The Riemann problem for the pressureless Euler system with the Coulomb-like friction term

The Riemann problem for the pressureless Euler system with the Coulomb-like friction term is solved constructively, whose solutions involve delta shock wave and vacuum state. The position, propagation speed and strength of the delta shock wave are calculated explicitly from the generalized Rankine–Hugoniot conditions of the delta shock wave which are derived in detail, such that the influence of the Coulomb-like friction term on the Riemann solutions can be seen clearly. Furthermore, the generalized Riemann problem with measure-valued initial data is also considered and four kinds of different structures are displayed for the Riemann solutions. It is shown that the Riemann solutions converge to the corresponding ones of the pressureless Euler system when the friction term vanishes. In addition, the balance relations associated with the mass and momentum transportation are established for the delta shock wave solution.