Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and applications

TL;DR

This paper develops a relative entropy framework for compressible Navier-Stokes equations with density-dependent viscosities, providing new insights into weak-strong uniqueness, convergence to Euler solutions, and justification of shallow water models.

Contribution

It introduces a novel relative entropy method for these equations and applies it to establish weak-strong uniqueness and convergence results, extending previous work with constant viscosities.

Findings

Established weak-strong uniqueness for the system.

Proved convergence of viscous shallow water solutions to inviscid models.

Justified the limit from viscous to inviscid shallow water equations.

Abstract

Recently, A. Vasseur and C. Yu have proved the existence of global entropy-weak solutions to the compressible Navier-Stokes equations with viscosities $\nu(\varrho)=\mu\varrho$ and $\lambda(\varrho)=0$ and a pressure law under the form $p(\varrho)=a\varrho^\gamma$ with $a>0$ and $\gamma>1$ constants. In this note, we propose a non-trivial relative entropy for such system in a periodic box and give some applications. This extends, in some sense, results with constant viscosities initiated by E. Feiersl, B.J. Jin and A. Novotny. We present some mathematical results related to the weak-strong uniqueness, convergence to a dissipative solution of compressible or incompressible Euler equations. As a by-product, this mathematically justifies the convergence of solutions of a viscous shallow water system to solutions of the inviscid shall-water system.