Inviscid dynamical structures near Couette flow

TL;DR

This paper investigates the nonlinear dynamics near Couette flow in inviscid fluids, revealing complex behaviors and the absence of nonlinear inviscid damping in certain vorticity neighborhoods, with a transition to simpler dynamics beyond a critical regularity.

Contribution

It demonstrates the existence of non-parallel steady flows near Couette flow in low regularity spaces and the absence of such flows in higher regularity spaces, highlighting a nonlinear transition at critical regularity.

Findings

Non-parallel steady flows exist in H^s with s<3/2 near Couette.

No non-parallel steady or unstable shears in H^s with s>3/2.

Rich long-term dynamics in low regularity spaces, simpler in higher regularity.

Abstract

Consider inviscid fluids in a channel {-1<y<1}. For the Couette flow v_0=(y,0), the vertical velocity of solutions to the linearized Euler equation at v_0 decays in time. At the nonlinear level, such inviscid damping has not been proved. First, we show that in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal period. This implies that nonlinear inviscid damping is not true in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow and for any horizontal period. Indeed, the long time behavior in such neighborhoods are very rich, including nontrivial steady flows, stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity) H^{s}(s>(3/2)) neighborhood of Couette, we show that there exist no non-parallel steadily travelling flows v(x-ct,y), and no unstable shears. This suggests that the long time dynamics in H^{s}(s>(3/2)) neighborhoods of Couette might be much simpler. Such contrasting dynamics in H^{s} 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^2.