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 where with shear and bulk viscosities (respectively and ) satisfying the BD relation and a pressure law (with a given constant) for any adiabatic constant . The non-linear shear viscosity satisfies some lower and upper bounds for low and high densities (our result includes the case with and 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].
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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. 1791–1837DOI 10.4171/JEMS/1143