,T0 ]×Rd )

=λ

d−2−d 2 p q



T0 0

 Rd

  | (∇u) t  , x |q dx

p/q

1/p dt 

.

Translation in time is a more delicate issue, although (1.15) implies that the H˙ 1 -norm is preserved under the action of eit0 Δ . The Strichartz norm of a free solution is also preserved under translation in time. More will be said later about translation in time. Proof of Theorem 3.7 The proof is slightly different for dimensions d ≥ 6 than for dimensions d = 3, 4, 5, so only the argument for d ≥ 6 will be presented here. Make a Littlewood–Paley decomposition. By H¨older’s inequality, 2d

|eitΔ fn | d−2 

∑

j1 ≤ j2

+

∑

|eitΔ Pj1 fn | |eitΔ Pj2 fn | |eitΔ P≥ j2 fn | d−2

∑

|eitΔ Pj1 fn | |eitΔ Pj2 fn | |eitΔ Pj1 ≤·≤ j2 fn | d−2 .

j1 ≤ j2

+

4

|eitΔ Pj1 fn | |eitΔ Pj2 fn | |eitΔ P≤ j1 fn | d−2

j1 ≤ j2

4

4

By Bernstein’s inequality, H¨older’s inequality, the Sobolev embedding

3.2 Profile Decomposition for the Energy-Critical Problem

75

theorem, and rearranging the order of summation, 2d  itΔ  d−2 e f n 

2d

Lt∞ Lxd−2 (R×Rd )



∑

j1 ≤ j2

× 

 itΔ  e P≤ j fn  1

2d Lt∞ Lxd−4

 itΔ  4 e Pk fn  d−2

∑

j1 ≤ j2

t

x

2d

Lt∞ Lxd−2 (R×Rd )

j1 ≤k≤ j2

∑

(R×Rd )

  P≥ j fn  ∞ 2 2 L L (R×Rd )

    P≤ j fn  ˙ 2 P≥ j fn  2 1 2 L H

 2   4    fn H˙ 1 sup eitΔ Pj fn  d−2

 itΔ  4 e Pk fn  d−2

∑

j1 ≤k≤ j2

.

2d

2d

Lt∞ Lxd−2 (R×Rd )

(3.27)

Lt∞ Lxd−2 (R×Rd )

j

4 Remark If d = 3, 4, 5 then d−2 > 1, which would necessitate handling the sums in (3.27) in a slightly different manner. The Sobolev embedding theorem    2d  H˙ 2 Rd → L d−4 Rd would also fail to hold in dimensions d = 3, 4, again necessitating a modification of (3.27).

Therefore, for some α (d) > 0,   sup eitΔ Pj fn 

2d Lt∞ Lxd−2

j

Remark

(R×Rd )

ε

 ε α A

.

(3.28)

The term α (d) may change from line to line during the proof.

By Bernstein’s inequality,  itΔ    e Pj fn  ∞ 2  2− j  fn H˙ 1 ≤ 2− j A, L L (R×Rd ) t

x

so by interpolation, (3.28) implies  ε α  (d−2)  . sup 2− j 2 Pj eitΔ Pj fn L∞ (R×Rd )  ε t,x A j Thus there exist some xn , tn , jn such that     ε α (d−2)  itn Δ  2 jn 2 .  e Pjn fn (xn )   ε A Taking λn = 2− jn and letting d−2

gn (x) = λn 2



 eitn Δ fn (λn x + xn ) ,

we get |P1 gn (0) |  ε

 ε α A

.

(3.29)

76

The Energy-Critical Problem in Higher Dimensions   By Alaoglu’s theorem, a bounded subset of H˙ 1 Rd is weakly compact, so   there exists φ ∈ H˙ 1 Rd such that after passing to a subsequence, d−2   λn 2 eitn Δ fn (λn x + xn ) = gn (x)  φ (x) ,

(3.30)

weakly in H˙ 1 , which proves (3.23). Moreover, (3.29) implies that there exists     α 2d  ψ ∈ L d+2 Rd ⊂ H˙ −1 Rd , ψ H˙ −1 ∼ Aε with Fourier support on |ξ | ∼ 1, such that for all n,  ε α | ψ , gn |  ε , A which implies  α  2 φ  ˙ 1 d  ε 2 ε . H (R ) A Furthermore, by (3.30) and the invariance of H˙ 1 under (3.25), (3.26), and the operator eit0 Δ , fn , fn H˙ 1 − fn − φn , fn − φn H˙ 1 − φn , φn H˙ 1 = φn , 2 ( fn − φn ) H˙ 1 = 2 φ , gn − φ H˙ 1 → 0.  Next let χ ∈ C0 Rd be a cutoff function: χ = 1 on |x| ≤ 1 and χ is supported on |x| ≤ 2. By Rellich’s theorem, (3.30) implies that for any R < ∞ and t ∈ R, x x eitΔ gn → χ eitΔ φ , χ R R   in L2 Rd , so therefore  ∞

eitΔ gn → eitΔ φ , pointwise almost everywhere in R × Rd . Then by Fatou’s lemma,     B B B lim eitΔ gn LB (R×Rd ) − eitΔ φ LB (R×Rd ) − eitΔ (φ − gn ) LB (R×Rd ) = 0, t,x t,x t,x n→∞ where here and below we write B = 2(d+2) d−2 for clarity. Therefore, (3.24) follows from  ε β  itΔ  e φ  B  ε , d Lt,x (R×R ) A which in turn follows directly from interpolation and the fact that  ε α  itΔ  e P1 gn  . ε 2d A Lt∞ Lxd−2 (R×Rd )

(3.31)

Making a computation similar to (2.68), (3.31) implies that for some tn ∈ R,  ε α  itΔ  e P1 gn  2d ε , (3.32) A L d−2 (Rd )

3.2 Profile Decomposition for the Energy-Critical Problem 77     β β for all t ∈ [tn − c Aε ,tn + c Aε ], where c > 0 is a small constant. Interpolating (3.32) with   itΔ e P1 gn  ∞ 2 ≤ A, (3.33) Lt Lx (R×Rd ) and the Littlewood–Paley theorem proves  itΔ  e φ 

2(d+2) Lt,xd−2

ε

2(d+2) d−2

 ε β A

(R×Rd )

.

(3.34)

This completes the proof of Theorem 3.7. Armed with Theorem 3.7, Keraani’s (2001) proved a profile decomposition result. We introduce the following terminology from that paper. Definition 3.8 (Asymptotic orthogonality) A sequence λn , for λn > 0, is called a scale. Call the sequence (zn ) = (tn , xn ), tn ∈ R, xn ∈ Rd a core. Two pairs (λ , z) and (λ  , z ) are called asymptotically orthogonal if ! λn λn |tn − tn | |xn − xn | + + + = +∞. lim n→∞ λn λn λn λn (λn λn )1/2   Theorem 3.9 (Profile decomposition) Let un ∈ H˙ 1 Rd be a bounded a sequence of sequence of functions.  d  un and  j j Then there exists a subsequence j 1 j ˙ scales λn , cores tn , xn , and functions φ j ∈ H R , with φ possibly equal to zero, such that for any J < ∞,   j J   d−2 j j −( 2 ) −itn Δ j x − xn e + wJn , φ (3.35) un = ∑ λn λnj j=1 with the remainder estimate   lim lim sup eitΔ wJn  J→∞ n→∞

2(d+2)

2d

= 0.

Moreover, for any 1 ≤ j ≤ J,  j   j  d−2  λn 2 eitn Δ wJn λnj x + xnj  0, weakly and

(3.36)

Lt,xd−2 ∩Lt∞ Lxd−2 (R×Rd )

! J    J 2  2 j 2      lim un H˙ 1 (Rd ) − ∑ φ H˙ 1 (Rd ) − wn H˙ 1 (Rd ) = 0. n→∞

(3.37)

(3.38)

j=1

    Finally, λnj , znj and λnk , zkn are asymptotically orthogonal when j = k, and φ j and φ k are nonzero.

78

The Energy-Critical Problem in Higher Dimensions

Proof

This proof may be used for any dimension d ≥ 3. If   lim inf eitΔ un  2(d+2) = 0, n→∞ Lt,xd−2 (R×Rd )

(3.39)

then simply take J = 0 and w0n = un , after passing to a subsequence such that (3.39) holds with lim inf replaced by lim. In this case   lim eitΔ un  = 0. (3.40) 2d n→∞ Lt∞ Lxd−2 (R×Rd ) Otherwise, (3.31)–(3.33) would imply the existence of a φ satisfying (3.34). Now suppose

  lim inf eitΔ un  n→∞

= ε > 0,

2(d+2)

Lt,xd−2

(R×Rd )

and pass to a subsequence satisfying

  lim ∇un L2 (Rd ) ≤ A, n→∞   lim eitΔ un  2(d+2) = ε. n→∞ Lt,xd−2 (R×Rd )

By Theorem 3.7, there exists φ 1 ∈ H˙ 1 such that after passing to a further subsequence,  α  1 φ  ˙ 1 d  ε ε H (R ) A and  1      1  d−2 λn 2 eitn Δ un λn1 x + xn1  φ 1 (x) ∈ H˙ 1 Rd , weakly. Let w1n

 − d−2 1 2 = un − λn1 e−itn Δ φ 1



x − xn1 λn1

.

Again by Theorem 3.7, possibly after passing to a subsequence,     2 2 2 lim ∇un L2 (Rd ) − ∇φ 1 L2 (Rd ) − ∇w1n L2 (R2 ) = 0 n→∞ and

 1    1  d−2 λn 2 eitn Δ w1n λn1 x + xn1  0, weakly.

  Now suppose (3.35), (3.37), and (3.38) hold for 1 ≤ j ≤ J0 − 1, and λnj , znj are pairwise asymptotically orthogonal. If   = 0, lim inf eitΔ wnJ0 −1  2(d+2) n→∞ Lt,xd−2 (R×Rd )

3.2 Profile Decomposition for the Energy-Critical Problem

79

then by (3.40), the proof is complete. Otherwise, after passing to a further subsequence,   lim eitΔ wnJ0 −1  2(d+2) = ε (J0 ) > 0 n→∞ Lt,xd−2 (R×Rd ) and

 J0   J  d−2  λn 0 2 eitn Δ wnJ0 −1 λnJ0 x + xnJ0  φ J0 (x) ,

(3.41)

weakly. By Theorem 3.7,  J 2 φ 0  ˙ 1 d  ε (J0 )2 H (R ) and



ε (J0 ) A

α

   2  2 2 lim wnJ0 −1 H˙ 1 − φ J0 H˙ 1 (Rd ) − wJn0 H˙ 1 (Rd ) = 0.

n→∞

Therefore, (3.38) continues to hold. Pairwise asymptotic orthogonality also continues to hold. By contradiction, suppose that for some 1 ≤ j < J0 ,

λnj λnJ0

+

λnJ0 λnj

J

+

|xnj − xn0 | J

(λnj λn 0 )1/2

J

+

|tnj − tn0 |

λnj λnJ0

has a bounded subsequence. Then there exists λ ∈ (0, ∞), x0 ∈ Rd , and t ∈ R such that along some further subsequence,

λnJ0 λnj By (3.37),

→ λ,

xnj − xnJ0

λnJ0

→ x0 ,

tnj − tnJ0 J

( λn 0 ) 2

→ t.

 j    j  d−2 λn 2 eitn Δ wnJ0 −1 λnj x + xnj  0, weakly,

while by (3.41), 

λnJ0

 J0   d−2  2 eitn Δ wnJ0 −1 λnJ0 x + xnJ0  φ J0 (x) , weakly.

Now by (1.42),   J0 j  j    J  d−2 i t −t Δ 2 0 e n n eitn Δ wnJ0 −1 λnJ0 x + xnJ0 (3.42) = λn    j    J  d−2 (tnJ0 − tnj ) itn Δ J0 −1 2 0 e = exp i Δ λ w λnJ0 x + xnJ0 n n J0 2 ( λn )

(3.42)

80

The Energy-Critical Problem in Higher Dimensions   J  j   J  d−2  (tn0 − tnj ) itn Δ J0 −1 2 0 = exp i e Δ T w λnJ0 x + xnj J0 λn j n J0 2 xn −xn ( λn ) J λn 0    j   j  d−2  (tnJ0 − tnj ) itn Δ J0 −1 2 Δ T w λnj x + xnj = exp i e J0 R J0 λn j n J0 2 xn −xn λn ( λn ) J0 j λn

λn

itΔ

 e Tx0 Rλ 0 = 0, weakly in H˙ 1 , where we have used (3.25) and (3.26). This gives a contradiction with (3.42) since φ J0 = 0. Therefore, pairwise asymptotic orthogonality of (λnj , znj ) must also hold for 1 ≤ j ≤ J0 . Now verify (3.37). Let    J − d−2  J  J x − xnJ0 J0 J0 −1 2 0 0 0 wn = wn − λn exp −itn Δ φ . λnJ0 It is clear from Theorem 3.7 that   J  d−2     λn 0 2 exp itnJ0 Δ wJn0 λnJ0 x + xnJ0  0. Also, by induction, for 1 ≤ j < J0 ,     d−2    lim λnj 2 exp itnj Δ wnJ0 −1 λnj x + xnj  0.

n→∞

Next, by pairwise asymptotic orthogonality up to J0 , 

λnj

 d−2 2

 exp i

λnJ0

(tnj − tnJ0 ) (λnj )2





Δ φ

J0

λnj x + xnj − xnJ0 λnJ0

  0,

J

weakly. Then (3.37) follows from the definition of wn0 . Finally, by (3.24), there exists some δ (J0 ) > 0 such that   lim sup eitΔ wJn0  n→∞

2(d+2)

Lt,xd−2 (R×Rd )

  < lim sup (1 − δ (J0 )) eitΔ wnJ0 −1  n→∞

2(d+2)

.

(3.43)

Lt,xd−2 (R×Rd )

Moreover, δ (J)  0 can only occur as J → ∞ if   lim lim sup eitΔ wJn 

J→∞ n→∞

2(d+2)

Lt,xd−2 (R×Rd )

= 0.

(3.44)

3.2 Profile Decomposition for the Energy-Critical Problem

81

Thus, combining (3.43), (3.44), and the discussion at the beginning of the proof immediately following (3.39) gives (3.36). Asymptotic orthogonality of individual profiles implies energy decoupling as n → ∞.   Theorem 3.10 Suppose u0,n is a sequence of functions bounded in H˙ 1 Rd and E (u0,n ) → E0 ∈ R. Then after making the profile decomposition of Theorem 3.9 and setting   j  j − d−2 j j −itn Δ j x − xn 2 un (0) = λn e φ λnj  !    j − d−2 tnj x − xnj j 2 = λn exp −i j Δ φ , ( λn ) 2 λnj J

∑E J→∞ n→∞ lim lim

   unj (0) + E wJn = E0 .



(3.45)

j=1

Proof

From the definition of the energy, E (u (t)) =

1 2



|∇u (t, x) |2 dx +

d −2 2d



2d

|u (t, x) | d−2 dx,

and (3.36), (3.38), it only remains to prove that ! J  2d/(d−2) 2d/(d−2)  j     lim lim un (0) L2d/(d−2) Rd − ∑ un (0) L2d/(d−2) Rd = 0. ( ) j=1 ( ) J→∞ n→∞

(3.46)

This follows from the pairwise asymptotic orthogonality of (λnj , znj ). If   j fact j tn /(λn )2  → +∞, then the dispersive estimates combined with the dominated  j    convergence theorem imply that  exp −i tnj Δ φ j  2d → 0. Now sup( λn ) 2

pose that

tnj /(λnj )2

and

tnk /(λnk )2

Lxd−2 (Rd )

converge to constants in R. In this case,

λnj λnk |xnj − xnk | + +  1/2  1/2 → +∞, λnk λnj λnj λnk and then the dominated convergence theorem implies that for fixed j, k,    

 d+2 j    1 k  d−2 1 x − x x − x    n n  uk 0,  j d−2 unj 0,  dx → 0. × d−2 n j k) d     ( λ R ( λn ) 2 ( λn ) n (λ k ) 2



n

82

The Energy-Critical Problem in Higher Dimensions

Also, (3.38) combined with the Sobolev embedding theorem implies that all the quantities in (3.46) are finite, so by definition of wJn , ! J  2d/(d−2) 2d/(d−2)  j     lim lim sup un (0) 2d/(d−2) − ∑ un (0) 2d/(d−2) (Rd )

Lx

J→∞ n→∞

   lim lim sup wJn  J→∞ n→∞

2d Lxd−2 (Rd )

(Rd )

Lx

j=1

  un (0) 

2d Lxd−2 (Rd )

  + wJn 

2d Lxd−2 (Rd )

! d+2 d−2

= 0.

This proves (3.45). Now let u0,n ∈ H˙ 1 be a sequence of initial data, and un  the  solution to (3.22)   with initial data u (0) = u0,n on R, with E (u0,n )  E0 and un 2(d+2) ≥ Lt,xd−2 (R×Rd ) 2n. Moreover, there exists some tn ∈ R such that, after translating in time u(t) → u(t − tn ),   un  2(d+2) ≥n (3.47) Lt,xd−2 ([0,∞)×Rd ) and

  un 

≥ n.

2(d+2)

Lt,xd−2

(3.48)

((−∞,0]×Rd )

Because E0 is the minimal energy such that C (E0 ) = ∞, if E0 < ∞,   lim lim sup E wJn = 0. J→∞ n→∞

Otherwise, by (3.45), u˜Jn,0 = 

J

∑ unj (0) ,

j=1



with E u˜Jn,0 < E0 − δ for some fixed δ > 0, independent of J. Then if u˜Jn is a solution to the initial value problem with initial data u˜J0,n ,  J u˜n  2(d+2) ≤ C (E0 − δ ) < ∞. Lt,xd−2 (R×Rd ) Then by (3.36) and the proof of Theorem 3.5, in particular (3.19)–(3.20), for n and J sufficiently large,   un  2(d+2) (3.49) ≤ C (E0 − δ ) + ε (J, n) , Lt,xd−2 (R×Rd ) with lim lim sup ε (J, n) = 0.

J→∞ n→∞

(3.50)

3.2 Profile Decomposition for the Energy-Critical Problem

83

This contradicts (3.47) and (3.48). Next, if E0 < ∞, then for each j such that φ j is nonzero, tnj /(λnj )2 must remain uniformly bounded. Without loss of generality suppose that it is not true when j = 1. Take J = 1 and let u˜n,0 = w1n . Then by (3.45), since φ 1 is nonzero,   lim sup E w1n < E0 − δ ,

(3.51)

n→∞

for some δ > 0. Therefore, in this case, if u˜n is the solution to (3.22) with initial data u˜n,0 , then   u˜n  2(d+2) (3.52) ≤ C (E0 − δ ) , Lt,xd−2 (R×Rd ) and by the dominated convergence theorem either   lim inf eitΔ φ 1  2(d+2) =0 n→∞ Lt,xd−2 ([0,∞)×Rd ) or

  lim inf eitΔ φ 1  n→∞

= 0.

2(d+2)

Lt,xd−2

(3.53)

(3.54)

((−∞,0]×Rd )

By Theorem 3.5 and (3.52), (3.53) contradicts (3.47) and (3.54) contradicts (3.48). Therefore, after passing to a subsequence, tnj (λnj )2

→ t j ∈ R.

(3.55)

It is possible to replace φ j with e−it j Δ φ j and set tnj ≡ 0, absorbing the error into wJn , since by Strichartz estimates, tnj → 0 implies that for any j,   j   lim eitΔ φ j − ei(t−tn )Δ φ j  2(d+2) = 0. (3.56) n→∞ Lt,xd−2 (R×Rd ) Finally, φ j = 0 for all j ≥ 2, possibly after an initial relabeling. Relabel so as j  ∞ and suppose that this was not that the energy of φ j is   decreasing true, and that in fact E φ 2 = δ > 0. Let u j be the solution to the initial value problem with initial data φ j . For each j, by (3.45),  j u  2(d+2) ≤ C (E0 − δ ) . (3.57) Lt,xd−2 (R×Rd )

84

The Energy-Critical Problem in Higher Dimensions   Also, by (3.45), E φ j ≥ ε0 for only finitely many j, so by Theorem 3.1, for j sufficiently large,  j 2   u  2(d+2) E φj . Lt,xd−2 (R×Rd ) Now by asymptotic orthogonality and the bilinear Strichartz estimate, when j = k, 

  1 t x − xnj   j lim sup   j d−2 u ,  n→∞  (λn ) 2 (λnj )2 λnj 

 d+2  k  d−2   1 t x − x n k   ×  → 0. (3.58) u , d−2   2 k k ˙1 λ k 2 ( λn ) n d (λn ) N (R×R ) Therefore, (3.57)–(3.58) combined with Theorem 3.5 imply   lim un  2(d+2)  C (E0 − δ ) . n→∞ Lt,xd−2 (R×Rd ) Putting all of this together, we have the following result. Theorem 3.11 If C (E) < ∞ for all E < E0 < ∞ and C (E0 ) = ∞, then there exists  a solution u (t) on a maximal interval of existence with initial data E φ 1 = E0 , such that     u 2(d+2) = u 2(d+2) = ∞. Lt,xd−2 (I∩[0,∞)×Rd ) Lt,xd−2 (I∩(−∞,0]×Rd ) The proof of Theorem 3.11 could  1  also be applied to u (tn ), for some sequence1 tn ∈ I. Because E (u (tn )) = E φ , a subsequence weakly converging in H˙ must converge in H˙ 1 -norm. Remark To completely mirror the proof of Theorem 3.11, this argument could be applied to a sequence (1 − εn ) u (tn ) for some εn  0. In conclusion, u (t) lies in a compact subset of H˙ 1 , modulo scaling and translation symmetries, so there exists λ (t) : I → (0, ∞) and x (t) : I → Rd such that (λ (t))

d−2 2

u (t, λ (t) x + x (t)) (3.59)  lies in some compact subset of H˙ Rd for all t ∈ I. Setting N (t) = λ 1(t) , by the Arzel`a–Ascoli theorem, this implies that for any η > 0, there exists C (η ) < ∞ such that 

|x−x(t)|≥C(η )λ (t)

 1

|∇u (t, x) |2 +



|x−x(t)|≥C(η )λ (t)

2d

|u (t, x) | d−2 dx < η

(3.60)

3.3 Global Well-Posedness and Scattering When d ≥ 5 and



|ξ | |uˆ (t, ξ ) | d ξ + 2

|ξ |≥C(η )N(t)



2

N(t)

|ξ |≤ C(η )

|ξ |2 |uˆ (t, ξ ) |2 d ξ < η .

85

(3.61)

Definition 3.12 (Almost-periodic solution) A solution for which (3.59) lies in a compact set for all t ∈ I is called an almost-periodic solution. Remark Compare the derivation of (3.61) to the derivation of (2.69) in Section 2.3. The relation (2.69) shows that either a nonzero fraction of the solution lies in a compact set modulo scaling symmetries (but not translation in space due to the radial symmetry of the solution) for a nonzero fraction of the time of its existence, or it is possible to apply induction on energy to prove scattering. The derivation of (3.61) carries this argument to its natural limit. Either,  at  the minimal energy, the entire solution must lie in a compact subset of H˙ 1 Rd , modulo symmetries, for the entire time of its existence, or it is possible to apply an induction on energy-type argument, as in (3.49).

3.3 Global Well-Posedness and Scattering for the Energy-Critical Problem When d ≥ 5 The quantities N (t) and x (t) in (3.60) and (3.61) may be chosen to be differentiable in time. Theorem 3.13

We can choose N (t) and x (t) in (3.59) to satisfy |N  (t) |  N (t)3

(3.62)

and |x (t) |  Moreover, for any t ∈ I,

  u (t) 

1 = λ (t) . N (t)

2d

Lxd−2 (Rd )

 1.

(3.63)

(3.64)

Proof Let K ⊂ H˙ 1 be a compact subset of H˙ 1 such that (3.59) lies in K for all t ∈ I. By Theorem 3.3, each u0 ∈ K is the initial data function for a local solution on [−T (u0 ) , T (u0 )], with   u 2(d+2) (3.65) = ε, Lt,xd−2 ([−T (u0 ),T (u0 )]×Rd ) where ε > 0 is a small, fixed number. Furthermore, by Theorem 3.5, T (u0 ) is continuous on H˙ 1 . Therefore, by compactness there must exist some T0 > 0

86

The Energy-Critical Problem in Higher Dimensions

such that T (u0 ) ≥ T0 for all u0 ∈ K. Equation (3.65) also implies that the linear solution dominates the Duhamel expression for a solution on [−T0 , T0 ]. Since eitΔ is a Fourier multiplier with absolute value 1, this implies N (t) ∼ 1

(3.66)

for all t ∈ [−T0 , T0 ] and u0 ∈ K. Moreover, by (1.28) and N (t) ∼ 1, for any u0 ∈ K, |x (t) |  1 for all t ∈ [−T0 , T0 ]. Then by rescaling, (3.66) implies (3.62), |x (t) |  1 and rescaling gives (3.63), and   u 2(d+2)   ≤ ε. (3.67) Lt,xd−2

t0 −

T0 T ,t0 + 0 2 N (t0 )2 N (t0 )

Let G be the group of symmetries generated by translation and scaling (see (3.59)). To prove that (3.22) is globally well posed and scattering, it suffices to show that the only almost-periodic solution to (3.22) is u ≡ 0. To prove this, it is enough to exclude the existence of a finite set of solutions which may be obtained by taking the limit of a generic almost-periodic solution. Indeed, if u is a generic almost-periodic solution,  u(tn ), tn ∈ I has a subsequence that converges in H˙ 1 /G to some u0 ∈ H˙ 1 Rd . Moreover, by Theorem 3.5, u0 is the initial condition to a blowup solution. Theorem 3.14 (Energy-critical scenarios) If there is a nonzero almostperiodic solution to the defocusing, energy-critical problem, then there exists a nonzero almost-periodic solution on an interval I such that one of two scenarios holds: I = R,

N (t) ≥ 1 for all t ∈ R

(3.68)

or sup (I) < ∞ and

lim N (t) = +∞.

(3.69)

t→sup(I)

Remark One may obtain (3.69) for a generic I = R by time reversal symmetry. Proof This argument follows the reduction in Killip et al. (2009). Define the quantity osc (T ) = inf

t0 ∈I

sup{N (t) : t ∈ I and |t − t0 | ≤ T N (t0 )−2 } inf{N (t) : t ∈ I and |t − t0 | ≤ T N (t0 )−2 }

.

First suppose lim osc (T ) < ∞.

T →∞

(3.70)

3.3 Global Well-Posedness and Scattering When d ≥ 5

87

In this case, there exists a sequence tn ∈ I such that lim

n→∞

sup{N (t) : t ∈ I and |t − tn | ≤ nN (tn )−2 } inf{N (t) : t ∈ I and |t − tn | ≤ nN (tn )−2 }

< ∞.

Moreover, after rescaling, (3.70) implies that N (t) ∼ 1 for all t ∈ I, and by (3.67), I = R. Now suppose lim osc (T ) = ∞.

T →∞

For any t0 ∈ I, define a (t0 ) =

N (t0 ) N (t0 ) + . sup{N (t) : t ∈ I and t ≥ t0 } sup{N (t) : t ∈ I and t ≤ t0 }

Suppose inf a (t0 ) = 0.

t0 ∈I

In that case, for any n, there exist tn− ,tn ,tn+ ∈ I, tn− < tn < tn+ , such that     N tn+ = N tn− = nN (tn ) . Choose t0,n satisfying tn− < t0,n < tn+ ,

N (t0,n ) =

inf N (t) .

t∈[tn− ,tn+ ]

  Then after rescaling and translating in space, u (t0,n ) converges in H˙ 1 Rd /G to some u0 . Moreover, u0 is the initial condition for an almost-periodic solution satisfying N (t) ≥ 1 for all t ∈ I. Either I = R, in which case (3.68) holds, or I = R, and after time reversal symmetry (3.69) holds. Finally, suppose inf a (t0 ) = 2ε > 0.

t0 ∈I

If there exist sequences tn  sup (I) and tn  inf (I) such that for all n, N (tn ) ≥ ε, sup{N (t) : t ∈ I,t ≤ tn }

N (tn ) ≥ ε, sup{N (t) : t ∈ I,t ≤ tn }

(3.71)

then (3.71), tn  sup (I), tn  inf (I) would imply N (t1 ) ∼ N (t2 ) for any t1 ,t2 ∈ I, which would imply that, after rescaling, N (t) ∼ 1 for all t ∈ I, I = R. On the other hand, by time reversal symmetry, suppose without loss of generality that there exists t0 ∈ I, such that for all tn ≥ t0 , tn ∈ I, N (tn ) ≥ ε. sup{N (t) : t ∈ I,t ≥ tn }

88

The Energy-Critical Problem in Higher Dimensions

Since limT →∞ osc (T ) = ∞, there exists T sufficiently large so that osc (T ) > ε2 . Now define a sequence tn  sup (I) in the following way. Start the sequence with t0 and rescale so that N (t0 ) = 1. For any n, define tn = tn + 4T N (tn )−2 . If N (tn ) ≤ 12 N (tn ) then take tn+1 = tn . If N (tn ) ≥ 12 N (tn ) then    −2   −2   ,tn + T N tn ⊂ tn ,tn + 8T N (tn )−2 . (3.72) tn − T N tn  Combining (3.72) with osc (T ) > ε2 implies that there exists tn+1 ∈ tn ,tn +  8T N (tn )−2 such that N (tn+1 ) ≤ 12 N (tn ). Arguing by induction, for every n, there exists tn ∈ I, tn > t0 , such that N (tn ) ≥ ε, sup{N (t) : t ∈ I,t ≥ tn }

t0 < tn < t0 + T 22n ,

(3.73)

and N (tn ) ≤ 2−n N (t0 ) .

(3.74)

Then by (3.62) and (3.73) and (3.74), we have N (t) ∼ (t − t0 )−1/2 for all t − t0 > 1. By the sequence  d(3.59),   d  u (tn ) has a subsequence that converges in 1 1 ˙ ˙ H R /G to u0 ∈ H R , which is the initial value of a blowup solution. Moreover, (3.74) implies N (t) ∼ t −1/2 for all t > 0, which is clearly a finite time blowup solution to (3.22). This completes the proof of Theorem 3.14. It is straightforward to rule out the existence of a finite time blowup solution to the energy-critical problem. Theorem 3.15 There does not exist an almost-periodic solution to the energy-critical problem that blows up in finite time.   Proof Let ψ ∈ C0∞ Rd be a cutoff function satisfying ψ (x) = 1 for |x| ≤ 1 and ψ (x) = 0 for |x| > 2. Without loss of generality suppose sup (I) < ∞ and let T = sup (I). For any R > 0, (3.60), H¨older’s inequality, and N (t)  ∞ imply     2 x   u (t, x)  dx = 0. (3.75) lim ψ d tT R R By (1.112) and integrating by parts,   x 2 d ψ |u (t, x) |2 dx dt Rd R  x 1 |∇u (t, x) | |u (t, x) |ψ  dx R |x|≤2R R

1/2     2 1/2  x 1   2 u (t, x)  dx |∇u (t, x) | dx . ≤ ψ R Rd R Rd

3.3 Global Well-Posedness and Scattering When d ≥ 5

89

In particular, by the fundamental theorem of calculus and (3.75), 

2 1/2 (T − t) E  x   0 u (t, x)  dx .  ψ R R Rd

Taking R → ∞ proves u (0, x) ≡ 0, which certainly can be continued to a global solution. Therefore, to prove scattering for (3.22), it remains to show that the only almost-periodic solution satisfying (3.68) is u ≡ 0. For radially symmetric data, it is enough to use a generalization of the Morawetz estimate (2.28). Theorem 3.16

The defocusing, energy-critical initial value problem 4

iut + Δu = F (u) = |u| d−2 u,

u (0, x) = u0 ∈ H˙ 1 (Rd )

is globally well posed and scattering when u0 is radial. Proof Radial symmetry combined with (3.60) implies x (t) ≡ 0. Generalizing (2.28), for any T , d, 2d  T  |u (t, x) | d−2

−T

|x|

dx dt  E (u0 ) T 1/2 .

(3.76)

However, by (3.60), (3.61), x (t) ≡ 0, N (t) ≥ 1, and (3.64), 1



2d

|u (t, x) | d−2 dx |x|

(3.77)

with a bound that is uniform in t. Since I = R, T may be chosen to be sufficiently large so that (3.77) contradicts (3.76) when u is not the identically zero solution. Remark A variation of this argument was actually introduced in Kenig and Merle (2006) to prove scattering for the radial, focusing, energy-critical problem. Remark The proof of Theorem 3.16 could also be applied to a solution that is symmetric across d linearly independent hyperplanes. For a general nonradial solution, the argument will require a few more deductions concerning an almost-periodic solution. Suppose u is an almostperiodic solution to iut + Δu = F (u) ,

90

The Energy-Critical Problem in Higher Dimensions

on a maximal interval I. By Duhamel’s principle, for any t0 ,t1 ∈ I, u (t1 ) = ei(t1 −t0 )Δ u (t0 ) − i

 t1 t0

ei(t1 −τ )Δ F (u (τ )) d τ .

