Flux approximation to the isentropic relativistic Euler equations

• The isentropic relativistic Euler equations under flux perturbations are studied.
• A family of delta-shock and U-shaped pseudo-vacuum solutions are constructed.
• The vanishing pressure and flux approximation limits are analyzed respectively.
• The flux approximations have their respective effects on the delta-shock and vacuum.

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 δδ-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.