Flexibility and rigidity for the Couette flow in the infinite channel

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 Hölder spaces $C^s$, we identify the index $s=1+\frac{1}{p}$ as the vorticity regularity threshold separating flexibility from rigidity. Specifically, for any $s<1+\frac{1}{p}$ we prove the existence of $C^{\mathrm{\infty }}$ 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 $s>1+\frac{1}{p}$ 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.