Kinetic-fluid boundary layers and acoustic limit for the Boltzmann equation with general Maxwell reflection boundary condition

Abstract: We prove the acoustic limit from the Boltzmann equation with hard sphere collisions and the Maxwell reflection boundary condition. Our construction of solutions include the interior fluid part and Knudsen-viscous coupled boundary layers. The main novelty is that the accommodation coefficient is in the full range $0<\alpha\leq 1$. The previous works in the context of classical solutions only considered the simplest specular reflection boundary condition, i.e. $\alpha=0$. The mechanism of the derivation of fluid boundary conditions in the case $\alpha=O(1)$ is quite different with the cases $\alpha=0$ or $\alpha=o(1)$. This rigorously justifies the corresponding formal analysis in Sone's books \cite{sone2002kinetic,sone2007molecular}. In particular, this is a smooth solution analogue of \cite{jiang2010remarks}, in which the renormalized solution was considered and the boundary layers were not visible.