Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities

Abstract

In this paper, we considerably extend the results on global existence of entropy-weak solutions to the compressible Navier–Stokes system with density dependent viscosities obtained, independently (using different strategies) by Vasseur–Yu [Invent. Math. 206 (2016) and arXiv:1501.06803 (2015)] and by Li–Xin [arXiv:1504.06826 (2015)]. More precisely, we are able to consider a physical symmetric viscous stress tensor $\sigma =2\mu (\rho )\mathbb{D}(u)+(\lambda (\rho )\mathrm{div}u-P(\rho )\mathrm{Id}$ where $\mathbb{D}(u)=[\nabla u+{\nabla }^{T}u]/2$ with shear and bulk viscosities (respectively $\mu (\rho )$ and $\lambda (\rho )$) satisfying the BD relation $\lambda (\rho )=2({\mu }^{\prime }(\rho )\rho -\mu (\rho ))$ and a pressure law $P(\rho )=a{\rho }^{\gamma }$ (with $a>0$ a given constant) for any adiabatic constant $\gamma >1$. The non-linear shear viscosity $\mu (\rho )$ satisfies some lower and upper bounds for low and high densities (our result includes the case $\mu (\rho )=\mu {\rho }^{\alpha }$ with $2/3<\alpha <4$ and $\mu >0$ constant). This provides an answer to a longstanding question on compressible Navier–Stokes equations with density dependent viscosities, mentioned for instance by F. Rousset [Bourbaki 69ème année, 2016–2017, exp. 1135].