Mathematical Analysis of Vortex Dynamics and Waterwave Problem.

PI提出研究流体力学的三个主题问题：一般两层流的界面运动、边界层问题和水波运动。一般两层流体流动界面的运动包括作为特例的涡片运动。在最近的一项工作中，PI表明，任意指定独立的位置和速度数据通常不会产生任何正时间的Sobolev级涡表。这项工作的一些关键假设是：流体是无粘性的，没有表面张力，界面在正时间保持规则表面。因此，一个令人感兴趣的问题是涡片运动的适定模型。PI建议将粘度重新引入流体中，并了解界面附近粘度的影响。这导致了两层粘性流体零粘度极限的研究。边界层问题是一个具有数学和实际意义的相关问题。问题是在具有固定非空边界的区域中求不可压缩Navier-Stokes流的零粘度极限。众所周知，困难在于边界层，在边界层内法向速度梯度通常变得非常大。PI的方法与通常的方法不同，因为PI将假设不知道可能的极限方程。该方法提出直接分析Navier-Stokes流，得到Navier-Stokes方程解的边界部分和内部部分的定性行为。方法将从谐波分析出发。期望在解决边界层问题中发展的技术和结果将为寻找旋涡片运动的适定模型提供见解。PI建议继续她在水波问题上的研究。最近，PI及时证明了水波问题局部解的存在唯一性。提出的研究集中在与水波的长时间行为有关的问题：解的全局存在性和唯一性，奇点前水波的寿命，以及解的奇点剖面。方法将从谐波分析和克利福德分析两方面进行。PI开发的解决水波问题的方法和技术在旋涡片问题中得到了应用。这个项目的成功将增强我们对波动、流体混合、边界层分离、湍流模型中声音和相干结构的产生的理解。

The PI proposes to study problems in three subjects of fluid dynamics: the motion of the interface of general two layered flow, the boundary layer problem, and the motion of water wave. The motion of the interface of general two layered fluid flow includes vortex sheet motion as a special case. In a recent work, the PI shows that arbitrarily specifying independent position and velocity data generally will yield no Sobolev class vortex sheet for any positive time. Some crucial assumptions in this work are: the fluids are inviscid, there is no surface tension, and the interface remains a regular surface at positive time. A problem of interest is therefore a well-posed model for the vortex sheet motion. The PI proposes to reintroduce viscosity into the fluids, and to understand the effect of the viscosity near the interface. This leads to the study of the zero viscosity limit of two layered viscous fluids. A related problem of both mathematical and practical importance is the boundary layer problem. The question is to find the zero viscosity limit of the incompressible Navier-Stokes flow in a domain with a fixed nonempty boundary. It is well-known that the difficulty is in the boundary layer, within which the normal velocity gradient generally becomes very large. The PI's approach is different from the usual one, in the sense that the PI will assume no knowledge of the possible limit equations. The PI proposes to analyze directly the Navier-Stokes flow, and to obtain the qualitative behavior of the boundary part and the interior part of the solutions of Navier-Stokes equation. The method will be from harmonic analysis. It is expected that the techniques and results developed in solving the boundary layer problem will provide insight in finding a well-posed model for the vortex sheet motion.The PI proposes to continue her study in the water wave problem. Recently, the PI proved the existence and uniqueness of solutions locally in time for the Water wave problem. The proposed research concentrates on issues relating to the long time behavior of the water wave: the global existenceand uniqueness of solutions, the lifespan of the water wave before singularity,and the singularity profile of the solution. The method will be from harmonic analysis and Clifford analysis.The methods and techniques developed by the PI in solving the water wave problemhas found applications in the vortex sheet problem. Success in this project will enhance our understanding of the wave motion, of the mixing of fluids, separation of boundary layers, generation of soundsand coherent structures in turbulence models.