Well-posedness of the Riemann problem with two shocks for the isentropic Euler system in a class of vanishing physical viscosity limits

TL;DR

This paper proves the stability and uniqueness of the two-shock Riemann solution for the 1D Euler system as a weak inviscid limit of Navier-Stokes solutions, advancing the understanding of shock interactions in fluid dynamics.

Contribution

It extends previous single-shock results to two shocks using weighted relative entropy and $a$-contraction theory, addressing challenges in controlling shock separation under perturbations.

Findings

Two-shock Riemann solutions are stable and unique in the inviscid limit.

The method ensures shock separation even with large perturbations.

Lays groundwork for analyzing interacting wave families and the Bianchini-Bressan conjecture.

Abstract

We consider the Riemann problem composed of two shocks for the 1D Euler system. We show that the Riemann solution with two shocks is stable and unique in the class of weak inviscid limits of solutions to the Navier-Stokes equations with initial data with bounded energy. This work extends to the case of two shocks a previous result of the authors in the case of a single shock. It is based on the method of weighted relative entropy with shifts known as $a$-contraction theory. A major difficulty due to the method is that very little control is available on the shifts. A modification of the construction of the shifts is needed to ensure that the two shock waves are well separated, at the level of the Navier-Stokes system, even when subjected to large perturbations. This work put the foundations needed to consider a large family of interacting waves. It is a key result in the program to solve the Bianchini-Bressan conjecture, that is the inviscid limit of solutions to the Navier-Stokes equation to the unique BV solution of the Euler equation, in the case of small BV initial values.