学术报告报告主题: On a classification of steady solutions to two-dimensional Euler equations
报 告 人: 桂长峰教授（澳门大学、珠海澳大科技研究院）
报告时间： 2025年8月21日（星期四）下午16:00-17:00
报告地点： 37号楼 3A02
邀 请 人: 温焕尧 教授
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数学学院
2025年8月19日
报告摘要：
In this talk, I shall 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 semilinear 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. This talk is based on joint works with David Ruiz, Chunjing Xie and Huan Xu.
报告人介绍：
桂长峰，澳门大学数学系讲座教授，数学系主任，澳大发展基金会数学杰出学者，博士生导师。1991年在美国明尼苏达大学获博士学位。桂长峰教授曾入选国家级人才计划和海外高层次人才，入选美国数学会首届会士，美国西蒙斯会士、美国科学促进会会士，曾获得过IEEE最佳论文奖、加拿大太平洋数学研究所研究成果奖、加拿大数学中心Andrew Aisensdadt奖等荣誉。桂长峰教授研究方向为非线性偏微分方程、图像分析和处理，特别是在Allen-Cahn方程的研究、Moser-Trudinger不等式最佳常数的猜想、De Giorgi猜想和Gibbons猜想等方面取得了一系列在国际上有重大影响的工作，在国际顶一流数学期刊上发表90余篇，其中包括《Annals of Mathematics》、《Inventiones Mathematicae》、《Communications on Pure and Applied Mathematics》等顶级期刊。