The Boltzmann equation for plane Couette flow

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 for a steady nonlinear Boltzmann equation driven by an external shear force under the homogeneous nonmoving diffuse reflection boundary condition. In the case of the Maxwell molecule collisions, we establish the existence of spatially inhomogeneous nonequilibrium 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 a polynomial tail at large velocities for the stationary distribution. Moreover, the large time asymptotic stability of the stationary solution with exponential convergence is also obtained and as a consequence the nonnegativity of the steady profile is justified.