Abstract:It is well known that the full compressible Navier-Stokes equations can be deduced via the Chapman-Enskog expansion from the Boltzmann equation as the first-order correction to the Euler equations with viscosity and heat-conductivity coefficients of order of the Knudsen number $\epsilon >0$. In the paper, we carry out the rigorous mathematical analysis of the compressible Navier-Stokes approximation for the Boltzmann equation regarding the initial-boundary value problems in general bounded domains. The main goal is to measure the uniform-in-time deviation of the Boltzmann solution with diffusive reflection boundary condition from a local Maxwellian with its fluid quantities given by the solutions to the corresponding compressible Navier-Stokes equations with consistent non-slip boundary conditions whenever $\epsilon >0$ is small enough. Specifically, it is shown that for well chosen initial data around constant equilibrium states, the deviation weighted by a velocity function is $O(\epsilon ^{1/2})$ in $L^\infty _{x,v}$ and $O(\epsilon ^{3/2})$ in $L^2_{x,v}$ globally in time. The proof is based on the uniform estimates for the remainder in different functional spaces without any spatial regularity. One key step is to obtain the global-in-time existence as well as uniform-in-$\epsilon$ estimates for regular solutions to the full compressible Navier-Stokes equations in bounded domains when the parameter $\epsilon >0$ is involved in the analysis.

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中文翻译：

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有界域中 Boltzmann 方程的可压缩 Navier-Stokes 近似

摘要：众所周知，完全可压缩的 Navier-Stokes 方程可以从 Boltzmann 方程通过 Chapman-Enskog 展开推导出，作为对具有 Knudsen 数阶的粘度和导热系数的 Euler 方程的一阶修正$\epsilon >0$。在本文中，我们针对一般有界域中的初边值问题，对玻尔兹曼方程的可压缩 Navier-Stokes 近似进行了严格的数学分析。主要目标是测量具有漫反射边界条件的玻尔兹曼解与局部麦克斯韦方程的均匀时间偏差，其流体量由具有一致防滑边界条件的相应可压缩纳维-斯托克斯方程的解给出$\epsilon > 0$ 足够小。具体来说，它表明，对于围绕恒定平衡状态精心选择的初始数据，速度函数加权的偏差是 $O(\epsilon ^{1/2})$ in $L^\infty _{x,v}$和 $O(\epsilon ^{3/2})$ 在 $L^2_{x,v}$ 中的全局时间。证明基于对不同功能空间中的余数的统一估计，没有任何空间规律。一个关键步骤是在涉及参数 $\epsilon >0$ 时，获得有界域中完全可压缩 Navier-Stokes 方程的正则解的全局时间存在性和统一的 $\epsilon$ 估计在分析中。v}$ 和 $O(\epsilon ^{3/2})$ 在 $L^2_{x,v}$ 中的全局时间。证明基于对不同功能空间中的余数的统一估计，没有任何空间规律。一个关键步骤是在涉及参数 $\epsilon >0$ 时，获得有界域中完全可压缩 Navier-Stokes 方程的正则解的全局时间存在性和统一的 $\epsilon$ 估计在分析中。v}$ 和 $O(\epsilon ^{3/2})$ 在 $L^2_{x,v}$ 中的全局时间。证明基于对不同功能空间中的余数的统一估计，没有任何空间规律。一个关键步骤是在涉及参数 $\epsilon >0$ 时，获得有界域中完全可压缩 Navier-Stokes 方程的正则解的全局时间存在性和统一的 $\epsilon$ 估计在分析中。

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