Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities
Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities
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DOI：
10.4171/jems/1143
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发表时间：
2019-05
影响因子：
2.6
通讯作者：
D. Bresch;A. Vasseur;Cheng Yu
中科院分区：
文献类型：
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作者：
D. Bresch;A. Vasseur;Cheng Yu
本文较大程度地推广了Vasseur-Yu和Li-Xin分别(使用不同的策略)获得的与密度相关的粘性可压缩N-S系统的整体弱解的存在性结果。更确切地说，我们能够考虑一个物理对称粘性应力张量$\Sigma=2\Mu(\Rho)\，{\mathbb{D}}(U)+\bigl(\lambda(\rho){\rm div}u-P(\rho)\bigr)\，其中${\mathbb D}(U)=[\nabla u+\nabla^Tu]/2$，剪切粘度和体积粘度(分别为$\mU(\rho)$和$\lambda(\rho)$)满足BD关系$\lambda(\rho)=2(\mU‘(\rho)\rho-\mU(\rho))$和压力定律$P(\rho)=a\rho^\γ$(其中$a>；对于任何绝热常数$\Gamma>；1$。对于低密度和高密度，非线性剪切粘度满足一些上下界(我们的数学结果包括$2/3 0不变的情况)。这为一个长期存在的关于具有密度相关粘度的可压缩Navier-Stokes方程的数学问题提供了答案，例如F.Rousset在The Bourbaki 69eme Anne，2016-2017，No 1135中提到的问题。
In this paper, we extend considerably the global existence results of entropy-weak solutions related to compressible Navier-Stokes system with density dependent viscosities obtained, independently (using different strategies), by Vasseur-Yu [Inventiones mathematicae (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 $\sigma=2\mu(\rho)\,{\mathbb{D}}(u)+\bigl(\lambda(\rho){\rm div}u -P(\rho)\bigr)\, {\rm Id}$ where ${\mathbb D}(u) = [\nabla u + \nabla^T u]/2$ with a 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 nonlinear shear viscosity $\mu(\rho)$ satisfies some lower and upper bounds for low and high densities (our mathematical result includes the case $\mu(\rho)= \mu\rho^\alpha$ with $2/3 0$ constant). This provides an answer to a longstanding mathematical question on compressible Navier-Stokes equations with density dependent viscosities as mentioned for instance by F. Rousset in the Bourbaki 69eme annee, 2016--2017, no 1135.