Topic: Classifications of steady solutions to two-dimensional Euler equations

Lecturer: Prof. Gui Changfeng, University of Macau

Time: April 2, 2026, 15:30, UTC+8

Venue: Room 208, Mathematics and Statistics Building

Biography: Gui Changfeng is a Chair Professor in the Faculty of Science and Technology and Head of the Department of Mathematics at the University of Macau, as well as a Distinguished Professor in Mathematics under the UM Development Foundation. He previously served as Assistant Professor and Associate Professor at the University of British Columbia, Canada, Professor at the University of Connecticut, USA, and Dan Palman Chair Professor in Applied Mathematics at the University of Texas at San Antonio, USA. His research focuses on nonlinear partial differential equations, image analysis and processing, and he has solved numerous challenging mathematical problems worldwide. He has published multiple papers in top-tier international mathematics journals such as Annals of Mathematics, Inventiones Mathematicae, and Communications on Pure and Applied Mathematics. He has received awards including the Pacific Institute for the Mathematical Sciences Research Prize, the Canadian Mathematical Society Aisensdadt Prize, the IEEE Signal Processing Society Best Paper Award, and the National Natural Science Foundation of China Overseas Cooperation Fund (Overseas Outstanding Young Scholar). He has been selected as a National Distinguished Expert and is a Fellow of the American Mathematical Society, Simons Fellow, and Fellow of the American Association for the Advancement of Science.

Abstract: This talk will provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the whole circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows appeared in the whole space case, there exists an additional class of steady flows for which the set of flow angles is either the upper or lower closed semicircles. This type of flows is proved to be the class of non-shear flows that have the least total curvature. A further classification of this type of solutions will also be discussed. As consequences, we obtain Liouville-type theorems for two-dimensional semi-linear elliptic equations with only bounded and measurable nonlinearity, and the structural stability of shear flows whose all stagnation points are not inflection points, including Poiseuille flow as a special case. Our proof relies on the analysis of some quantities related to the curvature of the streamlines. We shall also provide a complete classification of vanishing viscosity limits of the steady Navier-Stokes equations in certain planar domains. This talk is based on joint works with David Ruiz, Chunjing Xie and Huan Xu.

Rewritten by: Li Huihui

Edited by: Mei Mengqi, Li Tiantian

Source: School of Mathematics and Statistics