Flexibility and rigidity for the Couette flow in the infinite channel

Dengjun Guo; Xiaoyutao Luo; Guolin Qin

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arXiv · 2605.19971

Flexibility and rigidity for the Couette flow in the infinite channel

Dengjun Guo·Xiaoyutao Luo·Guolin Qin

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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.

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Dengjun Guo, Xiaoyutao Luo, Guolin Qin. 2026-05-19. Flexibility and rigidity for the Couette flow in the infinite channel. https://arxiv.org/abs/2605.19971

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