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

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 $\mathrm{\Omega }\in (-\mathrm{\infty },\inf \omega \mathrm{/}2]\cup [\sup \omega \mathrm{/}2,+\mathrm{\infty })$, in both the patch and smooth settings. This result extends the rigidity theorems established in \cite{Gom2021MR4312192} ($\mathit{\text{Duke}}\mathit{\text{Math.}}\mathit{\text{J.}}$,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.