Mathematical Analysis of Vortex Sheet and Water Wave Motion

题目：涡旋片与水波运动的数学分析题目：涡旋片问题与水波运动的数学分析题目：涡旋片问题与水波运动的数学分析题目：涡旋片问题是流体流动中涡度演化的一个原型。人们可以认为，例如尾流的翼型作为一个典型的问题，这种类型。这个问题可以用不可压缩欧拉方程来描述，其中初始涡度理想地是一个有限的氡测量，支持在曲线上。问题是确定这条曲线的演变。一个进一步的假设，旋涡片在稍后的时间保持曲线导致Birkhoff-Rott方程。PI的初步研究表明，涡旋片一般不可能是一条合理规则的曲线。另一方面，Delort的结果表明涡旋片适合作为欧拉方程的弱解（对于初始非负涡度）。然而，弱解似乎是一个太大的类别，无法描述涡旋片演化的具体性质。通过对相似螺旋解的研究、对黏度效应和涡层演化的理解，进一步研究涡旋片演化的针尖性质。水波是我们日常生活中最熟悉的经历之一。数学上的描述是不可压缩的、无旋转的欧拉方程，定义在运动的水域中。对水波的研究可以追溯到150多年前，其中PI最近确立了问题在局部时间上的适定性，即波浪从许多最初不自相交的波面在有限时间内不破裂地演变。提出的研究重点是大时间行为：光滑解的全局存在性，波浪破碎-导致波浪破碎的机制和破碎剖面。该建议是从现有的极限方程理论出发。通过对完整水波方程与极限方程的比较，pi旨在发展足够的机制和理解，从而进一步进行更广泛的研究。本文的研究将进一步加深我们对湍流模型中水波的波动和破波、流体的混合、边界层的分离、声音的产生和相干结构等自然现象的理解。它将对影响我们日常生活的科学技术产生直接影响。

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.