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.

本文对自由水面波的数学理论进行了三个方面的研究。 第一个方面是行波的存在性及其定性性质。 拓扑度理论和全局分支定理将被改进以适应非紧和奇异算子，并用于构造一般涡量类的大振幅周期性孤立行波。 将建立极值形式的斯托克斯波的存在性及其几何性质。 第二个方面是水波的柯西问题。 这个问题将被视为一个系统的非线性色散偏微分方程，和先验估计的长期行为的解决方案将获得。 第三个方面是水波平衡的水动力稳定性。 对于一般的自由表面重力剪切流和有限深度的小振幅旋转Stokes波，将得到线性化不稳定性的一个精确判据。 其他自由表面欧拉方程的稳定性和不稳定性，如广义涡补丁将建立。水波是应用数学的一个主要例子，描述了可能在海洋中观察到的波动，从涟漪到海啸或畸形（流氓）波的大小不等。水波数学问题的非线性特征表现出不同的行为，如卷起或崩溃，它们在数学分析以及海流和大气的工程研究中提出了巨大的挑战。拟议的研究的一个关键目标是开发新的方法和数学理论，在严格分析的数学问题，模型自由表面水波。该项目的结果将增强我们对海洋波浪流动力学的理解，并将有助于工程设计和数值模拟。在这里获得的数学进展将是有用的，在分析其他自由表面的问题所产生的涡运动的研究，这是在气候研究和材料科学相变的潜在重要性。