JournalsjemsVol. 24, No. 5pp. 1791–1837

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
Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities cover
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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 σ=2μ(ρ)D(u)+(λ(ρ)divuP(ρ)Id\sigma = 2 \mu(\rho) \,{\mathbb D}(u) +(\lambda(\rho) \operatorname{div} u - P(\rho) \operatorname {Id} where D(u)=[u+Tu]/2{\mathbb D}(u) = [\nabla u + \nabla^T u]/2 with shear and bulk viscosities (respectively μ(ρ)\mu(\rho) and λ(ρ)\lambda(\rho)) satisfying the BD relation λ(ρ)=2(μ(ρ)ρμ(ρ))\lambda(\rho)=2(\mu'(\rho)\rho - \mu(\rho)) and a pressure law P(ρ)=aργP(\rho)=a\rho^\gamma (with a>0a>0 a given constant) for any adiabatic constant γ>1\gamma>1. The non-linear shear viscosity μ(ρ)\mu(\rho) satisfies some lower and upper bounds for low and high densities (our result includes the case μ(ρ)=μρα\mu(\rho)= \mu\rho^\alpha with 2/3<α<42/3 < \alpha < 4 and μ>0\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