Existence and exponential growth of global classical solutions to the compressible Navier-Stokes equations with slip boundary conditions in 3D bounded domains
Existence and exponential growth of global classical solutions to the compressible Navier-Stokes equations with slip boundary conditions in 3D bounded domains
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DOI：
10.1512/iumj.2023.72.9591
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发表时间：
2021-02
影响因子：
1.1
通讯作者：
Guocai Cai;Jing Li
中科院分区：
文献类型：
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作者：
Guocai Cai;Jing Li
研究了三维单连通有界区域中带滑移边界条件的正压可压缩Navier-Stokes方程，该区域的光滑边界具有有限个二维连通分量.对于任意大于1的绝热指数，在发现了与滑移边界条件有关的边界积分的一些新的估计后，证明了只要初始能量适当小，该系统的初边值问题的弱解和经典解在时间上是全局存在的.此外，密度有很大的振荡，并包含真空状态。最后，它也表明，对于经典的解决方案，密度的振荡将增长无界的长期指数率提供真空出现（即使在一个点）最初。这是第一个关于可压缩Navier-Stokes方程在一般的三维有界光滑区域上具有含密度真空态经典解的整体存在性的结果。
We investigate the barotropic compressible Navier-Stokes equations with slip boundary conditions in a three-dimensional (3D) simply connected bounded domain, whose smooth boundary has a finite number of two-dimensional connected components. For any adiabatic exponent bigger than one, after discovering some new estimates on boundary integrals related to the slip boundary condition, we prove that both the weak and classical solutions to the initial-boundary-value problem of this system exist globally in time provided the initial energy is suitably small. Moreover, the density has large oscillations and contains vacuum states. Finally, it is also shown that for the classical solutions, the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially. This is the first result concerning the global existence of classical solutions to the compressible Navier-Stokes equations with density containing vacuum states initially for general 3D bounded smooth domains.