Compressible Navier-Stokes system with slip boundary from Boltzmann equations with reflection boundary: derivations and justifications

This is the first in a series of papers connecting the boundary conditions for the compressible Navier-Stokes system from the Boltzmann equations with the Maxwell reflection boundary. The slip boundary conditions are formally derived from the Boltzmann equation with both specular and almost specular reflection boundary conditions. That is, the accommodation coefficient $\alpha_\eps=O(\eps^\beta)$ with $\beta>0$ or $\alpha_\eps =0$. Here, the small number $\eps>0$ denotes the Knudsen number. The systematic formal analysis is based on the Chapman-Enskog expansion and the analysis of the Knudsen layer. In particular, for the first time, we employ the appropriate ansatz for the general $\beta>0$. This completes the program started in \cite{aoki2017slip}. In the second part, the compressible Navier-Stokes-Fourier approximation for the Boltzmann equation with specular reflection in general bounded domains is rigorously justified. The uniform regularity for the compressible Navier-Stokes system with the derived boundary conditions is investigated. For the remainder equation, the $L^2\mbox{-}L^6\mbox{-}L^\infty$ framework is employed to obtain uniform estimates in $\eps$.