Introduction to Nonlinear Dispersive Equations

by Felipe Linares, Gustavo Ponce

Overview

The aim of this textbook is to introduce the theory of nonlinear dispersive equations to graduate students in a constructive way. The first three chapters are dedicated to preliminary material, such as Fourier transform, interpolation theory and Sobolev spaces. The authors then proceed to use the linear Schröet;dinger equation to describe properties enjoyed by general dispersive equations. This information is then used to treat local and global well-posedness for the semi-linear Schröet;dinger equations. The end of each chapter contains recent developments and open problems, as well as exercises.

Product Details

ISBN-13:	9781493921812
Publisher:	Springer-Verlag New York, LLC
Publication date:	12/15/2014
Series:	Universitext
Sold by:	Barnes & Noble
Format:	eBook
File size:	10 MB

About the Author

Felipe Linares is a Researcher at the Instituto Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro, Brazil.

Gustavo Ponce is a Professor of Mathematics at the University of California in Santa Barbara.

Table of Contents

1 The Fourier Transform 1

1.1 The Fourier Transform in L1 (Rn) 1

1.2 The Fourier Transform in L2 (Rn) 6

1.3 Tempered Distributions 8

1.4 Oscillatory Integrals in One Dimension 13

1.5 Applications 17

1.6 Exercises 18

2 Interpolation of Operators. A Multiplier Theorem 25

2.1 The Riesz-Thorin Convexity Theorem 25

2.1.1 Applications 28

2.2 Marcinkiewicz Interpolation Theorem 29

2.2.1 Applications 33

2.3 The Stein Interpolation Theorem 37

2.4 A Multiplier Theorem 38

2.5 Exercises 39

3 Sobolev Spaces and Pseudo-Differential Operators 45

3.1 Basics 45

3.2 Pseudodifferential Operators 51

3.3 The Bicharacteristic Flow 53

3.4 Exercises 55

4 The Linear Schröet;dinger Equation 59

4.1 Basic Results 59

4.2 Global Smoothing Effects 64

4.3 Local Smoothing Effects 67

4.4 Comments 72

4.5 Exercises 84

5 The Nonlinear Schröet;dinger Equation. Local Theory 89

5.1 L2 Theory 91

5.2 H1 Theory 99

5.3 H2 Theory 103

5.4 Comments 106

5.5 Exercises 116

6 Asymptotic Behavior for NLS Equation 119

6.1 Global Results 119

6.2 Formation of Singularities 124

6.2.1 Case α ϵ (1+4/n, 1+4/(n-2)) 126

6.2.2 Case α=1+4/n 128

6.3 Comments 133

6.4 Exercises 137

7 Korteweg-de Vries Equation 139

7.1 Linear Properties 141

7.2 Modified Korteweg-de Vries Equation 146

7.3 Generalized Korteweg-de Vries Equation 149

7.4 Korteweg-de Vries Equation 155

7.5 Comments 166

7.6 Exercises 170

8 Asymptotic Behavior for k-gKdV Equations 173

8.1 Cases k=1,2,3 174

8.2 Case k=4 180

8.3 Comments 185

8.4 Exercises 188

9 Other Nonlinear Dispersive Models 191

9.1 Davey-Stewartson Systems 191

9.2 Ishimori Equation 193

9.3KP Equations 194

9.4 BO Equation 196

9.5 Zakharov System 200

9.6 Higher Order KdV Equations 203

9.7 Exercises 208

10 General Quasilinear Schröet;dinger Equation 211

10.1 The General Quasilinear Schröet;dinger Equation 211

10.2 Comments 230

10.3 Exercises 231

A Appendix 233

References 239

Index 255

Editorial Reviews

This is the second edition of a self-contained graduate level introduction to the results and methods in the well-posedness theory for initial-value problems of nonlinear dispersive equations with special focus on the nonlinear Schrödinger and Korteweg de Vries equations. … I strongly welcome this updated version and I can only recommend it warmly to anybody (both students and teachers) interested in this central area of analysis.” (G. Teschl, Monatshefte für Mathematik, Vol. 180, 2016)

From the Publisher