  We prove asymptotic stability of shear flows in a neighborhood of the Couette
flow for the 2D Euler equations in the domain $\T\times[0,1]$. More precisely
we prove that if we start with a small and smooth perturbation (in a suitable
Gevrey space) of the Couette flow, then the velocity field converges strongly
to a nearby shear flow. The vorticity, which is initially assumed to be
supported in the interior of the channel, will remain supported in the interior
of the channel, will be driven to higher frequencies by the linear flow, and
will converge weakly to $0$ as $t\to\infty$, modulo the shear flows (zero mode
in $x$).
