Problems in the Mathematical Theory of Water Waves

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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.