Study of Multidimensional Systems of Conservation Laws: Problems, Difficulties and Progress

In the study of multidimensional systems of conservation laws people confront more difficulties than that for one-dimensional systems. The difficulties include characteristic boundary, free boundary associated with unknown nonlinear waves, various nonlinear wave structure, mixed type equations, strong singularities, etc. Most of them come from the complexity of characteristics. We will give a survey on the progress obtained in the study of this topic with the applications in various physical problems, and will also emphasize some crucial points for the further development of this theory in future.