Hydrodynamic Limit with Geometric Correction of Stationary Boltzmann Equation

TL;DR

This paper investigates the hydrodynamic limit of the stationary Boltzmann equation in a bounded domain, demonstrating approximation by fluid equations with boundary layer corrections and challenging classical boundary layer assumptions.

Contribution

It establishes the validity of approximating solutions by Navier-Stokes-Fourier systems with geometric boundary layers and provides a counterexample to classical boundary layer theory.

Findings

Solution approximated by interior and boundary layer components

Boundary layer includes geometric correction

Counterexample to classical boundary layer behavior

Abstract

We consider the hydrodynamic limit of a stationary Boltzmann equation in a unit plate with in-flow boundary. We prove the solution can be approximated in $L^{\infty}$ by the sum of interior solution which satisfies steady incompressible Navier-Stokes-Fourier system, and boundary layer with geometric correction. Also, we construct a counterexample to the classical theory which states the behavior of solution near boundary can be described by the Knudsen layer derived from the Milne problem.