Mathematical aspects of surface water waves

本课题主要研究表面水波及相关界面波运动的数学理论中的三个方面：行波的存在及其性质；柯西问题及表面张力的色散性质；行波的稳定性和不稳定性。针对一般涡度构造了任意振幅的孤立水波。对于具有涡度的stokes类波，将得到一个先验边界。对于具有表面张力的水波问题和涡片问题，将建立小数据的长期存在性。研究了相不稳定性和爆破问题。研究了表面张力效应的色散特性及其后果。对于深水上的Stokes波，可以解析地理解Benjamin-Feir不稳定性。研究了广义涡旋斑块的稳定性和不稳定性。重点讨论了界面处波浪运动的大尺度动力学和非线性特性；这些研究最终取决于分析证明。表面水波表现为在海洋或河流表面可以观察到的各种自然现象；它们的范围从涟漪到海啸或巨浪。这一课题不断吸引着数学家、物理学家和工程师的注意。此外，相当一部分波浪运动的数学理论是在水波研究的基础上开创的。本项目的一个主要目标是发展表面水波和相关界面波分析研究的新方法和新理论。本文的研究结果将有助于为表面水波现象的数值模拟和工程设计提供基本原理。对涉及水波的特殊问题的深入分析将刺激新的数学思想和分析技术的发展，以解决其他高度非线性问题。

This project focuses on three aspects in the mathematical theory of surface water waves and related interfacial wave motions, (i) the existence of traveling waves and their properties, (ii) the Cauchy problem and dispersive properties due to surface tension, (iii) stability and instability of traveling waves. Solitary water waves of arbitrary amplitude are constructed for a general class of vorticity. A priori bounds will be obtained for Stokes-kind waves with vorticity. Long-time existence for small data will be established for the water wave problem and the vortex sheet problem with surface tension. The phase instability and blow-up will be investigated. Dispersive properties of the effect of surface tension and their consequences will be studied. The Benjamin-Feir instability will be analytically understood for Stokes waves on deep water. Stability and instability of generalized vortex patches will be investigated. Emphasis is taken on the large-scale dynamics and nonlinear behavior of the wave motions at interface; the studies ultimately hinge upon analytical proofs.Surface water waves are manifested in a variety of natural phenomena which may be observed on the surface of the ocean or the river; they range from ripples to tsunamis or rogue waves. The subject constantly attracts attention of mathematicians as well as physicists and engineers. Furthermore, a considerable part of the mathematical theory of wave motion has been pioneered on the basis of studies of water waves. A key objective of this project is to develop new methodologies and theories in the analytical studies of surface water waves and related interfacial waves. Results from this project will help to furnish underlying principles of numerical simulations and engineering designs for the surface water waves phenomena. Deep analysis of particular problems involving water waves will stimulate the development of new mathematical ideas and analytical techniques for solving other highly nonlinear problems.