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Global existence of entropy-weak solutions to the compressible Navier-Stokes equations with non-linear density dependent viscosities. (English) Zbl 1490.35299

Summary: 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 A. F. Vasseur and C. Yu [Invent. Math. 206, No. 3, 935–974 (2016; Zbl 1354.35115)] and by J. Li and Z. Xin [“Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities”, Preprint, arXiv:1504.06826]. 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 [Astérisque 407, Exp. No. 1135, 565–584 (2019; Zbl 1483.35155)].

MSC:

35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35D30 Weak solutions to PDEs
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