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

### Didier Bresch

CNRS Université Savoie Mont-Blanc, Le Bourget du lac, France### Alexis F. Vasseur

University of Texas at Austin and the Oden Institute, USA### Cheng Yu

University of Florida, Gainesville, USA

## Abstract

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

## Cite this article

Didier Bresch, Alexis F. Vasseur, Cheng Yu, Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities. J. Eur. Math. Soc. 24 (2022), no. 5, pp. 1791–1837

DOI 10.4171/JEMS/1143