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

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}$}(u)+\bigl(\lambda(\rho){\rm div}u -P(\rho)\bigr), {\rm Id}$where${\mathbb D}(u) = [\nabla u + \nabla^T u]/2$withashearandbulkviscosities(respectively$\mu(\rho)$and$\lambda(\rho)$)satisfyingtheBDrelation$\lambda(\rho)=2(\mu’(\rho)\rho - \mu(\rho))$andapressurelaw$P(\rho)=a\rho^\gamma$(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 $2\mathrm{/}3<\alpha <4$ 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 69`eme ann'ee, 2016–2017, no 1135.