Mathematical Analysis of Vortex Dynamics and Waterwave Problem.

PI建议研究流体动力学的三个学科：一般两层流的界面运动、边界层问题和水波运动。一般两层流体的界面运动作为特例包括涡片运动。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在任何正时间内都不会产生Soblev类涡片。这项工作中的一些关键假设是：流体是无粘性的，没有表面张力，界面在正时间保持规则的表面。因此，一个令人感兴趣的问题是一个适合涡片运动的模型。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。这就引出了两层粘性流体的零粘性极限的研究。一个既具有数学意义又具有实际意义的相关问题是边界层问题。问题是在具有固定非空边界的区域中求出不可压缩N-S流动的零粘性极限。众所周知，困难在于边界层，在边界层内，法向速度梯度通常会变得非常大。PI的方法不同于通常的方法，在某种意义上，PI将假定不知道可能的极限方程。PI建议直接分析N-S流动，得到N-S方程解的边界部分和内部部分的定性行为。该方法将从调和分析入手。期望在解决边界层问题中发展的技术和结果将为寻找适合涡片运动的模型提供帮助。PI建议继续她对水波问题的研究。最近，PI证明了水波问题及时局部解的存在唯一性。拟议的研究集中在与水波的长时间行为有关的问题上：解的全球存在性和唯一性、水波在奇点之前的寿命以及解的奇性轮廓。该方法将来自调和分析和Clifford分析。PI在解决水波问题中开发的方法和技术已在涡片问题中得到应用。这个项目的成功将加深我们对波动、流体混合、边界层分离、声音的产生和湍流模型中的相干结构的理解。

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.