Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities

Abstract

In this paper, we extend considerably the global existence results of entropy-weak solutions related to compressible Navier-Stokes system with density dependent viscosities obtained, independently (using different strategies), by Vasseur-Yu [Inventiones mathematicae (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 $\ddot{I}ƒ=2\hat{I}¼(\ddot{I})\mathbb{D}(u)+(\hat{I}»(\ddot{I})\mathrm{div}u-P(\ddot{I}))\mathrm{Id}$ where $\mathbb{D}(u)=[\nabla u+{\nabla }^{T}u]/2$ with a shear and bulk viscosities (respectively $\hat{I}¼(\ddot{I})$ and $\hat{I}»(\ddot{I})$) satisfying the BD relation $\hat{I}»(\ddot{I})=2(\hat{I}{¼}^{\prime }(\ddot{I})\ddot{I}-\hat{I}¼(\ddot{I}))$ and a pressure law $P(\ddot{I})=a\ddot{I}{}^3$ (with $a>0$ a given constant) for any adiabatic constant ${\hat{I}}^3>1$. The nonlinear shear viscosity $\hat{I}¼(\ddot{I})$ satisfies some lower and upper bounds for low and high densities (our mathematical result includes the case $\hat{I}¼(\ddot{I})=\hat{I}¼\ddot{I}{}^{\hat{I}}\pm$ with $2/3<\hat{I}\pm <4$ and $\hat{I}¼>0$ constant). This provides an answer to a longstanding mathematical question on compressible Navier-Stokes equations with density dependent viscosities as mentioned for instance by F. Rousset in the Bourbaki 69ème année, 2016--2017, no 1135.