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arXiv · 2506.05034

Remarks on radial symmetry of stationary and uniformly-rotating solutions for the 2D Euler equation

Boquan Fan·Yuchen Wang·Weicheng Zhan
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Abstract

We prove that any uniformly rotating solution of the 2D incompressible Euler equation with compactly supported vorticity $\omega$ must be radially symmetric whenever its angular velocity satisfies $\Omega \in (-\infty,\inf \omega / 2] \cup \, [ \sup \omega / 2, +\infty )$, in both the patch and smooth settings. This result extends the rigidity theorems established in \cite{Gom2021MR4312192} (\textit{Duke Math. J.},170(13):2957-3038, 2021), which were confined to the case of non-positive angular velocities and non-negative vorticity. Moreover, our results do not impose any regularity conditions on the patch beyond requiring that its boundary consists of Jordan curves, thereby refining the previous result to encompass irregular vortex patches.

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Boquan Fan, Yuchen Wang, Weicheng Zhan. 2025-06-05. Remarks on radial symmetry of stationary and uniformly-rotating solutions for the 2D Euler equation. https://arxiv.org/abs/2506.05034

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