Entropy solutions of the Euler equations for isothermal relativistic fluids

Abstract

We investigate the initial-value problem for the relativistic Euler equations of isothermal perfect fluids, and generalise an existence result due to LeFloch and Shelukhin for 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 covered solutions with bounded variation that avoid the vacuum state. Our new framework provides solutions in a larger function space and allows for the mass density to vanish and the velocity field to approach the light speed. 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 and takes advantage of a scaling invariance property of the Euler equations.