Then if t0  sup (I) or t0  inf (I), almost-periodicity of u implies ei(t1 −t0 )Δ u (t0 )  0,

(3.78)  weakly in H˙ Rd . Indeed, if N (t0 )  +∞ or N (t0 )  0 then (3.78) follows from (3.60). On the other hand, if N (t0 ) ∼ N (t1 ) as t0  sup (I) or t0  inf (I), Theorem 3.13 implies sup (I) = +∞ or inf (I) = −∞ respectively. Then by (1.19) and (3.60), it follows that (3.78) holds. Therefore, by Rellich’s theorem, for any 2 ≤ p < ∞, j ∈ Z, and t ∈ I,  t     i(t−τ )Δ Pj u (t)  p d  lim  e P F (u ( τ )) d τ   p d j Lx (R ) Lx (R ) t− inf(I) t− (3.79)   t+      i(t−τ )Δ   Pj u (t) L p (Rd )  lim  and e Pj F (u (τ )) d τ  p d . x Lx (R ) t+ sup(I) t  1

2d This fact shows that u (t) ∈ Lt∞ Lxp for some p < d−2 , which gives an important improvement over the Sobolev embedding theorem.

Theorem 3.17 Suppose u is an almost-periodic solution satisfying (3.68). Then when d ≥ 5, for any q satisfying 2d 2 (d + 2) 0, there exists j0 (η ) ∈ Z, j0 (η ) < 0, such that   P≤ j u (t)  ∞ ˙ 1 ≤ η. (3.81) 0 Lt H (R×Rd ) Now take

1 p

1 = 12 − d−2 . By the Sobolev embedding theorem, for any t ∈ R,

  

t+2−2 j

t−2−2 j

  ei(t−τ )Δ Pj F (u (τ )) d τ 

 jd   2 d−2 

t+2−2 j

t−2−2 j

2 jd

 2 d−2

p

Lx

  ei(t−τ )Δ Pj F (u (τ )) d τ 

 t+2−2 j  t−2−2 j

 Pj F (u (τ )) 

 4j   2 d−2 F (u (t)) 

Lx2

p

Lx

p

Lt∞ Lx (R×Rd )

.

dτ

3.3 Global Well-Posedness and Scattering When d ≥ 5 Meanwhile, by (1.20),  ∞  

  ei(t−τ )Δ Pj F (u (τ )) d τ 

t+2−2 j



91

p

Lx

 ∞

1

t+2−2 j

|t − τ |

d d−2

 4j   2 d−2 F (u (t))  Therefore, by (3.79),   Pj u (t)  ∞ p

Lt Lx (R×Rd )

  Pj F (u (τ ))  p

Lt∞ Lx (R×Rd )

p

Lx

dτ

.

 4j   2 d−2 Pj F (u (t)) 

p

Lt∞ Lx (R×Rd )

.

(3.82)

Split F as follows:

    4 d+2 F (u) = F (P≤ j u) + O |P> j u| |P≤ j u| d−2 + O |P≥ j u| d−2 .

When d = 5, Bernstein’s inequality implies         Pj F (P≤ j u)  6/5  2− j ∇F (P≤ j u)  6/5  2− j ∇u 2 P≤ j u4/3 . (3.83) L L4 Lx

Lx

x

x

Then if j ≤ j0 , for any small, fixed ε > 0,    3/4  1/4 ∑ Pk uL4  ∑ Pk uL6 Pk uL2 x

k≤ j

x

x

k≤ j

ε η 1/4 2−

3ε j 4

3j



24

 4k  sup 2ε k 2− 3 Pk uL6



sup 2 j≤ j0

 Pj F (P≤ j u) 

Lt∞ Lx

6/5

(R×Rd )

ε η

,

x

k≤ j0

and therefore by (3.81), (3.82), and (3.83), ε j

3/4

1/3



ε k − 4k 3 

sup 2 2

(3.84)

  Pk u L∞ L6 . x

t

k≤ j0

Also by Bernstein’s inequality and (3.84),   4/3 4/3     sup 2ε j P> j uL2 (Rd ) P≤ j uL4 Rd  2(ε −1) j ∇uL2 P≤ j uL4 ( ) x x x x j≤ j0

 4k  ε η 1/3 sup 2ε k 2− 3 Pk uL∞ L6 . t

k≤ j0

Finally, by Bernstein’s inequality,    4/7   3/7 2ε j F (P> j u)  6/5  2ε j ∑ Pk uL2  Pk uL6 + Lx

j≤k≤ j0

x

x

∑

k≥ j0



 4k   Cε η 4/3 sup 2ε k 2− 3 Pk uL6 +Cη . k≤ j0

x

  Pk u

x

7/3 14/5

Lx

92

The Energy-Critical Problem in Higher Dimensions

Therefore, by (3.82),  4j  sup 2ε j 2− 3 Pj uL∞ L6 (R×Rd ) t

j≤ j0

 Cη +Cε η

x



1/3

   Pj u L∞ L6 (R×Rd ) .

ε j − 43j 

sup 2 2

(3.85)

x

t

j≤ j0

Choosing η (ε ) > 0 sufficiently small,  4j  sup 2ε j 2− 3 Pj uL∞ L6 (R×Rd ) ε 1. t

j≤ j0

x

Since this estimate for any ε > 0, interpolating (3.85) with Bernstein’s   holds − j   inequality Pj u L2  2 and the Sobolev embedding theorem proves (3.80) x when d = 5. The computations are similar when d ≥ 6. By Bernstein’s inequality,     4j  4  2ε j Pj F (P≤ j u)  p  2ε j 2− d−2  |∇| d−2 F (P≤ j u)  p L x

Lx

 4    ∇P≤ j u d−2 P≤ j u p  2ε j 2 Lx Lx2

  4 4k ε η d−2 sup 2ε k 2− d−2 Pk uL p . 4j − d−2

(3.86)

x

k≤ j0

Furthermore, using Bernstein’s inequality once again,   4    |P> j u| |P≤ j u| d−2  p L x

4 4  4 4  4   d−2  d−2  d−2  1− d−2  1− d−2            P≤ j u L p ∑ Pk u L p Pk u L2 + ∑ Pk u L p Pk u L2 x

4

ε η d−2



x

j≤k≤ j0

x

x

k≥ j0

x



4 d−2   4k  4k  . sup 2ε k 2− d−2 Pk uL p + sup 2ε k 2− d−2 Pk uL p x

k≤ j0

x

k≤ j0

(3.87) Finally, by Bernstein’s inequality,   4   d−2 Pk u d+2 p + ∑ Pk uLd+2 2 L x

x

j≤k≤ j0

4

 Cε η d−2



∑

j≥ j0

4   d−2   d+2 Pk u 2 Pk u d+2 p L L x

d−2

x



 4k  sup 2ε k 2− d−2 Pk uL p +Cη . x

k≤ j0

Relations (3.86)–(3.88) imply that for any ε > 0,

 4k  sup 2ε k 2− d−2 Pk uL p ε 1, k≤ j0

d+2

x

(3.88)

3.3 Global Well-Posedness and Scattering When d ≥ 5

93

which proves (3.80) for d ≥ 6. Arguing by induction, and using Theorem 3.17 as a base case, it is possible to show that a solution satisfying (3.60) with N (t) ≥ 1 for all t ∈ R must be uniformly bounded in L p for some p < 2, which by duality and the Sobolev embedding theorem implies u ∈ Lt∞ H˙ −ε for some ε > 0. Theorem 3.18 Let d ≥ 5 and let u be a global solution to the energy-critical problem that is almost-periodic modulo symmetries. Suppose, in addition, that  −ε (d)  R × Rd for some ε = ε (d) > 0. In parinft∈R N (t) ≥ 1. Then u ∈ Lt∞ H˙ x ∞ 2 ticular,  d  this means that u ∈ Lt Lx , and indeed lies in a compact subset of 2 Lx R modulo the translation symmetry. Proof

Theorem 3.18 follows directly from an inductive proposition.

Proposition 3.19 (Inductive proposition) Let d ≥ 5 and let u be as in Theorem 3.18. Assume further that |∇|s F (u) ∈ Lt∞ Lxq for 2(d−2)(d+1) = q and some 0 ≤ d 2 +3d−6 s−s ∞ s ≤ 1. Then there exists some s0 (q, d) > 0 such that u ∈ Lt H˙ x 0 . By the fractional product rule,  s   |∇| F (u) 

q

Lx (Rd )

   4    |∇|s uL2 (Rd ) u d−2 2(d+1) x

Lx d−1

. (Rd )

Starting with Theorem 3.17 and iterating Proposition 3.19 proves Theorem 3.18. Proof of Proposition 3.19 Proposition 3.19 is equivalent to showing that for some s0 > 0, say s0 = dq − d+4 2 > 0, and all j ≤ 0,  s   |∇| Pj u ∞ 2  2 js0 . L L t

x

By time translation symmetry it suffices to show that   s  |∇| Pj u (0)  2  2 js0 . L x

The proof of this fact uses the double Duhamel argument. Lemma 3.20 (Double Duhamel lemma) If I is the maximal interval of existence and u is an almost-periodic solution, then for any t1 ∈ I, u (t1 ) , u (t1 ) H˙ 1 = − lim

" lim

t0 inf(I) t2 sup(I)

t1

t0

ei(t1 −τ )Δ F (u (τ )) d τ ,

 t2 t1

ei(t1 −t)Δ F (u (t)) dt

# . H˙ 1

(3.89)

94

The Energy-Critical Problem in Higher Dimensions By (3.78), for any t0 < t1 < t2 ,

Proof

u (t1 ) , u (t1 ) H˙ 1 "  = ei(t1 −t0 )Δ u (t0 ) − i

t1

t0

=

ei(t1 −τ )Δ F (u (τ )) d τ ,

#  t1 ei(t1 −t)Δ F (u (t)) dt ei(t1 −t2 )Δ u (t2 ) − i t2 H˙ 1 "  t1 lim ei(t1 −τ )Δ F (u (τ )) d τ , ei(t1 −t0 )Δ u (t0 ) − i

t2 sup(I)

t0

i "

= − lim

lim

t0 inf(I) t2 sup(I)

Remark

 t2

ei(t1 −t)Δ F (u (t)) dt

t1 t1

t0

ei(t1 −τ )Δ F (u (τ )) d τ ,

 t2

# H˙ 1

ei(t1 −t)Δ F (u (t)) dt

#

t1

. H˙ 1

Equation (3.89) also holds for the inner product Pj ·, Pj · H˙ 1 .

Integrating one term in (3.89) forward in time and the other term backward in time takes full advantage of the dispersive estimates, since in that case |t − τ | = 0 if and only if t = τ = t1 . By (3.89),   s   |∇| Pj u (0) 2 2 ≤ L x

By (1.19),

∞ 0 0

−∞

   Pj |∇|s F (u (t)) , ei(t−τ )Δ Pj |∇|s F (u (τ ))  dt d τ .

     Pj |∇|s F (u (t)) , ei(t−τ )Δ Pj |∇|s F (u (τ )) L2   |t − τ |

   d 12 − 1q   |∇|s F (u) 2 ∞ q . Lt Lx

Also, by the Sobolev embedding theorem,    2   2j d −d    Pj |∇|s F (u (t)) , ei(t−τ )Δ Pj |∇|s F (u (τ )) L2   2 q 2  |∇|s F (u) L∞ Lq . t

Since

d q

− d2 > 2, there exists s0 > 0 such that

 ∞ 0     s d d  −1 2 j q − 2  |∇| Pj u (0) 2 2   |∇|s F (u) 2 ∞ q min |t − τ | , 2 Lx Lt Lx 0 −∞ 2  s 4 js0  |∇| F (u) L∞ Lq . 2 t

x

This proves the proposition. Armed with Theorem 3.18, scattering follows directly.

x

3.3 Global Well-Posedness and Scattering When d ≥ 5

95

Theorem 3.21

The defocusing, energy-critical initial value problem   4 iut + Δu = F (u) = |u| d−2 u, (3.90) u (0, x) = u0 ∈ H˙ 1 Rd   is globally well posed and scattering for any u0 ∈ H˙ 1 Rd , d ≥ 5.

Proof By Theorem 3.18, if u is an almost-periodic solution satisfying (3.68) then there exists some ε (d) > 0 such that sup u (t)  For any η > 0,

−ε (d)

Hx

t∈R

(Rd )∩H˙ x1 (Rd )

< ∞.

  P≤c(η )N(t) u ˙ 1 d < η , H (R ) x

(3.91)

(3.92)

and interpolating (3.91) with (3.92), along with Bernstein’s inequality, then   1 P>c(η )N(t) u 2 d  Lx (R ) N (t) c (η ) implies u (t) L2 (Rd )  x

ε (d) 1 + η 1+ε (d) . N (t) c (η )

If lim supt+∞ N (t) = +∞, since η > 0 is arbitrary, conservation of mass implies that u (t) L2 (Rd ) = 0, and therefore u ≡ 0. Thus, if u is a nonzero, almost-periodic solution to (3.90), N (t) ∼ 1.

Lemma 3.22 When d ≥ 3, for any interval I ⊂ R,  3−d   3/4  1/4    ∇uL∞ L2 I×Rd uL∞ L2 I×Rd .  |∇| 4 u 4 d ( ) ) t x t x( Lt,x (I×R ) Postponing the proof of Lemma 3.22 and interpolating  3−d    0, then for d ≥ 3,  3−d 4    |∇| 4 u 4

Lt,x (

I×Rd

)

   3  uL∞ L2 (I×Rd ) uL∞ H˙ 1 (I×Rd ) . t

x

t

(3.93)

Proof Recall that the Morawetz estimate of (2.27) was quite useful in the study of radially symmetric problems, due to the presence of the weight 1/|x| combined with the fact that radiality implies that u must remain centered at the origin. However, for general nonradial problems there is no a priori reason for choosing one element of Rd as the origin. (Although see Kenig and Merle (2010) for (2.27) applied to a nonradial problem.) Instead, the approach of Colliander et al. (2004) was to replace (2.27) with an estimate that is centered around the solution u. Recall from (2.24)–(2.25) that if 

Mh (t) = then

∂t Mh (t) = and

∂tt Mh (t) =



h (x) |u (t, x) |2 dx,



h j (x) T0 j (t, x) dx

(3.94)

(−ΔΔh (x)) |u|2 + 4h jk Re (∂ j u¯∂k u) + 2h j {N , u} pj dx. (3.95)

Translating h (x), let Mhy (t)



=

h (x − y) |u (t, x) |2 dx.

(3.96)

For any weight function g (t, y) ∈ L1 (Rn ), take  

g (t, y) h (x − y) |u (t, x) |2 dx dy.

(3.97)

Then integrating by parts in time and following the argument in the proof of

3.4 Interaction Morawetz Estimate

97

Theorem 2.8, if u solves the cubic problem in three dimensions, then by (2.31), 8π

 T  0

g (t, y) δ (x − y) |u (t, x) |2 dx dy dt  T 

+4

g (t, y)

 1  |∇u (t, x) |2 − |∂r(y) u (t, x) |2 dx dy dt |x − y|

g (t, y)

|u (t, x) |4 dx dy dt |x − y|

0

 T 

+2 0

≤

 T

g (t, y) ∂tt



h (x − y) |u (t, x) |2 dx dy dt 

 T ∂t g (t, y) ∂t h (x − y) |u (t, x) |2 dx dy dt =− 0 

 2 + g (T, y) ∂t h (x − y) |u (t, x) | dx|t=T dy 

 2 − g (0, y) ∂t h (x − y) |u (t, x) | dx|t=0 dy, 0

(3.98)

where δ is the usual delta function, r (y) is the radial unit vector centered at y ∈ Rd , and

(x − y) (x − y) ·∇ . (3.99) ∂r(y) = |x − y| |x − y| The weight g (t, y) should depend on the size of u in some sense. In that case the Morawetz quantity will have the largest weight precisely where the solution u is located. Moreover, the term 

 T 2 ∂t g (t, y) ∂t h (x − y) |u (t, x) | dx dy dt (3.100) 0

is the only one in (3.98) that does not have a counterpart in the proof of Theorem 2.8. The presence of ∂t g (t, y) means that this term may be much more effectively estimated when g (t, y) is a conserved quantity. Momentum does not control the size of u; in fact, momentum can often be zero for nonzero u. This leaves two options for g (t, y): mass or energy. Colliander et al. (2004) chose g (t, y) = |u (t, y) |2 , proving Lemma 3.22 when d = 3. Subsequent work by Colliander et al. (2009), Planchon and Vega (2009), and Tao et al. (2007a) extended the interaction Morawetz estimates to other dimensions. There are a number of good reasons to choose g (t, y) = |u (t, y) |2 , not the least of which is because then in three dimensions the first term on the left-hand side of (3.98) is the L4 -norm of u in space and time. Remark It is useful to think of the standard Morawetz estimate in Theorem 2.8 as showing that a solution to a defocusing problem must spread out and

98

The Energy-Critical Problem in Higher Dimensions

move away from the origin. The appropriately named interaction Morawetz estimate shows that a solution must move away from itself. Let



M int (t) =

|u (t, y) |2 Mhy (t) dy.

(3.101)

By (1.112), (3.94), (3.95), (3.96), and the fact that h (x − y) is an even function, 1 ∂t Mhint (t) = 2



|u (t, y) |2 ∂t Mhy (t) dy

(3.102)

and 1 ∂tt Mhint (t) = 2



|u (t, y) |2 ∂tt Mhy (t) dy − 2



∂k T0k (t, y) ∂t Mhy (t) dy. (3.103)

Integrating by parts, 

(3.103) =

(−ΔΔh (x − y)) |u (t, x) |2 |u (t, y) |2 dx dy 

+4 −4

 

+2 

+4

(3.104)

h jk (x − y) Re (∂ j u¯∂k u) (t, x) |u (t, y) |2 dx dy

(3.105)

h jk (x − y) Im (u¯∂ j u) (t, x) Im (u¯∂k u) (t, y) dx dy

(3.106)

h j (x − y) {N , u} pj (t, x) |u (t, y) |2 dx dy

(3.107)

h j (x − y) Im (u¯∂ j u) (t, x) {N , u}m (t, y) dx dy.

(3.108)

Observe that (3.106) and (3.108) are the terms arising from (3.100). Decompose ∇ as follows: ∇ = ∇y + ∂r(y) , where ∂r(y) is given by (3.99), and refers to a radial derivative with origin shifted to y, and ∇ to angular derivatives with origin shifted to y. Since ∇y and

∂r(y) are orthogonal vectors, if h (x) = |x|, then h jk (x) =

δ jk |x|

−

x j xk , |x|3

and

h jk (x − y) Re (∂ j u¯∂k u) (t, x) |u (t, y) |2 − h jk (x − y) Im (u¯∂ j u) (t, x) Im (u¯∂k u) (t, y) =

2 1 1  |∇u (t, x) |2 |u (t, y) |2 − ∂ u (t, x)  |u (t, y) |2 |x − y| |x − y| r(y)

3.4 Interaction Morawetz Estimate

99

1 Im (u∇u) ¯ (t, x) · Im (u∇u) ¯ (t, y) |x − y|     1 + Im u¯∂r(y) u (t, x) · Im u¯∂r(x) u (t, y) |x − y|   = | ∇y u (t, x) |2 |u (t, y) |2 − Im u ¯ ∇y u (t, x) · Im (u ¯ ∇x u) (t, y) . −

(3.109)

By the Cauchy–Schwarz inequality,    u (t, x) ∇y u (t, x)  u (t, y) ∇x u (t, y)  1 1 ≤ | ∇y u (t, x) |2 |u (t, y) |2 + |u (t, x) |2 | ∇x u (t, y) |2 , 2 2

(3.110)

so (3.105) + (3.106) ≥ 0. Therefore,    I

(−ΔΔh (x − y)) |u (t, x) |2 |u (t, y) |2 dx dy dt   

+2 I

h j (x − y) {N , u} pj (t, x) |u (t, y) |2 dx dy dt

≤ ∂t M int (T ) − ∂t M int (0) −4

   I

(3.111) (3.112)

h j (x − y) Im (u¯∂ j u) (t, x) {N , u}m (t, y) dx dy dt, (3.113)

and we may proceed as in the proof of Theorem 2.8. Since |h j (x) | ≤ 1, we have       ∂t M int (T ) − ∂t M int (0)  ≤ 2u3 ∞ 2 u ∞ ˙ 1 . L H L L t

x

t

Also by (2.30), 8π

 

|u (t, x) | dx dt + 2

  

4

I

Rd

 3 ≤ 2u ∞

Lt Lx2 (I×R3 )

  

+4 I

  u

I

|u (t, y) |2

(x − y) · {N , u} p (t, x) dx dy dt |x − y|

Lt∞ H˙ 1 (I×Rd )

|{N , u}m (t, y) | |u (t, x) | |∇u (t, x) | dx dy dt.

(3.114) (3.115)

When d ≥ 4, we have



d −1 d −1 (d − 1) (d − 3) ∂r ∂rr + ∂r r = , − ΔΔ|x| = − ∂rr + r r r3

100

The Energy-Critical Problem in Higher Dimensions

and so in this case,   

|u(t, x)|2 |u(t, y)|2 dx dt |x − y|3 I

    (x − y) +2 |u (t, y) |2 · {N , u} p (t, x) dx dy dt |x − y| I  3   ≤ 2uL∞ L2 (I×Rd ) uL∞ H˙ 1 (I×Rd ) t x t

(3.116)

|{N , u}m (t, y) | |u (t, x) | |∇u (t, x) | dx dy dt.

(3.117)

(d − 1)(d − 3)

  

+4 I |u| p u,

When N = recalling (1.113), we get {N , u}m = 0. By (1.118) and integrating by parts,  

|u (t, y) |2 =

2p p+2

(x − y) j |x − y|

 

{N , u} pj dx dy

|u (t, y) |2

1 |u (t, x) | p+2 dx dy ≥ 0. |x − y|

Therefore, when d = 3,   

 4 1 |u (t, x) | p+2 dx dy dt + uL4 (I×R3 ) t,x |x − y| I    3  uL∞ L2 (I×R3 ) uL∞ H˙ 1 (I×R3 ) , t x t

2p p+2

|u (t, y) |2

and both terms in the left-hand side are positive. This proves Lemma 3.23 when d = 3. When d ≥ 4, by (3.116)–(3.117), 2p p+2

   I

|u (t, y) |2

1 |u (t, x) | p+2 dx dy dt |x − y|

+ (d − 1) (d − 3)

   I

|u (t, x) |2 |u (t, y) |2

   3 d uL∞ L2 (I×Rd ) uL∞ H˙ 1 (I×Rd ) . t

x

1 dx dy dt |x − y|3

t

By the Littlewood–Paley theorem, if K (x) is the Littlewood–Paley kernel,  3−d 4  2  j(3−d)    2   |∇| 4 f  4 d ∼d  ∑ 2 2 Pj f   2 d L (R ) L (R ) j    2   3 j(d+1)     =  ∑ 2 2  K 2 j (x − y) f (y) dy  2 d L (R ) j           j(d+3)     ∑ 2 jd |K 2 j (x − y) | dy 2 2 |K 2 j (x − y) | | f (y) |2 dy  2 d L (R ) j      j(d+3)    ∑2 2 |K 2 j (x − y) | | f (y) |2 dy 2 d . (3.118) L (R ) j

3.4 Interaction Morawetz Estimate By (1.33),

∑2

j(d+3) 2

  j  K 2 (x − y)  

j

1 |x − y|

d+3 2

101

.

Expanding the inner product,  2   1 2  | f (y) | dy d+3   2 d |x − y| 2 L (R ) % $  1 1 | f (y) |2 dy, | f (z) |2 dz = d+3 d+3 |x − y| 2 |x − z| 2 L2     1 1 ∼ ∑ j( d+32 ) k( d+32 ) |x−y|∼2 j , dx | f (y) |2 | f (z) |2 dy dz. (3.119) 2 j≤k 2 k |x−z|∼2

By the triangle inequality, 2k  |y − z|, so by H¨older’s inequality, when d > 3, (3.119)  

   

∑

2( j−k)(

d−3 2

) 2−3k | f (y) |2 | f (z) |2 dy dz

j≤k,2k |y−z|

1 | f (y) |2 | f (z) |2 dy dz. |z − y|3

This completes the proof of Lemma 3.23. Remark  

The identity  3−d 2 1   2 2 2 |u|2  |∇| |u (t, x) | |u (t, y) | dx dy ∼  d 2 R3 |x − y|3 Lt,x ( )

implies that the same argument also proves 2  3−d  3      |∇| 2 |u|2  2 d uL∞ L2 uL∞ H˙ 1 Lt,x

t

x

t

(3.120)

(3.121)

when d ≥ 3. See Tao et al. (2007a) for example. Colliander et al. (2009) and Planchon and Vega (2009) proved that (3.121) also holds in dimensions d = 1, 2; the latter’s proof used bilinear estimates, and is useful in the study of the mass-critical problem. We consider it further in the next chapter.

4 Mass-Critical NLS Problem in Higher Dimensions

4.1 Bilinear Estimates Interaction Morawetz estimates also yield bilinear estimates, an essential component to the profile decomposition in the mass-critical case. We begin with the one-dimensional bilinear result of Planchon and Vega (2009). Theorem 4.1 problems

Suppose that μ ≥ 0 and that u and v solve the nonlinear iut + Δu = F (u) = μ |u| p u,

u : R×R → C

(4.1)

ivt + Δv = F (v) = μ |v| p v,

v : R × R → C.

(4.2)

and

Then  

2  ¯ L 2 4∂x (uv)

pμ + |u (t, x) | p+2 |v (t, x) |2 dx dt t,x (R×R) p+2   pμ |u (t, x) |2 |v (t, x) | p+2 dx dt + p+2      2  uL∞ H˙ 1 (R×R) uL∞ L2 (R×R) vL∞ L2 (R×R) t t x t x      2       + v L∞ H˙ 1 (R×R) v L∞ L2 (R×R) u L∞ L2 (R×R) . t

Proof

t

x

t

x

(4.3)

Define the quantity  

M (t) =

|x − y| |v (t, x) |2 |u (t, y) |2 dx dy. 102

(4.4)

4.1 Bilinear Estimates

103

Taking a derivative in time, (1.112) yields  

(x − y) Im [u¯ (t, x) ∂x u (t, x)] |v (t, y) |2 dx dy |x − y|   (x − y) Im [v¯ (t, x) ∂x v (t, x)] |u (t, y) |2 dx dy. +2 |x − y|

∂t M (t) = 2

(4.5)

Then (1.112), (1.119), and (1.120) imply

∂tt M (t)

⎫   (x − y) 2 3 2 ⎪ |v (t, y) | ∂x |u (t, x) | dx dy ⎪ = ⎬ |x − y|     ⎪ (x − y) ⎭ + |u (t, y) |2 ∂x3 |v (t, x) |2 dx dy ⎪ |x − y|   ⎫   (x − y) ⎪ −4 |v (t, y) |2 ∂x |∂x u (t, x) |2 dx dy ⎪ ⎬ |x − y|     ⎪ (x − y) ⎭ −4 |u (t, y) |2 ∂x |∂x v (t, y) |2 dx dy ⎪ |x − y|   ⎫   2pμ (x − y) ⎪ − |v (t, y) |2 ∂x |u (t, x) | p+2 dx dy ⎪ ⎬ p+2 |x − y|     ⎪ 2pμ (x − y) ⎭ − |u (t, y) |2 ∂x |v (t, x) | p+2 dx dy ⎪ p+2 |x − y|   ⎫ (x − y) ⎪ −4 Im[u¯ (t, x) ∂x u (t, x)]∂y Im[v¯ (t, y) ∂y v (t, y)] dx dy ⎪ ⎬ |x − y|   ⎪ (x − y) ⎭ −4 Im[v¯ (t, x) ∂x v (t, x)]∂y Im[u¯ (t, y) ∂y u (t, y)] dx dy.⎪ |x − y|  

Approximating u and v with Schwartz functions, distribution in R2x,y . Distributionally, parts, we obtain (4.8) =

4pμ p+2

 

∂x (x−y) |x−y|

(x−y) |x−y|

(4.6)

(4.7)

(4.8)

(4.9)

may be treated as a

= 2δ (x − y), so integrating by

|v (t, x) |2 |u (t, x) | p+2 + |u (t, x) |2 |v (t, x) | p+2 dx dy ≥ 0. (4.10)

Also, (4.9) = −16 and (4.7) = 8

 

 

Im[u¯ (t, x) ∂x u (t, x)] Im[v¯ (t, x) ∂x v (t, x)] dx dy

|v (t, y) |2 |∂x u (t, x) |2 dx dy + 8

 

(4.11)

|u (t, y) |2 |∂x v (t, x) |2 dx dy. (4.12)

104

Mass-Critical NLS Problem in Higher Dimensions

Finally, integrating by parts, we get       (4.6) = −2 |v (t, x) |2 ∂x2 |u (t, x) |2 dx − 2 |u (t, x) |2 ∂x2 |v (t, x) |2 dx =4

 

∂x |v (t, x) |2

  ∂x |u (t, x) |2 dx.

Adding up yields



(4.11) + (4.12) + (4.13) = 8

(4.13)

|∂x (uv) ¯ (t, x) |2 dx ≥ 0.

(4.14)

Therefore, by the fundamental theorem of calculus, Theorem 4.1 follows. Corollary 4.2 If u solves (4.1) then   2  2    3 ∂x |u|  2  uL∞ H˙ 1 (R×R) uL∞ L2 (R×R) . L (R×R) t,x

Proof

t

t

(4.15)

x

Take u = v.

The estimate (4.15) is exactly (3.121) when d = 1. Also notice that Theorem 4.1 implies that if u and v solve (4.1) and (4.2) respectively with μ = 0, and if k ≤ j − 5,      (Pj u) Pk v = P˜j (Pj u) Pk v , (4.16) so by Bernstein’s inequality and (4.3),   2  j     −2 j k  2  (Pj u) Pk v 2 2 v  u 2  2 + 2 0 2 Lt,x (R×R) Lx (R) 0 Lx2 (R)  2  2  2− j u0 L2 (R) v0 L2 (R) . x

Then since

x

(4.17)

        (Pj u) (Pk v) 2 2 =  (Pj u) Pk v Pj u (Pk v)  1 , L L t,x

t,x

by (4.17) and H¨older’s inequality it is easy to see that   2    2  (Pj u) (Pk v) 2 2  2−2 j 2 j + 2k u0 L2 (R) v0 L2 (R) Lt,x (R×R) x x    2 − j  2   u0 L2 (R) v0 L2 (R) . 2 x

x

Such bilinear estimates are invariant under the Galilean transformation. Theorem 4.3 Let ξ (t) : I → R be a function of time. If u and v solve (4.1) and (4.2) respectively, then   2  2    ∂x (uv) ¯ L2 (I×R)  vL∞ L2 (I×R) e−ix·ξ (t) uL∞ H˙ 1 (I×Rd ) uL∞ L2 (I×R) t,x t x t t x  2    −ix·ξ (t)        + u L∞ L2 (I×R) e v L∞ H˙ 1 (I×Rd ) v L∞ L2 (I×R) . t

x

t

t

x

(4.18)

4.1 Bilinear Estimates Proof

Since

105

Split Im[u¯∂x u] as follows:    Im [u¯∂x u] = Im u¯∂x e−ix·ξ (t) u + ξ (t) |u|2 . (x−y) |x−y|

is an odd function, we get

ξ (t)

 

(x − y) |u (t, x) |2 |v (t, y) |2 dx dy |x − y|   (x − y) |v (t, x) |2 |u (t, y) |2 dx dy = 0, + ξ (t) |x − y|

and therefore |∂t M (t) | is bounded by the right-hand side of (4.18). Then (4.14) proves Theorem 4.3. For a solution to the linear Schr¨odinger equation, more may be said. Let χ ∈ C0∞ (R) be a partition of unity function, that is, a function supported on [−2, 2] such that for any x ∈ R,

∑ χ (x − m) = 1.

m∈Z

Furthermore, let Pm be the Fourier multiplier such that for any f ,   F (Pm f ) = χ 2− j (ξ − m) fˆ (ξ ) . Then by Theorem 4.3, we have the next result. Theorem 4.4 Suppose u and v solve (4.1) and (4.2) with initial data u0 and v0 respectively. Then for any j ∈ Z,       Pj (uv) ¯ L2 (R×R)  2− j/2 u0 L2 (R) v0 L2 (R) . t,x

Proof

By the triangle inequality and the Cauchy–Schwarz inequality,       Pj (uv) ¯ L2 (R×R)  2− j/2 ∑ Pm u0 L2 (R) Pn v0 L2 (R) t,x

|m+n|≤3

    u0 L2 v0 L2 .

