The asymptotic limits of Riemann solutions for the isentropic extended Chaplygin gas dynamic system with the vanishing pressure

Abstract

We construct the solutions to the Riemann problem for the isentropic extended Chaplygin gas dynamic system for all kinds of situations by the phase plane analysis. We investigate the asymptotic limits of solutions to this problem in detail when the pressure given by the state equation of the system becomes the one of pressureless gas. During the process of vanishing pressure, the two-shock Riemann solution tends to a delta shock solution, whereas the two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution with a vacuum state.

On the delta shockwave interactions for the isentropic Chaplygin gas system consisting of three scalar equations

“…The generalized Riemann problem with delta initial data was also considered by Wang and Zhang [46] with the help of the generalized Rankine-Hugoniot conditions. Let us also see such as [11,14,17,21,25,27,28,33,34,37,39,42,47] for some related results. In contrast to the traditional isentropic Chaplygin gas dynamic system, the solution to the Riemann problem (1.3) and (1.4) is either the combination of three contact discontinuities when u − − 1 ρ − < u + + 1 ρ + or the delta shock wave solution when u − − 1 ρ − > u + + 1 ρ + , which can be obtained by using more complicated generalized Rankine-Hugoniot conditions.…”

Construction of global solutions for a symmetric system of Keyfitz–Kranzer type with three piecewise constant states

“…. , n. It should be stressed that some well-known hyperbolic systems, such as the pressureless gas dynamics system [18,19], the isentropic Chaplygin gas dynamics system [20][21][22][23][24], and the macroscopic production model [25], are recovered by choosing suitable φ.…”