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Chapter Ⅰ．Ordinary differential equations 1

1.1．Introduction 1

1.2．Local existence and uniqueness for the Cauchy problem 1

1.3．Existence of solutions in the large 6

1.4．Generalized solutions 9

Chapter Ⅱ．Scalar first order equations with one space variable 13

2.1．Introduction 13

2.2．The linear case 13

2.3．Classical solutions of Burgers'equation 15

2.4．Weak solutions of Burgers'equation 17

2.5．General strictly convex conservation laws 28

Chapter Ⅲ．Scalar first order equations with several variables 36

3.1．Introduction 36

3.2．Parabolic equations 36

3.3．The conservation law with viscosity 41

3.4．The entropy solution of the conservation law 41

Chapter Ⅳ．First order systems of conservation laws with one space variable 46

4.1．Introduction 46

4.2．Generalities on first order systems 46

4.3．The lifespan of classical solutions 56

4.4．The Riemann problem 59

4.5．Glimm's existence theorem 63

4.6．Entropy pairs 70

Chapter Ⅴ．Compensated compactness 72

5.1．Introduction 72

5.2．Weak convergence in L∞ 72

5.3．Weak convergence of solutions of linear differential equations 77

5.4．A scalar conservation law with one space variable 80

5.5．Probability measures associated with a system of two equations 83

5.6．Existence of weak solutions for a system of two equations 87

Chapter Ⅵ．Nonlinear perturbations of the wave equation 89

6.1．Introduction 89

6.2．The linear wave equation 91

6.3．The energy integral method 96

6.4．Interpolation and Sobolev inequalities 106

6.5．Global existence theorems for nonlinear wave equations 117

6.6．The null condition in three dimensions 130

6.7．Global existence theorems by the conformal method 141

Chapter Ⅶ．Nonlinear perturbations of the Klein-Gordon equation 145

7.1．Introduction 145

7.2．Asymptotic behavior of solutions of the Klein-Gordon equation 145

7.3．L2．L∞ estimates for the Klein-Gordon equation 155

7.4．Existence theorems 162

7.5．Remarks on the lowest space dimensions 168

7 6．Alternative energy and Sobolev estimates 171

7.7．Existence theorems for Cauchy data of compact support 174

7.8．The method of Shatah 176

Chapter Ⅷ．Microlocal analysis 186

8.1．Introduction 186

8.2．The wave front set 186

8.3．Microlocal regularity of a product 189

8.4．Pseudo-differential operators 193

8.5．Composite functions 200

8.6．H？lder and Zygmund classes 201

Chapter Ⅸ．Pseudo-differential operators of type 1,1 211

9.1．Introduction 211

9.2．Some basic facts on pseudo-differential operators 212

9.3．Continuity in H(s)and in C？ 213

9.4．Adjoints 224

9.5．Composition 226

9.6．Symbols with additional smoothness 227

9.7．The sharp G？rding inequality 233

Chapter Ⅹ．Paradifferential calculus 235

10.1．Introduction 235

10.2．Regularisation of symbols and paradifferential calculus 235

10.3．Bony's linearisation theorem 240

Chapter Ⅺ．Propagation of singularities 246

11.1．Introduction 246

11.2．First order scalar differential equations 246

11.3．Linear pseudo-differential equations 251

11.4．Nonlinear differential equations 255

11.5．Second order hyperbolic equations 257

11.6．Beals'restriction theorem 263

Appendix on pseudo-Riemannian geometry 270

A.1．Basic definitions 270

A.2．Geodesic coordinates and curvature 270

A.3．Conformal changes of metric 275

A.4．A conformal imbedding of Minkowski space 277

References 283

Index of notation 286

Index 288