Long time inviscid damping near Couette in Sobolev spaces

TL;DR

This paper provides an elementary proof of long-term inviscid damping near the Couette flow for 2D Euler equations in Sobolev spaces, showing velocity decay over extended times for small perturbations.

Contribution

It introduces a simplified proof of inviscid damping in Sobolev spaces near Couette flow, extending the understanding of long-time behavior for perturbations.

Findings

Velocity damping persists up to time scales depending on initial perturbation size.

Damping time scales as $O( ext{perturbation}^{-rac{1}{3}})$ for $s o 1+$ and $O( ext{perturbation}^{-rac{1}{2}})$ for $s > 2$.

Elementary proof technique applicable for Sobolev perturbations.

Abstract

We give an elementary proof of long time inviscid damping for Sobolev perturbations near the Couette flow $(y,0)$ for the 2D Euler equations on $\mathbb{T} \times \mathbb{R}$. For any $s>1$ and any initial vorticity perturbation of size $O(\epsilon)$ in $H^s$, we obtain velocity damping estimates up to a time scale $ t = O(\epsilon^{-\delta_s} )$, where $\delta_s=1/3$ when $s\to 1+$ and $\delta_s=1/2$ for $s>2$.