The Exact Riemann Solutions to the Generalized Pressureless Euler Equations with Dissipation

The Riemann solutions for the generalized pressureless Euler equations with a dissipation term are constructed explicitly. It is shown that the delta shock wave appears in Riemann solutions in some situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established, and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the dissipation term makes contact discontinuities and delta shock waves bend into curves and the Riemann solutions are not self-similar anymore. Moreover, as the dissipation term vanishes, the Riemann solutions converge to the corresponding ones of the generalized pressureless Euler equations. Finally, we give the application of our results on two typical examples.