The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit

TL;DR

This paper analyzes the 3D kinetic Couette flow using the Boltzmann equation in the diffusive limit, showing the solution's approximation by the Navier-Stokes-Fourier system and its convergence to a steady state.

Contribution

It introduces a novel approach combining Fourier analysis, anisotropic function spaces, and decomposition techniques to study the diffusive limit of the Boltzmann equation for 3D Couette flow.

Findings

First-order approximation governed by Navier-Stokes-Fourier system

Solutions asymptotically converge to steady planar flow

Method effectively handles small Knudsen number singularity

Abstract

In the paper we study the Boltzmann equation in the diffusive limit in a channel domain $\mathbb{T}^2\times (-1,1)$ for the 3D kinetic Couette flow. Our results demonstrate that the first-order approximation of the solutions is governed by the perturbed incompressible Navier-Stokes-Fourier system around the fluid Couette flow. Moverover, in the absence of external forces, the 3D kinetic Couette flow asymptotically converges over time to the 1D steady planar kinetic Couette flow. Our proof relies on (i) the Fourier transform on $\mathbb{T}^2$ to essentially reduce the 3D problem to a one-dimensional one, (ii) anisotropic Chemin-Lerner type function spaces, incorporating the Wiener algebra, to control nonlinear terms and address the singularity associated with a small Knudsen number in the diffusive limit, and (iii) Caflisch's decomposition, combined with the $L^2\cap L^\infty$ interplay technique, to manage the growth of large velocities.