Flexibility and rigidity for the Couette flow in the infinite channel

TL;DR

This paper studies the existence and non-existence of steady and traveling wave solutions near Couette flow in a 2D infinite channel, identifying a critical regularity threshold that distinguishes flexible from rigid flow behaviors.

Contribution

It establishes the precise Sobolev and H"older space regularity thresholds for the existence of such solutions, revealing a sharp transition at a specific regularity index.

Findings

Existence of smooth, compactly supported steady states below the threshold.

Non-existence of relative equilibria above the threshold.

Flexible solutions are in every Gevrey class below analyticity.

Abstract

We investigate the existence of stationary and traveling wave solutions to the 2D Euler equations near the Couette flow in the infinite channel $\mathbb{R} \times [-1,1]$. For Sobolev spaces $W^{s,p}$ or H\"older spaces $C^s$, we identify the index $s= 1+ \frac1p $ as the vorticity regularity threshold separating flexibility from rigidity. Specifically, for any $s<1+ \frac1p$ we prove the existence of $C^\infty$ smooth, compactly supported steady states and traveling waves arbitrarily close to the Couette flow in all $W^{s,p}$ and $C^{1-}$. Conversely, we establish the non-existence of such relative equilibria in $ W^{s,p}$ with $s>1+ \frac1p$ or $C^{1+}$. A notable feature of the variational construction is that these flexible solutions belong to every Gevrey class strictly below the analytic threshold.