Entropy solutions of the Euler equations for isothermal relativistic fluids

We investigate the initial-value problem for the relativistic Euler equationsgoverning isothermal perfect fluid flows, and generalize an approach introducedby LeFloch and Shelukhin in the non-relativistic setting. We establish theexistence of globally defined, bounded measurable, entropy solutions witharbitrary large amplitude. An earlier result by Smoller and Temple for the samesystem covered solutions with bounded variation that avoid the vacuum state.The new framework proposed here provides entropy solutions in a larger functionspace and allows for the mass density to vanish and the velocity field toapproach the speed of light. The relativistic Euler equations become stronglydegenerate in both regimes, as the conservative or the flux variables vanish orblow-up. Our proof is based on the method of compensated compactness fornonlinear systems of conservation laws (Tartar, DiPerna) and takes advantage ofa scaling invariance property of the isothermal fluid equations. We also relyon properties of the fundamental kernel that generates the mathematical entropyand entropy flux pairs. This kernel exhibits certain singularities on theboundary of its support and we are led to analyze certain nonconservativeproducts (after Dal Maso, LeFloch, and Murat) consisting of functions ofbounded variation by measures.