Regularity Properties of Dispersive PDE

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Regularity Properties of Dispersive PDE Abstract of Proposed ResearchMarkus KeelThe project is to study the long-time evolution of nonlinear dispersive partial differential equations, including Korteweg-de Vries (KdV), certain nonlinear Schroedinger (NLS) equations, and nonlinear second order hyperbolic systems. The goals are a better understanding of the long time regularity of the solutions, of their qualitative properties such as how (or whether) the solution scatters, and of how energy is or isn't transported to higher frequencies as time passes. We shall explore several new approaches to these issues, including Fourier space and physical space methods.This research is motivated by a number of considerations. First, the model equations to be investigated here are well-known approximations, or symmetry reductions, of accepted theories. For example, KdV gives approximate descriptions of certain fluid flows; while NLS arises in the description of diverse physical phenomena -including Bose-Einstein condensates, and as a description of the envelope dynamics of a general dispersive wave in a weakly nonlinear medium. The wave maps equation describes certain symmetric solutions to the Einstein vacuum equations. A second motivation for the research is that the questions considered require careful mathematical analysis of the ways different components of the nonlinear waves interact with one another. This is currently not well understood. We expect that progress made in understanding this issue will provide useful mathematical tools to study physical theories with other, possibly quite different, nonlinearities.