Riemann problem and vanishing pressure limit of solutions to the isentropic relativistic Euler equations for logarithmic gas

We study the concentration and cavitation of Riemann solutions for the isentropic relativistic Euler equations (IREE) with logarithmic gas. The Riemann solutions containing rarefaction wave and shock wave are constructed first. Then, the limiting behavior of solution is investigated, and the formation of delta shock wave and vacuum state is observed as the pressure vanishes. It is shown that the two-shock-wave solution and the two-rarefaction-wave solution of the IREE converge to the delta-shock solution and vacuum solution of the corresponding pressureless relativistic Euler equations, respectively.