Speaker: 詹伟城副教授，厦门大学

Inviter: 曹道民研究员

Title: On the radial symmetry of stationary and uniformly rotating solutions of the 2D Euler equation

Language: Chinese

Time & Venue: 2025.03.10 10:00-11:00 南楼224

Abstract: In this talk, I will present recent results on the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler equation in the unit disc, considering both the smooth and patch regimes. In the patch setting, we establish that any uniformly rotating vortex patch with angular velocity $\Omega\le 0$ or $\Omega \ge 1/2$ must be radially symmetric, with both bounds being sharp. This result holds regardless of the smoothness or connectivity of the patch boundary, including cases where the boundary consists of multiple Jordan curves. For smooth solutions, we prove that a uniformly rotating solution $\omega_0$ is necessarily radially symmetric if its angular velocity satisfies $\Omega\le \inf \omega_0/2$ or $\Omega\ge \sup \omega_0/2$. Moreover, we will discuss analogous results in the entire plane. The proof relies on the symmetry properties of non-negative solutions to elliptic equations. To address the symmetries of non-negative solutions to piecewise coupled semilinear elliptic equations, we develop a novel approach tailored to this setting. This talk is based on joint work with Boquan Fan and Yuchen Wang.