Mathematical Analysis of Vortex Sheet and Water Wave Motion

提案DMS-0400643题目：涡面和水波运动的数学分析PI：Sijue Wu，美国密歇根大学摘要涡面问题作为流体流动中涡量演化的原型。例如，我们可以把翼型的尾流看作是这类问题的典型例子.这个问题可以用不可压缩的欧拉方程来描述，其中初始涡量理想地是一个有限的Radon测度，它被支撑在一条曲线上。问题是要确定这条曲线的演变。进一步假设涡面在以后的时间内保持弯曲，则导出伯克霍夫-罗特方程。PI的初步研究表明，涡量一般不可能是合理的规律性曲线。另一方面，Delort的结果表明涡面适合作为Euler方程的弱解（对于初始非负涡量）。然而，弱解似乎是一个太大的类来描述涡面演化的特殊性质.拟开展的研究重点是通过研究相似螺旋解，了解粘性效应和涡层的演化，进一步明确涡层演化的本质。水波是我们日常生活中最熟悉的体验之一。 数学推导是不可压缩的，无旋欧拉方程，定义在运动的水域。水波的研究可以追溯到150多年前，其中PI最近建立了该问题在时间上的局部适定性，即波在有限时间内从许多初始非自相交的波面演化而不破裂。本文的研究重点是大时间行为：光滑解的整体存在性，波的破碎--引起波破碎和破碎剖面的机制。本研究的目的是从现有的极限方程理论出发，通过对全水波方程和极限方程的比较，发展足够的理论和理解，使研究具有更大的普遍性。本研究将进一步加深我们对自然现象的理解，如水波的波动和破碎，流体的混合，边界层的分离，在湍流模型中产生声音和相干结构。它将对影响我们日常生活的科学和技术产生直接影响。

Proposal DMS-0400643Title: Mathematical analysis of vortex sheet and water wave motionPI: Sijue Wu, University of MichiganABSTRACTThe vortex sheet problem serves as a prototype for the evolution ofvorticity in fluid flows. One can think for example of the wake of an airfoil as a typical problemof this type. This problem can be described by the incompressible Euler equations, wherethe initial vorticity is ideally a finite Radon measure supported on a curve. The issue is to determinethe evolution of this curve. A further assumption that the vortex sheet remains acurve at a later time leads to the Birkhoff-Rott equation. The PI's initial studyshows that a vortex sheetin general can not be a curve of reasonable regularity. On the other hand, Delort's result shows that vortex sheetfits as a weak solution of the Euler equation (for initially non-negative vorticity). However weaksolutions seem to be a class too big to describe thespecific nature of the vortex sheet evolution. The proposed research focuses on further pin point thenature of the vortex sheet evolution, through studying similarity spiral solutions, understanding the viscosity effects andthe evolution of vortex layers.Water wave is one of our most familiar experiences in daily life. A mathematical descriptionis the incompressible, irrotational Euler equation, defined in the moving water domain. Study of waterwave can be traced back to more than 150 years, in which the PI recently established the well-posedness of the problem locally in time, that is, the wave will evolutewithout breaking for a finite time period, fromany initially non-self intersecting wave surface. The proposed research focuses on the large time behavior:the global existence of smooth solutions, the wave breaking-- the mechanisms that cause thewave breaking and breaking profiles. The proposal is to initiate from existing theories on limit equations.Through comparisons of the full water wave equation with the limit equations, thePI aims at developing enoughmachinery and understandingthat lead to further research with greater generality.The proposed research will further our understanding of the nature phenomena such as the water wavemotion and wave breaking, the mixing of fluids, separation of boundary layers, generation ofsounds and coherent structuresin turbulence models. It will have a direct impact on the science and technology that influence our daily life.