− j/2 

2

Theorem 4.4 may also be proved directly using Fourier analysis.   Proof using Fourier analysis Let F (t, x) L2 (R×R) = 1, such that Fˆ (t, ξ ) is t,x

supported on |ξ | ∼ 2 j . By H¨older’s inequality and the Plancherel identity it suffices to estimate  

Fˆ (t, ξ + η ) uˆ (t, η ) vˆ¯ (t, ξ ) d ξ d η dt  

=C

  F˜ −ξ 2 + η 2 , η + ξ uˆ0 (η ) vˆ¯0 (ξ ) d η d ξ .

(4.19)

106

Mass-Critical NLS Problem in Higher Dimensions

Making a change of variables,       F˜ η 2 − ξ 2 , η + ξ 2 d η d ξ ∼ Therefore,

Corollary 4.5 ∼ 2 j , then

    (4.19)  2− j/2 u0 L2 (Rd ) v0 L2 (Rd ) . If the supports of uˆ0 (ξ ) and vˆ0 (ξ ) are separated by distance   uv 2 L

t,x (R×R)

Proof

1   ˜ 2 ˜ F η˜ , ξ d ξ d η˜  2− j . |ξ + η |

Again write

     2− j/2 u0 L2 (R) v0 L2 (R) .

   2  2 uv 2 = uv¯uv ¯ L1 = uv¯L2 . L t,x

t,x

t,x

(4.20)

(4.21)

By the Fourier support properties of u and v we have ¯ , uv¯ = P j (uv)

(4.22)

and so by Theorem 4.4 the proof is complete. Theorem 4.4 and Corollary 4.5 may be extended to higher dimensions. Both the Fourier-analytic arguments of Bourgain (1998), Colliander et al. (2008), and Killip and Visan (2013), and the interaction Morawetz estimates of Planchon and Vega (2009) will be presented here, since both are useful in the study of the mass-critical problem. Theorem 4.6 Suppose u and v are solutions to (4.1) and (4.2) with μ = 0 (linear problem) with initial data u0 and v0 in dimension d ≥ 2. Suppose uˆ0 (ξ ) is supported on |ξ | ∼ 2 j and vˆ0 (ξ ) is supported on |ξ | ∼ 2k with k ≤ j − 5. Then       (Pj u) (Pk v)  2 +  (Pj u) Pk v L2 (R×Rd ) Lt,x (R×Rd ) t,x k( d−1 )    2 2  Pj u0  2 d Pk v0  2 d . (4.23) d L R L (R ) j/2 ( ) 2 Proof Because uˆ0 (ξ ) is supported on |ξ | ∼ 2 j , it is possible to decompose u0 such that u0 = u0,1 + · · · + u0,d , where uˆ0,i (ξ ) is supported on |ξi | d 2 j . For example, in the case d = 2, let χ : R → R be a cutoff function such that χ (ξ ) = 1 for 2 j−5 ≤ |ξ | ≤ 2 j+5 and χ is supported on 2 j−6 ≤ |ξ | ≤ 2 j+6 . Then let uˆ0,1 (ξ ) = χ (ξ1 ) uˆ0 (ξ )

and

uˆ0,2 (ξ ) = (1 − χ (ξ1 )) uˆ0 (ξ ) .

4.1 Bilinear Estimates

107

By construction, uˆ0,1 is supported on |ξ1 | ∼ 2 j . Also, since uˆ0,2 (ξ ) is supported on |ξ1 |  2 j and uˆ0 (ξ ) is supported on |ξ | ∼ 2 j , it follows that uˆ0,2 (ξ ) is supported on |ξ2 | ∼ 2 j . The decomposition for a general d > 2 is similar. Relabeling u0 = u0,1 and integrating in ξ1 , η1 ,   

Fˆ (t, ξ + η ) u (t, ξ ) v (t, η ) d ξ d η dt

    F |ξ |2 − |η |2 , ξ + η uˆ0 (ξ ) vˆ0 (η ) d ξ d η = C (d)        F |ξ |2 − |η |2 , ξ1 + η1 , ξ2 + η2 , . . . , ξd + ηd 

  × uˆ0 (ξ1 , ξ2 , . . . , ξd ) L2

ξ1

(R)

  vˆ0 (η1 , η2 , . . . , ηd ) 

Lξ2 ,η (R×R) 1 1

Lη2 1 (R)

d η d ξ , (4.24)

where for clarity we have written d η = d η2 · · · d ηd and d ξ = d ξ2 · · · d ξd . Then by the proof of Theorem 4.4, the support of vˆ0 (ξ ), and H¨older’s inequality, d−1

   2k( 2 )  u0  2 d v0  2 d . L R L (R ) j/2 ( ) 2 A similar argument may be made for u0,2 . Then by (4.21) and (4.22), the proof is complete. (4.24) d

For the second proof we will need the following definition. Definition 4.7 (Radon transform) form is given by

For a general function f , the Radon trans-

R ( f ) (s, ω ) =

 x·ω =s

f d μs,ω ,

where d μs,ω is the Lebesgue measure on the hyperplane {x : x · ω = s}. Second proof of Theorem 4.6 For any ω ∈ Sd−1 , let xω = x · ω and let ∂ω = ω · ∇. Then there exists some C (d) such that 

1 |x|, |x · ω | d ω = C (d)

and

 

M (t) =



xω 1 (ω · ∇) d ω = ∇, |xω | C (d)

|u (t, x) |2 |u (t, y) |2 |x − y| dx dy  

= C (d)

|u (t, x) |2 |u (t, y) |2



Sd−1

|xω − yω | d ω dx dy.

Now fix ω ∈ Sd−1 , and let

 

Mω (t) =

|xω − yω | |v (t, x) |2 |u (t, y) |2 dx dy.

(4.25)

108

Mass-Critical NLS Problem in Higher Dimensions

Without loss of generality suppose ω = (1, 0, 0, . . . , 0). Taking a derivative in time and following (4.4)–(4.14),  

(x1 − y1 ) Im [u¯ (t, x) ∂x1 u (t, x)] |v (t, y) |2 dx dy |x1 − y1 |   (x1 − y1 ) Im [v¯ (t, x) ∂x1 v (t, x)] |u (t, y) |2 dx dy, +2 |x1 − y1 |

∂t Mω (t) = 2

(4.26)

and

∂tt Mω (t)  

=

  (x1 − y1 ) |v (t, y) |2 ∂x1 Δ |u (t, x) |2 dx dy |x1 − y1 |     (x1 − y1 ) + |u (t, y) |2 ∂x1 Δ |v (t, x) |2 dx dy |x1 − y1 |   ⎫   (x1 − y1 ) ⎪ −4 |v (t, y) |2 ∂k Re ∂x1 u · ∂k u (t, x) dx dy ⎪ ⎬ |x1 − y1 | (4.27)     ⎪ (x1 − y1 ) ⎭ −4 |u (t, y) |2 ∂k Re ∂x1 v · ∂k v (t, x) dx dy ⎪ |x1 − y1 |   ⎫   2pμ (x1 − y1 ) ⎪ |v (t, y) |2 ∂x1 |u (t, x) | p+2 dx dy ⎪ − ⎬ p+2 |x1 − y1 | (4.28)     ⎪ 2pμ (x1 − y1 ) ⎪ 2 p+2 − |u (t, y) | ∂x1 |v (t, x) | dx dy ⎭ p+2 |x1 − y1 |  ⎫ (x1 − y1 ) ⎪ −4 Im[u¯ (t, x) ∂x1 u (t, x)]∂k Im[v¯ (t, y) ∂k v (t, y)] dx dy ⎪ ⎬ |x1 − y1 | (4.29)  ⎪ (x1 − y1 ) ⎪ ⎭ −4 Im[v¯ (t, x) ∂x1 v (t, x)]∂k Im[u¯ (t, y) ∂k u (t, y)] dx dy. |x1 − y1 |

Integrating by parts, since μ ≥ 0, 2pμ (4.28) = p+2

 

x1 =y1

|v (t, y) |2 |u (t, x) | p+2 + |u (t, y) |2 |v (t, x) | p+2 dx dy ≥ 0.

Remark When μ = 0 this term does not appear, but this computation with μ > 0 will prove to be useful in later study of the nonlinear Schr¨odinger equation. Adding up as in (4.14), (4.27) + (4.28) + (4.29)     2   =8 ∂x1 u (t, x1 , x2 , . . . , xd ) v (t, x1 , y2 , . . . , yd )  dx dy.

(4.30)

Since the same computations may be performed for any ω ∈ Sd−1 , by the

4.2 Mass-Critical Profile Decomposition

109

fundamental theorem of calculus, (4.30) gives an estimate for the Radon transform of u,  T ∞    2 ∂s R |u|2 (s, ω ) ds dt  |∂t Mω (T ) − ∂t Mω (0) |. (4.31) 0

Since

−∞

 T 0

 ∞

  2 ∂s R |u (t, x) |2 (s, ω ) ds dt d ω −∞ 2  d−3   = 2 |∇| 2 |u (t, x) |2  2 , Lt,x ([0,T ]×Rd )

Sd−1

(4.32)

it follows that (3.93) holds for all d ≥ 1. It is possible to wring even more information from (4.30). Suppose for example that u = Pj u and v = Pk v with k ≤ j − 5. Then by H¨older’s inequality, for any ω ∈ Sd−1 ,       2  2  2   ∂ω u (t, x) v (t, x + h)  dx dh dt  2 j Ru0 L2 v0 L2 . (4.33) h∈Rd :|h|≤R

¯ is supported on Averaging over ω ∈ Sd−1 , by (4.25) and the fact that F (uv) |ξ | ∼ 2 j , we obtain       2  2  2   ∇ u (t, x) v (t, x + h)  dx dh dt  2 j Ru0 L2 v0 L2 , h∈Rd :|h|≤R

and therefore by Bernstein’s inequality, and the support of F (uv), ¯      2   2  2    u (t, x) v (t, x + h)  dx dh dt  2− j Ru0  2 v0  2 . L

h∈Rd :|h|≤R

L

(4.34) By the Littlewood–Paley decomposition, 

u (t, x) v (t, x) = u (t, x)

Kk (x − y) v (t, y) dy,

and then by (1.34) and (4.34),  2  2  2 uv¯ 2  2− j 2k(d−1) u0 L2 v0 L2 . Lt,x (R×Rd )

(4.35)

This completes the second proof of Theorem 4.6.

4.2 Mass-Critical Profile Decomposition The profile decomposition for the mass-critical nonlinear Schr¨odinger equation,   4 (4.36) u (0, x) = u0 ∈ L2 Rd , iut + Δu = F (u) = |u| d u,

110

Mass-Critical NLS Problem in Higher Dimensions

has an additional complication over the energy-critical profile decomposition, namely that there is no canonical choice for the origin in frequency. For the energy-critical problem,   uH˙ 1 (Rd ) =  |ξ |uˆ (ξ ) L2 (Rd ) is a weighted L2 -space in frequency, and therefore has a clear, canonical choice of origin. This is not so for the mass-critical problem. Instead, a profile decomposition for the mass-critical problem must take into account the Galilean transform. While the Galilean transform will certainly change the energy of a solution to the nonlinear Schr¨odinger problem, and thus a solution will no longer be a minimal energy blowup solution after applying a Galilean transform, the Galilean transform will not change the L2 -norm of a solution, and therefore, the symmetry group for solutions of (4.36) must include the Galilean transformation. The existence of a profile localized around some ξn ∈ Rd may be quantified by showing that the interaction of u with itself is large. This is accomplished using a bilinear Strichartz estimate. Define the group action iθ ix·ξ0

gθ ,ξ0 ,x0 ,λ u = e e



x − x0 u d/2 λ λ 1

.

(4.37)

Remark The group action f → eiθ f is a compact group action, since S1 ⊂ C is compact. The other components ξ0 , x0 , and λ belong to the noncompact groups Rd , Rd , and (0, ∞). This group obeys the multiplication law gθ ,ξ0 ,x0 ,λ · gθ  ,ξ  ,x ,λ  = g 0 0

x ·ξ 

ξ

θ +θ  − 0λ 0 ,ξ0 + λ0 ,x0 +λ x0 ,λ λ 

and each element has the inverse  −1 gθ ,ξ0 ,x0 ,λ = g−θ −x0 ·ξ0 ,−λ ξ0 ,− x0 , 1 . λ

λ

,

(4.38)

(4.39)

By (1.39) and (1.42),  it Δ  eitΔ gθ ,ξ0 ,x0 ,λ u0 = gθ −t|ξ0 |2 ,ξ0 ,x0 +2t ξ0 ,λ e λ 2 u0 .

(4.40)

A profile may be extracted modulo these symmetries. Theorem 4.8 Suppose fn is a bounded sequence of functions such that   (4.41) lim  fn L2 (Rd ) = A n→∞

4.2 Mass-Critical Profile Decomposition and

  lim eitΔ fn 

n→∞

= ε > 0.

2(d+2)

Lt,x d

111

(4.42)

(R×Rd )

  Then passing to a subsequence, there exists φ ∈ L2 Rd , λn ∈ (0, ∞), ξn , xn ∈ Rd , tn ∈ R, and α (d) > 0, such that   d/2 λn e−iξn ·(λn x+xn ) eitn Δ fn (λn x + xn )  φ (x) , weakly, (4.43) and

 ε α   2  2 2 lim  fn L2 (Rd ) −  fn − φn L2 (Rd ) = φ L2 (Rd )  ε 2 , n→∞ A where



 x − xn 1 −itn Δ iξn ·x −itn Δ φn (x) = e g0,ξn ,xn ,λn φ = d/2 e φ e . λn λn Proof

(4.44)

First, generalize dyadic intervals to any dimension d.

Definition 4.9 (Dyadic cubes) Let D j be the set of all dyadic cubes of side length 2 j ,   D j = ∏dl=1 [2 j kl , 2 j (kl + 1)) ⊂ Rd : k ∈ Zd . Then let D=



D j,

j∈Z

and for any Q ∈ D, let fˆQ (ξ ) = χQ (ξ ) fˆ (ξ ) , where χQ (ξ ) is the characteristic function of Q. d Then, make a Whitney decomposition in R order utilize   in  to effectively  itΔ itΔ   e u0 e u0 L p , for some bilinear Strichartz estimates for estimating t,x 1 ≤ p ≤ ∞. By Lemma 1.15 it is possible to partition Rd × Rd into a union of pairs of dyadic cubes:

Rd × Rd =



Q × Q ,

(4.45)

Q∈D:Q∼Q

where Q ∼ Q if dist (Q, Q ) ∼ diam (Q) = diam (Q ). First consider the case when d = 1. Theorem 4.10 Suppose dist (Q, Q )  diam (Q) = diam (Q ). Then,       itΔ   itΔ    e fQ e fQ  13/5  |Q|−5/13  fˆ 13/8  fˆ 13/8  . Lt,x (R×R)

Lξ

(Q)

Lξ

(Q )

(4.46)

112

Mass-Critical NLS Problem in Higher Dimensions

Proof

By (1.11),   itΔ   itΔ        e fQ e fQ  ∞   fˆL1 (Q)  fˆL1 (Q ) . L (R×R) ξ

t,x

Also, by (4.20),   itΔ   itΔ   e fQ e fQ  2 L

t,x (R×R)

ξ

(4.47)

     |Q|−1/2  fˆL2 (Q)  fˆL2 (Q ) . ξ

ξ

Then (4.46) holds by interpolation. 13 13 The importance of the exponents 13 5 and 8 in (4.46) is that 5 < 3 and 13 13 < 2. Inequality (4.46) with 5 replaced by 3 and 8 replaced by 2 follows directly from H¨older’s inequality and Strichartz estimates. The fact that 13 5 d+3 d+1 ,       itΔ   itΔ   d+2  e u0 e u1  q  Rd− q u0 L2 (Rd ) u1 L2 (Rd ) . Lt,x (R×Rd ) q,d Proof

See Tao (2003).

This yields the analogue of Theorem 4.10 in dimensions d ≥ 2. Theorem 4.12 Suppose dist (Q, Q )  diam (Q) = diam (Q ). Then for some p < 2, 1       itΔ   itΔ   1− 2 −  e fQ e fQ  d 2 +3d+1 d,p |Q| p d 2 +3d+1  fˆL p (Q)  fˆL p (Q ) . ξ ξ d(d+1) Lt,x (R×Rd ) (4.48) Remark d 2 +3d+1 d(d+1)

Proof


  itΔ   itΔ    e fQ e fQ 

q

ξ

(4.49)

d+3 d+1 ,

Lt,x (R×Rd )

q |Q|

 1− d+2 dq 

   fˆL2 (Q)  fˆL2 (Q ) . ξ

ξ

(4.50)

4.2 Mass-Critical Profile Decomposition Therefore, (4.48) holds for

d 2 +3d+1

2( ) d 2 +3d+2

113

< p ≤ 2.

If the endpoint case (q = d+3 d+1 ) of Theorem 4.11 were proved then 2(d 2 +3d+1) (4.48) would hold for d 2 +3d+2 ≤ p ≤ 2. When p = 2, (4.48) follows directly from (4.50). If (4.50) also held for q = d+3 d+1 , then interpolating (4.50) (for q = 2(d 2 +3d+1) d+3 d+1 ) with (4.49) would yield (4.48) with p = d 2 +3d+2 .

Remark

Theorems 4.10 and 4.12 combined with the Whitney decomposition imply the following proposition. When d = 1,

2/15  13/15   − 21  itΔ    e fQ L∞ (R×Rd )  f L2 (R) sup |Q| . (R×R)

Proposition 4.13  itΔ  e f 

6 Lt,x

x

For d ≥ 2, let q =  itΔ  e f 

2(d 2 +3d+1) . d2

2(d+2)

Lt,x d

(R×Rd )

t,x

Q∈D

(4.51)

Then

  d+1 d  f  d+2 Lx2 (Rd )

sup |Q|

d+2 − 1 2 dq

 itΔ  e f Q 

q Lt,x

Q∈D



1 d+2

.

(R×Rd )

(4.52) Proposition 4.13 directly implies the analogue (3.27) for the mass-critical problem. As in the energy-critical case, α (d) may change from line to line.   By Alaoglu’s theorem, a ball in Lx2 Rd is weakly compact. When d = 1, (4.41), (4.42), and (4.51) directly imply that after passing to a subsequence there exists a sequence of cubes {Qn } ⊂ D such that

ε

 ε  132 A

   |Qn |−1/2 eitΔ ( fn )Qn L∞ (R×R) . t,x

When d ≥ 2, (4.41), (4.42), and (4.52) imply

ε d+2 A−(d+1)  lim |Qn | n→∞

d+2 − 1 2 dq

 itΔ  e ( fn )  q Qn Lt,x (R×Rd ) .

(4.53)

Let λn−1 be the side length of Qn and let ξn be the center of Qn . By H¨older’s inequality, (4.53), and Strichartz estimates, d − d+2 q

ε d+2 A−(d+1)  lim inf λn2 n→∞

Lt,x

Lt,x d

d − d+2 q

 lim inf λn2 n→∞

  d(d+2)   d+1  itΔ  2 +3d+1  itΔ  2  e ( fn )Qn  d 2(d+2)  e ( fn )Qn  d∞+3d+1  d+1 d(d+2)    2 A d 2 +3d+1 eitΔ ( fn )Qn  d∞+3d+1 . Lt,x

114

Mass-Critical NLS Problem in Higher Dimensions

Therefore, there exist sequences tn ∈ R, xn ∈ Rd such that    ε α −d   ε  lim inf λn 2 eitn Δ ( fn )Qn  (xn ) . n→∞ A By the Sobolev embedding theorem,      ∇ e−ix·ξn eitn Δ ( fn )Qn 

− d2 −1

Lx∞ (Rd )

 λn

A,

  and therefore there exists some ψ ∈ Lx2 Rd , ψ L2 = 1, such that $ %  x 1  −i(x+xn )·ξn itn Δ  lim inf  e e ( fn )Qn (x + xn ) , d/2 ψ  n→∞   λn λn    ε α    d/2  = lim inf  λn e−i(λn x+xn )·ξn eitn Δ fn (λn x + xn ) , ψ   ε . n→∞ A Therefore, g−xn ·ξn ,−λn ξn , −xn , λn

 1 λn

  it Δ  eitn Δ fn = g−1 e n fn 0,ξn ,xn ,λn

(4.54)

 d 2 hasa subsequence   ε αwhich converges weakly to some φ (x) ∈ Lx R with φ  2 d  ε . This proves (4.43). A Lx (R ) Weak convergence of (4.54) implies   it Δ  −1  it Δ  n g−1 e , g e n fn 2 f n 0,ξn ,xn ,λn 0,ξn ,xn ,λn L    it Δ    −1 −1 n − g0,ξn ,xn ,λn e fn − φ , g0,ξn ,xn ,λn eitn Δ fn − φ 2 L    2  it Δ  −1 = 2g0,ξn ,xn ,λn e n fn − φ , φ 2 → φ L2 . L

 2

 Since the L Rd inner product is invariant under the Galilean transform, translation in space, scaling, and the action of eitΔ , we have  2 fn , fn L2 − fn − φn , fn − φn L2 → φ L2 (Rd ) , which proves (4.44). To complete the proof of Theorem 4.8 it remains only to prove Proposition 4.13.   Proof of Proposition 4.13 Without loss of generality suppose  f L2 (Rd ) = 1. By (4.45), for almost every ξ , ξ  ∈ Rd × Rd ,   ∑ χQ (ξ ) χQ ξ  = 1. Q∼Q

4.2 Mass-Critical Profile Decomposition

115

Since dist (Q, Q ) ≤ 10 diam (Q), let Q = Q + Q = {ξ  : ξ  = ξ + ξ  , ξ ∈ Q, ξ  ∈ Q }. Clearly, diam (Q ) = diam (Q) + diam (Q ). Let c (Q ) be the center of Q . Adding two paraboloids together gives 1 1 |ξ | 2 + | ξ  | 2 = | ξ + ξ  | 2 + | ξ − ξ  | 2 2 2      1   1   = |c Q |2 + c Q · ξ + ξ  − c Q + |ξ + ξ  − c Q |2 2 2 1 + |ξ − ξ  |2 . 2 Then let 







R Q+Q =

* 1 τ − 12 |c (Q ) |2 − c (Q ) · [η − c (Q )] (τ , η ) : ≤ ≤1 . 8 diam (Q )2

If F (τ , ξ ) is the Fourier transform of F (t, x) in space and in time, then

     itΔ f  (τ , η ) ⊂ R Q + Q . e supp eitΔ f Q Q The supports of R (Q + Q ) are approximately disjoint. Lemma 4.14

For α ≤ 1.01, sup

∑

τ ,η (Q,Q )∈F

χα R(Q+Q ) (τ , η ) d 1.

Proof For (τ1 , ξ1 ), (τ2 , ξ2 ) on the paraboloid τ = |ξ |2 , ξ1 ∈ Q, ξ2 ∈ Q , Q ∼ Q , 1 1 τ1 + τ2 − |ξ1 + ξ2 |2 = |ξ1 − ξ2 |2 ∼ diam (Q)2 . 2 2 So for a given (τ , η ) it suffices to consider Q such that diam (Q)2 ∼ τ − 12 |η |2 and dist (Q, Q ) ∼ diam (Q). Since |Q| = |Q |,



1 1  dist η , Q , dist η , Q  diam (Q) . 2 2 Therefore, for a fixed τ , η , there are only finitely many (Q, Q ) ∈ F such that (τ , η ) ∈ R (Q + Q ).

116

Mass-Critical NLS Problem in Higher Dimensions

Therefore, by the Plancherel identity,    itΔ 4 e f  4 =   Lt,x

∑

itΔ

e

Q∼Q

∑





Q∼Q



itΔ

fQ e

  2 fQ   2

Lt,x

  itΔ   itΔ  2  e fQ e fQ  2 . L

(4.55)

t,x

Meanwhile, by the triangle inequality,  itΔ 2 e f  2 ≤ L t,x

  itΔ   itΔ   e fQ e fQ L1

∑

(4.56)

t,x

Q∼Q

and   itΔ 2   e f 

∞ Lt,x

  itΔ   itΔ   e fQ e fQ L∞



∑

(4.57)

t,x

Q∼Q

also hold. Interpolating (4.55) and (4.57) gives, when d = 1,   itΔ 2 3/2  e f  3

Lt,x (R×R)





∑

Q∼Q

  3/2 eitΔ fQ eitΔ fQ L3 (R×R) .

(4.58)

t,x

Therefore, by (4.46), (4.58)

∑



 itΔ 1/5  itΔ 1/5  13/10  13/10 e fQ  ∞ e fQ  ∞ |Q|−1/2  fˆ 13/8  fˆ 13/8 L (R×R) L (R×R) t,x

(Q,Q )∈F





sup |Q| Q∈D

 e

−1/2  itΔ

Lξ

t,x

 fQ 

2/5 ∞ (R×R) Lt,x

∑ |Q|

(Q)

 13/5 fˆ 13/8



−3/10 

Q∈D

Lξ

Lξ

(Q)

(Q )

.

Now for |Q| = 2 j for some j ∈ Z, make the following split: fˆ (ξ ) = fˆ j (ξ ) + fˆj (ξ ) , ˆ ˆ where fˆ j (ξ ) = fˆ (ξ ) when | fˆ (ξ ) | ≥ 2− j/2 and fˆj (ξ) =  f (ξ ) when | f (ξ ) | < − j/2   . By the Plancherel identity, the fact that f L2 = 1, and Young’s 2

4.2 Mass-Critical Profile Decomposition

117

inequality, we have

∑∑

13/5 3j  2− 10  fˆ j  13/8 Lξ

j Q∈D j

=∑

3j − 10

∑

(Q)



2

Q

j Q∈D j 3j − 10

∑

∑

2

∑

∑

2− 10



j Q∈D j 3j



j Q∈D j

∑

  j  fˆ (ξ ) 13/8 d ξ 

∑

{ξ ∈Q:2k ≤| fˆ(ξ )| 0 such that   itΔ  e u (t0 )   ≤ ε0 . (4.82) T2 (ε0 ) T2 (ε0 ) 2(d+2)/d Lt,x

t0 −

N (t0 )2

,t0 +

N (t0 )2

Also, for any nonzero u (t0 ),   itΔ  e u (t0 )  2(d+2)/d

×Rd



T (ε ) T (ε ) t0 − 2 02 ,t0 + 2 02 ×Rd N (t0 ) N (t0 )

Lt,x



> 0,

so since u (t0 ) lies in a precompact set modulo symmetries,   itΔ  e u (t0 )   ∼ ε0 . T2 (ε0 ) T2 (ε0 ) 2(d+2)/d Lt,x

t0 −

N (t0 )2

,t0 +

N (t0 )2

×Rd

(4.83)

 Inequality (4.82) and Strichartz estimates imply that for all t ∈ t0 − T2 2 ,t0 + N(t0 ) T2  , 2 N(t0 )

  u (t) − ei(t−t0 )Δ u (t0 ) 

Lx2 (Rd )

1+ d4

 ε0

.

(4.84)

By H¨older’s inequality, the Sobolev embedding theorem, and the Arzel`a– Ascoli theorem, there exists a constant C (u) such that 999m0 ≤ 1000

 |x−x(t)|≤



m20 ≤C 1000

C(m20 /1000) N(t)



|u (t, x) |2 dx

m2 N (t0 ) 1+ 4 + 0 +C (u) ε0 d . N (t) 1000

This implies that for ε0 sufficiently small, N (t)  N (t0 ) for all

 T2 (ε0 ) T2 (ε0 ) t ∈ t0 − ,t + . 0 N (t0 )2 N (t0 )2

(4.85)

126

Mass-Critical NLS Problem in Higher Dimensions

Time reversal symmetry proves that N (t)  N (t0 ) as well. Relations (4.83)– (4.85) imply (4.81) in the mass-critical case. The proof in the energy-critical case is similar, although Lemma 3.2 is used in place of the perturbative argument in (4.84). Additionally, for the mass-critical problem (4.84), + 2

, + 2

, m0 m0 ξ : |ξ − ξ (t0 ) | ≤ C N (t0 ) ∩ ξ : |ξ − ξ (t) | ≤ C N (t) 1000 1000 = 0. /

 Therefore, for all t ∈ t0 − This implies the following. Theorem 4.22

 T2 ,t0 + T2 2 , N(t0 )2 N(t0 )

we have |ξ (t) − ξ (t0 ) |  N (t).

Possibly after modifying C (η ) in (4.80) by a fixed constant,   d   N (t)  N (t)3 (4.86)  dt 

and

  d   ξ (t)  N (t)3 .  dt 

(4.87)

Theorem 4.21 also implies that if sup (I) < ∞, then lim N (t) = +∞.

tsup(I)

In fact N (t)  (sup (I) − t)−1/2 . The same holds if inf (I) > −∞. Now let us make a reduction similar to Theorem 3.14 to yield the next result. Theorem 4.23 (Mass-critical scenarios) If there is a nonzero almost-periodic solution to (4.36), then for at least one such solution, one of the following two scenarios holds: I = R,

N (t) ≤ 1 for all t ∈ R

(4.88)

or I = (0, ∞) ,

N (t) = t −1/2 .

(4.89)

We call (4.89) a self-similar solution. Proof The proof is nearly identical to Theorem 3.14. Once again define the quantity osc (T ) = inf

t0 ∈I

sup{N (t) : t ∈ I and |t − t0 | ≤ T N (t0 )−2 } inf{N (t) : t ∈ I and |t − t0 | ≤ T N (t0 )−2 }

.

4.2 Mass-Critical Profile Decomposition

127

If supT osc (T ) < ∞ it is possible to extract a solution satisfying N (t) ∼ 1. In the case that lim osc (T ) = ∞,

T →∞

modify a (t0 ) slightly, defining a (t0 ) =

inf{N (t) : t ∈ I and t ≥ t0 } inf{N (t) : t ∈ I and t ≤ t0 } + . N (t0 ) N (t0 )

This time, if inf a (t0 ) = 0,

t0 ∈I

it is possible to extract an almost-periodic solution satisfying N (t) ≤ 1 for all t ∈ I, rather than N (t) ≥ 1 for all t ∈ I, as was the case in the energy-critical problem. The condition N (t) ≤ 1 on I automatically implies I = R. Finally, if inf a (t0 ) = 2ε > 0,

t0 ∈I

it is possible to extract an almost-periodic solution satisfying N (t) ∼ t −1/2 for all t ∈ (0, ∞), as in Theorem 3.14. Because this is the only finite-time reduction for the mass-critical problem, (3.69) may be replaced by (4.89). The quantity N (t)3 occurs quite naturally in the study of the mass-critical nonlinear Schr¨odinger equation. We have already seen that        d  ξ (t) ,  d N (t)  N (t)3 . (4.90)    dt  dt It is convenient to split (4.88) into two scenarios. Definition 4.24 (Soliton-like solution)  ∞ 0

N (t)3 dt = ∞.

Definition 4.25 (Rapid cascade solution)  ∞ 0

N (t)3 dt < ∞.

(4.91)

2 bilinear Strichartz estimates of As we shall see in the next section, the Lt,x Theorem 2.12 will also introduce scaling like N (t)3 . These bilinear Strichartz estimates are fundamentally tied to the fact that the Schr¨odinger equation has two derivatives.

128

Mass-Critical NLS Problem in Higher Dimensions

Remark Notice that in case (4.91), ξ (t) can travel only a finite total distance. Additionally,   ∞    d N (t) dt < ∞,   dt 0 which implies limt→∞ N (t) = 0. For a more general function that satisfies -∞ 0 f (t) dt < ∞, all that can be said is lim inft→∞ f (t) = 0. Remark The reduction to three scenarios in Killip et al. (2009) divided (4.88) in a slightly different manner, separating it into the case when N (t) = 1, and the case when N (t) ≤ 1 and lim inft→±∞ N (t) = 0.

