Analytical and Computational Approaches to Free-Surface Problems in Fluid Dynamics

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Understanding the motion of fluids with free surfaces is a fundamental problem in fluid dynamics. Part of the current challenge lies in extending two-dimensional results to the three-dimensional setting. This project explores several free-surface problems, including well-posedness of three-dimensional problems for single fluids and for vortex-sheet interfaces between two shearing fluids, development of efficient numerical methods to deal with surface tension in three dimensions, and investigation of global existence of small solutions and of weak solutions.Predicting the motion of fluid interfaces is central to many branches of sciences and engineering. The aim of this project is to facilitate advances in fluid dynamics research through analysis of underlying fundamental mathematical structures. By investigating well-posedness of systems of partial differential equations, this work helps determine whether the underlying fluid-dynamical models can accurately describe physical reality. By developing efficient numerical methods, this work facilitates insight into long-time behavior and mechanisms through which singularities may form. The insight gained will significantly influence future analysis and experimental studies.