Flux approximation to the isentropic relativistic Euler equations

Abstract

The isentropic relativistic Euler equations for polytropic gas under flux perturbations are studied. The Riemann problem of the pressureless relativistic Euler equations with a flux approximation is firstly solved, and a family of delta-shock and U-shaped pseudo-vacuum state solutions are constructed. Then it is shown that, as the flux approximation vanishes, the limits of the family of delta-shock and U-shaped pseudo-vacuum solutions are exactly the delta-shock and vacuum state solutions to the pressureless relativistic Euler equations, respectively. Secondly, we study the Riemann problem of the isentropic relativistic Euler equations with a double parameter flux approximation including pressure term. We further prove that, as the pressure and two-parameter flux perturbation vanish, respectively, any two-shock Riemann solution tends to a delta-shock solution to the pressureless relativistic Euler equations, and the intermediate density between the two shocks tends to a weighted δ-measure which forms a delta shock wave; any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution to the pressureless relativistic Euler equations, and the nonvacuum intermediate state in between tends to a vacuum state.