Inviscid dynamical structures near Couette ow

Consider inviscid ‡uids in a channel f 1 < y < 1g. For the Couette ‡ow ~v0 = (y; 0), the vertical velocity of solutions to the linearized Euler equation at ~v0 decays in time. At the nonlinear level, such inviscid damping is widely open. First, we show that in any (vorticity) H s < 3 2 neighborhood of Couette ‡ow, there exist non-parallel steady ‡ows with arbitrary minimal horizontal period. This implies that nonlinear inviscid damping is not true in any (vorticity) H s < 3 2 neighborhood of Couette ‡ow and for any horizontal period. Indeed, the long time behavior in such neighborhoods are very rich, including nontrivial steady ‡ows, stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity) H s > 3 2 neighborhood of Couette, we show that there exist no non-parallel steadily travelling ‡ows ~v (x ct; y), and no unstable shears. This suggests that the long time dynamics in H s > 3 2 neighborhoods of Couette might be much simpler. Such contrasting dynamics in H spaces with the critical power s = 3 2 is a truly nonlinear phenomena, since the linear inviscid damping near Couette is true for any initial vorticity in L: