LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I Edited by T. Ozawa and Y. Tsutsumi Sapporo, 2004 Partially supported by JSPS • Grant-in-Aid for formation of COE “Mathematics of Nonlinear Structure via Singularities” • Grant-in-Aid for Scientific Research S(2) #16104002 (T. Ozawa) A(2) #15204008 (Y. Tsutsumi) PREFACE This volume, together with the next, is intended as the proceedings of expository lectures in Special Months “Nonlinear Dispersive Equations. ” 　Nonlinear dispersive equations, such as nonlinear Schr¨odinger equations, KdV equation, and Benjamin-Ono equation, are of mathematical and phys- ical importance. Expository courses in August 2004 are intended to cover a broad spectrum of the issues, from mathematical and physical backgrounds to the latest developments. 　We wish to express our sincere thanks to - J. Bona H. Koch, F. Planchon, P. Rapha¨el, and N. Tzvetkov for excellent lectures. - M. Ikawa and A. Ogino for effitient arrangements. T. Ozawa and Y. Tsutsumi CONTENTS Program J. Bona 　Derivationandsomefundamentalpropertiesofnonlineardispersivewaves equations F. Planchon 　Schr¨odinger equations with variable coefficients P. Rapha¨el 　On the blow up phenomenon for the L2 critical non linear Schr¨odinger Equation COE Program Special Months Lectures On Nonlinear Dispersive EquationsI Organizers: T. Ozawa (Hokkaido Univ.) Y. Tsutsumi (Kyoto Univ.) Period : 　August 23 - 27, 2004 Venue : 　Department of Mathematics, Hokkaido University 　Science Building #8 Room 309 Program : 10:30–12:00 13:30–15:00 15:30–17:00 Aug. 23(Mon) Bona(1) Bona(2) Planchon(1) 24(Tue) Bona(3) Bona(4) Planchon(2) 25(Wed) Bona(5) Rapha¨el(1) Planchon(3) 26(Thu) Rapha¨el(2) Rapha¨el(3) Planchon(4) 27(Fri) Rapha¨el(4) Rapha¨el(5) Planchon(5) • J. Bona (University of Illinois at Chicago) Derivationandsomefundamentalpropertiesofnonlineardispersivewavesequations • F. Planchon (Universit´e de Paris-Nord) Schr¨odinger equations with variable coefficients • P. Rapha¨el（Universit´e de Cergy-Pontoise） On the blow up phenomenon for the L2 critical non linear Schr¨odinger Equation −−COEProgramSpecialMonthsHomepage−− http://coe.math.sci.hokudai.ac.jp/news/special/index02.html.en Secretariat: Atsuko Ogino TEL: 011-706-4671 FAX: 011-706-4672 Derivation and some fundamental properties of nonlinear dispersive waves equations. Jerry Bona (University of Illinois at Chicago) Abstract This series of lectures aims to introduce some of the principal aspects of nonlinear dispersive wave theory. We start with an appreciation of the early history, and then introduce, within the original fluid mechanics context, the paradigm Korteweg-de Vries equation. Some of the more important properties of this equation are then outlined. These properties motivate and give direction to the further study of this and other nonlinear dispersive wave equations. Further issues to be addressed will be chosen from among the following topics. 　1. Existence theory for solitary waves 　2. Stability and instability of solitary waves 　3. Singularity formation 　4. Initial-value and initial-boundary-value problems 　5. Incorporation of damping into nonlinear dispersive wave equations 　6. Application of the theory to problems in mechanics Lectures notes on Schrödinger equations with variable coefficients COE Program Special Months : Nonlinear Dispersive Equations, Hokkaïdo University, Sapporo Fabrice Planchon Département de Mathématiques, Institut Galilée Université Paris 13, 93430 Villetaneuse, France August 2004 2 Introduction The aim of this series of lectures is to give anoverview of dispersive estimates fortheSchrödinger equation. These estimates areakey toolforvariousprob- lems, both linear and non-linear, and we will give examples along the way. We focus on the case where the domain Ω is the whole space Rn: other situa- tions (torus Tn, bounded domain with Dirichlet conditions) are significantly more intricate and the subject of active research; a good knowledge of the Rn case is anyway a prerequisite. As the title of the notes suggests, we would like to deal with variable coeffi- cients: namely, what happens if we replace the standard Laplacian by, say, a Laplace-Beltrami operator associated to a metric g ? As we will see in the ij first lecture, dispersive estimates for the flat case are obtained through har- monic analysis methods; these in turn rely heavily on the Fourier transform and admit no easy generalization to curved space. We will deal with the admittedly easiest case, namely n = 1: while some of the techniques which we will use are somewhat 1D specific, the problems one might encounter in the general case are already present. Moreover, we present much sharper results than those available at present for n ≥ 2. Finally, we will deal with an application of these results to the Benjamin-Ono family of equations. There exists a huge literature on the subject of dispersive equations. We have tried to give as many references as possible, but being exhaustive is an impossible task, so these references represent a snapshot of the author’s current knowledge rather than an accurate picture or historical account. We have tried to make the notes as self-contained as possible, assuming basic knowledge of functional analysis, distributions and Fourier analysis. There will be, however, blackboxes which won’t be detailed: interpolation theory, for which we refer to [7], [6] or [63] which already contains most of what we need. Another blackbox which we will only half-open is the zoo of functional spaces: we will merely use Besov spaces and refer to [72] for an exhaustive reference, [52] for building up intuition or [7] if one just needs a quick summary. For those who have an interest in harmonic analysis by itself, [63] and its companion [65] are classic if not up to date with current 3 trends. Recent books like [64] or [32] arecloser to a modern days perspective. Comments and suggestions are welcome, [email protected]. 4 Contents 1 Dispersive estimates for the flat Schrödinger equation 7 1.1 Dispersion and Strichartz estimates . . . . . . . . . . . . . . . 8 1.2 Strichartz and its connection with the restriction problem in harmonic analysis . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3 Smoothing estimates . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Maximal function estimates . . . . . . . . . . . . . . . . . . . 23 2 Smoothing for the 1D variable coefficients Schrödinger equa- tion 27 2.1 Functional spaces . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.1 Heuristic and Sobolev spaces . . . . . . . . . . . . . . . 30 2.1.2 Littlewood-Paley analysis . . . . . . . . . . . . . . . . 31 2.1.3 Sobolev’s embeddings . . . . . . . . . . . . . . . . . . . 34 2.2 Local smoothing . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.1 Reduction to smooth a . . . . . . . . . . . . . . . . . . 38 2.2.2 A resolvent estimate . . . . . . . . . . . . . . . . . . . 39 3 Strichartz and maximal function estimates 47 3.1 The estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Paraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 A singular metric: counterexamples 59 4.1 Construction of the metric . . . . . . . . . . . . . . . . . . . . 60 4.2 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . 62 5 A generalized Benjamin-Ono equation 65 5.1 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . 66 A Localization with respect to ∂ versus localization with re- x spect to (−∂x(a(x)∂x))12 75 A.1 The heat flow associated with −∂ (a(x)∂ . . . . . . . . . . . 75 x x 5