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

TL;DR

This paper proves the acoustic limit for the Boltzmann equation with Maxwell boundary conditions, including boundary layers and a full range of accommodation coefficients, extending previous results limited to specular reflection.

Contribution

It introduces a rigorous derivation of fluid boundary conditions for the Boltzmann equation with general Maxwell reflection, covering the full accommodation coefficient range.

Findings

Established the acoustic limit with Maxwell boundary conditions for all <1111111111111111111111111111111 boundary condition.

Constructed solutions include interior fluid and Knudsen-viscous boundary layers.

Provided a rigorous justification of formal analysis in kinetic theory literature.

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.