Flux-approximation limits of solutions to the relativistic Euler equations for polytropic gas

The flux-approximation problem of the relativistic Euler equations for polytropic gas in special relativity is studied. At first, we solve the Riemann problem of the pressureless relativistic Euler equations with a flux approximation, and obtain two kinds of solutions involving a family of delta shock wave and pseudo-vacuum state. Then, as the flux approximation vanishes, we show that the limits of the family of delta-shock and pseudo-vacuum solutions are exactly the delta-shock and vacuum state solutions of the pressureless relativistic Euler equations, respectively. Next, the Riemann problem of the relativistic Euler equations with a double parameter flux approximation including pressure is solved analytically. Furthermore, it is rigorously proved that, as the double parameter flux perturbation vanishes, any two-shock Riemann solution tends to a delta-shock solution to the pressureless relativistic Euler equations; 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.