Steady stratified water waves and asymptotic models

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ChenDMS-1613375 The theory of water waves is a fascinating topic. The first comprehensive mathematical understanding of a fluid goes back to Euler in the 1750s, and the equations named after him continue to be the basic model used by mathematicians and engineers today. Precise and rigorous mathematical analysis of these equations makes it possible to understand the qualitative structures and behaviors of the physical systems they represent. In this project, the investigator studies: oceanic internal solitary waves, which are recognized to be highly significant components of the dynamics of the coastal sea; the detailed dynamics of wave current interactions, which can help in design of offshore engineering structures as well as in submarine navigation; and the theory of approximate models that shed light on aspects of tsunami waves. The mathematical results generated from this project are expected to advance the fundamental understanding of ocean waves and currents. Graduate students are included in the work of the project. In this project, (1) existence of large-amplitude stratified solitary waves is established; (2) continuous dependence of surface and internal solitary waves on the streamline densities in a stratified fluid is examined; (3) the stability of solitary waves in some long-wave water wave models as well as a plasma model is studied; (4) the breakdown mechanism of smooth solutions for asymptotic model equations of full water waves is studied; (5) the well-posedness (including existence, uniqueness, and continuous dependence on initial data) of global weak solutions to a class of quasilinear dispersive equations is investigated. The techniques developed in carrying out this project are expected to be useful for other nonlinear water wave equations. Graduate students are included in the work of the project.