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© 2014

Dispersive Equations and Nonlinear Waves

Generalized Korteweg–de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps

  • Exposition of central ideas in dispersive equations

  • Basic techniques and function spaces

  • Coherent introduction to induction on energy, minimal blow up solutions and interaction Morawetz estimates

  • Introduction to gauge transform, choice of functions spaces, and control of interacting waves

Textbook

Part of the Oberwolfach Seminars book series (OWS, volume 45)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Nonlinear Dispersive Equations

    1. Front Matter
      Pages 1-1
    2. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 3-3
    3. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 5-22
    4. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 23-39
    5. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 41-71
    6. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 73-85
    7. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 87-109
    8. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 111-122
    9. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 123-126
    10. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 127-134
    11. Back Matter
      Pages 135-137
  3. Geometric Dispersive Evolutions

    1. Front Matter
      Pages 139-139
    2. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 141-142
    3. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 143-150
    4. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 151-160
    5. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 161-199
    6. Herbert Koch, Daniel Tataru, Monica Vişan
      Pages 201-218
    7. Back Matter
      Pages 219-222
  4. Dispersive Equations

    1. Front Matter
      Pages 223-224

About this book

Introduction

The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ideas and should provide graduate students with a stepping stone to this exciting direction of research.

Keywords

Fourier transform dispersion wave interaction wave propagation

Authors and affiliations

  1. 1.Institute of MathematicsUniversity of BonnBonnGermany
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

About the authors

Herbert Koch has been a professor at the University of Bonn, Germany since 2006, Daniel Tataru at the University of California in Berkeley, USA, since 2001 and Monica Vişan is an associate professor at UCLA, USA.

Bibliographic information

  • Book Title Dispersive Equations and Nonlinear Waves
  • Book Subtitle Generalized Korteweg–de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps
  • Authors Herbert Koch
    Daniel Tataru
    Monica Vişan
  • Series Title Oberwolfach Seminars
  • Series Abbreviated Title Oberwolfach Seminars(formerly:Sem.Dt.Math.Ver.)(Birkh.)
  • DOI https://doi.org/10.1007/978-3-0348-0736-4
  • Copyright Information Springer Basel 2014
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-0348-0735-7
  • eBook ISBN 978-3-0348-0736-4
  • Series ISSN 1661-237X
  • Series E-ISSN 2296-5041
  • Edition Number 1
  • Number of Pages XII, 312
  • Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
  • Topics Partial Differential Equations
  • Buy this book on publisher's site
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