On the isentropic compressible Navier-Stokes equation

TL;DR

This paper proves the stability of weak solutions to the isentropic compressible Navier-Stokes equations with density-dependent viscosity, including models for shallow water, in both two and three dimensions.

Contribution

It establishes stability results for weak solutions with vacuum-dependent viscosity coefficients, extending understanding to the Saint-Venant shallow water model.

Findings

Stability of weak solutions in 2D and 3D

Results valid for any gamma > 1

Includes shallow water Saint-Venant model

Abstract

We consider the compressible Navier-Stokes equation with density dependent viscosity coefficients, focusing on the case where those coefficients vanish on vacuum. We prove the stability of weak solutions both in the torus and in the whole space in dimension 2 and 3. The pressure is given by p=rho^gamma, and our result holds for any gamma>1. In particular, we obtain the stability of weak solutions of the Saint-Venant model for shallow water.