This project focuses on three aspects in the mathematical theory of surface water waves and related interfacial wave motions, (i) the existence of traveling waves and their properties, (ii) the Cauchy problem and dispersive properties due to surface tension, (iii) stability and instability of traveling waves. Solitary water waves of arbitrary amplitude are constructed for a general class of vorticity. A priori bounds will be obtained for Stokes-kind waves with vorticity. Long-time existence for small data will be established for the water wave problem and the vortex sheet problem with surface tension. The phase instability and blow-up will be investigated. Dispersive properties of the effect of surface tension and their consequences will be studied. The Benjamin-Feir instability will be analytically understood for Stokes waves on deep water. Stability and instability of generalized vortex patches will be investigated. Emphasis is taken on the large-scale dynamics and nonlinear behavior of the wave motions at interface; the studies ultimately hinge upon analytical proofs.
Surface water waves are manifested in a variety of natural phenomena which may be observed on the surface of the ocean or the river; they range from ripples to tsunamis or rogue waves. The subject constantly attracts attention of mathematicians as well as physicists and engineers. Furthermore, a considerable part of the mathematical theory of wave motion has been pioneered on the basis of studies of water waves. A key objective of this project is to develop new methodologies and theories in the analytical studies of surface water waves and related interfacial waves. Results from this project will help to furnish underlying principles of numerical simulations and engineering designs for the surface water waves phenomena. Deep analysis of particular problems involving water waves will stimulate the development of new mathematical ideas and analytical techniques for solving other highly nonlinear problems.
The problem of surface water waves concerns wave motions at the interface separating in two or three dimensions an incompressible inviscid fluid below a body of air, acted upon by gravity and possibly surface tension. Describing in an idealized fashion what may be observed in the ocean, water waves are a perfect specimen of applied mathematics. They host a wealth of wave phenomena, from ripples driven by surface tension to tsunamis and to rogue waves. They provide source and inspiration to several branches of mathematics. Furthermore, they impact outside of mathematics such as hydraulics and weather prediction. The water wave problem, however, presents profound and subtle difficulties for rigorous analysis, modeling and numerical simulations. Notably, boundary conditions at the, a priori unknown, fluid surface are nonlinear. The PI developed new methods in partial differential equations and applied mathematics, and extended and combined existing ones, to study several, theoretical aspects of water wave phenomena. Emphasis is placed upon the large scale dynamics and genuinely nonlinear behaviors, such as breaking and peaking, an acute understanding of which ultimately relies upon analytical proofs. Specifically, the PI explained regularizing effects of surface tension for the exact, water wave problem, and she illustrated finite time blowup and ill-posedness for various approximate models of surface water waves, whose proofs may be extended to the exact, water wave problem. The PI constructed periodic traveling waves in deep water for an arbitrary vorticity, from the zero wave to the extremal one, and she justified that no solitary wave forms exist in two dimensions in deep water. She discussed analytical and geometric properties of traveling water waves and their implications to the physical system. The PI developed a new framework to study stability and instability to sideband perturbations, for a general class of equations in wave motions, allowing nonlocal dispersion and nonlinearities. She unveiled, analytically and numerically, modulataional stability and instability for various models of water waves. Progress in the proposed research increased the body of knowledge and techniques in partial differential equations and fluid dynamics. Progress in the proposed research provided guiding principles of modeling and numerical simulations, and led to applications in related problems in applied mathematics and engineering. The PI was engaged in several applied mathematics activities at her host institution. She organized seminars and the Mathemtics in Science and Soceity colloquium, and conferences and meetings in the area of her research. She made numerous invited presentations at national and international institutions, and seminars and conferences. She gave a survey talk about her research for beginning graduate students in mathematics and wrote an expository article. The PI innovated the upper-level, undergraduate ordinary differential equations course and the first-year, graduate partial differential equations course; she complied the course contents and published online. She developed and taught a new graduate course in the area of her research. The PI mentored two PhD students and one masters student in the area of her research. The PI was involved with the women in math group at her host institution. She was a session speaker and a panelist at several women in math conferences.
