Some Mathematical Problems on Exact Solitary Water Waves

关于精确孤立水波的一些数学问题

• 批准号: 1210979
• 负责人: Shu-Ming Sun
• 金额: $ 14.57万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1210979&HistoricalAwards=false
• 关键词: Some; Mathematical; Problems; Exact; Solitary

The proposed research considers mathematical theory on the existence and stability of two- and three-dimensional traveling waves on water of finite depth moving under the influence of gravity and small (or zero) surface tension. The exact fully nonlinear equations governing the fluid flows will be used. The project includes three problems.The first one is to study the nonlinear stability of solitary waves on water with zero surface tension, based upon recent breakthrough on the linear asymptotic stability. The second problem intends to give a rigorous proof on the existence of multi-hump (or multi-solitary) waves for water with small surface tension, since such waves have been observed in experiments and derived using model equations. The third problem deals with the existence of three-dimensional waves bifurcating from two-dimensional solitary waves (also called dimension-breaking bifurcation) on water with small surface tension, which may confirm the phenomena observed from many large-scale experiments. Here, the main thrust is to use the exact equations, rather than approximate equations,to study the waves on water of finite depth. An interplay of the theories in fluid dynamics and applied analysis will be essential to this research.The mathematical theory of wave motions on a free surface over a body of water is a fascinating subject, with a long history in both applied and pure mathematical research, and with a continuing relevance to the enterprises of mankind having to do with rivers and oceans. Numerous observations in the real world, such as waves generated by boats in lakes or ships in oceans, have been studied experimentally, numerically,and mathematically. The mathematical study on the existence and stability of these surface waves is one of the important and difficult research subjects in this area. In particular, experiments and observations have confirmed that single-hump waves (called solitary waves) propagating along a channel or an open sea have a remarkable property of permanence. Yet, the stability of such waves remains an unsolved problem mathematically. Although multi-hump waves or three-dimensional waves bifurcating from two dimensional waves have been observed in many experiments, mathematical research on these waves is still lagging behind. The goal of proposed research is to make some significant mathematical advances on these problems. The research may have potential impact on many scientific research in mathematics, physics and engineering that involve fluid interfaces and wave propagation and interactions. For example, the theory may provide helpful forecasts and useful information on the propagation of tsunami waves in oceans caused by earthquakes, the prevention of giant waves generated from fast ferries that have been blamed for many boat accidents, and the properties of waves induced by storms or hurricanes in oceans which may cause tremendous damage to offshore oil rigs or marine structures.

拟议的研究考虑的数学理论的存在和稳定性的二维和三维行波的有限深度的水移动重力和小（或零）表面张力的影响下。将使用控制流体流动的精确的完全非线性方程。本课题主要包括三个方面的内容：第一，在线性渐近稳定性研究取得突破的基础上，研究零表面张力条件下孤立波的非线性稳定性。第二个问题旨在给出一个严格的证明存在的多峰（或多孤波）的水与小表面张力，因为这样的波已经在实验中观察到，并使用模型方程。第三个问题是关于在小表面张力的情况下，二维孤立波分叉成三维孤立波的存在性（也称为破维分叉），这可能证实了许多大型实验中观察到的现象。本文的主要目的是用精确方程而不是近似方程来研究有限水深水面上的波浪。流体动力学理论和应用分析理论的相互作用对这项研究是必不可少的。水体上自由表面波动的数学理论是一个迷人的课题，在应用和纯数学研究方面都有着悠久的历史，并且与人类与河流和海洋有关的事业持续相关。在真实的世界中的许多观测，例如湖中的船只或海洋中的船只产生的波浪，已经被实验、数值和数学研究。对这类表面波的存在性和稳定性的数学研究是这一领域的重点和难点之一。特别是，实验和观测已经证实，沿沿着水道或公海传播的单峰波（称为孤立波）具有显著的永久性。然而，这种波的稳定性仍然是一个数学上未解决的问题。虽然在许多实验中观察到了多峰波或二维波分叉成三维波的现象，但对这类波的数学研究还比较滞后。拟议研究的目标是在这些问题上取得一些重大的数学进展。该研究可能对数学、物理和工程中涉及流体界面、波传播和相互作用的许多科学研究产生潜在的影响。例如，该理论可以提供有关地震引起的海啸波在海洋中的传播的有用预测和有用信息，防止由快速渡轮产生的巨浪，这些巨浪被归咎于许多船只事故，以及海洋中风暴或飓风引起的波浪的性质，这些波浪可能对海上石油钻井平台或海洋结构造成巨大损害。