On Yudovich's conjecture: An overdetermined problem for constant-vorticity Euler flows with constant particle revolution period

Speaker: 徐欢 博士  澳门大学

Inviter: 秦国林 博士

Title: On Yudovich's conjecture: An overdetermined problem for constant-vorticity Euler flows with constant particle revolution period

Language: Chinese

Time & Venue: 2026.04.27 15:30-16:30  南楼620

Abstract: Let \(\Omega \subset \mathbb{R}^{2}\) be a bounded convex domain. A steady, incompressible, inviscid fluid flow with constant vorticity in \(\Omega\) is governed by the torsional problem \[ -\Delta u = 1 \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial \Omega.\] For each \(c \in (0, \max_{\Omega} u)\), the period of revolution of a fluid particle along the streamline \(\{u = c\}\) is defined by  \[ T(c) \coloneqq \int_{\{u = c\}} \frac{1}{|\nabla u|} \, ds.\] Yudovich conjectured that \(T(\cdot)\) is constant if and only if \(\Omega\) is an ellipse. In this talk, we confirm Yudovich's conjecture under the additional assumption that \(\Omega\) is symmetric with respect to both the \(x\)- and \(y\)-axes.