Problems in the Mathematical Theory of Water Waves

对自由表面水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。将拓扑度理论和全局分岔定理加以改进，以适应非紧性和奇异算子，并用于构造一般涡量的周期和孤立的大振幅行水波。建立极值形式Stokes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非线性色散偏微分方程系统，并将获得解的长期行为的先验估计。第三个方面是水波平衡的水动力稳定性。对于一般类型的自由表面重力剪切流和有限深度的小振幅旋转斯托克斯波，将得到线性化失稳的一个尖锐判据。建立其他自由曲面欧拉方程的稳定性和不稳定性，如广义涡斑。水波是应用数学描述海浪运动的一个主要例子，这种运动可以在海洋中观察到，从涟漪到海啸或反常（流氓）波浪的大小不等。水波数学问题的非线性特性表现出卷起或破裂等多种行为，这对数学分析以及洋流和大气的工程研究提出了巨大的挑战。提出的研究的一个关键目标是发展新的方法和数学理论，以严格分析模拟自由表面水波的数学问题。该计划的研究结果将有助我们了解海浪的动态，并有助于工程设计和数值模拟。在这里获得的数学进展将有助于分析在涡旋运动研究中出现的其他自由表面问题，这些问题在气候研究和材料科学的相变中具有潜在的重要性。

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.