Entropy solutions of the Euler equations for isothermal relativistic fluids

Abstract

Abstract. We investigate the initial-value problem for the relativistic Euler equations governing isothermal perfect fluid flows, and generalize an approach introduced by LeFloch and Shelukhin in the non-relativistic setting. We establish the existence of globally defined, bounded measurable, entropy solutions with arbitrary large amplitude. An earlier result by Smoller and Temple for the same system covered solutions with bounded variation that avoid the vacuum state. The new framework proposed here provides entropy solutions in a larger function space and allows for the mass density to vanish and the velocity field to approach the speed of light. The relativistic Euler equations become strongly degenerate in both regimes, as the conservative or the flux variables vanish or blow-up. Our proof is based on the method of compensated compactness for nonlinear systems of conservation laws (Tartar, DiPerna) and takes advantage of a scaling invariance property of the isothermal fluid equations. We also rely on properties of the fundamental kernel that generates the mathematical entropy and entropy flux pairs. This kernel exhibits certain singularities on the boundary of its support and we are led to analyze certain nonconservative products (after Dal Maso, LeFloch, and Murat) consisting of functions of bounded variation by measures.

Limits of Riemann Solutions to the Relativistic Euler Systems for Chaplygin Gas as Pressure Vanishes

“…Based on the results in [25], Hsu et al [26] proved the existence of global weak solutions to (1) with a more realistic equation of state and ∞ initial data containing vacuum state in the framework of compensated compactness. The existence of entropy solutions for problems without vacuum state was established by LeFloch and Yamazaki [27]. More results about entropy solutions to system (1) can be found in [28][29][30][31].…”

Numerical Study of Singularity Formation in Relativistic Euler Flows

“…Elementary nonlinear waves for the relativistic Euler equations have been analyzed [1,25,35,36]. Existence of entropy solutions for problems without vacuum state [33] as well as for problems with possible vacuum formation [22] has been established. The mechanism by which singularities form is, however, not fully understood.…”