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 $\sigma =2\mu (\rho )\mathbb{D}(u)+(\lambda (\rho )\mathrm{div}u-P(\rho ))\mathrm{Id}$ where $\mathbb{D}(u)=[\nabla u+{\nabla }^{T}u]/2$ with a 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 nonlinear shear viscosity $\mu (\rho )$ satisfies some lower and upper bounds for low and high densities (our mathematical 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 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.

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• Global existence of finite energy weak solutions to the Quantum Navier-Stokes equations with non-trivial far-field behavior
• Global existence of weak solutions to the drift-flux system for general pressure laws
• Global existence of weak solutions to the Navier-Stokes equations with temperature-depending viscosity coefficient
• On Hodge decomposition, effective viscous flux and compressible Navier-Stokes