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2 To show that there does not exist a self-similar solution, it is enough to prove that a self-similar solution has H˙ 1 regularity. Then by Theorem 1.30 this contradicts that the self-similar solution blows up in finite time. Theorem 4.26 (Additional regularity for a self-similar solution) Suppose u is a self-similar solution to (4.36), that is, u is of the form of (4.89). Then there exists some ξ∞ ∈ Rd such that for any 0 ≤ s < 1 + d4 , t ∈ R, we have  −ix·ξ  ∞ e u (t) H˙ s (Rd ) d,s,u t −s/2 . x

Proof Killip et al. (2008, 2009) and Tao et al. (2007b) proved Theorem 4.26 for radial data in dimensions d ≥ 2. The argument used the restriction theory of Shao (2009) for a radial solution to prove the base case of the inductive argument. The proof here will instead use the double Duhamel argument for the base case, and will hold for both radial and nonradial solutions. The inductive argument is the same as in Killip et al. (2008, 2009) and Tao et al. (2007b). By (4.90), ξ (t) converges to some ξ∞ ∈ Rd as t → +∞. Moreover, (4.90) implies |ξ (t) − ξ∞ |  t −1/2 ,

for all t ∈ (0, ∞) .

Therefore, after modifying C (η ) by a constant, we may choose ξ (t) = ξ∞ , and then without loss of generality make a Galilean transform so that ξ∞ = 0.

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2 Following Killip et al. (2009), define   M (k) = sup u>k− j/2 L∞ L2 ([2 j ,2 j+1 ]×Rd ) , t x j∈Z   S (k) = sup u>k− j/2  2(d+2) , j∈Z Lt,x d ([2 j ,2 j+1 ]×Rd )   . N (k) = sup P>k− j/2 F (u (t))  2(d+2) (d+4) j∈Z Lt,x ([2 j ,2 j+1 ]×Rd )

129

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.92)

By (4.81) we know M (k) + S (k) + N (k)  1, and for any j ∈ Z, Strichartz estimates imply   u  1, for d ≥ 3, 2d Lt2 Lxd−2 ([2 j ,2 j+1 ]×Rd )     u 8/3  1, uL4 L∞ ([2 j ,2 j+1 ]×R)  1. Lt Lx8 ([2 j ,2 j+1 ]×R2 ) t x

(4.93) ⎫ ⎪ ⎬ ⎪ ⎭

(4.94)

Next, by (4.80), lim M (k) = 0.

(4.95)

S (k)  M (k) + N (k) .

(4.96)

k∞

Also, by Strichartz estimates,

Make the following splitting:      P F (u)  2(d+2)  P>k− j F u≤k− j  2(d+2) >k− 2j 2 2 Lt,xd+4 ([2 j ,2 j+1 ]×Rd ) Lt,xd+4 ([2 j ,2 j+1 ]×Rd )   4   +  |u>k− j | |u| d  2(d+2) . 2 Lt,xd+4 ([2 j ,2 j+1 ]×Rd ) By Bernstein’s inequality, (4.94), and interpolation,    P F u≤k− j  2(d+2) >k− 2j 2 Lt,xd+4 ([2 j ,2 j+1 ]×Rd )   d4   1  u  2(d+2)  2(d+2) j ∇u≤k− j 2 2k− 2 Lt,x d ([2 j ,2 j+1 ]×Rd ) Lt,x d ([2 j ,2 j+1 ]×Rd ) 1   d+1   d+2 j u>l  ∞  2−k+ 2 ∑ 2l u>l  d+2 2(d+1) 2(d+1) Lt Lx2 ([2 j ,2 j+1 ]×Rd ) Lt d Lx d−1 ([2 j ,2 j+1 ]×Rd ) l≤k− 2j  d+1  1   j P>l u d+2  2−k+ 2 ∑ 2l P>l u d+2 2(d+1) 2(d+1) Lt∞ Lx2 ([2 j ,2 j+1 ]×Rd ) Lt d Lx d−1 ([2 j ,2 j+1 ]×Rd ) l≤k− 2j 1

 ∑ 2l−k M (l) d+2 . l≤k

(4.97)

130

Mass-Critical NLS Problem in Higher Dimensions

Expressions (4.95) and (4.97) imply limk∞ N (k) = 0, and therefore, by (4.96), lim M (k) + S (k) + N (k) = 0.

k∞

(4.98)

This decay can be upgraded to a quantitative decay estimate. When d ≥ 2, equation (4.98) has the quantitative decay

Theorem 4.27

k

M (k) + S (k) + N (k)  2− 2d ;

(4.99)

and when d = 1, the decay is k

M (k) + S (k) + N (k)  2− 4 . The proof of this theorem will be postponed. Theorem 4.27 may then be used as the base case in an inductive argument. Lemma 4.28 (Inductive lemma) Let N > 0, 0 < s < 1 + d4 , and d ≥ 3. For all k > 100, 0 < β ≤ 1,

∑

N (k) s

k1 ≤kβ −N

2(k1 −k)s S (k1 )

+ S

4/d kβ − N + S (kβ − N) S (kβ − N) 2 (d − 1)

+ 2−2kβ /d [M (kβ − N) + N (kβ − N)]. 2

(4.100)

Proof This lemma was proved in Killip et al. (2008) for d ≥ 3: their argument will be the one presented here. First, split the nonlinearity: F (u)

    4 = F u≤kβ − j/2−N + O |u≤kα − j/2−N | d |u>kβ − j/2−N |     4 4 + O |ukα − j/2−N≤·≤kβ − j/2−N | d |u>kβ − j/2−N | + O |u>kβ − j/2−N |1+ d ,

where α will be defined later. By Bernstein’s inequality, the fractional product rule, and the fractional chain rule of Visan (2007), for any 0 ≤ s < 1 + d4 ,    P>k− j/2 F u≤kβ − j/2−N  2(d+2) Lt,xd+4 ([2 j ,2 j+1 ]×Rd )   j   −k s   |∇|s F u≤kβ − j/2−N  2(d+2)  2 2 Lt,xd+4 ([2 j ,2 j+1 ]×Rd ) s

∑

k1 ≤kβ −N

2(k1 −k)s S (k1 ) .

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2 By H¨older’s inequality,   4    |ukα − j/2−N≤·≤kβ − j/2−N | d |u>kβ − j/2−N | 

131

2(d+2)

Lt,xd+4

([2 j ,2 j+1 ]×Rd )

4

 S (kα − N) d S (kβ − N) and

  4   |u>kβ − j/2−N |1+ d 

4

2(d+2) Lt,xd+4

 S (kβ − N)1+ d . ([2 j ,2 j+1 ]×Rd )

β Finally by the bilinear Strichartz estimates of (4.23) with α = 2(d−1) , (4.96), H¨older’s inequality in time, and the Christ–Kiselev lemma (see Christ and Kiselev (2001), and Smith and Sogge (2000)),    4   O |u≤kα − j/2−N | d |u>kβ − j/2−N |  2(d+2) Lt,xd+4 ([2 j ,2 j+1 ]×Rd )    82   u≤kα − j/2−N u>kβ − j/2−N  d2 j j+1 d Lt,x ([2 ,2 ]×R )   1− d82 4− 8 u≤kα − j/2−N  d2 d 2 × u>kβ − j/2−N  2(d+2) Lt,x ([2 j ,2 j+1 ]×Rd ) Lt,x d ([2 j ,2 j+1 ]×Rd )  8 − 1   d−1  82  2kβ − j/2−N 2 2kα − j/2−N 2 d [M (kβ − N) + N (kβ − N)] d 2 1− 82

× S (kβ − N) −k 2β2

2

d

d

2

  j d2 − 42 d

[M (kβ − N) + N (kβ − N)].

This proves the lemma. Lemma 4.28 may be used to prove increasing regularity by induction, since the second and third terms in (4.100) are of order 1 + d4 . Indeed, suppose that for some σ > 0, S (k) + N (k) + M (k)  2−σ k .

(4.101)

Expression (3.78) holds in L2 for almost-periodic solutions to the mass-critical problem, so     P>k u (1)  2 d  ∑ P>k F (u)  2(d+2) , Lx (R ) j≥0 Lt,xd+4 ([2 j ,2 j+1 ]×Rd ) and then by (4.92),



j M (k)  ∑ N k + . 2 j≥0

(4.102)

132

Mass-Critical NLS Problem in Higher Dimensions

Then by (4.96) and (4.102), M (k) + S (k) + N (k)  M (k) + N (k) 

∑N

j≥0



j k+ . 2

By Lemma 4.28, with β = 1 − 2d1 2 ,

j N k + ∑ 2 j≥0 s

∑

∑

  j≥0 k ≤ k+ j β −N 1 2



2

 k1 −k− 2j s

S (k1 )

 

! d4

k + 2j β j −N +S k+ +∑ S β −N 2 (d − 1) 2 j≥0



j ×S k+ β −N 2







  j j − k+ 2j 2β2 d +∑2 β −N +N k+ β −N . M k+ 2 2 j≥0 

(4.103) Plugging (4.101) into (4.103), (4.103) 

∑



∑

2

  j≥0 k ≤ k+ j β −N 1 2

+∑

(k+ j )β σ

 k1 −k− 2j s −σ k1

2 − ·4 − 2 2(d−1) d2

2

   k+ 2j β −N σ

j≥0

+∑

     − k+ 2j · 2β2 − k+ 2j β −N σ d 2 2

j≥0

2kβ − d(d−1) σ −kβ σ

N,s,σ 2kβ (s−σ ) 2−ks + 2

2

− 2k2β

+2

d

2−kβ σ .

(4.104)

Iterating (4.103), starting with (4.99), proves that M (k) s 2−sk for all 0 < s < 1 + d4 . When d = 2, by (4.23),   P>k− j/2 F (u)  4/3 Lt,x ([2 j ,2 j+1 ]×R2 )     F u>kβ − j/2−N  4/3 j j+1 2 Lt,x ([2 ,2 ]×R )

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2 133       2   u>kβ − j/2−N  4/3 j j+1 2 + O ukα − j/2−N≤·≤kβ − j/2−N Lt,x ([2 ,2 ]×R )    2     u>kβ − j/2−N  4/3 j j+1 2 + O u≤kα − j/2−N Lt,x ([2 ,2 ]×R )  S (kα − N)2 S (kβ − N) + [M (kβ − N) + N (kβ − N)]2 Take β =

12 13 ,

α=

6 13 .

k(α −β ) 2

.

If M (k) + N (k) + S (k)  2−σ k ,

then M (k) + N (k) + S (k)  2− 13 σ k + 2− 13 σ 2− 13 , 24

12k

3k

and therefore, when s < 3, by induction, M (k) + N (k) + S (k) s 2−sk . Finally, for d = 1, by (4.23),   P>k− j/2 F (u)  6/5 Lt,x ([2 j ,2 j+1 ]×R)    4  u> 20k − j/2 L6 ([2 j ,2 j+1 ]×R) u> 10k − j/2 L6 ([2 j ,2 j+1 ]×R) t,x t,x 21 21



5k 20k 20k + 2− 21 M +N , 21 21 which implies M (k) s 2−5k when s < 5. This completes the proof of Theorem 4.26. Proof of Theorem 4.27 method. As in (3.89),

Theorem 4.27 is proved using the double Duhamel

   P>k u (1) , P>k u (1) =

∞ 1 1

0

 e−itΔ P>k F (u (t)) , e−iτ Δ P>k F (u (τ )) dt d τ .

Recall from linear algebra that if A + B = A + B ,     A + B, A + B  |A|2 + |A |2 + B, B .

(4.105)

For some T > 2, set  1

A = 0, A =

 T 1

B= 0

e−iτ Δ P>k F (u (τ )) d τ ,

B =

 ∞ T

e−itΔ P>k F (u (t)) dt, e−iτ Δ P>k F (u (τ )) d τ .

134

Mass-Critical NLS Problem in Higher Dimensions

First consider d ≥ 4. By Strichartz estimates,   A  2 d L (R )

   P>k F (u) N 0 ([1,T ]×Rd )

   P>k F (u≤k ) N 0 ([1,T ]×Rd )    +  |u>k | u≤

k 2(d−1)

4  d  

  4   u k  d  |u | + .  2(d+2)  >k > 2(d−1) Lt1 Lx2 ([1,T ]×Rd ) Lt,xd+4 ([1,T ]×Rd ) (4.106)

By (4.23) and H¨older’s inequality,     |u>k | u≤

k 2(d−1)

4  d  

Lt1 Lx2 ([1,T ]×Rd )

  2  T 1− d  |u>k | u≤

k 2(d−1)

4  d  

2 Lt,x

([1,T ]×Rd )

 1− d4 u>k  ∞ 2

Lt Lx ([1,T ]×Rd )

 k k 4/d T 2− 2 2 4  2−k/d .

(4.107)

Next, since N (t) ≤ 1 for t ≥ 1,   u>

  u>k 

2(d+2) Lt,x d

k 2(d−1)

4/d  2(d+2) 

Lt,x ([1,T ]×Rd )

4/d k . T S (k) S 2 (d − 1)

d

([1,T ]×Rd )

Finally, by Bernstein’s inequality and the product rule,   P>k F (u≤k ) 

N0

(

[1,T ]×Rd

  −k   0 ∇F (u  2 ) ≤k N ([1,T ]×Rd ) )  4  2−k  |∇u≤k | |u≤k | d N 0 ([1,T ]×Rd ) .

By the triangle inequality, 4

|∇u≤k | |u≤k | d  

∑ |∇uk1 | |u≤k | d

4

k1 ≤k

| d + ∑ |∇uk1 | |u≤ k1 | d . k1 ∑ |∇uk1 | |u 2(d−1) ≤·≤k 2(d−1) 4

k1 ≤k

4

k1 ≤k

(4.108)

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2

135

Therefore, by (4.108), H¨older’s inequality, and (4.23),   4  2−k  |∇u≤k | |u≤k | d  0 N ([1,T ]×Rd )     u k 4/d  2−k ∑ ∇uk1  2(d+2) 2(d+2) 1 ≥ 2(d−1) Lt,x d ([1,T ]×Rd ) k1 ≤k Lt,x d ([1,T ]×Rd )  4/d  1− 4 2 + 2−k T 1− d ∑  |∇uk1 | |u≤ k1 | L2 [1,T ]×Rd ∇uk1  ∞ d2 Lt Lx ([1,T ]×Rd ) ) t,x ( 2(d−1) k1 ≤k

∑

T



4 d k1 k1 2k1 −k 2 4 2− 2 +

k1 ≤k k

T 2− d + S



k 4 (d − 1)

∑

k1 ≤k

4 d

2(k1 −k) S (k1 ) S



k1 2 (d − 1)

4 d

∑ 2(k1 −k) S (k1 ) .

(4.109)

k1 ≤k

Next, B, B L2 = = 

 1 ∞ 0

T

0

T

 1 ∞

 e−itΔ P>k F (u (t)) , e−iτ Δ F (u (τ )) d τ

P>k F (u (t)) , ei(t−τ )Δ P>k F (u (τ )) dt d τ

 1 0

 P>k F (u (t)) 

2d Lxd+4

(Rd )

  

∞

T

  ei(t−τ )Δ P>k F (u (τ )) d τ 

2d

Lxd−4 (Rd )

dt.

(4.110) By Bernstein’s inequality,   P>k F (u) 

2d Lxd+4

(Rd )

   P>k F (u≤k ) 

2d Lxd+4

   2−k ∇F (u≤k )  

(Rd )

  4  +  |u>k | |u| d 

2d

Lxd+4 (Rd )

2d

Lxd+4 (Rd )





2d

Lxd+4 (Rd )

  4  +  |u>k | |u| d 

∑ 2k1 −k P>k1 uLx2 (Rd ) .

(4.111)

k1 ≤k

Since N (t) ≤ 1 when t ≥ 1, the dispersive estimate implies that for T > 2,   

∞

T

  ei(t−τ )Δ P>k F (u (τ )) d τ   sup

 ∞

t∈[0,1] T

2d

Lt∞ Lxd−4 ([0,1]×Rd )

1 2

(t − τ )

∑ 2k1 −k M (k1 ) dt

k1 ≤k



1 T

∑ 2k1 −k M (k1 ) .

k1 ≤k

136

Mass-Critical NLS Problem in Higher Dimensions

Therefore, 1 (4.110)  T



∑2

(k1 −k)



  M (k1 ) P>k F (u (t)) 

2d

Lt1 Lxd+4 ([0,1]×Rd )

k1 ≤k

Next, by (4.92) and (4.111), for any j ≤ 0,   P>k F (u (t)) 

2d

Lt1 Lxd+4 ([2 j ,2 j+1 ]×Rd )

2

j

∑2

(k1 −k)

k1 ≤k

M

. (4.112)

j k1 + , 2

and so B, B  (4.112)



j 1 (k1 −k) j (k1 −k) M (k ) 2 2 2 M k +  1 1 ∑ ∑ T k∑ 2 j≤0 k1 ≤k 1 ≤k

2 1 M (k1 ) 2(k1 −k) . (4.113)  T k∑ ≤k 1

Therefore, by (4.109) and (4.113), there exists a constant C (T ) such that   P>k u (1)  2 d  L (R )

k 1 ∑ 2(k1 −k) M (k1 ) +C (T ) 2− d T 1/2 k1 ≤k

4 d k +C (T ) S ∑ 2(k1 −k) S (k1 ) . (4.114) 4 (d − 1) k1 ≤k   The rescaling u (t, x) → T d/4 u T t, T 1/2 x maps a self-similar solution to a selfsimilar solution, so in fact, (4.114) implies x

M (k) 

k 1 ∑ 2(k1 −k) M (k1 ) +C (T ) 2− d T 1/2 k1 ≤k

4 d k +C (T ) S ∑ 2(k1 −k) S (k1 ) . 4 (d − 1) k ≤k

(4.115)

1

By Strichartz estimates and rescaling, S (k)  M (k)+(4.106), so by (4.107)– (4.115), M (k) + S (k) 

k 1 ∑ 2(k1 −k) [M (k1 ) + S (k1 )] +C (T ) 2− d T 1/2 k1 ≤k

4

4 ! d d k k +C (T ) M +S 4 (d − 1) 4 (d − 1)

×

∑ 2(k1 −k) [M (k1 ) + S (k1 )].

k1 ≤k

(4.116)

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2

137

Setting 1

α (k) = sup 2− 2d | j−k| [M ( j) + S ( j)] j

and remembering (4.93) gives

α (k) 

1 T

k

α (k) +C (T ) 2− 2d +C (T ) α 1/2



k 4 (d − 1)

4 d

α (k) .

(4.117)

The estimate (4.98) implies α (k)  0 as k  ∞, so choosing T large, but fixed, for k ≥ K0 , K0 is a large, fixed constant, (4.117) implies k

α (k)  C (T ) 2− 2d .

(4.118)

This completes the proof of Theorem 4.27 in the case when d ≥ 4. When d = 3 the proof is quite similar, save for some technical modifications. For any j ≥ 0, by Bernstein’s inequality,   P>k F (u) 

Lt4 Lx1 ([2 j ,2 j+1 ]×R3 )

   P>k F (u≤k ) L4 L1 ([2 j ,2 j+1 ]×R3 ) x

t

 4/3   + u>k L∞ L2 ([2 j ,2 j+1 ]×R3 ) u 16/3

t x Lt Lx8 ([2 j ,2 j+1 ]×R3 )    2−k ∇F (u≤k ) L4 L1 ([2 j ,2 j+1 ]×R3 ) + M (k) t



∑

k1 ≤k

1

 4/3 × uL∞ L2 t

  1   Since  t 3/2 ∞

T

x

Lt,x

([2 j ,2 j+1 ]×R3 )

([2 j ,2 j+1 ]×R3 )

 1/6 uk  ∞ 1 L L2 t

x

([2 j ,2 j+1 ]×R3 )

+ M (k)

∑ 2k1 −k [M (k1 ) + S (k1 )].

(4.119)

k1 ≤k

3j

4/3

Lt

  

x

 5/6 2k1 −k uk  10/3

([2 j ,2 j+1 ])

 2− 4 , when T > 2 we find

  ei(t−τ )Δ P>k F (u (τ )) d τ 

Lx∞

( ) R3



1

∑ 2k1 −k [M (k1 ) + S (k1 )].

T 3/4 k1 ≤k

(4.120)

138

Mass-Critical NLS Problem in Higher Dimensions

Also, using (4.113) and (4.119), we have 



∑ P>k F (u) Lt,x1 ([2 j ,2 j+1 ]×R3 )

j≤0



j≤0



∑ 23 j/4

j≤0







∑ 23 j/4 P>k F (u) Lt4 Lx1 ([2 j ,2 j+1 ]×R3 )





 j j k1 −k 2 + + M k + S k 1 1 ∑ 2 2 k ≤k 1

∑ 2k1 −k [M (k1 ) + S (k1 )].

(4.121)

k1 ≤k

Then, by (4.120) and (4.121), we have

1



B, B L2 

T 3/4

∑2

k1 −k

k1 ≤k

2 [M (k1 ) + S (k1 )]

.

To estimate A , consider   P>k F (u)  0 N ([1,T ]×R3 )

    4   P>k F (u≤k ) N 0 ([1,T ]×R3 ) +  |u>k | |u≤ k | 3  4

  4  +  |u>k | |u≥ k | 3  4

10

Lt,x7 ([1,T ]×R3 )

Lt1 Lx2 ([1,T ]×R3 )

.

Strichartz estimates and N (t) ≤ 1 for t ≥ 1 imply   4   |u>k | |u≥ k | 3  4

10/7

Lt,x

([1,T ]×R3 )

T S (k) S

4 k 3 . 4

Next, by (4.23) and (4.94),   4   |u>k | |u≤ k | 3  4

Lt1 Lx2 ([1,T ]×R3 )

  2/3   T 1/6  |u>k | |u≤ k | L2 [1,T ]×R3 uL2 L6 ([1,T ]×R3 ) ( ) t x 4 t,x  k k 2/3 k T 2− 2 2 4  2− 6 .

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2

139

Therefore, following (4.106), by Bernstein’s inequality, (4.23), and (4.94),       A  2  P>k F (u≤k )   2−k ∇F (u≤k ) N ([1,T ]×R3 ) L N ([1,T ]×R3 )     u k 4/3  2−k ∑ ∇uk1  10/3 10/3 Lt,x ([1,T ]×R3 ) ≥ 41 Lt,x ([1,T ]×R3 ) k1 ≤k 2/3  + 2−k ∑  |∇uk1 | |u≤ k1 | L2 [1,T ]×R3 ) t,x ( 4 k1 ≤k   2/3 1/3 × ∇uk1 L2 L6 [1,T ]×R3 uL2 L6 [1,T ]×R3 ( ) ) t x t x( 4/3 k1 T ∑ 2k1 −k S (k1 ) S + 2−k/6 . 4 k ≤k 1

Following the argument in (4.113)–(4.118), M (k) + S (k)  2−k/6 . Next take d = 2. In this case the nonlinearity is algebraic, so for j ≥ 0,   P>k F (u)  2 1 j j+1 2 Lt Lx ([2 ,2 ]×R )  2    u>k−3 L∞ L2 ([2 j ,2 j+1 ]×R2 ) uL4 ([2 j ,2 j+1 ]×R2 )  M (k − 3) . t x t,x Then for T > 2,  ∞    ei(t−τ )Δ F (u (τ )) d τ  

∞ [0,1]×R2 Lt,x ( )

T



1 M (k − 3) . T 1/2

When j ≤ 0,   P>k F (u)  1 j j+1 2 Lt,x ([2 ,2 ]×R )

 2   j  2 j/2 u>k−3 L∞ L2 ([2 j ,2 j+1 ]×R2 ) uL4 ([2 j ,2 j+1 ]×R2 )  2 j/2 M k + , t x t,x 2 so B, B L2 

1 T 1/2

M (k − 3)

∑2

k1 ≤k

(k1 −k) 2

M (k1 ) .

Also,   P>k F (u) 

4/3 Lt,x ([1,T ]×R2 )       |u>k−3 | |u≤ k | L2 ([1,T ]×R2 ) u 4/3 Lt,x ([1,T ]×R2 ) t,x 2       + u>k−3 L4 ([1,T ]×R2 ) u> k L4 ([1,T ]×R2 ) uL4 ([1,T ]×R2 ) t,x t,x t,x 2

k k T 2− 4 + S (k − 3) S . 2

Again following the argument in (4.113)–(4.118), we obtain M (k) + S (k)  2−k/4 .

140

Mass-Critical NLS Problem in Higher Dimensions

When d = 1 the nonlinearity is also algebraic, so if j ≥ 0,   P>k F (u)  4/3 Lt Lx1 ([2 j ,2 j+1 ]×R)  4    P>k−3 uL∞ L2 ([2 j ,2 j+1 ]×R) uL16/3 L8 [2 j ,2 j+1 ]×R  M (k − 3) , ) t x x( t so for T > 2,   

∞ T

  ei(t−τ )Δ F (u (τ )) d τ 

∞ ([0,1]×R) Lt,x



1 M (k − 3) . T 1/4

When j ≤ 0,   P>k F (u)  1 j j+1 Lt,x ([2 ,2 ]×R)

 4   j j/4  j/4    P>k−3 u L∞ L2 ([2 j ,2 j+1 ]×R) u L16/3 L8 [2 j ,2 j+1 ]×R  2 M k + 2 , ) t x x( t 2 so B, B L2  Also,   P>k F (u) 

1 T 1/4

M (k − 3)

∑2

k1 ≤k

(k1 −k) 4

M (k1 ) .

6/5

Lt,x (θ )

 1/2  1/2  1/2  3   |u>k−3 | |u≤ k | L2 (θ ) u>k−3 L6 (θ ) u≤ k L∞ (θ ) uL6 (θ ) t,x t,x 2 2 t,x t,x  3     + u>k−3 L6 (θ ) u> k L6 ([1,T ]×R2 ) uL6 ([1,T ]×R2 ) t,x t,x t,x 2

k k T 2− 4 + S (k − 3) S , 2 where we have written θ = [1, T ] × R for clarity. Again following the argument in (4.113)–(4.118), it follows that M (k) + S (k)  2−k/4 . This completes the proof of Theorem 4.27, which in turn completes the proof of Theorem 4.26. Having shown that an almost-periodic, self-similar solution does not exist, it only remains to prove that the only almost-periodic solution in the form of (4.88) is u ≡ 0. Killip et al. (2008, 2009) and Tao et al. (2007b) proved this for d ≥ 2 when u is radial. Theorem 4.29 tion

The defocusing, mass-critical nonlinear Schr¨odinger equa-

  u (0) = u0 ∈ L2 Rd   is globally well posed and scattering for all u0 ∈ L2 Rd radial. 4

iut + Δu = F (u) = |u| d u,

(4.122)

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2 Proof

141

This also follows from proving additional regularity.

Theorem 4.30 (Additional regularity) If u (t, x) is a radial, almost-periodic global solution to the mass-critical problem with N (t) ≤ 1 for all t ∈ R, then for 0 ≤ s < 1 + d4 , d ≥ 2,   u (t, x) ∈ Lt∞ H˙ xs R × Rd . Killip et al. (2008, 2009) and Tao et al. (2007b) used the in/out decomposition to prove this. If f is a radial function, then standard Fourier analysis implies 

f (x) = C (d) and fˆ (ξ ) = C (d)

fˆ (|ξ |)



f (|y|)

 π /2 −π /2

 π /2 −π /2

ei|x| |ξ | sin(θ ) (cos (θ ))

d−2 2

d θ d|ξ |

ei|y| |ξ | sin(θ ) (cos (θ ))

d−2 2

d θ d|y|.

The Bessel function 

π 2

− π2

ei|y| |ξ | sin(θ ) (cos (θ ))

d−2 2

dθ

(4.123)

is then split into two pieces, the incoming piece and the outgoing piece, [P+ + P− ] f = f .

(4.124)   Remark When d > 4, P+ and P− need not be bounded operators in Lx2 Rd . By time-translation invariance, assume without loss of generality that t1 = 0. When calculating the norm of P± u (0) in some Hilbert space, it suffices to calculate the norms of P+ u (0) = −i and P− u (0) = i

 ∞

e−itΔ P+ F (u (t)) dt

(4.125)

e−itΔ P− F (u (t)) dt.

(4.126)

0

 0 −∞

Equation (4.125) pulls back outgoing waves, while (4.126) pushes forward incoming waves, so it is possible to utilize the decay in space of a radially symmetric function to estimate (4.125) and (4.126). Remark When d = 1, the radial Sobolev embedding does not give any decay as |x| → ∞ for radially symmetric functions. This is why Theorem 4.30 is restricted to d ≥ 2.

142

Mass-Critical NLS Problem in Higher Dimensions

Here, the proof will use the double Duhamel method, which is similar, although in this case the solution is not partitioned into two pieces that do not lie in Lx2 Rd , as in (4.124). Recall that Theorem 1.7 implies that the solution to eitΔ Pk propagates with speed ∼ 2k . The radial Sobolev embedding theorem gives good estimates outside the cone |x| = c2k |t| for some 0 < c < 1. The propagation speed ∼ 2k also implies that when ei(τ −t)Δ P k acts on the part of F (u (t)) with support restricted to inside the cone, say when t < 0, all waves will become outgoing waves in positive times, and must lie outside the cone |x| = c2k |τ | when τ > 0. A similar argument can be made for waves inside the cone when t > 0, and τ < 0. This fact can be utilized to prove additional regularity for an almost-periodic solution to (4.122). Turning to the details, modifying (4.92), define   M (k) = sup P>k u (t) L2 (Rd ) , t∈R   , S (k) = sup u>k  2(d+2) T ∈R Lt,x d ([T,T +1]×Rd ) and

  N (k) = sup P>k F (u)  T ∈R

.

2(d+2)

Lt,xd+4

([T,T +1]×Rd )

As in (4.93), (4.95), and (4.96), M (k) + S (k) + N (k)  1,

lim M (k) = 0,

k→∞

(4.127)

and S (k)  M (k) + N (k) . In fact, S (k) is controlled by the frequency envelope controlling N (k). Lemma 4.31 For any d ≥ 1, 0 < s < 1 + d4 , and δ > 0,   u>k  2(d+2) Lt,x d ([T,T +δ ]×Rd )    M (k) + η (δ ) ∑ 2(k1 −k)s u>k1  2(d+2) , Lt,x d ([T,T +δ ]×Rd ) k1 ≤k where η (δ )  0 as δ  0. Proof By (4.127), the Sobolev embedding theorem, Strichartz estimates, H¨older’s inequality in time, and interpolation, it follows that   = 0. (4.128) lim sup u 2(d+2) δ 0 T ∈R Lt,x d ([T,T +δ ]×Rd )

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2

143

Then by (4.128) and Bernstein’s inequality, for any 0 < s < 1 + d4 ,   P>k F (u) 

2(d+2)

Lt,xd+4

   u>k 

([T,T +δ ]×Rd )

2(d+2) Lt,x d

 4/d u 2(d+2)

([T,T +δ ]×Rd ) Lt,x d ([T,T +δ ]×Rd )   + P>k F (u≤k )  2(d+2) Lt,xd+4 ([T,T +δ ]×Rd )    η (δ )4/d u>k  2(d+2) Lt,x d ([T,T +δ ]×Rd )   + η (δ )4/d ∑ 2(k1 −k)s u>k1  2(d+2) . Lt,x d ([T,T +δ ]×Rd ) k1 ≤k

(4.129)

  Proof of Theorem 4.30 Let χ (t, y) ∈ C0∞ Rd be a smooth function, χ (t, y) = 1 on |y| ≤ c2k |t| for some small constant c > 0, and χ (t, y) be supported on |y| ≤ c2k+1 |t|. Following (4.105), let A=

 δ 0

B=

 ∞ δ

e−itΔ P>k F (u (t)) dt +

 ∞ δ

e−itΔ (1 − χ ) (t, y) P>k F (u (t)) dt,

e−itΔ χ (t, y) P>k F (u (t)) dt,

and let A , B be the corresponding integrals in the negative time direction. The dual to the radial Strichartz estimate  d−1       u0 L2 (Rd )  |x| 2 eitΔ u0  4 ∞ d x Lt Lx (R×R ) implies   

δ

∞

  e−itΔ (1 − χ ) (t, y) P>k F (u) (t) dt  2 d Lx (R )     1   (1 − χ ) (t, y) P>k F (u) (t)   d−1   (c2k t) 2

. 4/3 1 Lx

Lt

(4.130)

([δ ,∞)×Rd )

Also, (4.102) and (4.127) imply  d−1     |x| 2 u

Lt4 Lx∞ ([n,n+1]×Rd )

 1.

