The Incompressible Euler Limit of the Boltzmann Equation with Accommodation Boundary Condition

TL;DR

This paper investigates the convergence of solutions from the Boltzmann equation with boundary conditions to the incompressible Euler equations, establishing conditions under which this limit holds and connecting kinetic and fluid boundary behaviors.

Contribution

It proves the incompressible Euler limit of the Boltzmann equation with accommodation boundary condition assuming small accommodation parameter, and relates kinetic boundary conditions to fluid slip boundary conditions.

Findings

Solutions of Navier-Stokes with slip boundary condition converge to Euler solutions.

The Boltzmann equation with Maxwell's accommodation condition leads to Navier-Stokes with slip boundary condition.

Incompressible Euler limit of Boltzmann equation established under small accommodation parameter.

Abstract

The convergence of solutions of the incompressible Navier-Stokes equations set in a domain with boundary to solutions of the Euler equations in the large Reynolds number limit is a challenging open problem both in 2 and 3 space dimensions. In particular it is distinct from the question of existence in the large of a smooth solution of the initial-boundary value problem for the Euler equations. The present paper proposes three results in that direction. First, if the solutions of the Navier-Stokes equations satisfy a slip boundary condition with vanishing slip coefficient in the large Reynolds number limit, we show by an energy method that they converge to the classical solution of the Euler equations on its time interval of existence. Next we show that the incompressible Navier-Stokes limit of the Boltzmann equation with Maxwell's accommodation condition at the boundary is governed by the Navier-Stokes equations with slip boundary condition, and we express the slip coefficient at the fluid level in terms of the accommodation parameter at the kinetic level. This second result is formal, in the style of [Bardos-Golse-Levermore, J. Stat. Phys. 63 (1991), 323-344]. Finally, we establish the incompressible Euler limit of the Boltzmann equation set in a domain with boundary with Maxwell's accommodation condition assuming that the accommodation parameter is small enough in terms of the Knudsen number. Our proof uses the relative entropy method following closely the analysis in [L. Saint-Raymond, Arch. Ration. Mech. Anal. 166 (2003), 47-80] in the case of the 3-torus, except for the boundary terms, which require special treatment.