Pressure and flux-approximation to the isentropic relativistic Euler equations for modified Chaplygin gas

The formation of delta shock waves and vacuum states in solutions to the isentropic relativistic Euler equations for modified Chaplygin gas is analyzed as the double-parameter pressure and the triple-parameter flux approximation vanish, respectively. The Riemann problems under pressure perturbation and flux approximation are analytically solved. Then, it is shown that as the pressure and flux approximation, respectively, vanish, 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 that 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. Especially in the process of three parameters decreasing, a U-shaped pseudo-vacuum state appears in solutions to the flux-approximation system.