  We prove asymptotic stability of shear flows close to the planar Couette flow
in the 2D inviscid Euler equations on $\Torus \times \Real$. That is, given an
initial perturbation of the Couette flow small in a suitable regularity class,
specifically Gevrey space of class smaller than 2, the velocity converges
strongly in L^2 to a shear flow which is also close to the Couette flow. The
vorticity is asymptotically driven to small scales by a linear evolution and
weakly converges as $t \rightarrow \pm\infty$. The strong convergence of the
velocity field is sometimes referred to as inviscid damping, due to the
relationship with Landau damping in the Vlasov equations. This convergence was
formally derived at the linear level by Kelvin in 1887 and it occurs at an
algebraic rate first computed by Orr in 1907; our work appears to be the first
rigorous confirmation of this behavior on the nonlinear level.
