### Entropy solutions of the Euler equations for isothermal relativistic fluids.

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**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.

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**Date:** 2007-01-17