(4.131)

144

Mass-Critical NLS Problem in Higher Dimensions

When d = 2, by (4.131) and the support properties of 1 − χ , we have    1   1 (1 − χ ) (t, y) P>k F (u (t))    4/3 1 1/2 k/2 1/2 c 2 t Lt Lx ([δ ,∞)×R2 )    1   1/2 k/2 P>k−3 uL∞ L2 uL∞ L2 x t t x c 2

4/3 3/4 ∞   1  (1 − χ ) u 4 ∞ × ∑ 2 Lt Lx ([δ n,δ (n+1)]×R ) 1/2 δ 1/2 n=1 n 

2−k/2 M (k − 3) . δ 3/4

(4.132)

Therefore, for any s < 3, (4.129) and (4.132) imply     A 2 , A  2 L L s

  2−k/2 M (k − 3) + η (δ )2 ∑ 2(k1 −k)s P>k1 uL4 ([0,δ ]×R2 ) . (4.133) 3/4 t,x δ k1 ≤k

When d = 3, Bernstein’s inequality implies         P>k F (u)  1  2−11k/6  |∇|11/6 F (u≤k )  1 + u>k  2 u4/3 8/3 , L L L x

x

x

Lx

so     A 2 , A  2 L L   1 1   (1 −  k χ ) (t, y) P F (u (t)) >k   4/3 1 c2 t Lt Lx ([δ ,∞)×R3 )

 1/3   1    k 1/3 uL∞ L2 P>k uL∞ L2 sup  (1 − χ ) uL4 L∞ [n,n+1]×R3 ) t x t x t x( c2 δ n∈Z    4/3 1 + 17k/6 1/4  |∇|11/6 u≤k L∞ L2 u≤k  ∞ 8/3 x t Lt Lx c2 δ 2−k 1  1/3 M (k) + 7k/3 1/4 ∑ 211k1 /6 M (k1 ) . (4.134) δ 2 δ k1 ≤k When d ≥ 4, interpolating (4.130) with the usual Strichartz estimates,  ∞        −itΔ A 2 , A  2   (1 − χ ) (t, y) P>k F (u (t)) dt   δ e  2 d L L Lx (R )     1   2d  k 3/2 P>k F (u (t))  4/3 d+4 (c2 t) Lt Lx ([δ ,∞)×Rd ) k1 −k 1 (4.135)  3/2 3k/2 3/4 ∑ 2 2 M (k1 ) . c 2 δ k1 ≤k

4.3 Radial Mass-Critical Problem in Dimensions d ≥ 2 145   Now take B, B L2 . The cutoff function χ ∈ C0∞ Rd is a Schwartz function, so for any M,  k d c2 t χˆ (t, ξ ) M (4.136) M. (1 + c2k t|ξ |) Therefore, by the Parseval identity, (4.136), Strichartz estimates, and (4.127), for k ≥ 0, we have, since |ξ |  2k , for k > 0,  ∞   −δ  e−iτ Δ χ (τ , y) P>k F(u (τ )) d τ , e−itΔ χ (t, y) P>k F(u (t)) dt B, B L2 =  = d

−∞ −δ −∞



δ

ei(t−τ )Δ χ (τ , y) P>k F (u (τ )) d τ , 

∞

δ

c2k t

 ∞

2

d

d+6 (1 + c2k t|ξ |)

dt

δ

χ (t, y) P>k F (u (t)) dt

 δ −10 2−10k .



(4.137)

Take d = 2. By (4.132), (4.133), (4.137), and time-translation invariance, M (k) s δ −5 2−5k + + η (δ )2

1

M (k − 3)

  2(k1 −k)s sup P>k1 uL4 ([T,T +δ ]×R2 ) . t,x T ∈R

2k δ 3/4

∑

k1 ≤k

(4.138)

Now, for any s < s, define the frequency envelopes M  (k) = and

S  (k) =



∑ 2(k1 −k)s M (k1 )

k1 ≤k

∑

2(k1 −k)s

k1 ≤k





  sup P>k1 u

T ∈R

2(d+2) Lt,x d

. ([T,T +δ ]×Rd )

Then by Lemma 4.31 and (4.138), choosing δ (s, s ) > 0 small enough yields M  (k) s,s and

S  (k) s,s

δ −5 1 M  (k) + η (δ )2 S  (k)  + 2s k δ 3/4 2k M  (k) + η (δ )2 S  (k) . 

So when d = 2, for any s < 3, we have M (k) ≤ M  (k) s 2−s k . Similarly, when d = 3, Lemma 4.31, (4.134), and (4.137) imply that for s < s < 73 , 

M (k) ≤ M  (k) s,s 2−s k . Finally, when d ≥ 4, Lemma 4.31, (4.135), and (4.137) imply that for s < s < 1 + d4 , 

M (k) ≤ M  (k) s,s 2−s k . This completes the proof of Theorem 4.30.

146

Mass-Critical NLS Problem in Higher Dimensions

Proof of Theorem 4.29 Theorem 4.30 directly implies that if u is an almostperiodic solution to the defocusing, mass-critical nonlinear Schr¨odinger equation of the form (4.88), then u ≡ 0. Since u is radial, ξ (t) = 0. So for an almost-periodic solution in the form of (4.88), by the interaction Morawetz estimate in (4.32), then   3−d   < ∞. (4.139)  |∇| 2 |u|2  2 Lt,x (R×Rd ) When d = 2, the Sobolev embedding theorem implies   u 4 8 < ∞. Lt Lx (R×Rd )

(4.140)

If N (t) ∼ 1 and u is radial, then x (t) = 0, so u (t) lies in a compact subset of Lx2 (Rd ), which implies that if u is a nonzero solution to (4.122), then uLx8 ≥ δ (u) > 0, contradicting (4.140), forcing u ≡ 0. When d ≥ 3, interpolating (4.139) with the Sobolev embedding theorem and product rule implies   u < ∞. (4.141) 2(d+1) Ltd+1 Lx d−1 (R×Rd ) 2 d Once again, since u(t) must  lie in a compact subset of L (R ) for all t, by H¨older’s inequality, u (t)  2(d+1)/(d−1) ≥ δ (u) > 0 if u is a nonzero solution to Lx (4.122), which contradicts (4.141). Now suppose that lim inft→±∞ N (t) = 0. In this case, (4.80), ξ (t) = 0, the Sobolev embedding, and Theorem 4.30 imply that for any η > 0,

E (u (t))  C (η ) N (t) + η ε (d) , for some ε (d) > 0. Since η > 0 is arbitrary and lim inft→∞ N (t) = 0, then by conservation of energy we have E (u (t)) ≡ 0 for all t ∈ R, and thus u ≡ 0.

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3 For a nonradial, almost-periodic solution to the mass-critical problem,   iut + Δu = F (u) = |u|4/d u, u (0, x) = u0 ∈ Lx2 Rd , (4.142) satisfying N (t) ≤ 1 for all t ∈ R, instead of proving additional regularity, a solution will be estimated using a frequency-localized interaction Morawetz estimate. This truncation will introduce some error terms, which will be estimated using long-time Strichartz estimates. Long-time Strichartz estimates were first introduced in Dodson (2012) for the mass-critical nonlinear Schr¨odinger equation with d ≥ 3, although they

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3

147

were inspired by the frequency-localized interaction Morawetz estimates of Colliander et al. (2008) and the almost-Morawetz estimates used in the Imethod (Colliander et al. (2007), Colliander and Roy (2011), Dodson (2011)). Dodson (2016a,b) extended the work of Dodson (2012) to dimensions one and two, Killip and Visan (2012) and Visan (2012) applied these estimates to simplify scattering proofs of scattering for the energy-critical problem for dimensions three and four, and Murphy (2014, 2015) used the long-time Strichartz estimates for intercritical and energy supercritical problems. The heuristic behind these estimates is that at frequencies far away from N (t), the solution to (4.142) will be dominated by the free solution for long periods of time. Let u be an almost-periodic solution to (4.142). Define three constants 0 < η3  η2  η1 < 1,

(4.143)

according to the following rule. First, by (4.86) and (4.87) there exists some η1 > 0 such that |ξ  (t) | + |N  (t) | ≤ −1/4d

Writing Φ = 2−20 η3 

Φ |x−x(t)|≥ N(t)

N (t)3 . η1

(4.144)

for clarity, by (4.80), choose η2 and η3 such that

|u (t, x) |2 dx +



|ξ −ξ (t)|>ΦN(t)

|uˆ (t, ξ ) |2 d ξ ≤ η22

(4.145)

and

η3 < η210 . Let k0 be a positive integer and let [a, b] be a compact interval satisfying  b a

|u (t, x) |2(d+2)/d dx dt = 2k0 .

(4.146)

Rescaling using (1.81), it is possible to have N (t) satisfying  b a

N (t)3 dt = η3 2k0 .

(4.147)

Remark Strictly speaking, after rescaling using (1.81), the compact interval would rescale to [ λa2 , λb2 ]. However, since a, b, and λ are constants, there is no problem with calling λa2 the new a and λb2 the new b. Definition 4.32 (Galilean Littlewood–Paley projection) For j > 0, let Pj be the Littlewood–Paley operator defined by Definition 1.5. Let   Pξ0 , j f = eix·ξ0 Pj e−ix·ξ0 f .

148

Mass-Critical NLS Problem in Higher Dimensions

Abusing notation slightly, let Pξ0 ,0 = Pξ0 ,≤0 , and for j < 0, let Pξ0 , j = 0. For 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞, define the norm        q d  Pξ (t), j f  p q P = f (t, x) . (4.148)  d ξ (t), j Lt Lx (I×R ) Lx (R ) L p (I) t The long-time Strichartz estimate in Dodson (2012) proved that for all 0 ≤ j ≤ k0 ,   k0 − j Pξ (t),> j u 2 2 , 2d Lt2 Lxd−2 ([a,b]×Rd ) where crucially the implicit constant did not depend on k0 . Definition 4.33 (Long-time Strichartz seminorm) When d ≥ 3, define  2 2  u = sup 2 j−k0 Pξ (t),≥ j u . (4.149) 2d X ([a,b]×Rd ) L2 L d−2 ([a,b]×Rd ) 0≤ j≤k t

0

x

Theorem 4.34 (Long-time Strichartz estimate) If u is an almost-periodic solution to (4.142) for d ≥ 3, and u satisfies (4.146) and (4.147), then   u  1, (4.150) X ([a,b]×Rd ) with constant independent of k0 . Proof

Partition [a, b] in two different but related ways.

Definition 4.35 (Small intervals) Divide [a, b] into consecutive, disjoint intervals Jl , l = 0, . . . , 2k0 − 1, such that    u (t) 2(d+2)/d dt = 1. 2(d+2)/d Lx (Rd ) Jl Remark

Let N (Jl ) = supt∈Jl N (t). By (4.81) and (4.86), 

Jl

N (t)3 ∼ sup N (t) = N (Jl ) ∼ inf N (t) . t∈Jl

t∈Jl

(4.151)

Definition 4.36 Divide [a, b] into 2k0 consecutive, disjoint intervals Jα , α = 0, . . . , 2k0 − 1 such that    2(d+2)/d  N (t)3 + η3 u (t)  2(d+2)/d d dt = 2η3 . (4.152) Lx (R ) Jα For any integer 0 ≤ j < k0 , let Gkj

(k+1)2 j −1

=



α =k2 j

Jα ,

(4.153)

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3

149

and for j ≥ k0 , let Gkj = [a, b].

(4.154)

  ˜ let ξ Giα = ξ (a). If Giα = [a, ˜ b], ˜ Define ξ (Jl ) in a similar manner. Remark For N (t) constant, there is an upper bound on the number of Jl intervals that intersect any one Jα , and an upper bound on the number of Jα intervals that intersect any one Jl . For a general N (t), the partitions Jα and Jl can be quite different. Remark

By (4.144) and (4.152), for all t ∈ Giα , we have    |ξ (t) − ξ Giα | ≤

Giα

η1−1 N (t)3 dt ≤ η3 η1−1 2i ,

so for all t ∈ Giα ,       ξ : 2i−1 ≤ |ξ − ξ (t) | ≤ 2i+1 ⊂ ξ : 2i−2 ≤ |ξ − ξ Giα | ≤ 2i+2 . (4.155) Now fix some j and Gkj ⊂ [a, b] according to (4.153) and (4.154). For any t,t0 ∈ Gkj , Pξ (t),≥ j u (t) = ei(t−t0 )Δ Pξ (t),≥ j u (t0 ) − i

 t t0

ei(t−τ )Δ Pξ (t), j F (u (τ )) d τ .

Then by the appropriate choice of t0k for each Gkj , conservation of mass, the number of Gkj intervals, Strichartz estimates, and (4.155),   Pξ (t),≥ j u (t) 2 2d Lt2 Lxd−2 ([a,b]×Rd ) 2   2k0 − j + inf Pξ (t),≥ j−1 u (t) L2 (Rd ) x t∈[a,b]  t  2   2d   . (4.156) + ∑ Pξ (t),≥ j ei(t−τ )Δ F (u (τ )) d τ  2 d−2 j d k j

Gk ⊂[a,b]

Gk ×R

Lt Lx

t0

Also by (4.155), Strichartz estimates, and the fact that Pj is given by the convolution with a function whose L1 -norm is uniformly bounded,  t  2   i(t−τ )Δ 2d   P e F (u ( τ )) d τ   2 d−2 ∑ ξ (t),≥ j k j d j

Gk ⊂[a,b]

Gk ×R

Lt Lx

t0

 2  Pξ (τ ),≥ j−1 F (u (τ )) 

2d

Lt2 Lxd+2 ([a,b]×Rd )

.

(4.157)

Let n1 be an integer satisfying η1 ∼ 2n1 . Consistent with Definition 4.32, Pξ (t),≥a refers to Pξ (t),≥sup(a,0) , and so on.

150

Mass-Critical NLS Problem in Higher Dimensions

Remark Remember that a lowercase Latin letter refers to (1.24) and an uppercase Latin letter refers to (1.25). By Bernstein’s inequality, H¨older’s inequality, and u having finite mass, let  1 if j ≤ k0 , σ= 2 0 if j > k0 . Then

   Pξ (τ ),≥ j−1 F Pξ (τ ),≤ j+n u (τ )  1

2d

Lt2 Lxd+2 ([a,b]×Rd )

   2− j  (∇ − iξ (τ )) Pξ (τ ),≤ j+n1 u

2d

Lt2 Lxd−2 ([a,b]×Rd )

  1/2 (4.158)  η1 2σ (k0 − j) uX ([a,b]×Rd ) .   4 4 Next, by (4.145), since F (u) − F (ul )  |uh | |uh | d + |ul | d when u = ul + uh ,    F (u) − F Pξ (τ ),≤ j+n u  2d 1 Lt2 Lxd+2 ([a,b]×Rd )   4/d  P  Pξ (τ ),> j+n1 u uL∞ L2 [a,b]×Rd 1 2d − 4d ) t x( N(τ ) Lt2 Lxd−2 ([a,b]×Rd ) ξ (τ ),≥2−20 η3  4/d     u +  Pξ (τ ),> j+n1 u P  1 2d − 4d −20 ξ (τ ),≤2 η3 N(τ ) Lt2 Lxd+2 ([a,b]×Rd ) −1 4    2σ (k0 − j) η1 2 η2d uX ([a,b]×Rd )  4/d     +  Pξ (τ ),> j+n1 u P u .  1 2d − ξ (τ ),≤2−20 η3 4d N(τ ) Lt2 Lxd+2 ([a,b]×Rd ) (4.159) Now let χ (t, x) be a smooth function. We have ⎧ ⎨1 if |x − x (t) | ≤ 2−20 η −1/4d /N (t) , 3 χ (t, x) = ⎩0 if |x − x (t) | ≥ 2−19 η −1/4d /N (t) . 3 Again by (4.145),     Pξ (τ ),> j+n1 u P

−1/4d ξ (τ ),≤2−20 η3 N(τ )

4/d   u 

     Pξ (τ ),> j+n1 u P   + Pξ (τ ),> j+n1 u

−1/4d ξ (τ ),≤2−20 η3 N(τ ) 2d

Lt2 Lxd−2 ([a,b]×Rd )

2d

Lt2 Lxd+2 ([a,b]×Rd )

4/d  u χ (τ , x) 

2d

Lt2 Lxd+2 ([a,b]×Rd )

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3 151  4/d    × P u (1 − χ (τ , x))  ∞ 2 −1/4d ξ (τ ),≤2−20 η3 N(τ ) Lt Lx ([a,b]×Rd )     4/d      Pξ (τ ),> j+n1 u P u χ ( τ , x)  2d −1/4d ξ (τ ),≤2−20 η3 N(τ ) Lt2 Lxd+2 ([a,b]×Rd )   −1/2 4/d (4.160) + η1 η2 2σ (k0 − j) uX ([a,b]×Rd ) . Finally,   4/d 2     χ (τ , x) |Pξ (τ ),> j+n1 u| Pξ (τ ),≤2−20 η −1/4d N(τ ) u  2 =

∑

Jl ⊂[a,b]

2d

Lt Lxd+2 ([a,b]×Rd )

3

  4/d 2     χ (τ , x) |Pξ (τ ),> j+n1 u| Pξ (τ ),≤2−20 η −1/4d N(τ ) u  2

2d

Lt Lxd+2 (Jl ×Rd )

3

.

(4.161) When d = 3, (4.23) implies  4/3        χ (τ , x) Pξ (τ ),> j+n1 u Pξ (τ ),≤2−20 η −1/12 N(τ ) u  2 6/5 Lt Lx (Jl ×R3 ) 3         χ (τ , x)  ∞ 6   Pξ (τ ),> j+n1 u P u  2 −1/12 −20 Lt Lx (Jl ×R3 ) ξ (τ ),≤2 η3 N(τ ) Lt,x (Jl ×R3 ) 

−1/2 −1/8 η3 N (Jl )1/2 . 2 j/2

η1

Summing over intervals Jl in l 2 , by (4.152) and (4.153), we get −1/2 3/8 (k0 − j)/2 η3 2 .

(4.161)  η1

(4.162)

When d ≥ 4, (4.23) and H¨older’s inequality imply     χ (τ , x) |Pξ (τ ),> j+n1 u| P

− 1 ξ (τ ),≤2−20 η3 4d

4/d    u 

2d/(d+2)

(Jl ×Rd ) 4/d  u  2 − 1 Lt,x (Jl ×Rd ) ξ (τ ),≤2−20 η3 4d N(τ )  1−(4/d)     χ (τ , x)  d 2 /2(d−2) × Pξ (τ ),> j+n1 u 2 2d/(d−2) Lt∞ Lx Lt Lx (Jl ×Rd )

     Pξ (τ ),> j+n1 u P

−2/d



2−40(d−1)/d η1

N(τ )

Lt2 Lx

−(d−1)/2d 2

N (Jl )2(d−1)/d η3 22 j/d

 1−4/d × Pξ (τ ),> j+n1 u 2 2d/(d−2) Lt Lx

−(d−2)/2d 2

2−40(d−2)/d η3 (Jl ×Rd )

N (Jl )2(d−2)/d

. (4.163)

152

Mass-Critical NLS Problem in Higher Dimensions

When d = 4,



∑ (4.163)2

1/2

−1

11

 η1 2 η332 2

k0 − j 2

.

(4.164)

Jl

Applying H¨older’s inequality in time when d ≥ 5,

1/2 2−8/d N (Jl )4/d  −(2d−3)/d 2 −4/d   Pξ (τ ),≥ j+n1 u 2 2d/(d−2) η η1 ∑ 4 j/d Lt Lx (Jl ×Rd ) 3 J ⊂[a,b] 2 l

d 2 −2d+3 2d 2

 2σ (k0 − j) η3

1−4/d uX [a,b]×Rd . ( )

−2/d  

η1

(4.165)

Therefore, by (4.156), (4.158), (4.159), (4.160), (4.162), (4.164), and (4.165),   u  1. X ([a,b]×Rd ) This completes the proof of Theorem 4.34. Long-time Strichartz estimates may be used to prove scattering for the nonradial, mass-critical problem. Theorem 4.37 The defocusing, mass-critical initial problem (4.142) is  value  globally well posed and scattering for all u0 ∈ L2 Rd , d ≥ 3. Proof To prove this it suffices to prove that if u is an almost-periodic solution to (4.142) satisfying N (t) ≤ 1 for all t ∈ R, then u ≡ 0. Because u lies in L2 but need not lie in H˙ 1 , in order to use (3.120) and (3.121), a frequency-localized interaction Morawetz estimate will be utilized. Let [−T, T ] be an interval such -T -T N (t)3 dt = K and −T |u (t, x) |2(d+2)/d dx dt = 2k0 . Let λ = η3 2k0 /K that −T and rescale:  T /λ 2

  uλ (t, x) = λ d/2 u λ 2t, λ x ,

−T /λ 2

Nλ (t)3 dt = η3 2k0 .

(4.166)

Then by Theorem 4.34,   uλ   X



− T2 , T2 ×Rd λ

  1.

λ

2d

Recall from (4.149) that the X-norm is constructed from Lt2 Lxd−2 -norms, which are invariant under (4.166). Define the Fourier multiplier I : L2 (Rd ) → H 1 (Rd ),

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3     given by m ηK3 ξ , where m ∈ C∞ Rd is a radial function:  1, |ξ | ≤ 1, m (ξ ) = 0, |ξ | > 2.

153

This Fourier multiplier contains the frequency support of most of u, since by (4.144),  T

−T

|ξ  (t) | dt ≤ η1−1 K  η3−1 K.

Following (3.102), let

∂t M (t) =

 (x − y) j

|x − y|

2   Im Iu (t, x) ∂ j Iu (t, x) Iu (t, y)  dx dy.

Remark This is a slight abuse of notation since a truncated interaction Morawetz potential need not be a time derivative of a function in the form of (3.101). Relation (4.80) gives sufficiently good decay for u (t, x) at high frequencies to bound the endpoints of the interaction Morawetz estimate as we now show. Lemma 4.38

If N (t) ≤ 1 on R, |∂t M (t) |  o (K) .

Here o (K) is a quantity that satisfies Proof

o(K) K

→ 0 as K  ∞.

By (4.80),  (x − y)  j

   Iu (t, y) 2 Im Iu (t, x) (∂ j − iξ j (t)) Iu (t, x) dx dy |x − y|    3  IuL∞ L2 (R×Rd )  (∇ − iξ (t)) Iu (t, x) L∞ L2 (R×Rd ) t x t x   k   ∑ 2 Pξ (t),k u (t, x) L∞ L2 (R×Rd ) t x −1 2 k ≤ η3 K

    + η3−1 K P|ξ −ξ (t)|≥η −1 K u (t, x)  3

Lt∞ Lx2 (R×Rd )

 o (K) .

Also, following (4.19),  (x − y)  j

|x − y| =

   Iu (t, y) 2 Im |Iu (t, x) |2 iξ j (t) dx dy

 (x − y)  j

|x − y|

which proves Lemma 4.38.

   Iu (t, x) 2 Iu (t, y) 2 ξ j (t) dx dy = 0,

154

Mass-Critical NLS Problem in Higher Dimensions

Lemma 4.39

For any d, if u satisfies (4.80), 2  3−d   N (t)3   |∇| 2 |Iu (t, x) |2  2

Lx (Rd )

.

(4.167)

Proof First suppose d ≤ 3. By conservation of mass and the Sobolev embedding theorem,    P≤η N(t) |Iu (t, x) |2 2 2 d  η d N (t)d . Lx (R ) By H¨older’s inequality, |ξ (t) |  η3−1 K, and (4.80), for N (t) ≤ 1, there exists C (m0 , d) sufficiently large so that m40 ≤ 4



2 |Iu (t, x) | dx ,d ) 2

C(m0 |x−x(t)|≤ N(t)



 C (m0 , d)d  Iu (t, x) 4 4 d , Lx (R ) d N (t)

  where m0 is the L2 -norm u (t, x) L2 = m0 and m0 > 0. Then by Bernstein’s inequality,  C (m0 , d)d  Iu (t, x) 4 4 d Lx (R ) d N (t)     C (m0 , d)d  P>η N(t) |Iu|2 2 2 + P≤η N(t) |Iu|2 2 2  Lx Lx N (t)d

 3−d 2 C (m0 , d)d  d−3  d d−3 d 2 |Iu|2  |∇| η N (t) + η N (t)   Lx2 N (t)d 2 C (m0 , d)d   3−d  2 |Iu|2  . |∇|  C (m0 , d)d η d +  Lx2 N (t)3 Taking η (u) sufficiently small implies the lemma for d ≤ 3. For d > 3, (3.120) implies  3−d 2    |∇| 2 |Iu (t, x) |2  2 ∼



Lx (Rd )

1 |Iu (t, x) |2 |Iu (t, y) |2 dx dy |x − y|3

 N (t)3





|x−x(t)| |y−x(t)| ≤C(m0 ,d)N(t) ≤C(m0 ,d)N(t)

|Iu (t, x) |2 |Iu (t, y) |2 dx dy ∼ N (t)3 .

Inequality (4.167) implies a lower bound for

d2 M (t) dt 2

as follows.

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3 When μ = +1,

Theorem 4.40

-T

−T

Proof

N (t)3 dt = K, and d ≥ 3, we have

−T

 T 2 d

dt 2

155

M (t) dt  K.

Plugging the Fourier multiplier I into (4.142) yields

∂t Iu = iΔIu − iF (Iu) + iF (Iu) − iIF (u) . If

∂t Iu = iΔIu − iF (Iu) , then (4.167) combined with the proof of Lemma 3.23 would imply  T 2 d −T

dt

2  3−d   2 |Iu (t, x) |2  |∇| M (t) dt   2 2

Lt,x ([0,T ]×Rd )

 K.

Therefore, it only remains to control the errors arising from iF (Iu) − iIF (u) .

(4.168)

Let E (t) denote such an error. Then E (t) =

 T  (x − y) j

|x − y|

−T

+

|Iu (t, y) |2 Re

 T  (x − y) j

|x − y|

−T

+2

  |Iu (t, y) |2 Re Iu · ∂ j (F (Iu) − IF (u)) (t, x) dx dy dt

 T  (x − y) j −T

    IF (u) ¯ − F Iu · ∂ j Iu (t, x) dx dy dt

|x − y|

  Im Iu · IF (u)] (t, y) Im[ Iu · ∂ j Iu (t, x) dx dy dt. (4.169)

Similarly to (4.18), the error E (t) is Galilean invariant. Indeed,  (x − y) j

|x − y| + =2

|Iu (t, y) |2 ξ j (t) Im

 (x − y) j

|x − y|

 (x − y) j

|x − y|

     F Iu − IF (u) ¯ · Iu (t, x) dx dy

  |Iu (t, y) |2 ξ j (t) Im Iu · (IF (u) − F (Iu)) (t, x) dx dy

  |Iu (t, y) |2 ξ j (t) Im Iu · IF (u) (t, x) dx dy.

(4.170)

156

Mass-Critical NLS Problem in Higher Dimensions

Then by the antisymmetry of

2

  (x − y) j

|x − y| +2

(x−y) j |x−y|

we have

  |Iu (t, y) |2 ξ j (t) Im IuIF (u) (t, x) dx dy

 (x − y) j

|x − y|

  ξ j (t) Im IuIF (u) (t, y) |Iu (t, x) |2 dx dy = 0.

This implies E (t) =

 T  (x − y) j

|x − y|

−T

    |Iu (t, y) |2 Re IF (u) ¯ − F Iu (∂ j − iξ j (t)) Iu (t, x) dx dy dt (4.171)

 T  (x − y) j

+

|x − y|

−T

  |Iu (t, y) |2 Re Iu (∂ j − iξ j (t)) (F (Iu) − IF (u)) (t, x) dx dy dt (4.172)

+2

 T  (x − y) j −T

|x − y|

    Im Iu IF (u) (t, y) Im Iu (∂ j − iξ j (t)) Iu (t, x) dx dy dt. (4.173)

First take (4.171). Lemma 4.41

 (x−y)    Since  |x−y|j  ≤ 1, it follows that

 T  (x − y) j −T

|x − y|

|Iu (t, y) |2 Re

    IF (u) ¯ − F Iu (∂ j − ξ j (t)) Iu (t, x) dx dy dt

 o (K) . Proof

(4.174)

To simplify notation, let (t) = ∇ − iξ (t) ∇

and

(t) = ∇ + iξ (t) . ∇ 2d

Combining (4.150) with the scale invariance of Lt2 Lxd−2 implies that for j ≤ 0,     P|ξ −ξ (t)|>2 j η −1 K u 3

2d

Lt2 Lxd−2 ([−T,T ]×Rd )

 2− j/2 .

(4.175)

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3

157

Let ul = P|ξ −ξ (t)|≤η −1 K/4 u, 3

and let u = ul + uh . Then IF (ul ) − F (Iul ) = IF (ul ) − F (ul ) = (I − 1) F (ul ) . By Bernstein’s inequality, since I − 1 is supported on |ξ | ≥ η3−1 K,    (I − 1) F (ul )  

2d

Lt2 Lxd+2 ([−T,T ]×Rd )

   η3     . (4.176) ∇ul  P η −1 K |ul |4/d  ∞ d/2 2d 3 d−2 2 K Lt Lx ([−T,T ]×Rd ) d Lt Lx ([−T,T ]×R ) > 4

Remark

It is true that

    |u|4/d u = |u|4/d ∇u + u∇|u|4/d ∇

  2 (4/d)−2 2   2 + |u| . = 1+ u ∇u |u|4/d ∇u d d

(4.177)

However, the only characteristic of |u|4/d that will be used is that F (x) = |x|4/d is a H¨older continuous function of order 4/d when d ≥ 4, and is differentiable when d = 3. Since this is also true of |x|(4/d)−2 x2 , it will be enough to simply treat the first term in (4.177). Again by (4.150),    ∇Iu

2d Lt2 Lxd−2

(

[−T,T ]×Rd

)

 η3−1 K ∑ 2 j/2  η3−1 K.

(4.178)

j≤0

Splitting, IF (u) − F (Iu) = IF (u) − IF (ul ) + F (ul ) − F (Iu) + IF (ul ) − F (ul ) . (4.179) Then by Taylor’s theorem,   and F (u) − F (ul ) = uh O |u|4/d

  F (Iu) − F (ul ) = (Iuh ) O |Iu|4/d .

158

Mass-Critical NLS Problem in Higher Dimensions

By the fundamental theorem of calculus, for |ξ2 |  |ξ |, |m (ξ + ξ2 ) − m (ξ ) |  |ξ2 | |∇m (ξ ) | 

| ξ2 | . |ξ |

(4.180)

Therefore, by Bernstein’s inequality, (4.180), and the fractional chain rule of Visan (2007),   IF (u) − IF (ul ) + F (ul ) − F (Iu)  2d/(d+2) Lt2 Lx ([−T,T ]×Rd )    4/d 4/d = I uh |Iu| − (Iuh ) |Iu|  2d (d+2) Lt2 Lx ([−T,T ]×Rd ) 

3/d    3/d η3  uh   |∇| |Iu|4/d  ∞ 2 . 2d Lt Lx ([−T,T ]×Rd ) 3/d K Lt2 Lxd−2 ([−T,T ]×Rd )

(4.181)

Combining (4.145), (4.176), (4.178), and (4.181) gives (4.174)



   η3    uh  2d/(d−2) + ∇Iu 2 2d/(d−2) Lt2 Lx ([−T,T ]×Rd ) K Lt Lx ([−T,T ]×Rd )   3/d   4/d η3  3/d 4/d  |∇| |Iu|  d/2 × + uh L∞ L2 [−T,T ]×Rd Lt∞ Lx ([−T,T ]×Rd ) ) t x( K 3/d

 η3−1 K

  η3−1 K

3/d   4/d η3   |∇|3/d |Iu|4/d  d/2   ∞L d ) + uh L∞ L2 ([−T,T ]×Rd ) 3/d L [−T,T ]×R ( x t x t K



 o (K) . The last inequality follows from the fact that η3 (u) > 0 is fixed. Letting K → ∞, N (t) ≤ 1, then the H¨older continuity of |u|4/d combined with Bernstein’s inequality and (4.80) implies that, when K  ∞,     −3/2d  4/d  P u L∞ L2 |ξ −ξ (t)|>K 1/2 L∞ L2 ([−T,T ]×Rd ) + K t

x

t

x

([−T,T ]×Rd )

→ 0.

Next consider (4.172). Lemma 4.42  T  (x − y) j −T

|x − y|  o (K) .

  |Iu (t, y) |2 Re Iu (∂ j − iξ j (t)) (F (Iu) − IF (u)) (t, x) dx dy dt (4.182)

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3 Proof

159

Integrating by parts,

(4.182) =−

−

 T   −T

  1 |Iu (t, y) |2 Re (F (Iu) − IF (u)) Iu (t, x) dx dy dt |x − y| (4.183)

 T   (x − y) j

|Iu (t, y) |2 |x − y|   × Re (F (Iu) − IF (u)) (∂ j + iξ j (t)) Iu (t, x) dx dy dt. (4.184)

−T

The last line, (4.184), is equal to the complex conjugate of (4.174). To show that (4.183)  o (K), by the Hardy–Littlewood–Sobolev theorem,   (4.183)  F (Iu) − IF (u)  2d Lt2 Lxd+2 ([−T,T ]×Rd )    2 Iu × Iu 2d 2d d−5/6 d−7/3 4 d Lt8 Lx Lt Lx ([−T,T ]×Rd ) ([−T,T ]×R )  o (K) .

