Mathematical Analysis of Water Waves and Other Fluid Models

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Despite their ubiquity and importance in physics and engineering, water waves remain very difficult to analyze or to predict, largely because of the presence of the unknown, time-varying fluid domain and the nonlinear way the spatiotemporal patterns affect the transport of energy and fluid particles. While many analytical tools currently exist for studying water waves of small amplitude, this project focuses on developing new machinery to construct large-amplitude waves and to investigate their ability to persist under small perturbations. Another objective of this project is to understand how energy evolves in response to the interaction between a fluid and a solid. Results from this project will advance the mathematical theory of water waves and contribute to understanding of other mathematical models in fluid mechanics and other related branches of applied science and engineering. This research also involves training and collaboration with graduate students and postdoctoral researchers.This project will develop a rigorous study of some nonlinear partial differential equations arising from water waves and other fluid models, extending some of the analytic techniques to related contexts, and integrating the research to the education of undergraduate and graduate students. Specifically, the proposed research addresses three separate directions regarding the existence and qualitative properties of the solutions to the water wave problem as well as other physically significant systems: (1) using a novel global bifurcation theoretic machinery to construct large-amplitude front-type solutions and solitary wave solutions to a two-phase fluid system in a channel, and extend this global theory to handle traveling waves that evolve according to more general symmetry groups; (2) proving the spectral stability of solitary waves to some long-wave water wave model which are indefinite energy saddles; (3) establishing exact energy equality for a fluid-interaction model and deriving sufficient conditions for the weak inviscid limit of Navier-Stokes solutions in domains with boundary. The main ingredients and techniques involved in the study include methods from elliptic partial differential equations, bifurcation and degree theory and stability analysis, together with energy estimates and Fourier analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.