Delta shocks in the relativistic full Euler equations for a Chaplygin gas

TL;DR

This paper investigates delta shock waves in the relativistic Euler equations for a Chaplygin gas, providing a constructive solution to the Riemann problem and clarifying the formation, conditions, and uniqueness of these shocks.

Contribution

It introduces a novel type of delta shock wave where both density and energy contain Dirac delta functions, expanding understanding of shock solutions in relativistic fluids.

Findings

Constructive solution to the Riemann problem with delta shocks.

Clarification of formation mechanism and entropy conditions for delta shocks.

Proof of existence and uniqueness of delta shock solutions.

Abstract

The relativistic full Euler equations for a Chaplygin gas are studied. The Riemann problem is solved constructively. There are two kinds of Riemann solutions, in which one consists of three contact discontinuities and the other involves a delta shock wave on which both state variables the rest mass density and the proper energy density simultaneously contain the Dirac delta functions. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. The formation mechanism, generalized Rankine-Hugoniot relation and entropy condition are clarified for this type of delta shock wave. Under the generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta shock solutions are also established.