(4.185)

The last estimate follows from the same argument estimating (4.179), which shows that   F (Iu) − IF (u)  ≤ oK (1) . 2d Lt2 Lxd+2 ([−T,T ]×Rd )   Also, by (4.145), Pξ (t), j uL∞ L2 ([−T,T ]×Rd ) → 0 as j  ∞. Interpolating this t x fact with 1/2    K η3−1 Pξ (t),> j u  2d 2j Lt2 Lxd−2 ([−T,T ]×Rd ) implies that for any 2 < p < ∞, s > 1p , if (p, q) ∈ Ad is an admissible pair, then  s     |∇| Iu p q  o K s η3−s . Lt Lx ([−T,T ]×Rd ) Finally consider (4.173). Lemma 4.43

For d ≥ 3,

 T   (x − y) j −T

|x − y|  o (K) .

    Im Iu · IF (u) (t, y) Im Iu (∂ j − ξ j (t)) Iu (t, x) dx dy dt

160

Mass-Critical NLS Problem in Higher Dimensions

Proof

 2(d+2)  Because Im (uF ¯ (u)) = Im |u| d = 0, it follows that

 0 = Im (1 − I) u¯ · (1 − I) F (u) + (1 − I) u¯ · IF (u) + I u¯ · (1 − I) F (u)

 +I u¯ · IF (u) ,

and therefore   Im I u¯ · IF (u)   = Im − (1 − I) u¯ · (1 − I) F (u) − (1 − I) u¯ · IF (u) − I u¯ · (1 − I) F (u) . By (4.175) and (4.176), we have  T   (x − y) j

  Im (1 − I) u¯ · (1 − I) F (u) (t, y)

|x − y|   × Im Iu (∂ j − ξ j (t)) Iu (t, x) dx dy dt  o (K) .

−T

Next, integrating by parts, by the Fourier support of (1 − I),  T   (x − y) j

  Im (1 − I) u¯ · IF (u) + I u¯ · (1 − I) F (u) (t, y)

|x − y|   × Im Iu (∂ j − ξ j (t)) Iu (t, x) dx dy dt

−T

=−

 T   (x − y) j −T

|x − y|

Im

y ∇ y IF (u) (1 − I) u¯ · ∇ y Δ

! y ∇ y I u¯ · +∇ (1 − I) F (u) (t, y) y Δ

  × Im Iu (∂ j − ξ j (t)) Iu (t, x) dx dy dt    T    ∇y  1  O  (1 − I) u¯ |IF (u) | + |x − y| −T Δy  

 ∇y   + |I u| ¯  (1 − I) F (u)  (t, y) y Δ   × |Iu (t, x) |  ∇Iu (t, x)  dx dy dt.

(4.186)

Then by (4.150), (4.175), (4.176), (4.185), the chain rule ∇F (Iu) = F  (Iu) ∇Iu,

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3

161

and the Hardy–Littlewood–Sobolev inequality,    1    (4.186)  P>η −1 K u ∇Iu 2d 2d d−2 3 2 K Lt Lx Lt2 Lxd−2 ([−T,T ]×Rd ) ([−T,T ]×Rd )    1+4/d ∇IuL∞ L2 ([−T,T ]×Rd ) × uL∞ L2 [−T,T ]×Rd  ) t x t x(       IF (u) − F (Iu)  + P>η −1 K u 2d 2d 3 Lt2 Lxd+2 ([−T,T ]×Rd ) Lt2 Lxd−2 ([−T,T ]×Rd )     × uL∞ L2 ([−T,T ]×Rd )  ∇IuL∞ L2 ([−T,T ]×Rd ) t x t x    1    + P>η −1 K F (u)  ∇Iu 2d 2d 3 K Lt2 Lxd+2 ([−T,T ]×Rd ) Lt2 Lxd−2 ([−T,T ]×Rd )     ∇IuL∞ L2 ([−T,T ]×Rd ) × uL∞ L2 ([−T,T ]×Rd )  t x t x    1   IF (u)  + P>η −1 K u 2d 2d d−2 3 2 K d Lt4 Lxd+1 ([−T,T ]×Rd ) Lt Lx ([−T,T ]×R )     × uL∞ L2 ([−T,T ]×Rd )  ∇Iu 2d t x Lt4 Lxd−1 ([−T,T ]×Rd )    1   Iu + P>η −1 K F (u)  2d 2d d+2 3 2 K d Lt Lx Lt4 Lxd−3 ([−T,T ]×Rd ) ([−T,T ]×R )     ∇Iu × uL∞ L2 ([−T,T ]×Rd )  2d t x Lt4 Lxd−1 ([−T,T ]×Rd )  o (K) . This completes the proof of Lemma 4.43. Combining Lemmas 4.41–4.43 completes the proof of Theorem 4.40. Corollary 4.44 There exists no nonzero almost-periodic solution to (4.142) satisfying N (t) ≤ 1 for all t ∈ R, and  ∞

−∞

Proof

N (t)3 dt = ∞.

By Theorem 4.40 and Lemmas 4.41–4.43, if

(4.187) -T

−T

N (t)3 dt = K, then

K  o (K) . This implies that there must be an upper bound on K. -

T Now given the uniform bound −T N (t)3 dt = K ≤ K0 < ∞ for any T < ∞ and ξ (0) = 0, (4.144) implies that there exist |ξ+ |, |ξ− |  η1−1 K such that

lim ξ (t) = ξ± .

t→±∞

162

Mass-Critical NLS Problem in Higher Dimensions

Make a Galilean transformation mapping ξ+ → 0. Then (4.144) and (4.187) also imply that limt→±∞ N (t) = 0, so by (4.80), for any j,   lim Pj u (t) L2 (Rd ) = 0. (4.188) x t→±∞ Therefore, for any t ∈ [T, +∞),     P> j u (t)  2 d  P> j F (u (t))  2d Lx (R ) L2 L d+2 x

t

Also by Theorem 4.34 and a scaling argument, if 2 j ≤ η3−1 K,   Pξ (t),> j F (u (t)) 

-∞ T

then for any N (t)3 dt = K,

 2d

Lt2 Lxd−2 ([T,∞)×Rd )



. ([T,∞)×Rd )

η −1 K 3 2j

1/2 .

Now let j0 ∈ Z be the integer satisfying 2 j0 ∼ η3−1 K. For j ≥ 0, as in the proof of Theorem 4.34, (4.188) implies   P> j+ j F (u (t))  2d 0 Lt2 Lxd+2 ([T,∞)×Rd )  4/d       χ (t, x) P> j+ j0 +n1 u P|ξ −ξ (t)|≤η −1 N(t) u  2d 3 Lt2 Lxd+2 ([T,∞)×Rd )  4/d  + η2 P> j+ j0 +n1 u 2d Lt2 Lxd−2 ([T,∞)×Rd ) 1/2    2k Pξ (t),>k u + . (4.189) 2d ∑ −1 j Lt2 Lxd−2 ([T,∞)×Rd ) k≤ j+ j0 +n1 2 η3 K Then, arguing by induction,   Pj+ j F (u (t))  0

2d

Lt2 Lxd+2 ([T,∞)×Rd )

 2− j/2 .

In particular, this implies that for any j,   P> j u (t)  2 d ≤ C (d, η1 , η3 ) K 1/2 2− j/2 . Lx (R )

(4.190)

Applying the same calculation as in (4.97) to any small interval Jl ⊂ [T, ∞),   P> j F (u (t)) 

2(d+2) Lt,xd+4

1

(Jl ×Rd )

j

2(d+2) 2− 2(d+2) . K

(4.191)

Plugging (4.190) and (4.191) back in to the bilinear estimate in (4.189),   u (t)  1/2 1/2 . (4.192) K Lt∞ H˙ x ([T,∞)×Rd )

4.4 Nonradial Mass-Critical Problem in Dimensions d ≥ 3

163

The same is true for the interval (−∞, −T ]. By the dominated convergence theorem T∞ N (t)3 dt → 0, so by (4.192),  

lim

T →±∞

|u (T, y) |2

  (x − y) · Im u∇u ¯ (T, x) dx dy = 0, |x − y|

and therefore by (3.121) and (4.19), 2  3−d    |∇| 2 |u (t, x) |2  2

Lt,x (R×Rd )

= 0,

and thus u ≡ 0. This completes the proof of Theorem 4.37.

5 Low-Dimensional Well-Posedness Results

5.1 The Energy-Critical Problem in Dimensions Three and Four Proof of scattering for the defocusing, energy-critical Schr¨odinger problem when d = 3, 4 will also use frequency-localized interaction Morawetz estimates. In this case, u ∈ H˙ 1 , but need not lie in L2 . In fact, the focusing, nonlinear Schr¨odinger equation 4

iut + Δu = −|u| d−2 u

(5.1)

has a soliton solution W (x) = 

1 |x| 1 + d(d−2) 2

 d−2 , 2

  and W ∈ L2 Rd if and only if d ≥ 5. Examining the proof of Theorem 3.18, it is clear that the proof could be applied equally well to an almost-periodic solution to the focusing equation (5.1) as to the defocusing equation. In fact, Killip and Visan (2010) proved scattering for the focusing equation. Thus it is impossible  to prove that an almost-periodic solution in dimensions d = 3, 4 lies in L2 Rd using only linear methods. The proof of scattering for the radially symmetric energy-critical problem used the Morawetz estimate (2.7), which was truncated in space. Then, since    2d  u (t) ∈ L d−2 Rd , it follows that u (t) truncated in space lies in Lx2 Rd . However, for the nonradial, energy-critical problem, this estimate is much less useful (see (3.60)), hence the use of a frequency-localized interaction Morawetz estimate. At this point, a discussion on the chronology of the results presented in 164

5.1 The Energy-Critical Problem in Dimensions Three and Four

165

this book is in order. The global well-posedness and scattering results for the three-dimensional, energy-critical Schr¨odinger equation in Colliander et al. (2008), and also Kenig and Merle (2010), were the first scattering results for nonradial problems with critical regularity. Subsequently, the four-dimensional energy-critical result of Ryckman and Visan (2007) was proved, followed by the higher-dimensional energy-critical results and mass-critical results presented in Chapters 3 and 4. Remark Scattering for the defocusing, energy-critical problem when d ≥ 5 was proved in Visan (2007, 2006) and prior to Killip and Visan (2010), using frequency-localized interaction Morawetz estimates. The proof in Visan (2007, 2006) is quite technical, and will not be presented in this book. The ordering of topics in this book is for expository reasons. Put simply, the proof in Colliander et al. (2008) was a very deep argument, introducing frequency-localized interaction Morawetz estimates, a version of long-time Strichartz estimates, and the double Duhamel argument to the study of nonlinear Schr¨odinger equations. In the author’s opinion, it was better to introduce these concepts individually, for problems that only required one or two such tools. The reader should think of Sections 5.1 and 5.2 as the “final exam” for this book. Theorem 5.1

The nonlinear Schr¨odinger equation 4

iut + Δu = F (u) = |u| d−2 u,

(5.2) u (0, x) = u0 ,   is globally well posed and scattering for any u0 ∈ H˙ 1 Rd , for d = 3 or d = 4. Proof Recall from Theorems 3.14 and 3.15 that it suffices to show that if u is an almost-periodic solution to (5.2) on R with N (t) ≥ 1 for all t ∈ R, then u ≡ 0. The proof will follow five steps: 1. Show that the frequency-localized interaction Morawetz estimates must hold on some closed interval I with positive measure. 2. Prove Strichartz estimates on an interval I ⊂ R on which the bound on the interaction Morawetz estimate holds. 3. Show that there is an upper bound on N (t) on the interval I. As N (t)  ∞, (3.60) and (3.61) imply that for any N∗ , the mass at frequencies higher than N∗ must go to zero. Therefore, to prove that N (t) is bounded above it is enough to show that there is a lower bound on the mass at high frequencies. 4. Bootstrap Strichartz estimates on I with defocusing localized interaction Morawetz estimate bounds on I to prove that I must also be open and closed in R, and therefore I = R.

166

Low-Dimensional Well-Posedness Results

5. Make an argument similar to the proof of Theorem 3.21, proving that if u is an almost-periodic solution to (5.2) with N (t) ∼ 1 and with frequencylocalized interaction Morawetz estimates, then u ≡ 0. Take the constants 0 < η5  η4  η3  η2  η1  1. The quantity η j will depend on a power (that depends only on the dimension) c(d) of η j−1 of the form η j  η j−1 . Proposition 5.2 (Frequency-localized interaction Morawetz estimate) Suppose that u is an almost-periodic solution to (5.2) on an interval I ⊂ R and N∗ < C(ηη3 ) · mint∈I N (t), for C (η3 ) > 0 defined in (3.60) and (3.61). Choose 3 j∗ = sup{ j ∈ Z : 2 j ≤ N∗ }. Then, when d = 3,   I

|u≥ j∗ (t, x) |4 dx dt +

  I

|u≥ j∗ (t, y) |2 |u≥ j∗ (t, x) |6 dx dy dt  η1 N∗−3 , |x − y| (5.3)

and when d = 4,    I

|P≥ j∗ u (t, x) |2 |P≥ j∗ u (t, y) |2 dx dy dt |x − y|3

  

+ I

|P≥ j∗ u (t, x) |2 |P≥ j∗ u (t, y) |4 dx dy dt  η1 N∗−3 . |x − y|

(5.4)

Proof By scaling symmetry normalize N∗ = 1 and N (t) ≥ C(ηη33 ) for all t ∈ I, where C (η3 ) is defined in (3.60) and (3.61). Then         (5.5) + ≤ η3 . u< 1  ∞ ˙ 1 u 1 2d < η  ∞ d−2 η3 Lt H (I×Rd ) 3 Lt Lx (I×Rd ) Remark

Here u
0, for η3 (δ ) > 0 sufficiently small,       P≤ j u ˙1 P≤0 u (t)  ˙ 1 d + 2 j( 32 −δ )C1/2 η 1/2 . (5.25) inf  d δ 0 1 H (R ) S (I×R ) t∈I

Proof By Strichartz estimates and Bernstein’s inequality, for any t0 ∈ I,       P≤ j u (t)  ˙1  P≤ j u (t0 ) H˙ 1 (Rd ) + F (u≤ j ) N˙ 1 (I×Rd ) S (I×Rd )   + 2 j P≤ j [F (u) − F (u≤ j )] . (5.26) 2d Lt2 Lxd+2 (I×Rd )   Now since j ≤ 0, P≤ j uL∞ H˙ 1 (I×Rd )  η3 , so by interpolation, t    4  F (u≤ j )  ˙ 1  ∇u≤ j  d−2 ∞ 2 N (I×Rd )

Lt Lx (

I×Rd

)

  ∇u≤ j 

4    η3d−2 u≤ j S˙1 (I×Rd ) .

2d

Lt2 Lxd−2 (I×Rd )

(5.27)

Next, decompose:       4 4 d+2 F (u) − F (u≤ j ) = O |u≥ j | |u| d−2  O |u≥ j | |u≤ j | d−2 + O |u≥ j | d−2       4 d+2 d+2  O |u≥ j | |u≤ j | d−2 + O |u j≤·≤0 | d−2 + O |uh | d−2 . (5.28)

5.1 The Energy-Critical Problem in Dimensions Three and Four

171

For some 0 < δ < 12 , define the frequency envelope

∑

αj =

j≤k≤0

  3 2( 2 −δ )( j−k) P≤k uS˙1 (I×Rd ) .

Then by (5.6), (5.26), (5.27), (5.28), interpolation, and the Sobolev embedding theorem, when d = 3 or d = 4,     3 α j  ∑ 2( 2 −δ )( j−k) inf u≤0 (t) H˙ 1 (Rd ) t∈I j≤k≤0

4

∑

+ η3d−2

j≤k≤0

+

∑

  3 2( 2 −δ )( j−k) P≤k uS˙1 (I×Rd )

  3 2( 2 −δ )( j−k) 2k P≤k [F (u) − F (u≤k )]

2d

Lt2 Lxd+2 (I×Rd )

j≤k≤0

.

(5.29)

By the Sobolev embedding theorem and (5.28),   3 ∑ 2( 2 −δ )( j−k) 2k P≤k [F (u) − F (u≤k )] 2d

Lt2 Lxd+2 (I×Rd )

j≤k≤0











4   d−2 u≤k  ∞ −1 Lt H˙ 1 (I×Rd ) (I×Rd ) j≤k≤0 t x 3  4 1  3 3k  uk≤·≤0  d−2 −2d2 + ∑ 2( 2 −δ )( j−k) 2 2 uk≤·≤0  2 6d j≤k≤0 Lt3 Lx3d−4 (I×Rd ) Lt∞ Lxd−2 (I×Rd )     d+2  3  + ∑ 2( 2 −δ )( j−k) 2k P≤k |uh | d−2  . 2d Lt2 Lxd+2 (I×Rd ) j≤k≤0 2d ∑ 2( 2 −δ )( j−k) 2k u≥k Lt∞ Lx2 (I×Rd )∇u≤k L2 L d−2 3

Bernstein’s inequality, (5.6), and interpolation yield

∑

 3 3 3k 2 2 2( 2 −δ )( j−k) uk≤·≤0  2

6d

Lt3 Lx3d−4

j≤k≤0 1

 η32

∑

3 2( 2 −δ ) j 2δ k

j≤k≤0 1

δ η32

∑

j≤k1 ≤0

∑

k≤k1 ≤0

 k1  2− 2 uk1 

  3 2( 2 −δ )( j−k1 ) ∇uk1 

2d

Lt2 Lxd−2

2d

Lt2 Lxd−2

.

Therefore, (5.29) δ



 4 −1 4   inf u≤0 (t) H˙ 1 (Rd ) + η3d−2 α j + η3d−2 α j t∈I    d+2  3  + ∑ 2( j−k)( 2 −δ ) 2k P≤k |uh | d−2  . 2d Lt2 Lxd+2 (I×Rd ) j≤k≤0

(5.30)

For η3 (δ ) > 0 sufficiently small, the second and third terms on the right-hand

172

Low-Dimensional Well-Posedness Results

side of (5.30) can be absorbed into the left-hand side. Finally, by the Sobolev embedding theorem, (5.8), (5.9), and (5.12), we get      d+2  d+2  3  3  1/2 1/2 2k P≤k |uh | d−2   2 2 k  |uh | d−2   2 2 kC0 η1 . 2d 2d d+2 d+3 2 2 d d Lt Lx Lt Lx (I×R ) (I×R ) (5.31) Plugging (5.31) into (5.30) gives     3 1/2 1/2 α j  inf u≤0 (t) H˙ 1 (Rd ) + 2 j( 2 −δ )C0 η1 , t∈I

(5.32)

which proves Lemma 5.5. Lemma 5.5 implies an upper bound on N (t) when d = 4. Proposition 5.6 Suppose u is an almost-periodic solution to the energycritical initial value problem with N (t) ≥ C (η3 ) /η3 for all t ∈ R and d = 4. Then for all t ∈ I, we have N (t)  C (η5 ) Nmin . Proof Since N (t) was normalized so that N (t) ≥ C (η3 ) /η3 on I, there exists some tmin ∈ I such that N (tmin ) ≤ 2C (η3 ) /η3 . Then by Bernstein’s inequality and (3.60), if u is nonzero, for some η > 0 small,   PN(t )c(η ) 0,       3 1/2 1/2 P≤ j u ˙1   P  u (t) min + 2 j( 2 −δ )C0 η1 . ≤0 H˙ 1 (R3 ) S (I×R3 ) δ t∈I

Suppose there exist tmin ,tevac ∈ I such that N (tmin ) ≤ implies   uh (tmin )  2 ≥ η η3 , L C (η3 )

2C(η3 ) η3 , which by (5.33)

(5.57)

and N (tevac ) ≥ C (η5 ). Then (5.34) holds at tevac and thus by the continuity of N (t), there exists tmin < t1 < tevac such that N (t1 ) = C1 (η1 , η3 , η4 ) and N (t) ≤ C1 (η1 , η3 , η4 ) for all tmin ≤ t ≤ t1 .

5.2 Three-Dimensional Energy-Critical Problem

179

Again let uh = P≥ j u, u = uh + ul  for some j ≤ 0. By (1.112), d dt



|uh (t, x) |2 dx =



{Ph F (u) , uh }m dx



{Ph (F (u) − F (uh ) − F (ul  )) , uh }m dx

= −



{Pl  F (uh ) , uh }m dx +



{Ph F (ul  ) , uh }m dx. (5.58)

Let J = [tmin ,t1 ]. By Bernstein’s inequality and Lemma 5.10, for any δ > 0,   J

|{Ph F (ul  ) , uh }m | dx dt    5 δ 2− j ul  S˙1 (I×R3 ) uh L∞ L2 (I×R3 ) t

 2−2 j Next,

  J



  inf u≤0 (t)  ˙ 1 t∈I

H

x

 (R3 )

+C0 η1 2 j(3/2−δ ) 1/2 1/2

5

.

(5.59)

|{Ph (F (u) − F (uh ) − F (ul  )) , uh }m | dx dt  

  J

  J

|uh |2 |ul  | |u|3 dx dt |uh |5 |ul  | dx dt +

  J

|uh |2 |ul  |4 dx dt.

(5.60)

1/2

Choose j such that 2 j ∼ η4 and j5 such that 2 j5 ∼ η5 . Splitting ul  = u< j5 + u j5 ≤·≤ j and uh = u j p , which we do next. Theorem 5.23

If p < q, V p ⊂ U q.

5.3 The Mass-Critical Problem When d = 1

199

Theorem 5.23 was proved in Hadac et al. (2009). Proof of Theorem 5.23 Lemma 5.24 n ≥ 0,

First we decompose v ∈ V p .

Suppose v ∈ V p and vV p (I×Rd ) = 1. Then for all integers n,

1. there exists a partition Zn such that Z0 ⊂ Z1 ⊂ · · · and Zn ≤ 21+np , 2. there exists un (t) subordinate to Zn , that is, un (t) is constant on each subinterval of Zn , such that   un (t)  ∞ 2 ≤ 21−n , Lt Lx (I×Rd )   3. there exists vn (t) ∈ V p I × Rd such that   vn (t)  ∞ 2 ≤ 2−n , Lt Lx (I×Rd ) 4. and finally we have vn = un+1 + vn+1 , u0 = 0, v0 = v. Proof The proof is by induction. Without loss of generality suppose I = [0, T ]. Let u0 = 0, v0 = v, and Z0 = {0, T }. Then Z0 = 1. Next, for any partition Zn = {tn,k }, 0 define the refinement Zn+1 = {tn+1,k }. Set tn+1,k = tn,k . For j ≥ 1, let     j−1  j j−1  2 d > 2−n−1 . = inf t : tn+1,k < t ≤ tn,k+1 : vn (t) − vn tn+1,k tn+1,k L (R ) (5.115) j If no such t exists then move on to the next k. Then relabel the tn+1,k to obtain the increasing sequence tn+1,k . Let Kn = Zn , Kn+1

un+1 (t) =

∑ 1[tn+1,k−1 ,tn+1,k ) vn

  tn+1,k−1 ,

and

vn+1 (t) = vn (t) − un+1 (t) .

k=1

By induction,

  vn (t)  ∞ 2 ≤ 2−n , L L (R×Rd ) x

t

so

  un+1 (t)  ∞ 2 ≤ 2−n . L L (R×Rd ) t

Next, by definition of the partition,   vn+1 (t)  ∞

x

Lt Lx2 (R×Rd )

≤ 2−n−1 .

200

Low-Dimensional Well-Posedness Results

Finally, by the definition of the V p -norm and (5.115),  p 1 = vV p ≥ (Kn+1 − Kn ) 2−(n+1)p . By induction this implies Kn+1 = Zn+1 ≤ 2(n+1)p + Kn ≤ 2(n+1)p + 21+np ≤ 21+(n+1)p , which closes the induction, proving Lemma 5.24. Then for almost every t ∈ R, Lemma 5.24 implies ∞

v (t) =

∑ un (t)

n=0

and

  un (t) 

U q (R×Rd )

Therefore, if

p q

≤2

1+np q

21−n .

< 1,   ∑ un (t)  ∞

n=0

  n qp −1

≤ 4∑2 U q (R×Rd )

 p 1, q

which proves Theorem 5.23. This completes the proof of Theorem 5.21. Now we prove Theorem 5.20 in dimension d = 1. First, a lemma. Lemma 5.25 (Intermediate lemma) Suppose l ≥ i − 5. Then when d = 1,        Pξ (Giα ),l u |Pξ (t),≤i−10 u|2  2 i Lt,x (Gα ×R)      (i−l)/2  u i 2 . Pξ (Giα ),l u 2 i X (Gα ×R) UΔ (Gα ×R) Proof   Without loss of generality make a Galilean transformation so that ξ Giα = 0. Expanding gives    |Pl u| |Pξ (t),≤i−10 u|2 2 2 i Lt,x (Gα ×R)     ∑  (Pl u) Pξ (t),m2 u Lt,x2 (Giα ×R) m3 ≤m2 ≤i−10  2    ×  (Pl u) Pξ (t),m3 u L2 (Gi ×R) Pξ (t),≤m3 uL∞ (Gi ×R) . α α t,x t,x

5.3 The Mass-Critical Problem When d = 1

201

By the Sobolev embedding theorem, conservation of mass, bilinear Strichartz estimates, and Definition 5.18, we continue:      2  ∑ 2m3 −l Pl uUΔ2 (Giα ×R) Pξ (t),m2 uUΔ2 (Giα ×R) Pξ (t),m3 uUΔ2 (Giα ×R) m3 ≤m2 ≤i−10  2  2  2i−l Pl uU 2 (Gi ×R) uX (Gi ×R) . α α Δ Remark The same calculations also prove        Pξ (Giα ),l u |PC0 N(t)≤|ξ −ξ (t)|≤2i−10 u|2  2 i Lt,x (Gα ×R)         u u i   2(i−l)/2 Pξ (Gi ),l u 2 i . X (Gα ×R) |ξ −ξ (t)|≥C0 N(t) Lt∞ Lx2 (Giα ×R) α U (Gα ×R) Δ

(5.116) Proof of Theorem 5.20 tilinear estimates.

The proof of Theorem 5.20 will be split into two mul-

Suppose d = 1 and fix Giα ⊂ Gkj .   1/2 For i ≥ 10, l ≥ i − 5, m ≥ i − 2, and aiα ∈ Giα , if N Giα ≤ η3 2i−5 ,  t         i ei(t−τ )Δ Pm (Pl u) Pξ (τ ),≥i−10 u u3 (τ ) d τ  2 i UΔ (Gα ×R) aα     8/3 4/3  η2 2(m−l)/4 Pl uU 2 (Gi ×R) uX Gi ×R ( α ) α Δ     10/3  2/3 11/6  13/6 + 2(i−l)/2 2(i−m)/4 Pl uU 2 (Gi ×R) η2 uX Gi ×R + η2 uX Gi ×R . ( α ) ( α ) α Δ (5.117)

Theorem 5.26 (First multilinear estimate)

ˆ ) = 1 and f (t, ξ ) is −1/4 supported on |ξ | ∼ 2m . By (4.23), since 2l  η3 2−20 N (t), for l ≥ 1,   2     (Pl u) P|ξ −ξ (t)|≤η −1/4 2−20 N(t) u  2 i Lt,x (Gα ×R) 3

1/2   −1/4 −20 −l Pl u 2 i  ∑ N (Jk ) η3 2 2 U (G ×R)

Proof

By Theorem 5.21 choose f such that  f U 3 (Gi Δ

α ×R

α

Δ

Jk ∩Giα =0/

  3/8  η3 2(i−l)/2 Pl uU 2 (Gi Δ

).

(5.118)

α ×R

Then by (4.145), (5.116), and (5.118),      (Pl u) Pξ (t),≤i−10 u 2  2 i Lt,x (Gα ×R)     i−l  3/8  2 2 Pl uU 2 (Gi ×R) η3 + η2 uX (Gi Δ

α

α ×R)

 .

