The Boltzmann equation for plane Couette flow

TL;DR

This paper establishes the existence and stability of non-equilibrium stationary solutions to the Boltzmann equation modeling plane Couette flow of a rarefied gas, revealing polynomial tails and exponential convergence for small shear rates.

Contribution

It provides the first rigorous proof of stationary solutions and their stability for the Boltzmann equation under shear flow with Maxwell molecules, using advanced perturbation techniques.

Findings

Existence of stationary solutions for small shear rates.

Polynomial tail behavior at large velocities.

Exponential convergence to steady state.

Abstract

In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at $y=\pm L$ moving relative to each other with opposite velocities $(\pm \alpha L,0,0)$ along the $x$-direction. Assuming that the stationary state takes the specific form of $F(y,v_x-\alpha y,v_y,v_z)$ with the $x$-component of the molecular velocity sheared linearly along the $y$-direction, such steady flow is governed by a boundary value problem on a steady nonlinear Boltzmann equation driven by an external shear force under the homogeneous non-moving diffuse reflection boundary condition. In case of the Maxwell molecule collisions, we establish the existence of spatially inhomogeneous non-equilibrium stationary solutions to the steady problem for any small enough shear rate $\alpha>0$ via an elaborate perturbation approach using Caflisch's decomposition together with Guo's $L^\infty\cap L^2$ theory. The result indicates the polynomial tail at large velocities for the stationary distribution. Moreover, the large time asymptotic stability of the stationary solution with an exponential convergence is also obtained and as a consequence the nonnegativity of the steady profile is justified.