Problems in the Mathematical Theory of Water Waves

水波数学理论问题

• 批准号: 0707647
• 负责人: Vera Mikyoung Hur
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707647&HistoricalAwards=false
• 关键词: Problems; Mathematical; Theory; Water; Waves

Three aspects in the mathematical theory of free surface water waves are being studied. The first aspect is the existence of traveling waves and their qualitative properties. Topological degree theory and the global bifurcation theorem will be refined to adapt to non-compact and singular operators and employed to construct periodic and solitary traveling water waves of large amplitude for a general class of vorticity. The existence of Stokes waves of extremal form and their geometric properties will be established. The second aspect is the Cauchy problem for water waves. This problem will be viewed as a system of nonlinear dispersive partial differential equations, and a priori estimates for long-time behavior of solutions will be obtained. The third aspect is the hydrodynamic stability of equilibria of water waves. A sharp criterion for linearized instablility will be obtained for a general class of free-surface gravity shear flows and small-amplitude rotational Stokes waves of finite depth. Stability and instability of other free-surface Euler equations such as generalized vortex patches will be established.Water waves are a prime example of applied mathematics describing wave motions of the kind which may be observed in the ocean, ranging in size from ripples to tsunamis or freak (rogue) waves. Nonlinearities characteristic of the mathematical problem for water waves demonstrate diverse behaviors such as rollup or breakdown, and they pose great challenges in mathematical analysis as well as engineering studies of ocean currents and the atmosphere. A key objective of the proposed research is to develop new methodologies and mathematical theories in the rigorous analysis of the mathematical problem which models free-surface water waves. Results from the proposed project will enhance our understanding of the dynamics of the ocean wave currents, and they will help engineering designs and numerical simulations. Mathematical advances obtained here will be useful in the analysis of other free-surface problems arising in the study of vortex motions, which are of potential importance in climate studies and phase transitions in material science.

正在研究自由地表水波的数学理论中的三个方面。 第一个方面是存在波浪及其定性特性。 拓扑学理论和全球分叉定理将被完善，以适应非压缩和奇异算子，并用于构建一般涡流的周期性和孤立的流动水波。 将建立极端形式及其几何特性的Stokes波的存在。 第二个方面是水波的凯奇问题。 该问题将被视为非线性分散偏微分方程的系统，并且将获得有关解决方案长期行为的先验估计。 第三个方面是水波平衡的流体动力稳定性。 对于一般的自由表面重力剪切流动和小振幅旋转的有限深度的波浪，将获得有关线性稳定性的尖锐标准。 将建立其他自由表面Euler方程（例如广义涡流斑块）的稳定性和不稳定。水是应用数学的一个主要例子，描述了可以在海洋中观察到的波动运动的波动，从波纹到Tsunamis或Tsunamis或Freak（Roge）（Rogue）波浪。数学问题的非线性特征表明，诸如卷或分解等各种行为，它们在数学分析以及洋流和大气的工程研究中构成了巨大挑战。拟议的研究的一个关键目的是在对自由表面水波建模的数学问题的严格分析中开发新的方法和数学理论。拟议项目的结果将增强我们对海浪电流动态的理解，它们将有助于工程设计和数值模拟。此处获得的数学进步将在分析涡流运动研究中引起的其他自由表面问题的分析，这在材料科学中的气候研究和相位过渡中具有潜在的重要性。