CAREER: Quasilinear Dispersive Evolutions in Fluid Dynamics

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

Nonlinear dispersive equations model physical phenomena arising in fluid dynamics (oceanography), quantum mechanics, plasma physics, nonlinear optics, all the way to general relativity. The purpose of this project is to improve the mathematical and scientific understanding of those equations via (i) concrete research projects aimed at studying the long-time behavior of solutions to such equations, and (ii) an educational component aimed at introducing such problems to a new and diverse generation of mathematicians. A primary focus of the principal investigator will be the analysis of the two-dimensional water wave equations, which govern the evolution of a free fluid surface, or of the interface between two fluids. The goal is to provide a better description of both the local dynamics (e.g. low regularity solutions and formation of singularities) and of global dynamics in fluid flows. Here, by singularities in free boundary problems, the principal investigator means evolutions where the interface loses smoothness, possibly forming a corner-like singularity followed by "wave breaking''. Such a behavior is exhibited in many physically important phenomena, like turbulence in ocean waves, tsunami formation, just to mention a few. While this wave breaking is easy to observe and has very strong manifestations in nature, its scientific understanding is rather poor, and the mathematical justification of the phenomena based on the constitutive equations is rather difficult and is fully open at this time. This project aims to tackle a selection of key open problems related to singularity formation and to long-time properties of solutions to several classes of dispersive and hyperbolic equations that arise from a physical or geometric context, largely motivated by fluid dynamics. The project includes an educational component aimed at raising the interest of a younger generation of researchers in those fundamental problems through workshops, seminars, REU projects, etc. One of the main goals of the project is to understand the long-time behavior of solutions of the water waves. This includes lifespan estimates as well as global in time dynamics and scattering properties, where possible. From this perspective there are two key properties that play a role: one is the "dispersive decay'', and the other is the ``resonance analysis''. Some of the key tools developed by the PI and collaborators (the "quasilinear modified energy method'', and the "testing with wave packets method'') will contribute to a better understanding of the proposed problems, but nevertheless, improvements of the existing methods and the need to develop new and robust techniques remain essential in order to describe, for instance, the singularity formation mechanism. The quasilinear nature of these equations plays a crucial role in the difficulty associated with the analysis of the long- time behavior and properties of the solutions. Water waves and related models are effective equations for the ocean dynamics that are derived in the physics literature from heuristic considerations and have very important implications on the long-time behavior of dispersive partial differential equations. 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.