北航数学论坛学术报告

--- 分析、偏微分方程与动力系统讨论班(2024秋季第2讲)

On a classification of steady solutions to two-dimensional Euler equations

桂长峰 教授

(澳门大学)

时间：2024年9月25日（周三下午）16：00-17：00

地点：沙河主E404

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. 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 a joint work with Huan Xu and Chunjing Xie.

报告人简介: 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，于2013年当选美国数学会首届会士，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。他主要从事偏微分方程理论研究，特别在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，在国际一流数学学术期刊发表论文80余篇，其中包括Annals of Mathematics, Inventiones Mathematicae等顶级期刊。

邀请人：戴蔚 彭发

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