Mathematical Analysis of the Water Wave Problem

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The mathematical problem of water wave concerns the motion of the interface separating an inviscid, incompressible and irrotational fluid, under the influence of gravity, from a region of zero density (i.e. air). It is assumed that the surface tension is zero. In previous works, the PI established the local in time well-posedness of the water wave problem in Sobolev spaces of sufficient regularity. The focuses of this project are on the understanding of the nonlinear effect, the effect of the bottom topography to the long time behavior of the surface wave motions, and on finding some appropriate functional space settings describing not so smooth water wave motions. Water waves are one of our most familiar experiences in daily life. The causes of phenomena such as the rogue waves and wave breaking are still subjects of active research. Their understanding is of great importance to marine life and oceanography. This project aims at providing some qualitative understanding of water wave motion.