Introduction

Abstract

The occurrence and propagation of shock waves are general phenomena in the motion of continuous media. The characteristic feature of shock waves is a medium, which greatly changes its state in a very thin layer. If the viscosity of the medium is neglected, the shock can be considered as a surface with zero width, while the parameters describing the state of the media have discontinuity on this surface. Hence people have to study the theory on the solutions with discontinuity to partial differential equations, governing the motion of the medium. The expected solution takes discontinuity on some special surfaces, which is to be determined with the unknown solution together. In the theory of partial differential equations such solutions are called generalized solutions. Now the theory describing the shocks and other nonlinear waves of continuous media has greatly developed, and has formed a field containing profound and rich contents. When a shock hits an obstacle and then is reflected, the reflection is often powerful and produce severe damage, so that the reflection of shock waves attracted many people’s attention in their research. Although there have been many results and success in experiment investigation and numerical simulations on this subject, the theoretical study on mathematical analysis is not enough and even much lags behind the requirement of engineering. The main aim and the subject of this book is to show the recent development in the study of shock reflection and related problems by using the theory and technique of partial differential equations. In the first chapter we will describe the physical background of these problems and formulate them to boundary value problems of partial differential equations. Meanwhile, we will also solve the problem on reflection of planar shock by a straight surface, which is the simplest case of shock reflection and is also the basis of the further analysis in subsequent chapters.