**CBMS Regional Conference Series in Mathematics**

Volume: 106;
2006;
373 pp;
Softcover

MSC: Primary 35;

Print ISBN: 978-0-8218-4143-3

Product Code: CBMS/106

List Price: $57.00

Individual Price: $45.60

**Electronic ISBN: 978-1-4704-2466-4
Product Code: CBMS/106.E**

List Price: $57.00

Individual Price: $45.60

#### Supplemental Materials

# Nonlinear Dispersive Equations: Local and Global Analysis

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*Terence Tao*

A co-publication of the AMS and CBMS

Among nonlinear PDEs, dispersive and wave equations form an important class of equations. These include the nonlinear Schrödinger equation, the nonlinear wave equation, the Korteweg de Vries equation, and the wave maps equation. This book is an introduction to the methods and results used in the modern analysis (both locally and globally in time) of the Cauchy problem for such equations.

Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.

As the subject is vast, the book does not attempt to give a comprehensive survey of the field, but instead concentrates on a representative sample of results for a selected set of equations, ranging from the fundamental local and global existence theorems to very recent results, particularly focusing on the recent progress in understanding the evolution of energy-critical dispersive equations from large data. The book is suitable for a graduate course on nonlinear PDE.

#### Table of Contents

# Table of Contents

## Nonlinear Dispersive Equations: Local and Global Analysis

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface ix10 free
- Chapter 1. Ordinary differential equations 118 free
- Chapter 2. Constant coefficient linear dispersive equations 5572
- Chapter 3. Semilinear dispersive equations 109126
- 3.1. On scaling and other symmetries 114131
- 3.2. What is a solution? 120137
- 3.3. Local existence theory 129146
- 3.4. Conservation laws and global existence 144161
- 3.5. Decay estimates 153170
- 3.6. Scattering theory 162179
- 3.7. Stability theory 171188
- 3.8. Illposedness results 180197
- 3.9. Almost conservation laws 186203

- Chapter 4. The Korteweg de Vries equation 197214
- Chapter 5. Energy-critical semilinear dispersive equations 231248
- Chapter 6. Wave maps 277294
- Chapter A. Appendix: tools from harmonic analysis 329346
- Chapter B. Appendix: construction of ground states 347364
- Bibliography 363380
- Back Cover Back Cover1394

#### Readership

Graduate students and research mathematicians interested in nonlinear partial differential equations.

#### Reviews

Tao certainly succeeds in writing a vivid and instructional text on nonlinear dispersive partial differential equations. It touches on topics of recent research interest and is a valuable source both for the beginning graduate student and, to some extent, for the advanced researcher.

-- Mathematical Reviews

The work is well suited for a graduate level course on nonlinear PDE, and it is to be thoroughly recommended.

-- Alan Jeffrey for Zentralblatt MATH