Global solutions to the two-dimensional Riemann problem for a system of conservation laws

We study the global solutions to the two-dimensional Riemann problem for a system of conservation laws. The initial data are three constant states separated by three rays emanating from the origin. Under the assumption that each ray in the initial data outside of the origin projects exactly one planar contact discontinuity, this problem is classified into five cases. By the self-similar transformation, the reduced system changes type from being elliptic near the origin to being hyperbolic far away in self-similar plane. Then in hyperbolic region, applying the generalized characteristic analysis method, a Goursat problem is solved to describe the interactions of planar contact discontinuities. While, in elliptic region, a boundary value problem arises. It is proved that this boundary value problem admits a unique solution. Based on these preparations, five explicit solutions and their corresponding criteria can be obtained in self-similar plane.