Riemann Problems for a Class of Coupled Hyperbolic Systems of Conservation Laws

We study a class of coupled hyperbolic systems of conservation laws which contain the one-dimensional zero-pressure gas dynamics as a prototypical example. The Riemann problems are solved constructively. The Riemann solutions exactly include two kinds: delta-shock wave solutions and vacuum solutions. Under the generalized Rankine–Hugoniot relation and entropy condition, all of the existence, uniqueness, and stability of solutions to viscous perturbations are proved. Two typical examples are presented finally.