Mathematical aspects of surface water waves

表面水波的数学方面

• 批准号: 1008885
• 负责人: Vera Mikyoung Hur
• 金额: $ 14.32万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1008885&HistoricalAwards=false
• 关键词: Mathematical; aspects; surface; water; waves

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.

该项目侧重于表面水波和相关界面波运动的数学理论中的三个方面，（i）行波的存在及其性质，（ii）Cauchy问题和表面张力引起的色散性质，（iii）行波的稳定性和不稳定性。任意振幅的孤立波构造的一般类涡。先验边界将得到斯托克斯类波涡。对于水波问题和具有表面张力的涡面问题，将建立小数据的长时间存在性。我们将研究相不稳定性和爆破。将研究表面张力的影响及其后果的分散性能。Benjamin-Feir不稳定性将被分析理解为Stokes波在深水中。将研究广义涡斑的稳定性和不稳定性。重点放在大尺度动力学和非线性行为的波动在界面上;研究最终取决于分析证明。表面水波表现为各种自然现象，可以在海洋或河流表面观察到;它们的范围从涟漪到海啸或流氓波。这个问题不断吸引着数学家以及物理学家和工程师的注意。此外，波浪运动的数学理论的相当一部分是在水波研究的基础上开创的。该项目的一个主要目标是发展新的方法和理论，在表面水波和相关的界面波的分析研究。本计画之研究结果将有助于提供水面波浪现象之数值模拟与工程设计之基本原理。对涉及水波的特殊问题的深入分析将促进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。