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 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)D(u) + (lambda(rho) div u - P(rho)) Id where D(u) = [del u + del(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) = ap(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 69eme armee, 2016-2017, exp. 1135].