Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities
Didier Bresch, Alexis Vasseur, Cheng Yu
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].