Long time inviscid damping near Couette in Sobolev spaces

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 ${\delta }_s=1\mathrm{/}3$ when $s\to 1+$ and ${\delta }_s=1\mathrm{/}2$ for $s>2$.