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    Mathematical Aspects of Nonlinear Dispersive Equations
    Mathematical Aspects of Nonlinear Dispersive Equations
    Mathematical Aspects of Nonlinear Dispersive Equations
    Ebook316 pages30 minutes

    Mathematical Aspects of Nonlinear Dispersive Equations

    By Jean Bourgain and Sergiu Klainerman

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    About this ebook

    This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers.


    The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrödinger operators, nonlinear Schrödinger and wave equations, and the Euler and Navier-Stokes equations.

    LanguageEnglish
    PublisherPrinceton University Press
    Release dateJan 10, 2009
    ISBN9781400827794
    Mathematical Aspects of Nonlinear Dispersive Equations

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      Book preview

      Mathematical Aspects of Nonlinear Dispersive Equations - Jean Bourgain

      Annals of Mathematics Studies Number 163

      Jean Bourgain, Carlos E. Kenig, and S. Klainerman, Editors

      PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2007

      Copyright (c) 2007 by Princeton University Press

      Published by Princeton University Press, 41William Street, Princeton, New Jersey 08540

      In the United Kingdom: Princeton University Press, 3 Market Place,Woodstock,

      Oxfordshire OX20 1SY

      All Rights Reserved

      Library of Congress Cataloging-in-Publication Data

      Mathematical aspects of nonlinear dispersive equations / Jean Bourgain, Carlos E. Kenig, and S. Klainerman, editors.

      p. cm. — (Annals of mathematics studies; no. 163)

      Includes bibliographical references and index

      eISBN: 978-1-40082-779-4

      1. Differential equations, Nonlinear–Congresses. 2. Nonlinear partial differentialoperators–Congresses. I. Bourgain, Jean, 1954– II. Kenig, Carlos E., 1953– III. Klainerman, Sergiu, 1950–

      QA372.M387 2007

      515'.355–dc22 2006050254

      British Library Cataloging-in-Publication Data is available

      This book has been composed in Times Roman in LATEX

      Printed on acid-free paper.∞

      press.princeton.edu

      Printed in the United States of America

      10 9 8 7 6 5 4 3 2 1

      Contents

      Preface

      Chapter 1. On Strichartz’s Inequalities and the Nonlinear Schrödinger Equation on Irrational Tori

      J. Bourgain

      Chapter 2. Diffusion Bound for a Nonlinear Schrödinger Equation

      J. Bourgain and W.-M.Wang

      Chapter 3. Instability of Finite Difference Schemes for Hyperbolic Conservation Laws

      A. Bressan, P. Baiti, and H. K. Jenssen

      Chapter 4. Nonlinear Elliptic Equations with Measures Revisited

      H. Brezis, M. Marcus, and A. C. Ponce

      Chapter 5. Global Solutions for the Nonlinear Schrödinger Equation on Three-Dimensional Compact Manifolds

      N. Burq, P. Gérard, and N. Tzvetkov

      Chapter 6. Power Series Solution of a Nonlinear Schrödinger Equation

      M. Christ

      Chapter 7. Eulerian-Lagrangian Formalism and Vortex Reconnection

      P. Constantin

      Chapter 8. Long Time Existence for Small Data Semilinear Klein-Gordon Equations on Spheres

      J.-M. Delort and J. Szeftel

      Chapter 9. Local and GlobalWellposedness of Periodic KP-I Equations

      A. D. Ionescu and C. E. Kenig

      Chapter 10. The Cauchy Problem for the Navier-Stokes Equations with Spatially Almost Periodic Initial Data

      Y. Giga, A. Mahalov, and B. Nicolaenko

      Chapter 11. Longtime Decay Estimates for the Schrödinger Equation on Manifolds

      I. Rodnianski and T. Tao

      Chapter 12. Dispersive Estimates for Schrödinger Operators: A Survey

      W. Schlag

      Contributors

      Preface

      This book contains the written accounts of a number of lectures given during the CMI/IAS Workshop on mathematical aspects of nonlinear PDEs in the spring of 2004 at the Institute for Advanced Study in Princeton, New Jersey. Several of them have an expository nature, describing the state of the art and research directions. Topics that are discussed in this volume are new developments on Schrödinger operators, non-linear Schrödinger and wave equations, hyperbolic conservation laws, and the Euler and Navier-Stokes equations. There has been intensive activity in recent years in each of these areas, leading in several cases to very significant progress and almost always broadening the subject.

      The workshop is the conclusion of a year-long program at IAS centered around the analysis of nonlinear PDEs and the emergence of new analytical techniques. That year is marked by at least two important breakthroughs. The first is in the understanding of the blowup mechanism for the critical focusing Schrödinger equation. The other is a proof of global existence and scattering for the 3D quintic equation for general smooth data. Both cases illustrate in a striking way the role of hard analysis in addition to the more geometric approach and the role of energy estimates. This point of view is also reflected through the material presented in this volume.

      The articles are written in varying styles. As mentioned, some are mainly expository and meant for a broader audience. They are not an overall survey but present more focused perspectives by a leader in the field. Others are more technical

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