The Riemann problem and interaction of waves in two-dimensional steady zero-pressure adiabatic flow

Abstract

The Riemann problem for the system of conservation laws of mass, momentum and energy in two-dimensional steady zero-pressure adiabatic flow is solved completely. The Riemann solutions contain two kinds: vacuum states and delta shock waves, on which both density and internal energy simultaneously contain the Dirac delta function. This is quite different from the previous ones on which only one state variable contains the Dirac delta function. The formation mechanism, generalized Rankine–Hugoniot relation and entropy condition are clarified for this type of delta shock wave. Under the suitable generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solution is established. In addition, the interactions of delta shock waves and vacuum states are analyzed by solving the Riemann problems with initial data of three piecewise constant states case by case, and the global structure of solutions with four different configurations is constructed.