Chapter Ⅰ．Ordinary differential equations

1.1．Introduction

1.2．Local existence and uniqueness for the Cauchy problem

1.3．Existence of solutions in the large

1.4．Generalized solutions

Chapter Ⅱ．Scalar first order equations with one space variable

2.1．Introduction

2.2．The linear case

2.3．Classical solutions of Burgers'equation

2.4．Weak solutions of Burgers'equation

2.5．General strictly convex conservation laws

Chapter Ⅲ．Scalar first order equations with several variables

3.1．Introduction

3.2．Parabolic equations

3.3．The conservation law with viscosity

3.4．The entropy solution of the conservation law

Chapter Ⅳ．First order systems of conservation laws with one space variable

4.1．Introduction

4.2．Generalities on first order systems

4.3．The lifespan of classical solutions

4.4．The Riemann problem

4.5．Glimm's existence theorem

4.6．Entropy pairs

Chapter Ⅴ．Compensated compactness

5.1．Introduction

5.2．Weak convergence in L∞

5.3．Weak convergence of solutions of linear differential equations

5.4．A scalar conservation law with one space variable

5.5．Probability measures associated with a system of two equations

5.6．Existence of weak solutions for a system of two equations

Chapter Ⅵ．Nonlinear perturbations of the wave equation

6.1．Introduction

6.2．The linear wave equation

6.3．The energy integral method

6.4．Interpolation and Sobolev inequalities

6.5．Global existence theorems for nonlinear wave equations

6.6．The null condition in three dimensions

6.7．Global existence theorems by the conformal method

Chapter Ⅶ．Nonlinear perturbations of the Klein-Gordon equation

7.1．Introduction

7.2．Asymptotic behavior of solutions of the Klein-Gordon equation

7.3．L2．L∞ estimates for the Klein-Gordon equation

7.4．Existence theorems

7.5．Remarks on the lowest space dimensions

7 6．Alternative energy and Sobolev estimates

7.7．Existence theorems for Cauchy data of compact support

7.8．The method of Shatah

Chapter Ⅷ．Microlocal analysis

8.1．Introduction

8.2．The wave front set

8.3．Microlocal regularity of a product

8.4．Pseudo-differential operators

8.5．Composite functions

8.6．H？lder and Zygmund classes

Chapter Ⅸ．Pseudo-differential operators of type 1,1

9.1．Introduction

9.2．Some basic facts on pseudo-differential operators

9.3．Continuity in H(s)and in C？

9.4．Adjoints

9.5．Composition

9.6．Symbols with additional smoothness

9.7．The sharp G？rding inequality

Chapter Ⅹ．Paradifferential calculus

10.1．Introduction

10.2．Regularisation of symbols and paradifferential calculus

10.3．Bony's linearisation theorem

Chapter Ⅺ．Propagation of singularities

11.1．Introduction

11.2．First order scalar differential equations

11.3．Linear pseudo-differential equations

11.4．Nonlinear differential equations

11.5．Second order hyperbolic equations

11.6．Beals'restriction theorem

Appendix on pseudo-Riemannian geometry

A.1．Basic definitions

A.2．Geodesic coordinates and curvature

A.3．Conformal changes of metric

A.4．A conformal imbedding of Minkowski space

References

Index of notation

Index