(5.119)

202

Low-Dimensional Well-Posedness Results

Furthermore, by (5.116) and (5.118), if uˆ0 is supported on |ξ | ∼ 2m , by H¨older’s inequality, then  itΔ     e u0 Pξ (t),≤i−10 u  3 i Lt,x (Gα ×R)   itΔ   1/2 2 1/2   e u0 Pξ (t),≤i−10 u L2 Gi ×R eitΔ u0 L6 Gi ×R ( ) ) α t,x ( α t,x   1/2     i−m 3/8  2 4 u0 L2 η3 + η2 uX (Gi ×R) , (5.120) α so since  f U 3 = 1, (5.120) implies Δ

    f Pξ (t),≤i−10 u 

3 Gi Lt,x ( α

2 ×R)

i−m 4

   3/8 η3 + η2 uX (Gi

1/2

α ×R

)

Moreover, by (4.23),     1  f (Pl u)  3 i  2− 4 |m−l| Pl uU 2 (Gi ×R) . Lt,x (Gα ×R) α Δ

. (5.121)

(5.122)

Also,   Pξ (t),≥i−10 u3 6 i Lt,x (Gα ×R)       ∑  Pξ (t),m1 u Pξ (t),m3 u Lt,x3 (Giα ×R) Pξ (t),m2 uLt,x6 (Giα ×R) i−10≤m1 ≤m2 ≤m3  2 (5.123)  η2 uX (Gi ×R) . α Therefore, by (5.119)–(5.123),  Giα



   f , (Pl u) Pξ (t),≥i−10 u u3 dt

 4    Pξ (t),≥i−10 uL6  (Pl u) f L3 t,x t,x    2         f Pξ (t),≤i−10 u  3 Pξ (t),≥i−10 u 6 + (Pl u) Pξ (t),≤i−10 u 2 Lt,x Lt,x Lt,x    |m−l| 4/3  8/3  2− 4 η2 Pl uU 2 (Gi ×R) uX Gi ×R ( α ) α Δ

 (i−l) i−m  10/3  2/3 11/6  13/6 + 2 2 + 4 Pl uU 2 (Gi ×R) η2 uX Gi ×R + η2 uX Gi ×R . ( α ) ( α ) α Δ

This proves Theorem 5.26. The estimate (5.117) is in the form of (5.107), so Theorem 5.20 follows directly from (5.117) and the following second multilinear estimate. Theorem 5.27 (Second multilinear estimate) For d = 1, |l − m| ≤ 5, l ≥ i − 5,

5.3 The Mass-Critical Problem When d = 1 



203

1/2

and N Giα ≤ η3 2i−5 ,   

  4   ei(t−τ )Δ Pξ (Gi ),m Pξ (Gi ),l u Pξ (τ ),≤i−10 u d τ  2 i α α i UΔ (Gα ×R) aα     3  2(i−l)/6 η2 Pl uU 2 (Gi ×R) uX ([0,T ]×R) . t

Δ

α

k0

Indeed, decompose      4     + Pm O (Pl u) Pξ (t),≥i−10 u u3 . Pm (Pl u) u4 = Pm (Pl u) Pξ (t),≤i−10 u Thus, with Theorem 5.27, the proof of Theorem 5.20 will be complete. Proof of Theorem 5.27 Because Theorem 5.27 concerns only the interval Giα and not Gkj , to economize notation it is convenient to relabel and consider the interval Gkj and prove  t   4     j ei(t−τ )Δ Pm Pξ G j ,l u Pξ (τ ),≤i−10 u d τ  ak

   η2 2( j−l)/6 P

k

  j UΔ2 Gk ×R



 3   u  u   j j Xk0 ([0,T ]×R) ξ Gk ,l U 2 G ×R Δ

k

   m   1/2 when N Gkj ≤ η3 2 j−5 . In this case, ξ Gkj = 0, but ξ Gα 3 need not be m zero for Gα 3 ⊂ Gkj . Lemma 5.28 For any 0 ≤ m3 ≤ j − 10, let Zm3 be a partition of Gkj that is m the union of the endpoints of the subintervals Gα 3 ⊂ Gkj and akj . Then if u is u, replaced by P −20 −1/4 |ξ −ξ (t)|≥2

∑

 t 

0≤m3 ≤m2 ≤ j−10



j ak

η3

N(t)

  ei(t−τ )Δ (Pl u) Pξ (τ ),m2 u  2    × Pξ (τ ),m3 u Pξ (τ ),≤m3 u d τ     u2  . j X j ([0,T ]×R) UΔ2 Gk ×R

   2 j−l η22 Pl u

  j VΔ1 Zm3 ;Gk ×R

Also,

∑j

Jl ∩Gk =0/

  4     e−iτ Δ (Pl u) P|ξ −ξ (τ )|≤η −1/4 2−20 N(τ ) u d τ  Jl

3

 3/8   2 j−1 η3 Pl u

 . j UΔ2 Gk ×R

Lx2 (R)

204 Proof

Low-Dimensional Well-Posedness Results   First replace u with P u. If u0 L2 = 1 and u0 has −20 −1/4 |ξ −ξ (t)|≥2

η3

N(t)

Fourier transform supported on |ξ | ∼ 2l , then by (4.144) and (4.145),      2  u0 , m e−iτ Δ (Pl u) Pξ (τ ),m2 u Pξ (τ ),m3 u Pξ (τ ),≤m3 u d τ Gα 3         eitΔ u0 Pξ (Gm3 ),m −2≤·≤m +2 u Pξ (Gm3 ),≤m +2 u  2 m3 3 3 3 α α Lt,x (Gα ×R)       ×  (Pl u) Pξ (t),m2 u Pξ (Gm3 ),≤m +2 u  2 m3 3 α Lt,x (Gα ×R)      η2 2(m3 −l)/2 Pξ (Gm3 ),m −2≤·≤m +2 u 2 m3 3 3 α UΔ (Gα ×R)        ×  (Pl u) Pξ (t),m2 u Pξ (Gm3 ),≤m +2 u  2 m3 . 3 α Lt,x (Gα ×R) The last line used the Sobolev embedding theorem and (4.23). Summing   j up, using the Cauchy–Schwarz inequality and the definition of the X Gk × R norms, combined with an argument similar to the proof of Lemma 5.25 u), (remember that u was replaced with P −20 −1/4 |ξ −ξ (t)|≥2

∑

∑

m3 ≤m2 Gm3 ⊂G j α k ≤ j−10

 η2

N(t)

m

UΔ2 (Gα 3 ×R)

    ×  (Pl u) Pξ (t),m2 u Pξ (Gm3 ),≤m   2(m3 −l)/2 Pξ (t),m3 u

∑

  u  3 +2

α

m

2 G 3 ×R Lt,x ( α )

  j UΔ2 Gk ×R

m3 ≤m2 ≤ j−10

    ×  (Pl u) Pξ (t),m2 u Pξ (t),≤m3 u  2  j  Lt,x Gk ×R     j ∑ ∑  (Pl u) Pξ (t),m u

 X Gk ×R

 η2 u

   η22 u

η3

    η2 2(m3 −l)/2 Pξ (Gm3 ),m −2≤·≤m +2 u 3 3 α

m3 ≤ j−10



 j X Gk ×R

∑

m3 ≤ j−10

2

m3 ≤m2 ≤ j−10

2 1/2   × Pξ (t),≤m3 u  2  j  Lt,x Gk ×R   ∑ 2(m3 −l)/2 Pl u 2  j

 UΔ Gk ×R

m3 ≤m2 ≤ j−10

  × Pξ (t),m2 u    η22 Pl u

 j UΔ2 Gk ×R

 



2 1/2

   u2  . j j X Gk ×R UΔ2 Gk ×R

  Next, combining a bilinear estimate of  (Pl u) P

−1/4 −20 2 N(t)

|ξ −ξ (t)|≤η3

2  u L2

t,x

5.3 The Mass-Critical Problem When d = 1

205

with the Sobolev embedding theorem gives

∑j

Jl ∩Gk =0/

  4     e−iτ Δ (Pl u) Pξ (τ ),≤η −1/4 2−20 N(τ ) u d τ  Jl

3

  −1/4  N Gkj η3 +

∑j

−1/4



  Pl u

  j UΔ2 Gk ×R

N (Jl ) η3

Jl ⊂Gk



(5.124)

Lx2 (R)

(5.125)

  3/4 η3 2 j−l Pl uU 2 G j ×R . Δ

(5.126)

k

Inequality (5.126) follows from (4.151), which implies

∑ j N (Jl )  η3 2 j ,

(5.127)

Jl ⊂Gk

as well as the fact that there may be two intervals Jl that overlap Gkj but are not   1/2 contained in Gkj . However, because N (t) ≤ N Gkj ≤ η3 2 j−5 for any t ∈ Gkj , 1/2

their contribution is controlled by N (Jl )  η3 2 j−5 . Because VΔ1 ⊂ UΔ2 , to complete the proof of Theorem 5.27, it suffices to prove the following. Lemma 5.29

If u is replaced by P

−1/4

|ξ −ξ (t)|≥2−20 η3



∑

m3 ≤m2 ≤ j−10

 η2 2

  

∑ m

j

Gα 3 ⊂Gk

3( j−l) 4

t

m aα 3

UΔ2

u, then

   ei(t−τ )Δ (Pl u) Pξ (τ ),m2 u Pξ (τ ),m3 u

 2  2 × Pξ (τ ),≤m3 u d τ  2

  Pl u

N(t)

 



1/2 (5.128)

m

UΔ (Gα 3 ×R)

  u3  j X Gk ×R

(5.129)

j Gk ×R

and

∑j

Jl ∩Gk =0/

  4 2    ei(t−τ )Δ (Pl u) P|ξ −ξ (τ )|≤η −1/4 2−20 N(τ ) u d τ  2 3/8

Proof

UΔ (Jl ×R)

3

 η3 2

j−l 2

1/2

  Pl u

 . j UΔ2 Gk ×R

(5.130)

First suppose u is replaced with P

−1/4

|ξ −ξ (t)|≥2−20 η3

N(t)

u. By Theorem

5.21 and the Cauchy–Schwarz inequality, there exists some f such that fˆ is

206

Low-Dimensional Well-Posedness Results

supported on |ξ | ∼ 2l and  f 

12/5

UΔ

 

∑m 

Gα 3

m

Gα 3



m

(Gα 3 ×R)

j 3 = 1 for some Gm α ⊂ Gk , such that

   2  2 ei(t−τ )Δ (Pl u) Pξ (τ ),m2 u Pξ (τ ),m3 u Pξ (τ ),≤m3 u d τ  2

m

UΔ (Gα 3 ×R)

   2   sup  f Pξ (Gm3 ),m −2≤·≤m +2 u Pξ (Gm3 ),≤m +2 u  2 3 3 3 α α

m

Lt,x (Gα 3 ×R)

m

Gα 3

    2 ×  (Pl u) Pξ (t),m2 u Pξ (t),≤m3 u L2

(5.131)

 . j Gk ×R



t,x

(5.132)

By an argument similar to the proof of Lemma 5.25,      (Pl u) Pξ (t),m u Pξ (t),≤m u 2 2  j 2

 Lt,x Gk ×R

3

 2  2  η22 2 j−l 2m3 −m2 Pl uU 2 G j ×R uX G j ×R . Δ

To estimate

k

k

     f Pξ (t),m u Pξ (t),≤m u 2 2 m3 , 3 3 Lt,x (Gα ×R)

use 







∑ Pξ (t),i uLt,x12 (Gmα 3 ×R)  ∑ 2i/4 Pξ (t),i uUΔ2 (Gmα 3 ×R)

i≤m3

i≤m3

   2m3 /4 uX (Gm3 ×R) .

(5.133)

α

Therefore, by (5.133) and bilinear Strichartz estimates,      2  f Pξ (Gmα 3 ),m3 −2≤·≤m3 +2 u Pξ (t),≤m3 u  2 m3 Lt,x (Gα ×R)   2     Pξ (t),≤m u2 12 m3   f Pξ (Gm3 ),m −2≤·≤m +2 u  12/5 m3 3 Lt,x (Gα ×R) 3 3 α Lt,x (Gα ×R)    2 m3 −l  2 u m3  2 2 Pξ (Gm3 ),m −2≤·≤m +2 u 2 m3 X (Gα ×R) 3 3 α UΔ (Gα ×R) m3 −l  4  2 2 uX (Gm3 ×R) . α Therefore,   (5.131)1/2  η2 2(m3 −l)/4 2(m3 −m2 )/2 2( j−l)/2 Pl u

UΔ2

Summing over 0 ≤ m3 ≤ m2 ≤ j − 10 yields (5.129).

 



 u3 . X j ([0,T ]×R)

j Gk ×R

5.3 The Mass-Critical Problem When d = 1

207

Similarly, to derive (5.130), recall that the proof of (5.126) implies   2   2  (Pl u) Pξ (t),≤2−20 η −1/4 N(t) u  2  j  Lt,x Gk ×R

3

 2  2 3/4  η3 2 j−l Pj uU 2 G j ×R uX G j ×R . Δ

Because Jl is a small interval,    f P −20 ξ (t),≤2

−1/4 η3 N(t)

k

k

2   u 

2 (J ×R) Lt,x l

 1.

This proves (5.130) and Lemma 5.29. This completes the proof of Theorem 5.27. The proof of Theorem 5.20 is therefore now complete. As in dimensions d ≥ 3, the long-time Strichartz estimates of Theorem 5.19 yield frequency-localized interaction Morawetz estimates. Theorem 5.30 If u is an almost-periodic solution to (4.142) in dimension d = 1 on R with N (t) ≤ 1 for all t ∈ R, and [0, T ] is an interval satisfying  T

N (t)3 dt = K,

(5.134)

0

then  T   0

 8  P≤η −1 K u (t, x)  dx dt + 3

T

 

2 2   ∂x P≤η −1 K u (t, x)  dx dt  o (K) , 3

0

(5.135) o(K) K

→ 0 as K  ∞.  6 Proof Suppose [0, T ] is an interval such that uL6 ([0,T ]×R) = 2k0 . After t,x rescaling,   η3 2k0 . (5.136) u (t, x) → λ 1/2 u λ 2t, λ x , with λ = K Inequality (5.135) is equivalent to proving where o (K) is a quantity satisfying

 T 0

|P≤k0 u (t, x) |8 dx dt +

 T  0

∂x |P≤k0 u (t, x) |2

2

dx dt  o (K)

η3 2k0 . K

Now by (4.10) and (4.18) with μ = 1 and u = v, if u solves (4.142), then  T 0

 T  2 |u (t, x) |8 dx dt + ∂x |u (t, x) |2 dx dt 0  −ix·ξ (t)   3    e u L∞ H˙ 1 ([0,T ]×R) uL∞ L2 ([0,T ]×R) . t

t

x

208

Low-Dimensional Well-Posedness Results

Therefore, letting I = P≤k0 ,  T 0

|Iu (t, x) |8 dx dt +

 T 0

∂x |Iu (t, x) |2

2

dx dt

   3   e−ix·ξ (t) IuL∞ H˙ 1 ([0,T ]×R) IuL∞ L2 ([0,T ]×R) + t

t

x

0

T

E (t) dt,

where, as in (4.169), E (t) =

 

    (x − y) |Iu (t, y) |2 Re IF (u) ¯ − F Iu (∂x − iξ (t)) Iu (t, x) dx dy |x − y| (5.137)     (x − y) |Iu (t, y) |2 Re Iu (∂x − iξ (t)) (F (Iu) − IF (u)) (t, x) dx dy − |x − y|      (x − y)  Im IuIF (u) (t, y) Im Iu (∂x − iξ (t)) Iu (t, x) dx dy. +2 |x − y|

Also, as in Lemma 4.38,  −ix·ξ (t)  e Iu

Lt∞ H˙ 1 ([0,T ]×R)

 3 η3 2k0 Iu ∞ 2 o (K) .  Lt Lx ([0,T ]×R) K

Let ∂ x = ∂x − iξ (t). Lemma 5.31 If u is an almost-periodic solution to (4.142) that satisfies the above conditions, then  T  (x − y) 0

|x − y|  o (K)

|Iu (t, y) |2 Re

    IF (u) ¯ − F Iu (∂x − iξ (t)) Iu (t, x) dx dy dt

η3 2k0 . K

(5.138)

Proof Using the Galilean Littlewood–Paley projection, let ul = uξ (t),≤k0 −5 and uh + ul = u. By Fourier support arguments, IF (ul ) − F (Iul ) = 0.

(5.139)

Next, since I is a Fourier multiplier whose symbol satisfies |∂ξ m (ξ ) |  for |η | ≤ |ξ | we have |m (ξ + η ) − m (ξ ) | 

|η | , |ξ |

1 , |ξ |

(5.140)

5.3 The Mass-Critical Problem When d = 1

209

so    4    I uh u − (Iuh ) u4 ∂ x Iu  l

   2−k0  (uh ) u2  l

1 ([0,T ]×R) Lt,x

l

 1/2 × ∂ x ul L∞ L2 ([0,T ]×R) x

t

    ∂ x ul u2 1/2 2 ([0,T ]×R) l Lt,x   ∂ x Iu 4 ∞ .

2 ([0,T ]×R) Lt,x

(5.141)

Lt Lx ([0,T ]×R)

By Theorem 5.19, Lemma 5.25, (5.134), and (5.136),  2 uh u  2 l L ([0,T ]×R)  1.

(5.142)

t,x

j Similarly, partitioning[0, T ] into  Gα intervals and applying Lemma 5.25 and j the definition of the X Gα × R -norm on each subinterval gives    k0 /2 − j/2  Pξ (t), j u u2  2 2 . (5.143) l L ([0,T ]×R)  2 t,x

Therefore, again by Theorem 5.19,     ∂ x Iu u2  l

2 ([0,T ]×R) Lt,x



∑2

k0 − j 2

2 j  2k0 .

(5.144)

η3 2 k 0 K

for all t ∈ [0, T ],

j≤k0

Also, by definition of X-norms and the fact that N (t) ≤   ∂ x Iu 4 ∞  2k0 L L ([0,T ]×R) t

and

x

  ∂ x ul 

Lt∞ Lx2 ([0,T ]×R)

 o (K)

η3 2k0 . K (5.145)

Combining (5.141)–(5.145) we obtain (5.141)  o (K)

η3 2k0 . K

Next,  2 3  u u ∂ x Iu  1 h l Lt,x ([0,T ]×R)  2 3/2 2/3 1/3  1/2    uh ul L2 ([0,T ]×R) uh L6 ([0,T ]×R) ∂ x IuL4 L∞ ([0,T ]×R) ∂ x IuL∞ L2 ([0,T ]×R) t,x

 o (K)

t,x

x

t

x

η3 , K

 3 2  u u ∂ x Iu  1 h l Lt,x ([0,T ]×R)  2  2  uh ul L2 ([0,T ]×R) uh L6 t,x

 o (K)

t

2k0

η3 . K 2k0

t,x ([0,T ]×R)

    ∂ x Iu2/3 ∂ Iu1/3 L∞ L2 ([0,T ]×R) L4 L∞ ([0,T ]×R) x t

x

t

x

210

Low-Dimensional Well-Posedness Results

Finally,  4   u u ∂ x Iu  1 h Lt,x ([0,T ]×R)  4 4/5 1/5      uh L5 L10 ([0,T ]×R) uL∞ L2 ([0,T ]×R) ∂ x IuL4 L∞ ([0,T ]×R) ∂ x IuL∞ L2 ([0,T ]×R) t

 o (K)

x

t

x

t

t

x

x

η3 . K 2k0

This completes the proof of (5.138). Turning to the second term in (5.137), we have the following lemma. Lemma 5.32  T (x − y) 0

|x − y|  o (K)

Proof

  |Iu (t, y) |2 Re Iu (∂x − iξ (t)) (F (Iu) − IF (u)) (t, x) dx dy dt  η3 2k0  η3 2k0 1/4  Iu6 8 + o (K) . Lt,x ([0,T ]×R) K K

(5.146)

Integrating by parts,

(5.146) = −

 T  (x − y)

|x − y|

0

 |Iu (t, y) |2 Re (F (Iu) − IF (u))

 × (∂x + iξ (t)) Iu (t, x) dx dy dt (5.147)  T   − |Iu (t, x) |2 Re (F (Iu) − IF (u)) Iu (t, x) dx dt. (5.148) 0

Expression (5.147) is equal to (5.138), so by Lemma 5.31, we find (5.147)  k0 o (K) η3K2 . Since IF (ul ) − F (Iul ) = 0, |IF (u) − F (Iu) |  |ul |5 |uh | + |uh |6 , so  2  6 (5.148)  IuL∞ ([0,T ]×R) uh L6 ([0,T ]×R) t,x t,x  6       + Iu L8 ([0,T ]×R) uh 16 t,x

Lt 3 Lx8 ([0,T ]×R)

  Iu 16 8 . L L ([0,T ]×R) t

x

(5.149)

By the Sobolev embedding theorem,    2   Iu ∞  ∂ x IuL∞ L2 ([0,T ]×R) IuL∞ L2 ([0,T ]×R) L ([0,T ]×R) t,x

t

 o (K)

x

η3 , K 2k0

t

x

(5.150)

5.3 The Mass-Critical Problem When d = 1

211

and therefore  2  6 Iu ∞ u  L ([0,T ]×R) h L6

η3 2k0 . t,x t,x K Again using the Sobolev embedding theorem and also interpolation, we get      3/4 Iu 16 8 ∂ x Iu1/4 Iu ∞ 2  4 ∞ Lt Lx ([0,T ]×R) Lt Lx ([0,T ]×R) Lt Lx ([0,T ]×R)

1/4 k 0 η3 2  o (K) . (5.151) K  o (K) ([0,T ]×R)

This completes the derivation of (5.146). Finally, take the third term in (5.137). Lemma 5.33

 T  (x − y)

|x − y|

0

 o (K) Proof

    Im IuIF (u) (t, y) Im Iu (∂x − iξ (t)) Iu (t, x) dx dy dt



1/4  6 η3 2k0 η3 2k0 Iu 8 + o (K) . Lt,x ([0,T ]×R) K K

Because IuF (Iu) = |Iu|6 , it follows that     Im IuIF (u) = Im Iu [IF (u) − F (Iu)] .

By a careful analysis of Fourier supports, we obtain          I uh u4l − u4l Iuh = I P>k0 −2 u u4l − u4l P>k0 −2 Iu . Therefore,

       = 0, P≤k0 −10 ul I uh u4l − u4l Iuh

and integrating by parts yields  T  (x − y)

     Im Iu · I u4l uh − (Iuh ) u4l (t, y) |x − y| 0   × Im Iu (∂x − iξ (t)) Iu (t, x) dx dy dt  T      ∂y P>k0 −10 Im Iu I u4l uh − (Iuh ) u4l (t, y) =− Δy 0   × Im Iu (∂x − iξ (t)) Iu (t, y) dy dt. (5.152)

Using analysis similar to that for (5.149)–(5.151),

1/4  6 η3 2k0 Iu 8 . RHS of (5.152)  o (K) Lt,x ([0,T ]×R) K

212

Low-Dimensional Well-Posedness Results

For terms with at least two uh , by Lemma 5.25 we have  2 2  6  2 4 uh u  2 uh  6 u u  1  + h l L ([0,T ]×R) L ([0,T ]×R) L t,x

t,x ([0,T ]×R)

t,x

 1.

Using the second inequality in (5.150) completes the proof of Lemma 5.33.

Combining Lemmas 5.31–5.33 proves Theorem 5.30. Corollary 5.34 There exists no nonzero almost-periodic solution to (4.142) with N (t) ≤ 1 on R and  ∞

−∞

N (t)3 dt = ∞.

Proof If u is an almost-periodic solution to (4.142) with nonzero mass, then by (4.167), and (5.135), if u is nonzero,  T

K= 0

N (t)3 dt 

 T 

∂x |P≤η −1 K u (t, x) |2 3

0

2

dx dt  o (K) ,

with bound independent of T . This implies an upper bound on K as T  ∞. The negative time direction follows from time-reversal symmetry. Theorem 5.35 Suppose u is an almost-periodic solution to (4.142) when d = 1, on R with N (t) ≤ 1 for all t ∈ R. Suppose also that  ∞ −∞

N (t)3 dt < ∞.

(5.153)

Then u ≡ 0. Proof By (4.144), (5.153), and the fundamental theorem of calculus, there exist ξ+ , ξ− ∈ R such that ξ (t) → ξ± as t → ±∞ respectively. Suppose T∞ N (t)3 dt = K. Again by (4.144), |ξ (t) − ξ+ |  η1−1 K for all t ∈ [T, ∞). Without loss of generality make a Galilean transformation mapping ξ+ → 0. We now need the following intermediate theorem. -

Theorem 5.36 If T∞ N (t)3 dt = K and u is an almost-periodic solution to (4.142) when d = 1, and ξ+ = 0, then   u 1/2  K 1/2 . Lt∞ H˙ x ([T,∞)×Rd )

5.3 The Mass-Critical Problem When d = 1 Proof For any T1 < ∞, k0 > 0,

- T1 T

N (t)3 dt < K. Choose T1 such that for some integer

 T1 

|u (t, x) |6 dx dt = 2k0 .

T

Taking λ = η3 2k0 /

- T1 T

213

N (t)3 dt and rescaling u → uλ gives   uλ   T T    1. X k0

, 1 λ2 λ2

×R

Furthermore, Theorem 5.20 implies that for k ≥ k0 , we have   P>k uλ   T T   UΔ2 , 1 ×R λ2 λ2      T1     P>k uλ + ε P>k−5 uλ  2  T T1   .  2 UΔ , ×R λ λ2 λ2 Lx2 (R)

(5.154)

Rescaling back, since UΔ2 and L2 are scale-invariant norms, if 2k ≥ η3−1 K,       P>k u 2  P>k uλ (T1 ) L2 (R) + ε P>k−5 uλ U 2 ([T,T ]×R) . (5.155) U ([T,T ]×R) Δ

1

Δ

x

Now, N (t) → 0 and ξ (t) → 0 as t → ∞, so by (4.80),   lim P>k u (T1 ) L2 (R) = 0. T1 →+∞

x

1

(5.156)

Since the ε > 0 in (5.154) may be made arbitrarily small, in particular so that Cε ≤ 2−5 , where C is the implicit constant in (5.155), for any t ∈ [T, ∞), it follows that (5.155) and (5.156) imply that for 2k ≥ η3−1 K,   P>k u (t)  2 d  2−k K. Lx (R ) This combined with conservation of mass proves Theorem 5.36. Note that |ξ (t) − ξ+ | ≤ η1−1 K also implies   −ix·ξ (t) e u (t) H˙ 1/2 (R)  K 1/2 , (5.157) for ξ (t) : R → R and ξ (0) = 0. By the dominated convergence theorem,  ∞

lim

T →∞ T

N (t)3 dt = 0,

so for a general ξ+ ∈ R, (5.157) implies   lim e−ix·ξ (t) u (t) H˙ 1/2 (Rd ) = 0. t→+∞

214

Low-Dimensional Well-Posedness Results

The same analysis can be done as t → −∞. Therefore, by (4.18),   2  ∂x |u|  2 = 0, L (R×R) t,x

and therefore u ≡ 0, completing the proof of Theorem 5.35.

5.4 The Two-Dimensional Mass-Critical Problem Proof of scattering for the two-dimensional, mass-critical problem will also use the function space from Definition 5.18. In addition, the proof will utilize a seminorm that measures the size of an almost-periodic solution at frequency scales well separated from the almost-periodic scale. If Gkj ⊂ [a, b] define

Definition 5.37   u

j Y (Gk ×R2 )

=

∑

5max{ j,5}:

 2   Pξ (G j ),i−2≤·≤i+2 u 2

j

UΔ (Gk ×R2 )

k

.

1/2

j

N(Gk )≤2i−5 η3

Define uY ([a,b]×R2 ) in the manner analogous to (5.99). l

−1/2

By a calculation similar to (5.102)–(5.103), since 2i ≥ 32η3 t ∈ Giα , if aiα is as in (5.101), then (4.145) implies    2  ∑ 2i− j ∑ j Pξ (Giα ),i−2≤·≤i+2 u aiα Lx2 (Rd ) i 5max{ j,5}: j

  j  2  Pξ (G j ),i−2≤·≤i+2 u ak  2 k

Lx (Rd )

 η22 .

(5.157)

1/2

N(Gk )≤2i−5 η3

Remark Because of the projective properties of P0 , compared with those of Pi for i > 0 (see Definition 4.32), it is important that i = 0 not be included in the Y -seminorm. If i = 0 were included, the bound for (5.157) would be 1. Let us resume the two-dimensional analysis that began in the previous section and paused after the proof of Theorem 5.23.

5.4 The Two-Dimensional Mass-Critical Problem

When d = 2 and u is an almost-periodic solution to (4.142),

Theorem 5.38  2 u j d X(Gk ×R )

∑

 1+

j i> j:N(Gk ) 1/2 i−5 ≤2 η3

∑

+

215

 t  i(t−τ )Δ  je j

ak ξ (Gk ),i−2≤·≤i+2

∑j

2i− j

0≤i≤ j

  

Giα ⊂Gk :

t aiα

2  F (u (τ )) d τ  2 

j

UΔ Gk ×Rd



2  ei(t−τ )Δ Pξ (Gi ),i−2≤·≤i+2 F (u (τ )) d τ  2 i d α U (Gα ×R ) Δ

1/2 N(Giα )≤2i−5 η3

and  2 u j d Y (G ×R ) k



 t 2   η22 +  j ei(t−τ )Δ Pξ (G j ),i−2≤·≤i+2 F (u (τ )) d τ  2 j d UΔ (Gk ×R ) k a k i>max{ j,5}:

∑

1/2

j

N(Gk )≤2i−5 η3

+

∑

∑

2i− j

5 0 small in (5.107) will require uY to be small, which will follow along from Theorem 5.20, l (5.158), (5.159), and the fact that analogously to (5.111),     u ≤ 2uY ([0,T ]×R2 ) . (5.160) Yl+1 ([0,T ]×R2 ) l First compute  t        ei(t−τ )Δ Pm (Pl u) Pξ (τ ),≥i−10 u u d τ   aiα

UΔ2 (Giα ×R2 )

.

(5.161)

By (5.112), to estimate (5.161) it suffices to estimate the inner product of the Duhamel term with a generic f such that fˆ (t, ξ ) is supported on |ξ | ∼ 2m and  f  5/2 i 2 = 1. UΔ (Gα ×R )

216

Low-Dimensional Well-Posedness Results

  1/2 Make a bilinear Strichartz estimate and use (4.145) and N Giα ≤ 2i−5 η3 to give, if i ≥ 11,      f (Pl u) Pξ (t),≥i−10 u u 1 i 2 Lt,x (Gα ×R ) 1/2   1/2   f (Pl u)  5/2 5/3 i 2  f  5/2 10 i 2 Lt Lx (Gα ×R ) Lt Lx (Gα ×R )  1/2        × Pl u 5/2 10 i 2 Pξ (t),≥i−10 u 5/2 10 i 2 uL∞ L2 (Gi ×R2 ) L L G ×R α ) t x Lt Lx (Gα ×R ) x ( α t     |m−l| 5/6  2− 5 Pl uU 2 (Gi ×R2 ) Pξ (t),≥i−10 u 25/12 50 i 2 α Lx (Gα ×R ) Lt Δ  1/6 × Pξ (t),≥i−10 uL∞ L2 Gi ×R2 ) t x( α  5/6  |m−l| 1/6   2− 5 η2 Pl uU 2 (Gi ×R2 ) uX Gi ×R2 . (5.162) ( α ) α Δ For 0 ≤ i ≤ 10,

  (Pl u) Pξ (t),≥i−10 u u = (Pl u) u2 .

Since m ≥ i − 2 and uL4 (Gi ×R2 )  1, by Strichartz estimates and (4.145), α t,x   1/2  5/2     Pξ (Giα ),m (Pl u) u2  1 2 i 2  P|ξ −ξ (t)|>2m−5 uL∞ L2 u 5/2 10 t x Lt Lx Lt Lx (Gα ×R ) 1/2

 η2 . Meanwhile, by the analysis in (5.162),      |m−l|   f (Pl u) Pξ (t),≥i−10 u u 1 i 2  2− 5 Pl u 2 i 2 . Lt,x (Gα ×R ) UΔ (Gα ×R ) Then by interpolation,         |m−l|  f (Pl u) Pξ (t),≥i−10 u u 1 i 2  2− 10 η 1/4 1 + Pl u 2 i 2 . 2 Lt,x (Gα ×R ) UΔ (Gα ×R ) Therefore, after making the same relabeling and Galilean transform as in the proof of Theorem 5.27, it only remains to estimate, for j ≥ 11,  t  2    (5.163)  j ei(t−τ )Δ (Pl u) Pξ (τ ),< j−10 u d τ  2  j 2  . UΔ Gk ×R

ak

The estimate of (5.163) will utilize the bilinear estimate     |Pl u| |Pξ (t),< j−10 u|   j 2 2 Lt,x Gk ×R

2

j−l 2

  Pl u

UΔ2



j Gk ×R2





2   1 + uX ([0,T ]×R2 ) , j

5.4 The Two-Dimensional Mass-Critical Problem

217

which, by the definition of UΔ2 atoms, follows directly from    v Pξ (t),< j−10 u 



j

2 G ×R2 Lt,x k



2

j−l 2

 2     v0  2 2 1 + u , X ([0,T ]×R2 ) L (R ) j

when v solves i∂t v − Δv = 0,

v (0, x) = v0 ,

(5.164)

and vˆ0 (ξ ) is supported on |ξ | ∼ 2l , l ≥ j − 5. Theorem   5.39 (Two-dimensional bilinear Strichartz estimate) Suppose v0 ∈ L2 R2 has Fourier transform vˆ0 (ξ ) supported on |ξ | ∼ 2l , l ≥ j − 5, and j ≥ 11. Also suppose that uY j ([0,T ]×R2 ) is small. Then for any 0 ≤ l2 ≤ j −10, Glβ2 ⊂ Gkj ,    v Pξ (t),≤l u 2 2 2 L



l2 2 t,x Gβ ×R



    2  2l2 −l v0 L2 (R2 ) 1 + u



j Xl2 Gk ×R2



4

.

(5.165) Proof

  First observe that by (4.144) and the fact that ξ Glα2 = 0,   v (t, x) Pξ (t),≤l2 u

(5.166)

is supported on |ξ |  2l . Therefore, following the analysis in (4.33)–(4.35),    v Pξ (t),≤l u 2 2 2 L l2 −2l

2



t,x

l Gβ2



l

Gβ2 ×R2



Sd−1



  xω =yω

    ∂ω v (t, x) Pξ (t),≤l u (t, y) 2 dx dy d ω dt. 2

To simplify notation let w (t, x) = Pξ (t),≤l2 u. Also by (4.25), 

f (y)

 (x − y)ω  g (x) ∂ω h (x) d ω = C | (x − y)ω |



f (y)

(x − y) · [g (x) ∇h (x)]. |x − y|

Therefore, by μ = 1, (4.26)–(4.30), (5.164), the direct calculation    d  Pξ (t),≤l2 u ∂t w = ∂t Pξ (t),≤l2 u = Pξ (t),≤l2 (iΔu − iF (u)) + dt    d  = iΔw − iF (w) + iF Pξ (t),≤l2 u − iPξ (t),≤l2 F (u) + Pξ (t),≤l2 u, dt

218

Low-Dimensional Well-Posedness Results

and the fundamental theorem of calculus,        ∂ω v (t, y) w (t, x) 2 dx dy d ω dt 2l2 −2l l Gβ2

 2l2 −2l

xω =yω

   d  sup  M (t)  dt l2

t∈Gβ

+ 2l2 −2l

+ 2l2 −2l



 

l Gβ2



 

l Gβ2



+ 2l2 −2l

 l Gβ2

(5.167)

|v (t, y) |2

  (x − y) · Re N¯ (∇ − iξ (t)) w (t, x) dx dy dt |x − y| (5.168)

|v (t, y) |2

  (x − y) · Re w¯ (∇ − iξ (t)) N (t, x) dx dy dt |x − y| (5.169)

    (x − y) · Im v¯ (∇ − iξ (t)) v (t, x) dx dy dt, Im wN ¯ (t, y) |x − y| (5.170)

where

   d  N = iF Pξ (t),≤l2 u − iPξ (t),≤l2 F (u) + Pξ (t),≤l2 u dt  d  = N1 + P u = N1 + N2 dt ξ (t),≤l2

and

 

  (x − y) · Im v¯ (∇ − iξ (t)) v (t, x) dx dy |x − y|     (x − y) · Im w¯ (∇ − iξ (t)) w (t, x) dx dy + |v (t, y) |2 |x − y|    2  2 l  2   (5.171)  2 w ∞ 2 v0 2  2l v0  2 .

∂t M (t) =

|w (t, y) |2

Lt Lx

Lx

L

Remark Here, there is again an abuse of the notation in (4.26)–(4.30) since (5.171) need not be the time derivative of any quantity that is in the form of the M (t) in (4.26)–(4.30). It is easy to see that supt∈Giα 2l2 −2l |∂t M (t) | is bounded by the right-hand side of (5.165). Now for (5.168)–(5.170), to simplify notation, every Ltp Lxq -norm will be taken over Glβ2 × R2 unless otherwise stated. Since Pξ (t),≤l2 is a Fourier multi  plier whose symbol is φ ξ −2ξl2(t) , we have



 d ξ − ξ (t) ξ − ξ (t) ξ (t) φ (5.172) = ∇φ · l , dt 2l2 2l2 22

5.4 The Two-Dimensional Mass-Critical Problem

219

so (5.168)

  2  2−2l vL∞ L2 t

    |ξ  (t) |Pξ (t),l2 −5≤·≤l2 +5 u (t) L2  (∇ − iξ (t)) wL2 dt

l

Gβ2

x

 2   vL∞ L2 N1 

l2 −2l 

+2

t

3/2

Lt

x

(5.173)

    6/5 (∇ − iξ (t)) w

Lt3 Lx6

Lx



.

