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

Didier Bresch,Cheng Yu,Alexis F Vasseur

doi:10.4171/jems/1143

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].

Highlights

When a fluid is governed by the barotropic compressible Navier-Stokes equations, the existence of global weak solutions, in the sense of J
Lions [35] when the pressure law in terms of the density is given by P (ρ) = aργ where a and γ are two strictly positive constants. He has presented in 1998 a complete theory for P (ρ) = aργ with γ ≥ 3d/(d + 2) allowing to obtain the result of global existence of weak solutionsa la Leray in dimension d = 2 and 3 and for general initial data belonging to the energy space
First we prove that weak solutions of the approximate solution are renormalized solutions of the system, in the sense of [31]

Summary

Introduction

When a fluid is governed by the barotropic compressible Navier-Stokes equations, the existence of global weak solutions, in the sense of J. Note that in the last paper [34] by Li-Xin, they consider more general viscosities satisfying the BD relation but with a non-symmetric stress diffusion (σ = μ(ρ)∇u+(λ(ρ)divu−P (ρ))Id) and more restrictive conditions on the shear μ(ρ) viscosity and bulk viscosity λ(ρ) and on the pressure law P (ρ) compared to the present paper The objective of this current paper is to extend the existence results of global entropyweak solutions obtained independently (using different strategies) by Vasseur-Yu [47] and. To prove global existence of weak solutions of the compressible Navier-Stokes equations, we follow the strategy introduced in [31, 47]. We pass to the limit with respect to the parameters r, r0, r1, r2 and δ to recover a renormalized weak solution of the compressible Navier-Stokes equations and prove our main theorem. This will justify in some sense the two-velocities formulation introduced in [12] with the extra velocity linked to ∇μ(ρ)

The first level of approximation procedure

From weak solutions to renormalized solutions to the approximation