Mathematical Analysis of Vortex Dynamics and Waterwave Problem.

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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.