摘要：This talk is related to steady planar flow of an ideal fluid in a bounded simply connected domain and focuses on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. Here we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vortex patch problem has a unique solution.

简介：彭双阶，华中师范大学数学与统计学学院院长、教授。2011年获得国家杰出青年科学基金，2012年入选首批"湖北省高端人才引领培养计划"。曾获得教育部自然科学二等奖和湖北省自然科学奖一等奖, 国家级教学成果奖二等奖。正主持国家自然科学基金重点项目和教育部“长江学者与创新团队”发展计划项目。研究领域为非线性偏微分方程与非线性泛函分析，特别是改进了Lyapunov-Schmidt约化方法的应用框架、发展出基于局部Pohozaev恒等式的Blow-up技巧并用来研究非线性椭圆问题的高维集中解、二维不可压流体的稳态等离子问题及涡补丁问题，得到了一些列结果。共发表学术论文80余篇，其中多篇论文发表在Adv.Math.、Arch.Ratinal.Mech.Anal.、Proc. London.Math.Soc.,Math.Ann、Trans. AMS.、Indiana Univ. Math. J、J.Math.Pures Appl.、 Ann.I.H.Poincaré- AN、 J Funct. Anal.、Comm.PDEs等学术期刊上。

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