Mathematical Analysis of the Water Wave Motion

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The PI will continue her study of the motion of water waves. The focus is on the singularities in the water waves motion and the interaction of the free interface with a fixed rigid boundary. The goal is to understand the mechanism behind the phenomena of waves with angled crests, and to understand the interactions of waves with solid boundaries, such as the coast. This investigation will lead to better understandings of the structures of the equations, which will eventually lead to better modeling and more efficient numerical simulations of the phenomena. The project will involve graduate students and post-doctoral researchers. In her earlier work, the PI has shown that the two and three dimensional water wave problems are locally wellposed in smooth regime; she has also obtained the almost global and global wellposedness of the two and three dimensional water wave equations for small, smooth and localized initial data. Recently, she has been focusing on understanding the singularities in the water waves motion. She has constructed a class of self-similar solutions for the two dimensional water wave equation, and proved the local in time existence of solutions in a class of water waves that include interfaces with angled crests. Problems in the current project include the uniqueness of the solutions; the long time dynamics for water waves with angled crested interfaces; the extension of the two dimensional results to three space dimensions; the role of surface tension in non smooth water waves; and the interaction of the free interface with a fixed boundary. Besides classical tools from theories of partial differential equation and harmonic analysis, and from earlier work, the PI plans to develop new techniques to study the proposed problems.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.