Mathematical Analysis of Water Waves

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Despite the ubiquity of wave phenomena, the mathematical analysis of the dynamics of such free boundary problems still presents great challenges of significant current interest to the research community, understanding the mechanisms of singularities, wave breaking and rogue waves is of great importance to science and engineering. The PI proposes to continue her study on the evolution of water waves. The focus in this funding period is on understanding singularities as well as long time behaviors of the water wave motion. The PI will use rigorous analytical tools to tackle the problems. Through involving graduate students and post doctors, the project will provide a training ground for a younger generation of researchers. The mathematical problem of water waves concerns the motion of the interface separating an inviscid, incompressible, irrotational fluid, under the influence of gravity, from a region of zero density in two and three space dimensions, neglecting surface tension. In prior funding periods, the PI showed that the Taylor sign condition always holds, and obtained the local well-posedness of the two and three dimensional water wave problems for arbitrary (smooth) data. And the PI showed that the nature of the nonlinearity of the water wave problems are of cubic or higher orders, and obtained the almost global well-posedness for the two dimensional and global well-posedness for the three dimensional water wave problems for small and smooth initial data. Furthermore, to understand singular behaviors in the water wave motion, the PI constructed a class of self-similar solutions for the two dimensional water wave equation in the regime where convection is in dominance. The PI proposes to continue her study of singularities as well as the long time behaviors of the water wave motion. Particular problems include the stability of the self-similar solution she constructed and the long time evolution for a broader class of initial data. The objective is to achieve good understanding of a basic type of waves--waves with angled crests, and its role in the long time water wave motion, and to obtain better understanding of dispersion and the structure of the water wave equations.