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

TL;DR

This paper proves that uniformly rotating solutions of the 2D Euler equation with compact vorticity are radially symmetric under certain angular velocity conditions, extending previous rigidity results to irregular vortex patches without regularity constraints.

Contribution

It extends existing rigidity theorems to include irregular vortex patches and broadens the angular velocity conditions for radial symmetry in 2D Euler solutions.

Findings

Uniformly rotating solutions with certain angular velocities are radially symmetric.

The result applies to both patch and smooth vorticity settings.

No regularity conditions beyond Jordan curve boundaries are required.

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.