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

TL;DR

The paper analyzes the Knudsen boundary layer for the Boltzmann equation in a half-space with Maxwell reflection ($0<\alpha<1$) under hard-sphere angular cutoff, proving existence, uniqueness, and exponential-type decay of solutions in $L^{\infty}_{x,v}$ for both linear and nonlinear Knudsen layer problems. It introduces an auxiliary, damped framework and a finite-slab approach to obtain uniform estimates, then passes to the half-space limit while characterizing the asymptotic state $q^{\infty}\in\mathcal{N}$ and proving independence from auxiliary data. A key outcome is a rigorous description of the vanishing sources set and a constructive method to derive boundary conditions for fluid limits (e.g., incompressible Navier–Stokes) from the Knudsen layer, including explicit slip coefficients that depend on $\alpha$. The results extend prior work to the regime $0<\alpha<1$ in $L^2_{x,v}\cap L^{\infty}_{x,v}$ with general source terms, and demonstrate how Knudsen-layer symmetry properties yield boundary data in hydrodynamic expansions. Overall, the work provides a robust, quantitative bridge between kinetic boundary layers and macroscopic fluid boundaries under Maxwell-type diffusion conditions.

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 $0<α<1$. 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 $α\in(0,1]$ and $α=O(1)$. 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}

Theorems & Definitions (34)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.2
• Lemma 1.1
• Definition 2.1
• Lemma 2.1: A priori $L^\infty_{x,v}$ estimates
• proof
• Lemma 2.2: A priori $L^2_{x,v}$ estimates
• proof