Incompressible Euler limit from the Boltzmann equation with Maxwell reflection boundary condition in the half-space

TL;DR

This paper rigorously justifies the incompressible Euler limit from the Boltzmann equation with Maxwell boundary conditions in a half-space, employing conservation laws and analytic methods to handle boundary layer complexities.

Contribution

It introduces a novel approach using conservation laws to control derivative loss, enabling the justification of the Euler limit with boundary layers in the Boltzmann framework.

Findings

Established uniform estimates for the remainder equations.

Validated the incompressible Euler limit in shear flow scenarios.

Handled boundary layer effects without loss of regularity.

Abstract

In this paper, we rigorously justify the incompressible Euler limit of the Boltzmann equation with general Maxwell reflection boundary condition in the half-space. The accommodation coefficient $\alpha \in (0,1]$ is assumed to be $O(1)$. Our construction of solutions includes the interior fluid part and Knudsen-Prandtl coupled boundary layers. The corresponding solutions to the nonlinear Euler and nonlinear Prandtl systems are taken to be shear flows. Due to the presence of the nonlinear Prandtl layer, the remainder equation loses one order normal derivative. The key technical novelty lies in employing the full conservation laws to convert this loss of the normal derivative into the loss of tangential spatial derivative, avoiding any loss of regularity in time. By working within an analytic $L^2 \mbox{-} L^\infty$ framework, we establish the uniform estimate on the remainder equations, thus justify the validity of the incompressible Euler limit from the Boltzmann equation for the shear flow case.