Mathematical Analysis of the Water Wave Motion

这个项目涉及到水波在二维和三维空间中的运动。正在研究的特殊数学问题是，在重力的影响下，将二维或三维的、无粘性、不可压缩和无旋的流体从零密度区域分离出来的界面的运动，忽略了表面张力的影响。首席研究人员试图回答的问题集中在开放海洋中最初较小的海浪扰动的表面波后来的行为。具体地说，她的主要目标是(1)确定那些导致界面有限时间爆破的初始数据，并了解在界面上发展的奇点的性质，以及(2)识别导致永远避免爆破的情况的最大初始数据集。首席研究人员还试图严格证明二维和三维水波问题的调制近似是合理的。流浪是在深海中观察到的相对较大的表面波。它们经常在完全晴朗的天气中毫无征兆地出现，它们被指责为许多大型船只和远洋客轮的损坏和莫名其妙的失踪的罪魁祸首。随着这个项目中正在研究的问题的解决，首席研究员希望对水波动的计算和实验室模拟的有效设计提供见解，从而促进我们对与这种波有关的自然现象的原因的理解，包括无赖波的行为。为开展这项研究而开发的分析工具也可能有助于揭示数学物理中的相关问题。这项拟议的研究将涉及研究生。

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.