Delta-shocks for a 2 × 2 balance system of Keyfitz–Kranzer type with varying Chaplygin gas

Abstract

The motivation of this study is to derive the solutions of the Riemann problem for a 2 × 2 balance non-symmetric system of Keyfitz–Kranzer type with varying Chaplygin gas. What varying Chaplygin gas means is that the fluid obeys the pressure–density–time relation where the pressure is negative and is the product of a function of time and the inverse of the density. It includes the Chaplygin gas as a special case. Using variable substitution, the solutions with two kinds of different structures involving delta-shocks in two cases are constructed. The generalized Rankine–Hugoniot relation and entropy condition of the delta-shocks are clarified. Furthermore, the position, strength, and propagation speed of the delta-shocks are calculated explicitly. Because of the presence of the source terms, the Riemann solutions are non-self-similar. It is shown that the contact discontinuities and the delta-shocks are either curves or straight lines. A new and interesting phenomenon is that even when both the contact discontinuities and the delta-shock are straight lines, the weight of the delta-shock is no longer linear function of the time t. In this sense, the source term kρ appearing in the governing equation plays a role in adjusting the weights of the delta-shocks.