Speaker：Huan Xu (University of Macau)

Time：2026-9-28 15: 00

Location：Conference Room C610 at Administration Building at Haiyun Campus

Abstract:

We study two-dimensional steady incompressible Euler flows with finite total curvature. A distinguished family of such flows arises from finite-Morse-index solutions to semilinear elliptic equations in two dimensions. We establish three main properties of finite-total-curvature flows in the \(L^\infty \cap H^1_{\text{loc}}\) class. First, in the plane and the half-plane, every such flow without essential stagnation points is a parallel shear flow. Second, in the plane, the half-plane, and the infinite strip, the pressure converges uniformly to a constant at each end at infinity. Third, in the plane, the half-plane, the infinite strip, and the periodic strip, we establish sharp lower bounds for the total curvature in terms of the pressure oscillation.The rigidity result in fact extends to a broader class of flows: in the plane and the half-plane, any steady Euler flow without stagnation points is a parallel shear flow provided that \(\nabla P \in L^q\) for some \(1 \le q \le 2\). Combined with standard elliptic estimates for the pressure, this result yields a new proof of Hamel and Nadirashvili's rigidity theorems, with the regularity assumption sharpened optimally to \(H^1_{\text{loc}}\).The pressure serves as a unifying quantity throughout the analysis.