Radial symmetry of stationary and uniformly-rotating solutions to the 2D Euler equation in a disc

TL;DR

This paper investigates the radial symmetry of stationary and rotating solutions to the 2D Euler equations in a disc, establishing sharp conditions under which solutions must be radially symmetric, using novel elliptic symmetry techniques.

Contribution

It proves sharp symmetry results for rotating patches and smooth solutions of the 2D Euler equations in a disc, introducing new methods for elliptic symmetry analysis.

Findings

Rotating patches with certain angular velocities are necessarily radial.

Smooth solutions with specific angular velocity bounds are radially symmetric.

New approach for symmetry of solutions to coupled elliptic equations.

Abstract

We study the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler equation in the unit disc, both in the smooth setting and the patch setting. In the patch setting, we prove that every uniformly rotating patch with angular velocity $\Omega\le 0$ or $\Omega \ge 1/2$ must be radial, where both bounds are sharp. The conclusion holds under the assumption that the rotating patch considered is disconnected, with its boundaries consisting of several Jordan curves. We also show that every uniformly rotating smooth solution $\omega_0$ must be radially symmetric if its angular velocity $\Omega\le \inf \omega_0/2$ or $\Omega\ge \sup \omega_0/2$. The proof is based on the symmetry properties of non-negative solutions to elliptic problems. A newly tailored approach is developed to address the symmetries of non-negative solutions to piecewise coupled semi-linear elliptic equations.