On a regularization of the compressible Euler equations for an isothermal gas

We consider a Leray-type regularization of the compressible Euler equations for an isothermal gas. The regularized system depends on a small parameter α > 0. Using Riemann invariants, we prove the existence of smooth solutions for the regularized system for every α > 0. The regularization mechanism is a nonlinear bending of characteristics that prevents their finite-time crossing. We prove that, in the α → 0 limit, the regularized solutions converge strongly. However, based on our analysis and numerical simulations, the limit is not the unique entropy solution of the Euler equations. The numerical method used to support this claim is derived from the Riemann invariants for the regularized system. This method is guaranteed to preserve the monotonicity of characteristics.