Fourier Analysis and Dispersive Equations

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The questions the proposer addresses in her research are the following: given a dispersive equation, how much regularity does one have to assume for the initial profile (initial data) in order to be able to insure existence and uniqueness of the wave solution at later times? What are the conditions on the initial profile that guarantee ``a long life'' for the wave? And if the wave does ``live'' for a long time, which of its initial properties are preserved? A satisfactory analysis of these phenomena requires answering questions on long time existence and uniqueness for the solution of the associated Cauchy problem, as well as regularity properties of the solution. It requires also analyzing continuity with respect to the initial profiles, possible blow-up of some energies in finite time, and rate of blow-up. A mathematically rigorous approach to the questions of long time existence and blow-up is very difficult. Certainly numerical methods provide a guide for theoretical results. But it is believed that the analytic techniques available at the moment are not fine enough to recover the predictions of the numerical work. The techniques that proposer uses are purely analytical. The tools that she employs have been recently developed in the general area of Fourier Analysis and Harmonic Analysis. As the tools are new, the investigation is more likely to produce truly novel results. These methods may bring new insights into well studied theoretical and empirical issues. The proposer main field of interest is Partial Differential Equations. In particular, she concentrates her research on Dispersive nonlinear PDEs, so called because their solutions tend to be waves which spread out spatially. Two well known equations belong to this class: the Schrodinger equation and the Korteweg-de-Vries equation. These equations and their combinations with the wave equation, have been proposed as models for many basic wave phenomena in Physics. Examples of these phenomena are: the propagations of signals in optic fibers, nonlinear ionic-sonic waves in plasma in magnetic field and long waves in plasma.