Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases

Abstract

Keywords:Relativistic Euler equations in special relativity Pressureless relativistic Euler equations Delta shock waves Vacuum Vanishing pressure limits The Riemann solutions for the Euler system of conservation laws of energy and momentum in special relativity for polytropic gases are considered. It is rigorously proved that, as pressure vanishes, they tend to the two kinds of Riemann solutions to the corresponding pressureless relativistic Euler equations: the one includes a delta shock, which is formed by a weighted δ-measure, and the other involves vacuum state.

The Convergence of Riemann Solutions to the Modified Chaplygin Gas Equations with a Coulomb-Like Friction Term as the Pressure Vanishes

“…Sheng, Wang, and Yin [35] studied the vanishing pressure limit of the generalized Chaplygin gas dynamics system. Yin and Sheng [36] considered the delta shocks and vacuum states in vanishing pressure limit of solutions to the relativistic Euler equations for polytropic gases. For inhomogeneous equations, Guo, Li, and Yin [37,38] considered the vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term and the limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term.…”

Vanishing pressure and magnetic field limit of solutions to the nonisentropic magnetogasdynamics

“…Li considered the zero temperature limit for the isothermal case. Then, the method and results were extended to various models, see Shen for the isentropic magnetogasdynamics, Yang and Wang for the modified Chaplygin gas equations, Yin and Sheng for the relativistic fluid dynamics, Yang and Liu for the Euler equations for isentropic fluids and nonisentropic fluids by the flux approximation, etc. One of the main goals of this paper is to analyze rigorously the formation of delta‐shocks and vacuum states in solutions to the nonisentropic magnetogasdynamics as the pressure P and the transverse magnetic field both vanish.…”

Vanishing viscosity limit for Riemann solutions to zero-pressure gas dynamics with flux perturbation

“…See also Li [36] for the isothermal Euler equations with zero temperature. Now, the vanishing pressure limit method has been widely used and the results were extended to the relativistic Euler equations by Yin et al [37][38][39], to the perturbed Aw-Rascle model by Shen and Sun [40], to the modified Chaplygin gas equations by Yang and Wang [41,42], etc. It is clear that the flux-approximation method is indeed a natural generalization of the vanishing pressure limit method.…”