(5.174)

By (4.144), (4.147), and conservation of mass,  2 (5.173)  22l2 −2l v0 L2 . x

= ∇ − iξ (t), Next, by definition of w, again adopting the convention ∇

5/6      1/6 k        P ∇w 2 u ∇w ∞ 2 l l 5/2 ∑ ξ (t),k Lt3 Lx6 Gβ2 ×R2

Lx10 Gβ2 ×R2

Lt

0≤k≤l2

   2l2 uX .

Lt Lx

(5.175)

l2

Therefore,

 2   (5.168)  22l2 −2l v0 L2 u  l2 2  Xl2 Gβ ×R     × Pξ (t),≤l2 F (u) − F Pξ (t),≤l2 u 

3/2 6/5 Lx

Lt

.

Split u as uh + ul , where ul = Pξ (t),≤l2 −5 u. Following the same analysis as in (5.139),   (5.176) Pξ (t),≤l2 F (ul ) − F Pξ (t),≤l2 ul = 0. Next, as in (5.140)–(5.141),        Pξ (t),≤l u2 uh − Pξ (t),≤l ul 2 Pξ (t),≤l uh  3/2 6/5 l 2 2 2 Lt Lx  2       −l2        uh L3 L6 ∇w L3 L6 u L∞ L2  uX . 2 t

Finally,

 2 uu  h

x

3/2 6/5 Lt Lx

Therefore,

x

t

l2

x

t

 2    2  uh L3 L6 uL∞ L2  uX . t

z

t

    2 (5.174)  22l2 −2l v0 L2 (R2 ) 1 + u 

l Xl2 Gβ2 ×R2

and

(5.178)

l2

x

   2  (5.168)  22l2 −2l v0 L2 1 + u 

j X j Gk ×R2





3

(5.177)

3

.

(5.179)

220

Low-Dimensional Well-Posedness Results

The right-hand side of this term is clearly bounded by the right-hand side of (5.165). Turning now to (5.169), integrating by parts in space gives l2 −2l

(5.169) = (5.168) − 2



  l Gβ2

|v (t, y) |2

1 Re[wN ¯ ] (t, x) dx dy dt. |x − y|

Using the Hardy–Littlewood–Sobolev theorem, Hardy’s inequality, the Sobolev embedding theorem, (4.144), (4.147), (5.172), (5.177), and (5.178), we get 2l2 −2 j



 

l Gβ2

−2 j

2

 l

Gβ2

  1 |v (t, y) |2 Re wN ¯ (t, x) dx dy dt |x − y| 2      |ξ  (t) |v (t) L2  ∇w (t) L2 Pξ (t),l2 −5≤·≤l2 +5 w (t) L2 dt

        + 2l2 −2 j vL∞ L2 vL3 L6 N1  3/2 2 wL∞ L3 Lt Lx t x t x t x  2    2/3  1/3 2l2 −2 j  2 v0 2 + v0  2  2 ∇N1  3/2 6/5 N1  3/2 L

L

  3  2   22l2 −2 j v0 L2 1 + uX .

Lt

Lx

Lt

6/5 Lx

 1/3  2/3  ∇w ∞ 2 w ∞ 2 Lt Lx

l2

Lt Lx

(5.180)

This term is also clearly bounded by the right-hand side of (5.165). Now consider (5.170). Again by conservation of mass, (4.144), and (4.147), 

 

    (x − y) · Im v¯ (∇ − iξ (t)) v (t, x) dx dy dt Im N2 w¯ (t, y) |x − y|

     2   Pξ (t),l −5≤·≤l +5 u (t)  2 w (t)  2 dt  2− j v0 L2 | ξ (t) | l 2 2 L L

2l2 −2 j

l Gβ2

 2  2l2 − j v0  2 .

Gβ2

L

(5.181)

To analyze the contribution of N1 , first observe that   Im Pξ (t),≤l2 F (u) Pξ (t),≤l2 w      = Im Pξ (t),≤l2 F (u) − F Pξ (t),≤l2 u Pξ (t),≤l2 u . Recall from (5.176) that   Pξ (t),≤l2 F (ul ) − F Pξ (t),≤l2 ul = 0. Also, as in the one-dimensional case,     2    Pξ (t),≤l2 u2l uh − Pξ (t),≤l2 ul Pξ (t),≤l2 uh Pξ (t),≤l2 ul

(5.182)

5.4 The Two-Dimensional Mass-Critical Problem

221

is supported on |ξ | ∼ 2l2 , so as in (5.152), using the Hardy–Littlewood– Sobolev inequality and the Sobolev embedding theorem, yields 2l2 −2l



  l Gβ2

   Im [Pξ (t),≤l2 u2l uh

2     Pξ (t),≤l2 uh ] Pξ (t),≤l2 ul (t, y) − Pξ (t),≤l2 ul   (x − y) × · Im v¯ (∇ − iξ (t)) v (t, x) dx dy dt |x − y|    1 (t, x) | dx dy dt |v (t, x) | |∇v  2−2l l |uh (t, y) | |ul (t, y) |3 2 |x − y| Gβ    3    1/2  1/2  2−2l uh  3 6 ul  6 v ∞ 2  ∇v 3 6  ∇v ∞ 2 Lt Lx

Lt Lx

Lt,x

Lt Lx

Lt Lx

   1/2 2      1/2  1/2  2−2l uh L3 L6  |∇| ul L4 ul L∞ L2 vL∞ L2  ∇vL3 L6  ∇vL∞ L2 t x t,x t x t x t x t x     l2 −l  2  3 v0 2 u . 2 (5.183) L

Xl2

This term is bounded by the right-hand side of (5.165). Then 2l2 −2l



 

    (x − y) · Im v¯ (∇ − iξ (t)) v (t, x) dx dy dt Im u2h u2 (t, y) |x − y|    4  2  2l2 −2l uh ul L2 (Gl2 ×R2 ) + uh L4 (Gl2 ×R2 ) t,x β t,x β         × v ∞ 2 l2 2 ∇v ∞ 2 l2 2 Lt Lx (Gβ ×R ) Lt Lx (Gβ ×R )    4    2 2 (5.184)  2l2 −l v0 L2 uh ul L2 (Gl2 ×R2 ) + uh L4 (Gl2 ×R2 ) . l Gβ2

t,x

β

t,x

β

By direct computation,  2  4 2l2 −l v0 L2 uh L4

l

2 2 t,x (Gβ ×R )

 2  4  2l2 −l v0 L2 uX

l2

(5.185)

and  2   2  2l2 −l v0 L2 u2h Pξ (t),≥ j−15 u 

l

1 (G 2 ×R2 ) Lt,x β

2  2  2   2  4  2l2 −l v0 L2 uh L4 Pξ (t),≥ j−15 uL4  2l2 −l v0 L2 uX . t,x

t,x

l2

(5.186) Therefore, fixing l2 = j − 10 and Glβ2 = Gkj (which is an acceptable abuse of

notation since Gkj is the union of a bounded number, 210 , of Gβj−10 intervals)

222

Low-Dimensional Well-Posedness Results

and collecting (5.179), (5.180), (5.181), (5.184), (5.185), and (5.186) implies    2 2l− j  (Pl u) Pξ (t),≤ j−10 u L2 (G j ×R2 ) t,x k  4    2  Pl uU 2 (G j ×R2 ) 1 + uX (G j ×R2 ) j k Δ k  2    2 + Pl uU 2 (G j ×R2 )  Pξ (t),≥ j−5 u Pξ (t),≤ j−15 u L2 (G j ×R2 ) t,x k Δ k  4  2    Pl uU 2 (G j ×R2 ) 1 + uX (G j ×R2 ) j k Δ k  2   2 + Pl uU 2 (G j ×R2 ) sup sup  Pξ (t),≥l2 u Pξ (t),≤l2 −10 u L2 Δ

k



l2 2 t,x (Gα ×R )

0≤l2 Gl2 ⊂G j k ≤ j−5 α

.

Therefore, sup

sup

0≤ j≤k0

Gk ⊂[0,T ]:

    2 sup 2l− j  Pξ (G j ),l u Pξ (t),≤ j−10 u  2

j

j

j

Lt,x (Gk ×R2 )

k

l≥ j−5 1/2

N(Gk )≤2 j−10 η3

 4      2 2  uY ([0,T ]×R2 ) 1 + uX ([0,T ]×R2 ) + uY ([0,T ]×R2 ) k0 k0 k0     2 × sup sup sup 2k−l2  P l2 u Pξ (t),≤l2 −10 u  2 ξ (Gα ),k

0≤l2 ≤ Gl2 ⊂[0,T ] k≥l2 −5 α k0 −5

l

Lt,x (Gα2 ×R2 )

.

(5.187)   1/2 Also, trivially, by (4.144), (4.145), and (4.151), if N Glα2 ≥ 2l2 −10 η3 , then   uh ul 2 2 L

l2 2 t,x (Gα ×R )

 4  uL4

l2 2 t,x (Gα ×R )

 1.

(5.188)

Plugging (5.188) into the right-hand side of (5.187) if N(Glα2 ) ≥ 2l2 −10 η3 , then   a bootstrap argument combined with (5.186)–(5.188) implies that when u is small we have Y ([0,T ]×R2 ) 1/2

j

sup

sup

    2 sup 2l− j  Pξ (G j ),l u Pξ (t),≤ j−10 u  2

0≤ j≤k0 G j ⊂[0,T ] l≥ j−5 k

k

j

Lt,x (Gk ×R2 )

4     1 + uX ([0,T ]×R2 ) . k0

Combining (5.189) and (5.184) completes the proof of Theorem 5.39. Theorem 5.39 may be easily upgraded to an l 2 summation theorem.

(5.189)

5.4 The Two-Dimensional Mass-Critical Problem

223

Theorem 5.40 (Second bilinear Strichartz estimate) For any 0 ≤ j ≤ k0 ,

∑

0≤l2 ≤ j−10

 itΔ     e v0 Pξ (t),≤l u 2 2 2 L

j 2 t,x (Gk ×R )

6    2   2 j−l v0 L2 1 + uX (G j ×R2 ) . j

k

Proof Again use the bilinear interaction Morawetz estimate. Recall (5.167)– (5.170). It is straightforward to verify that as in (5.171),

∑

0≤l2 ≤ j−10

2l2 −2l sup |∂t M (t) |  j

t∈Gk

∑

0≤l2 ≤ j−10

 2  2 2l2 −l v0 L2  2 j−l v0 L2 .

Next, since there are 2 j−l2 intervals Glα2 contained in Gkj ,

∑

∑

0≤l2 ≤ j−10 Gl2 ⊂G j α k

3  2    (5.179)  2 j−l v0 L2 1 + uX ([0,T ]×R2 ) j

and

∑

∑

0≤l2 ≤ j−10 Gl2 ⊂G j α k

3    2  (5.180)  v0 L2 1 + uX ([0,T ]×R2 ) . j

This takes care of the terms generated by (5.168) and (5.169). Now consider the terms generated by (5.170). First take (5.184). By Definition 5.18, since l ≥ j − 5,

∑

0≤l2 ≤ j−10

4  2  2l2 −l v0 L2 Pξ (t),≥l2 −5 uL4

j 2 t,x (Gk ×R )

 2  4  v0 L2 uX ([0,T ]×R2 ) . j

Also, by (5.165), the fact that there are 2m−l2 intervals Glα2 ⊂ Gm β when m ≤ j, Definition 5.18, and Young’s inequality,

∑

0≤l2 ≤ j−10



 2     2 2l2 −l v0 L2  Pξ (t),≥l2 −5 u Pξ (t),≤l2 −5 u L2

∑

0≤l2 ≤ j−10

t,x

 2 v0  2 L

∑

m≥l2 −5

j

Gk ×R2



 4   2  2l2 −l Pξ (t),m uU 2 G j ×R2  1 + uX ([0,T ]×R2 )

6    2   2 j−l v0 L2 1 + uX ([0,T ]×R2 ) . j



Δ

k

j

224

Low-Dimensional Well-Posedness Results

This estimates (5.184). Next, summing (5.183) gives  3       2−2l ∑ Pξ (t),≥l2 −5 uL3 L6 Pξ (t),≤l2 −5 uL6  ∇vL6 L3 vL∞ L2  2−l  2−l

t,x

t

x

t

x

∑

   2   v0  2 Pξ (t),≥l −5 u 3 6 j 2 Pξ (t),≤l −5 u3 6 2 2 L L L L (G ×R )

∑

 2 v0  2 L

0≤l2 ≤ j−10 0≤l2 ≤ j−10

∑

l2 −5≤m≤ j Gm ⊂G j

∑

0≤l2 ≤ j−10

j 2 t,x (Gk ×R )

k

  Pξ (t),m u

2 Lt3 Lx6 (Gm α ×R )

k

 3 × Pξ (t),≤l2 −5 uL6 (Gm ×R2 ) t,x α    2   v0  2 Pξ (t),≥ j u 3 6 j 2 Pξ (t),≤l −5 u3 6 m 2 2 L L (G ×R ) L L (G ×R ) t

 2 v0  2 L

0≤l2 ≤ j−10

+ 2−l

∑

α

0≤l2 ≤ j−10

∑

x

t

∑

+ 2−l  2−l

x

t

0≤l2 ≤ j−10

∑

x

α

t,x

k

    Pξ (t),m u 2 j 2 Pξ (t),≤l −5 u3 6 2 L U (G ×R ) Δ

l2 −5≤m≤ j

j 2 t,x (Gk ×R )

k

   2   v0  2 Pξ (t),≥ j u 2 j 2 Pξ (t),≤l −5 u3 6 2 L L U (G ×R ) Δ

j 2 t,x (Gk ×R )

k

.

(5.190) Now by Young’s inequality and Definition 5.18,     l2  2 v 2−l 2 0 L2 ∑ ∑ Pξ (t),m u 0≤l2 ≤ j−10

2

2

∑

l2 −5≤m≤ j

 2  2 v0 L2 uX ([0,T ]×R2 )

0≤l2 ≤ j−10



2

j

 2 v0 L2

−l2 

2

j UΔ2 Gk ×R2

j−l 

and −l





∑

0≤m≤l2 −5

1/3  2m/3 Pξ (t),m u 2

j

UΔ (Gk ×R2 )

1/3    × sup Pξ (t),m uU 2 (Gm ×R2 ) Pξ (t),m uL∞ L2 j

Gm α ⊂Gk

Δ

α

t

6

x

 2  6  2 j−l v0 L2 uX ([0,T ]×R2 ) . j

Therefore, by the Cauchy–Schwarz inequality and the Sobolev embedding theorem,  2  6 (5.190)  2 j−l v0 L2 uX ([0,T ]×R2 ) . j It only remains to calculate

∑

0≤l2 ≤ j−10

l2 −2l

2



  j Gk

   (x − y) Im N2 Pξ (t),≤l2 u (t, y) |x − y|   × Im v¯ (∇ − iξ (t)) v (t, x) dx dy dt.

5.4 The Two-Dimensional Mass-Critical Problem Now by (5.172),

225

   Im N2 Pξ (t),≤l2 −10 u

is supported on |ξ | ∼ 2l2 . Therefore, again as in (5.183), 

 

   (x − y) Im N2 Pξ (t),≤l2 −10 u (t, y) |x − y| 0≤l2 ≤ j−10   × Im v¯ (∇ − iξ (t)) v (t, x) dx dy dt    1 |N2 (t, y) | |Pξ (t),≤l2 −10 u (t, y) |  ∑ 2−2l G j |x − y| 0≤l ≤ j−10 k

∑

2l2 −2l

j Gk

2

× |v (t, x) | | (∇ − iξ (t)) v (t, x) | dx dy dt, which, by Hardy’s inequality and (5.172),     2   −l 2 ∑ v0 L2 j  ∇Pξ (t),≤l2 −10 u (t) L2 Pξ (t),l2 −5≤·≤l2 +5 u (t) L2 dt Gk

0≤l2 ≤ j−10

 2  2 j−l v0 L2 . Also by Bernstein’s inequality,

∑

2l2 −2l

0≤l2 ≤ j−10

 2−l



 

j Gk

∑

0≤l2 ≤ j−10

 2  2 j−l v0 L2 .

   (x − y) Im N2 Pξ (t),l2 −10≤·≤l2 u (t, y) |x − y|

× Im[v¯ (∇ − iξ (t)) v] (t, x) dx dy dt   2   v0  2  ∇Pξ (t),≤l2 −10 u (t) L2 L j Gk

  × Pξ (t),l2 −5≤·≤l2 +5 u (t) L2 dt

Therefore, by the Cauchy–Schwarz inequality,

∑

0≤l2 ≤ j−10

 2 (5.181)  2 j−l v0 L2 .

This completes the proof of Theorem 5.40. Turning now to estimating (5.163), choose a constant C0 sufficiently large −1/2 and n (t) ∈ Z≥0 so that if i ≤ n (t) +C0 , then 2i ≤ 64N (t) η3 , 2n(t) ∼ N (t), −1/2

and C0 ∼ η3 . Then decompose:   2    Pξ (t),≤ j−10 u = ∑ Pξ (t),l2 u · Pξ (t),≤l2 u 0≤l2 ≤ j−10

=

∑

n(t)+C0 k uλ   T T   UΔ2 , 1 ×R2 λ2 λ2      T1        (5.209)  P>k uλ λ 2  2 2 + ε P>k−5 uλ UΔ2 T2 , T12 ×R2 . λ λ Lx (R ) -

Rescaling back, since UΔ2 and L2 are scale-invariant norms and TT1 N (t)3 dt < K, if 2k ≥ η3−1 K,       P>k u 2  P>k uλ (T1 ) L2 (R2 ) + ε P>k−5 uλ U 2 ([T,T ]×R2 ) . UΔ ([T,T1 ]×R2 ) 1 x Δ (5.210) Since N (t) → 0 and ξ (t) → 0 as t → ∞ then by (4.80),   lim P>k u (T1 ) L2 (R) = 0. (5.211) T1 →+∞

x

Since the ε > 0 in (5.209) may be made to be arbitrarily small, in particular so that Cε ≤ 2−5 , where C is the implicit constant in (5.210), then for any t ∈ [T, ∞), (5.210) and (5.211) imply that for 2k ≥ η3−1 K,   P>k u (t)  2 d  2−k K. Lx (R ) This combined with conservation of mass proves Theorem 5.47. For general ξ+ ∈ R2 , Theorem 5.47 and (4.144) imply  −ix·ξ (t)  e uL∞ H˙ 1/2 ([T,∞)×R2 )  K 1/2 . t

5.4 The Two-Dimensional Mass-Critical Problem

235

Now by the dominated convergence theorem,  ∞

lim

T →∞ T

so

N (t)3 dt = 0,

  lim e−ix·ξ (t) u (t) H˙ 1/2 (R2 ) = 0.

t→+∞

The same analysis can be done as t → −∞. Therefore, by (4.18), (4.31), (4.32), and the Sobolev embedding theorem, 2      4 = 0. u L4 L8 (R×R2 )   |∇|1/2 |u|2  2 t x Lt,x (R×R2 ) Therefore, u ≡ 0.

References

Berestycki, H. and P. L. Lions (1979). Existence d’ondes solitaires dans des probl`emes nonlin´eaires du type Klein–Gordon. C. R. Acad. Sci. Paris S´er. A–B, 288(7), A395–A398. Bergh, J. and J. L¨ofstrom (1976). Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, Berlin-New York: Springer. Bourgain, J. (1998). Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices, 1998(5), 253–283. Bourgain, J. (1999). Global well-posedness of defocusing critical nonlinear Schr¨odinger equation in the radial case. J. Amer. Math. Soc., 12(1), 145–171. Cazenave, T. and F. B. Weissler (1988). The Cauchy problem for the nonlinear Schr¨odinger equation in H 1 . F.B. Manuscripta Math., 61, 477–494. Christ, M., J. Colliander, and T. Tao (2003). Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math., 125(6), 1235–1293. Christ, M., J. Colliander, and T. Tao (2008). A priori bounds and weak solutions for the nonlinear Schr¨odinger equation in Sobolev spaces of negative order. J. Funct. Anal., 254(2), 368–395. Christ, M. and A. Kiselev (2001). Maximal functions associated to filtrations. J. Funct. Anal., 179(2), 409–425. Colliander, J., M. Grillakis, and N. Tzirakis (2007). Improved interaction Morawetz inequalities for the cubic nonlinear Schr¨odinger equation on R2 . Internat. Math. Res. Notices, 23(1), 90–119. Colliander, J., M. Grillakis, and N. Tzirakis (2009). Tensor products and correlation estimates with applications to nonlinear Schr¨odinger equations. Comm. Pure Appl. Math., 62(7), 920–968. Colliander, J., M. Keel, G. Staffilani, H. Takaoka, and T. Tao (2004). Global existence and scattering for rough solutions of a nonlinear Schr¨odinger equation on R3 . Comm. Pure Appl. Math., 21, 987–1014. Colliander, J., M. Keel, G. Staffilani, H. Takaoka, and T. Tao (2008). Global wellposedness and scattering for the energy-critical nonlinear Schr¨odinger equation on R3 . Ann. of Math. (2), 167, 767–865.

236

References

237

Colliander, J. and T. Roy (2011). Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on R2 . Commun. Pure Appl. Anal., 10(2), 397–414. Conway, J. B. (1990). A Course in Functional Analysis, 2nd edn, Graduate Texts in Mathematics, vol. 96, New York: Springer. Dodson, B. (2011). Improved almost Morawetz estimates for the cubic nonlinear Schr¨odinger equation. Commun. Pure Appl. Anal., 10(1), 127–140. Dodson, B. (2012). Global well-posedness and scattering for the defocusing L2 -critical nonlinear Schr¨odinger equation when d ≥ 3. J. Amer. Math. Soc., 25(2), 429–463. Dodson, B. (2015). Global well-posedness and scattering for the mass critical nonlinear Schr¨odinger equation with mass below the mass of the ground state. Adv. Math., 285, 1589–1618. Dodson, B. (2016a). Global well-posedness and scattering for the defocusing L2 -critical nonlinear Schr¨odinger equation when d = 1. Amer. J. Math., 138(2), 531–569. Dodson, B. (2016b). Global well-posedness and scattering for the defocusing L2 -critical nonlinear Schr¨odinger equation when d = 2. Duke Math. J., 165(18), 3435–3516. Duyckaerts, T., J. Holmer, and S. Roudenko (2008). Scattering for the non-radial 3D cubic nonlinear Schr¨odinger equation. Math. Res. Lett., 15(5–6), 1233–1250. Ginibre, J. and G. Velo (1992). Smoothing properties and retarded estimates for some dispersive evolution equations. Comm. Math. Phys., 144(1), 163–188. Grafakos, L. (2004). Classical Fourier Analysis, 3rd edn, Graduate Texts in Mathematics, vol. 249, New York: Springer. Grillakis, M. (2000). On nonlinear Schr¨odinger equations. Comm. Partial Differential Equations 25(9–10), 1827–1844. Guevara, C. (2014). Global behavior of finite energy solutions to the focusing nonlinear Schr¨odinger equation in d dimensions. Appl. Math. Res. Express, 2014, 177–243. Hadac, M., S. Herr, and H. Koch (2009). Well-posedness and scattering for the KPII equation in a critical space. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 26(3), 917–941. Holmer, J., R. Platte, and S. Roudenko (2010). Blow-up criteria for the 3D cubic nonlinear Schr¨odinger equation. Nonlinearity, 23(4), 977–1030. Holmer, J. and S. Roudenko (2008). A sharp condition for scattering of the radial 3D cubic nonlinear Schr¨odinger equation. Comm. Math. Phys., 282(2), 435–467. Keel, M. and T. Tao (1998). Endpoint Strichartz estimates. Amer. J. Math., 120(4–6), 945–957. Kenig, C. and F. Merle (2006). Global well-posedness, scattering, and blow-up for the energy-critical, focusing nonlinear Schr¨odinger equation in the radial case. Invent. Math., 166(3), 645–675. Kenig, C. and F. Merle (2010). Scattering for H˙ 1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions. Trans. Amer. Math. Soc., 362(4), 1937–1962. Keraani, S. (2001). On the defect of compactness for the Strichartz estimates of the Schr¨odinger equations. J. Differential Equations, 175(2), 353–392. Killip, R., T. Tao, and M. Visan (2009). The cubic nonlinear Schr¨odinger equation in two dimensions with radial data. J. Eur. Math. Soc. (JEMS), 11(6), 1203–1258. Killip, R. and M. Visan (2010). The focusing energy-critical nonlinear Schr¨odinger equation in dimensions five and higher. Amer. J. Math., 132(2), 361–424.

238

References

Killip, R. and M. Visan (2012). Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. Anal. PDE, 5(4), 855–885. Killip, R. and M. Visan (2013). Nonlinear Schr¨odinger equations at critical regularity. In Clay Math. Proceedings, vol. 17, Providence, RI: American Mathematical Society, pp. 325–437. Killip, R., M. Visan, and X. Zhang (2008). The mass-critical nonlinear Schr¨odinger equation with radial data in dimensions three and higher. Anal. PDE, 1(2), 229– 266. Koch, H. and D. Tataru (2007). A priori bounds for the 1D cubic NLS in negative Sobolev spaces. Internat. Math. Res. Notices, 16, Art. ID rnm053, 36 pp. Koch, H. and D. Tataru (2012). Energy and local energy bounds for the 1-D cubic NLS equation in H −1/4 . Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 29(6), 955–988. Lieb, E. and M. Loss (2001). Analysis, 2nd edn, Graduate Studies in Mathematics, vol. 14, Providence, RI: American Mathematical Society. Lin, J. and W. Strauss (1978). Decay and scattering of solutions of a nonlinear Schr¨odinger equation. J. Funct. Anal., 30(2), 245–263. Montgomery-Smith, J. S., (1998). Time decay for the bounded mean oscillation of solutions of the Schr¨odinger and wave equations. Duke Math. J., 91(2), 393–408. Murphy, J. (2014). Inter-critical NLS: critical H˙ s bounds imply scattering. SIAM J. Math. Anal., 46(1), 939–997. Murphy, J. (2015). The radial, defocusing, nonlinear Schr¨odinger equation in three dimensions. Comm. Partial Differential Equations, 40(2), 265–308. Muscalu, C. and W. Schlag (2013). Classical and Multilinear Harmonic Analysis, Cambridge Studies in Advanced Mathematics, vol. 137, Cambridge: Cambridge University Press. Planchon, F. and L. Vega (2009). Bilinear virial identities and applications. Ann. Sci. ´ Ecole Norm. Sup. (4), 42(2), 261–290. Ryckman, E. and M. Visan (2007). Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr¨odinger equation in R1+4 . Amer. J. Math., 129(1), 1–60. Shao, S. (2009). A note on the restriction conjecture in the cylindrically symmetric case. Proc. Amer. Math. Soc., 137(1), 135–143. Smith, H. F. and C. D. Sogge (2000). Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Partial Differential Equations, 25(11–12), 2171–2183. Sogge, C. D. (1993). Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge: Cambridge University Press. Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions, Princeton, NJ: Princeton University Press. Stein, E. M. (1993). Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton, NJ: Princeton University Press. Strichartz, R. S. (1977). Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J., 44(3), 705–714. Sulem, C. and P. L. Sulem (1999). The Nonlinear Schr¨odinger Equation, Applied Mathematical Sciences, vol. 139, New York: Springer.

References

239

Tao, T. (2000). Spherically averaged endpoint Strichartz estimates for the twodimensional Schr¨odinger equation. Comm. Partial Differential Equations, 25(7– 8), 1471–1485. Tao, T. (2003). A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal., 13(6), 1359–1384. Tao, T. (2005). Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schr¨odinger equation for radial data. New York J. Math., 11, 57–80. Tao, T. (2006). Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, vol. 104, Washington, DC: Conference Board of the Mathematical Sciences. Tao, T. and M. Visan (2005). Stability of energy-critical nonlinear Schr¨odinger equations in high dimensions. Electron. J. Differential Equations, 118, 28 pp. Tao, T., M. Visan, and X. Zhang (2007a). The nonlinear Schr¨odinger equation with combined power-type nonlinearities. Comm. Partial Differential Equations, 32(7– 9), 1281–1343. Tao, T., M. Visan, and X. Zhang (2007b). Global well-posedness and scattering for the defocusing mass-critical nonlinear Schr¨odinger equation for radial data in high dimensions. Duke Math. J., 140(1), 165–202. Tao, T., M. Visan, and X. Zhang (2008). Minimal-mass blowup solutions of the masscritical NLS. Forum Math., 20(5), 881–919. Tataru, D. (1998). Local and global results for wave maps I. Comm. Partial Differential Equations, 23(9–10), 1781–1793. Taylor, M. E. (2000). Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, vol. 81, Providence, RI: American Mathematical Society. Taylor, M. E. (2006). Measure Theory and Integration, Graduate Studies in Mathematics, vol. 6, Providence, RI: American Mathematical Society. Taylor, M. E. (2011). Partial Differential Equations I–III, 2nd edn, Applied Mathematical Sciences, vol. 115, New York: Springer. Visan, M. (2006). The defocusing energy-critical nonlinear Schr¨odinger equation in dimensions five and higher. Ph.D. thesis, UCLA. Visan, M. (2007). The defocusing energy-critical nonlinear Schr¨odinger equation in higher dimensions. Duke Math. J., 138, 281–374. Visan, M. (2012). Global well-posedness and scattering for the defocusing cubic nonlinear Schr¨odinger equation in four dimensions. Internat. Math. Res. Notices 2, 5, 1037–1067. Weinstein, M. (1982). Nonlinear Schr¨odinger equations and sharp interpolation estimates. Comm. Math. Phys., 87(4), 567–576. Weinstein, M. (1989). The nonlinear Schr¨odinger equation–singularity formation, stability and dispersion. In The Connection between Infinite-Dimensional and Finite-Dimensional Dynamical Systems (Boulder, CO, 1987), Contemporary Mathematics, vol. 99, Providence, RI: American Mathematical Society, pp. 213– 232. Yajima, K. (1987). Existence of solutions for Schr¨odinger evolution equations. Comm. Math. Phys., 110(3), 415–426.

240

References

Yosida, K. (1980). Functional Analysis, 6th edn, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Berlin: Springer.

Index

N space, 23 S space, 22 U p spaces, 191 UΔp spaces, 192 VΔp spaces, 197 admissible pair, 12 almost-periodic solution energy-critical, 85 mass-critical, 122 asymptotic orthogonality, 77 bilinear Strichartz estimate, 56 first proof, 106 second proof, 107 conservation of energy, 36 conservation of mass, 34 conservation of momentum, 35 core, 77 critical Sobolev space, 41 dispersive estimate, 5 double Duhamel lemma, 93 dyadic cubes, 111 dyadic interval, 18 first bilinear estimate in two dimensions, 217 Fourier inversion for Schwartz functions, 3 Fourier transform, 1 fractional product rule, 42 frequency-localized interaction Morawetz estimate, 166 Galilean invariant bilinear estimate, 104 Galilean Littlewood–Paley projection, 147 Galilean norm, 192 Galilean transformation, 9 interaction Morawetz estimate, 96 inverse Fourier transform, 2

linear Schr¨odinger equation, 1 Littlewood–Paley decomposition, 6 Littlewood–Paley kernel, 8 Littlewood–Paley theorem, 7 local conservation, 34 local well-posedness, 70 long-time perturbations, 71 long-time Strichartz estimate, 148 long-time Strichartz seminorm, 148 long-time Strichartz spaces, 193 mass bracket, 34 mass density, 34 momentum bracket, 34 momentum density, 34 Morawetz estimate, 49 Parseval identity, 4 perturbation lemma, 43, 67 perturbation lemma for the mass-critical problem, 26 Plancherel identity, 4 profile decomposition, energy-critical, 77 pseudoconformal conservation law, 37 pseudoconformal transformation, 39 Radon transform, 107 rapid cascade solution, 127 scale, 77 scaling, 10, 41 scattering, 25 scattering size function, energy-critical, 73 scenarios, energy-critical, 86 scenarios, mass-critical, 126 Schwartz space, 2 second bilinear estimate in two dimensions, 223 self-similar solution, 126

241

242 small data scattering for the energy-critical problem, 64 small intervals, 148 soliton-like solution, 127

Index Strichartz estimates, 14 Strichartz space, 22 well-posedness, 24 Whitney decomposition, 18