摘要：

A long-standing question in fluid dynamics asserts that uniformly rotating vortex patch solutions of the 2D incompressible Euler equation must be radially symmetric when the angular velocity satisfies $\Omega \in (-\infty,0] \cup[ \frac{1}{2},+\infty)$. In a groundbreaking work, Gómez-Serrano, Park, Shi, and Yao resolved this question completely for the Euler equation on the entire plane. However, extending their methodology as sharply to the unit disk presents significant technical challenges due to the presence of boundary effects. In this talk, we present our recent work concerning the radial symmetry of uniformly rotating vortex patches in the disk. By employing continuous Steiner rearrangement, we establish a sharp rigidity theorem that characterizes the radial symmetry of such patches. Moreover, our approach also recovers prior rigidity results in the whole plane under milder regularity assumptions on the patch boundary. This presentation is based on joint works with Boquan Fan and Weicheng Zhan.