CAREER: Analysis of Surface Water Waves

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The PI will develop new technical tools in partial differential equations and other branches of mathematics, and she will extend and combine existing tools, in order to tackle several long-standing open problems in theoretical aspects of water waves. They include (1) the global regularity versus finite-time blowup for the initial value problem, (2) the existence of traveling waves and their classification, (3) the stability and instability of traveling waves. Emphasis is placed upon the large scale dynamics and genuinely nonlinear behaviors, an acute understanding of which ultimately hinges upon analytical proofs. Emphasis is placed upon the use of the Euler equations in hydrodynamics rather than simple approximate models such as the Korteweg-de Vries equation. The PI proposes to foster applied mathematics at her host institution. She will continue organizing seminars and conferences, and she will disseminate her research through conference presentations, seminars and colloquia. The PI proposes to enhance the undergraduate ODE curriculum and develop new graduate courses. She plans to involve undergraduate and graduate students in her research and mentor graduate students and postdoctoral researchers. The PI will encourage women and minorities to pursue careers in mathematics, science and engineering, and improve the pipeline for women research mathematicians. The problem of water waves concerns the wave motion 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 an ocean or a lake, water waves are a perfect specimen of applied mathematics. They host a wealth of wave phenomena, ranging in length scale 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, from hydraulics to weather prediction. The water wave problem, notwithstanding, presents profound and subtle difficulties for rigorous analysis, modeling and numerical simulations. For one thing, the interface between the water and the air is a priori unknown and to be determined as part of the solution, namely a free boundary. Incidentally, free boundaries are mathematically challenging in their own right and they occur in numerous situations such as the melting of ice and the stretching of a flexible membrane over an obstacle. To make things worse, boundary conditions at the free surface are severely nonlinear. This project will develop new tools to advance understanding of these challenging problems.