Mathematical Analysis of Vortex Sheet and Water Wave Motion

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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.