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Global Smooth Solutions for 1D Barotropic Navier–Stokes Equations with a Large Class of Degenerate Viscosities
Journal of Nonlinear Science ( IF 3 ) Pub Date : 2020-03-11 , DOI: 10.1007/s00332-020-09622-z
Moon-Jin Kang , Alexis F. Vasseur

We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier–Stokes system with degenerate viscosity \(\mu (\rho )=\rho ^\alpha \). We establish that the smooth solutions have possibly two different far-fields, and the initial density remains positive globally in time, for the initial data satisfying the same conditions. In addition, our result works for any \(\alpha >0\), i.e., for a large class of degenerate viscosities. In particular, our models include the viscous shallow water equations. This extends the result of Constantin et al. (Ann Inst Henri Poincaré Anal Non Linéaire 37:145–180, 2020, Theorem 1.6) (on the case of periodic domain) to the case where smooth solutions connect possibly two different limits at the infinity on the whole space.

中文翻译:

具有一类退化粘度的一维正压Navier-Stokes方程的整体光滑解

我们证明了退化黏度为\(\ mu(\ rho)= \ rho ^ \ alpha \)的一维正压Navier–Stokes系统光滑解的整体存在性和唯一性。我们建立了光滑解可能具有两个不同的远场,并且对于满足相同条件的初始数据,初始密度在时间上全局保持为正。此外,我们的结果适用于任何\(\ alpha> 0 \)即,对于大量的简并粘度。特别是,我们的模型包括粘性浅水方程。这扩展了Constantin等人的结果。(Ann Inst HenriPoincaréAnal NonLinéaire,37:145–180,2020年,定理1.6)(在周期域上)与光滑解可能在整个空间的无穷大处连接两个不同极限的情况有关。
更新日期:2020-03-11
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