Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits

Abstract

This paper investigates the Knudsen layer equation in half-space, arising from the hydrodynamic limit of the Boltzmann equation to fluid dynamics. We consider the Maxwell reflection boundary condition with accommodation coefficient $0<\alpha<1$. We restrict our attention to hard sphere collisions with angular cutoff, proving the existence, uniqueness, and asymptotic behavior of the solution in $L^{\infty}_{x,v}$. Additionally, we demonstrate the application of our theorem to the hydrodynamic limit through a specific example. In this expample, we derive the boundary conditions of the fluid equations using our theorem and the symmetric properties of the Knudsen layer equation for $\alpha\in(0,1]$ and $\alpha=O(1)$. These derivations differs significantly from the cases of specular and almost specular reflection. This explicitly characterizes the {\em vanishing sources set} defined in \cite{jiang2024knudsenboundarylayerequations}