Mathematical aspects of surface water waves

本项目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：(1)行波的存在及其性质；(2)柯西问题及其表面张力的频散性质；(3)行波的稳定性与不稳定性。对于一般的涡度类，构造了任意振幅的孤立水波。给出了具有涡度的Stokes类波动的先验界。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。将研究相的不稳定性和爆破。将研究表面张力对色散特性的影响及其后果。本杰明-费尔不稳定性将被解析地理解为深水上的斯托克斯波。我们将研究广义涡斑的稳定性和不稳定性。重点放在界面波动的大尺度动力学和非线性行为上；研究最终取决于分析证据。表面水波表现在海洋或河流表面可能观察到的各种自然现象中；它们的范围从涟漪到海啸或流浪。这个问题一直吸引着数学家、物理学家和工程师的注意。此外，波浪运动的数学理论有相当一部分是在研究水波的基础上开创的。该项目的一个关键目标是开发分析研究表面水波和相关界面波的新方法和新理论。该项目的结果将有助于为地表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解决其他高度非线性问题的新的数学思想和分析技术的发展。

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.