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

In this paper, we considerably extend the results on global existence of entropy-weak solutions to 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)] Li–Xin \[arXiv:1504.06826 (2015)]. More precisely, are able consider a physical symmetric viscous stress tensor $\sigma = 2 \mu(\rho) ,{\mathbb D}(u) +(\lambda(\rho) \operatorname{div} u - P(\rho) \operatorname {Id}$ where ${\mathbb \[\nabla + \nabla^T u]/2$ shear bulk (respectively $\mu(\rho)$ $\lambda(\rho)$) satisfying BD relation $\lambda(\rho)=2(\mu'(\rho)\rho \mu(\rho))$ pressure law $P(\rho)=a\rho^\gamma$ (with $a>0$ given constant) for any adiabatic constant $\gamma>1$. The non-linear viscosity satisfies some lower upper bounds low high densities (our result includes case $\mu(\rho)= \mu\rho^\alpha$ $2/3 < \alpha 4$ $\mu>0$ constant). This provides an answer longstanding question equations viscosities, mentioned instance F. Rousset \[Bourbaki 69ème année, 2016–2017, exp. 1135].