Problems in the Mathematical Theory of Water Waves

对自由面水波数学理论中的三个方面进行了研究。第一个方面是行波的存在及其定性性质。拓扑度理论和整体分支定理将被改进以适应非紧算子和奇异算子，并被用于构造一类一般涡度的周期和孤立的大振幅行波。我们将建立极值形式的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。该问题将被视为一个非线性色散偏微分方程组，并将得到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Stokes波，将得到线性化不稳定性的精确判据。将建立其他自由表面欧拉方程的稳定性和不稳定性，如广义涡片。水波是描述海洋中可能观察到的那种波动的应用数学的一个主要例子，从涟漪到海啸或反常(无赖)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，如卷曲或破裂，这给数学分析以及洋流和大气的工程研究带来了巨大的挑战。拟议研究的一个关键目标是在严格分析模拟自由水面水波的数学问题方面发展新的方法和数学理论。该项目的结果将加深我们对海浪流动力学的了解，并将有助于工程设计和数值模拟。这里所取得的数学进展将有助于分析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意义。

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.