  We study the Stokes-transport system in a two-dimensional channel with
horizontally moving boundaries, which serves as a reduced model for
oceanography and sedimentation. The density is transported by the velocity
field, satisfying the momentum balance between viscosity, pressure, and gravity
effects, described by the Stokes equation at any given time. Due to the
presence of moving boundaries, stratified densities with the Couette flow
constitute one class of steady states. In this paper, we investigate the
asymptotic stability of these steady states. We prove that if the stratified
density is close to a constant density and the perturbation belongs to the
Gevrey-3 class with compact support away from the boundary, then the velocity
will converge to the Couette flow as time approaches infinity. More precisely,
we prove that the horizontal perturbed velocity decays as $\frac{1}{\langle
t\rangle^3}$ and the vertical perturbed velocity decays as $\frac{1}{\langle
t\rangle^4}$.
