Global structure of admissible solutions of multi-dimensional non-homogeneous scalar conservation law with Riemann-type data

We investigate the global expression and structure of admissible weak solutions of an n dimensional non-homogeneous scalar conservation law with the initial data that has two constant states, separated by an n − 1 dimensional smooth manifold. We obtain the unique global existence of non-self-similar solutions. It is the first result about the global structure of non-self-similar shock waves and rarefaction waves of n dimensional non-homogeneous scalar conservation law. The shock wave and the rarefaction wave can be directly expressed and studied by a global implicit function. Finally, we give some applications to discover some interesting phenomena.