Boundary Layers and Incompressible Navier-Stokes-Fourier Limit of the Boltzmann Equation in Bounded Domain I

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• 作者：Ning Jiang and Nader Masmoudi
• 刊名：Communications on Pure and Applied Mathematics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：70
• 期：1
• 页码：90-171
• 全文大小：546K
• ISSN：1097-0312

文摘

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.