Dispersive Equations and Nonlinear Waves

1. Front Matter

   Pages i-xii

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2. Nonlinear Dispersive Equations

   1. Front Matter

      Pages 1-1

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   2. Introduction

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 3-3

   3. Stationary phase and dispersive estimates

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 5-22

   4. Strichartz estimates and small data for the nonlinear Schrödinger equation

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 23-39

   5. Functions of bounded p-variation

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 41-71

   6. Convolution of measures on hypersurfaces, bilinear estimates, and local smoothing

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 73-85

   7. Well-posedness for nonlinear dispersive equations

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 87-109

   8. Appendix A: Young’s inequality and interpolation

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 111-122

   9. Appendix B: Bessel functions

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 123-126

   10. Appendic C: The Fourier transform

       • Herbert Koch, Daniel Tataru, Monica Vişan

       Pages 127-134

3. Back Matter

   Pages 135-137

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4. Geometric Dispersive Evolutions

   1. Front Matter

      Pages 139-139

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   2. Introduction

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 141-142

   3. Maps into manifolds

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 143-150

   4. Geometric pde’s

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 151-160

   5. Wave maps

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 161-199

   6. Schrödinger maps

      • Herbert Koch, Daniel Tataru, Monica Vişan

      Pages 201-218

5. Back Matter

   Pages 219-222

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6. Dispersive Equations

   1. Front Matter

      Pages 223-224

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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.​

• Fourier transform
• dispersion
• wave interaction
• wave propagation
• partial differential equations

• Institute of Mathematics, University of Bonn, Bonn, Germany

  Herbert Koch

• Department of Mathematics, University of California, Berkeley, USA

  Daniel Tataru

• Department of Mathematics, University of California, Los Angeles, USA

  Monica Vişan

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.

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