This book is the outcome of several courses and seminar talks held at the Instituto de Matematica Pura e Aplicada (IMPA) over the years.It is a greatly modified version of a previous work by the authors,Equacoes Diferenciais Parciais, Uma lntroducao, (Projeto Euclides, IMPA,1978). It has a twofold purpose, namely to introduce the student to the basic concepts of Fourier analysis and provide illustrations of recent applications where these concepts were used to study various properties of the solutions of some important nonlinear evolution equations.

Preface

Part One: Fourier Series and Periodic Distributions

1 Preliminaries

1.1 Basic Definitions and Examples

1.2 Classification into Types

1.3 Boundary and/or Initial Value Problems

1.4 Separation of Variables: Heat Flow in a Bar

1.5 The Schrodinger and the Wave Equations

1.6 The Dirichlet Problem in the Unit Circle

1.7 Maximum Principles and Uniqueness

2 Fourier Series: Basic Theory

2.1 Spaces of Periodic Functions and Sequences

2.2 The Fourier Transform

2.3 Geometric Interpretation

2.4 Decay and Differentiability

2.5 The Inversion Formula: Pointwise Convergence

2.6 Periodic Heat Flow

2.7 Approximate Identities and Summability

2.8 Cesaro Summability

3 Periodic Distributions and Sobolev Spaces

3.1 C Peiodic Funcitons

3.2 Perodic Distributions

3.3 Topological Remakes

3.4 Fourier Series in

3.5 The Convolution in

3.6 Sobolev Spaces

Part Two:Applications to Partial Dfferential Equations

4 Linear Equations

5 Nonlinear Evolution Equations

6 The Korteweg-de Vries Equation

7 Distributions,Fourier Transform and Linear Equations

8 KdV,BO and Friends

Appendix A Tools from the Theory of ODEs

Appendix B Commutator Estimates

Bibliography

Index