Delta-shocks and vacuums in the relativistic Euler equations for isothermal fluids with the flux approximation

We demonstrate the flux-approximation problem of the isothermal relativistic Euler equations describing a perfect fluid flow in special relativity. First, the Riemann problem of the isothermal relativistic Euler equations under flux perturbation is discussed, and four kinds of solutions are obtained. Second, we rigorously prove that, as the flux perturbation vanishes, 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. Correspondingly, 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.