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 $σ=2μ(ρ)\,{\mathbb{D}}(u)+\bigl(λ(ρ){\rm div}u -P(ρ)\bigr)\, {\rm Id}$ where ${\mathbb D}(u) = [\nabla u + \nabla^T u]/2$ with a shear and bulk viscosities (respectively $μ(ρ)$ and $λ(ρ)$) satisfying the BD relation $λ(ρ)=2(μ'(ρ)ρ- μ(ρ))$ and a pressure law $P(ρ)=aρ^γ$ (with $a>0$ a given constant) for any adiabatic constant $γ>1$. The nonlinear shear viscosity $μ(ρ)$ satisfies some lower and upper bounds for low and high densities (our mathematical result includes the case $μ(ρ)= μρ^α$ with $2/3 < α< 4$ and $μ>0$ 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.