  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.
