Mathematical Analysis of the Water Wave Motion

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This project is concerned with the motion of water waves in both two and three space dimensions. The particular mathematical problem under investigation is that of the motion of the interface separating a two- or three-dimensional, inviscid, incompressible, and irrotational fluid, under the influence of gravity, from a region of zero density, neglecting the effects of surface tension. The questions the principal investigator seeks to answer focus on the later-time behavior of surface waves in the open ocean for initially small wave disturbances. In particular, her primary objectives are (1) to identify those initial data that lead to finite time blow-up of the interface and to understand the nature of the singularities that develop in the interface and (2) to identify the largest set of initial data that lead to situations in which blow-up is avoided for all time. The principal investigator also seeks to justify rigorously the modulation approximations of the two- and three-dimensional water wave problems.Rogue waves are relatively large surface waves that are observed in deep ocean waters. Often appearing in perfectly clear weather without warning, they are blamed for the damage and inexplicable disappearance of numerous large ships and ocean liners. With the resolution of the questions under study in this project, the principal investigator hopes to provide insights into effective designs of computational and laboratory simulations of water wave motion and thereby to advance our understanding of the causes of natural phenomena associated with such waves, including the behavior of rogue waves. The analytical tools developed to carry out the research may also serve to shed new light on related problems in mathematical physics. The proposed research will involve graduate students.