Boundary Layers and Incompressible Navier‐Stokes‐Fourier Limit of the Boltzmann Equation in Bounded Domain I
Boundary Layers and Incompressible Navier‐Stokes‐Fourier Limit of the Boltzmann Equation in Bounded Domain I
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DOI：
10.1002/cpa.21631
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发表时间：
2015-10
影响因子：
3
通讯作者：
Ning Jiang;N. Masmoudi
中科院分区：
文献类型：
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作者：
Ning Jiang;N. Masmoudi
我们建立了不可压缩的Navier-Stokes-Fourier极限的解决方案，玻尔兹曼方程的一般截止碰撞核在一个有界区域。适当规模的家庭的DiPerna‐Lions（‐Mischler）重整化解决方案与麦克斯韦反射边界条件显示有波动收敛的克努森数为0。每个极限点都是Navier-Stokes-Fourier方程组的弱解，其边界条件取决于调节系数和Knudsen数之间的比值。本文的主要新结果是，这种收敛性是强的Dirichlet边界条件的情况下。事实上，我们证明，声波阻尼立即，即，他们阻尼在边界层的时间。这种阻尼是由于空间中粘性和动力边界层的存在。因此，我们也证明了对无穷小麦克斯韦的第一次修正，即从具有Navier-Stokes标度的Chapman-Enskog展开中获得的修正。
We establish the incompressible Navier‐Stokes‐Fourier limit for solutions to the Boltzmann equation with a general cutoff collision kernel in a bounded domain. Appropriately scaled families of DiPerna‐Lions(‐Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to 0. Every limit point is a weak solution to the Navier‐Stokes‐Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of the Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately; namely, they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman‐Enskog expansion with Navier‐Stokes scaling.