本书是以作者1986年～1987年在Lund大学三个学期授课的讲义为基础，经改写而成的，主要论述了非线性双曲型偏微分方程解的全局存在性或“爆破”(blowup)，以及解的奇异性传播。书中所用的方法是基于对波方程或Yang-Mills方程的非线性摄动研究中采用的保角变换，以及对非线性方程解的余法向奇异性的传播。

目次：常微分方程；一个空间变量的一阶标量方程；多个空间变量的一阶标量方程；一个空间变量的一阶守恒律系统；补偿列紧性；波方程的非线性摄动；Klein-Gordon方程的非线性摄动；微局部分析；拟微分算子；仿微分计算；奇异性的传播。

读者对象：本书可作为大学生在学习基础的分布理论、测度论和泛函分析等课程之后，进一步学习非线性双曲型偏微分方程的教科书。

Preface

Contents

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 Burges'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 Loo

 5.3 Weak convergence of solutions of linear differential equations

 5.5 Probability measures associated with a system of two equetions

 5.6 Existence of weak solutions for a system of two equations

Chapter Ⅵ Nonlinear perturbations of the wave equation

Chapter Ⅶ Nonlinear perturbations of the Klein-Gordon equation

Chapter Ⅷ Microlocal analysis

Chapter Ⅸ Pseudo-differential operators of type1,1

Chapter Ⅹ Paradifferential calculus

Chapter Ⅺ Propagation of singularities

Appendix on pseudo-Riemannian geometry

References

Index of notation

Index