Stability of Riemann Shocks for isothermal Euler by Inviscid limits of global-in-time large Navier-Stokes flows

TL;DR

This paper proves the stability of Riemann shocks in isothermal Euler equations by analyzing the inviscid limit of large, global-in-time solutions to the Navier-Stokes system, including degenerate viscosities and large perturbations.

Contribution

It establishes the global existence of strong solutions to the isothermal Navier-Stokes system without smallness assumptions, and demonstrates the stability of Riemann shocks via inviscid limits.

Findings

Global existence of strong solutions for large initial data

Uniform estimates for viscous shocks with respect to viscosity

Stability of Riemann shocks in the inviscid limit

Abstract

In this paper, we study the isothermal gas dynamics. We first establish the global existence of strong solutions to the one-dimensional isothermal Navier-Stokes system for smooth initial data without any smallness conditions, assuming that the initial density has strictly positive lower bound. The existence result allows for possibly degenerate viscosity coefficients and admits different asymptotic states at the far fields. We then prove a contraction property for the strong solutions perturbed from viscous shocks, yielding uniform estimates with respect to the viscosity coefficients. This covers any large perturbations, and consequently, we establish the inviscid limits and their stability estimate. In other words, we demonstrate the stability of Riemann shocks to the one-dimensional isothermal Euler system in the class of vanishing viscosity limits of the associated Navier-Stokes system.