Mathematical Analysis of Fluid Free Boundary Problems

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The project is concerned with modeling the motion of water waves and other related fluid free-boundary problems. The focus of the research is on the development of mathematical tools to understand the long-time behavior of water waves in two and three dimensions, with and without surface tension. The long-time behavior of solutions of the free-boundary problem of a self-gravitating incompressible fluid will also be investigated. Another research topic is devoted to the interaction of water waves with a fixed rigid boundary, such as the ocean's waves with the coast. This work will lead to better modeling and more efficient numerical simulations of the phenomena. An integral part of the project is the involvement of graduate students and postdoctoral fellows, which is crucially important for the preparation of the next generation workforce. The project will study the quartic integrability of the water wave equations and the long-time behaviors of water waves in two and three dimensions, with and without surface tension. It will also investigate the quartic and higher order integrability and long-time behaviors of solutions of the free boundary problem of a self-gravitating incompressible fluid. The goals are to unearth hidden structures of the equations, to yield novel information on the behavior of solutions, and to solve open questions such as the long-time existence and regularity of solutions under certain smallness conditions, and the long-time existence and regularity of solutions with initial interfaces of arbitrary large steepness. The project will also investigate the interaction of water waves with an arbitrary fixed rigid boundary. This project is an integral part of the PI's long-term goal of understanding behaviors of water waves and other related fluid motions, including the singularity formations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.