Regularity Properties of Dispersive PDE

本课题主要研究非线性色散偏微分方程的长时间演化问题，包括Korteweg-de Vries （KdV）方程、某些非线性薛定谔（NLS）方程和非线性二阶双曲型方程组。目标是更好地理解溶液的长时间规律性，它们的定性性质，如溶液如何（或是否）散射，以及随着时间的推移能量如何或不如何传输到更高的频率。我们将探索解决这些问题的几种新方法，包括傅里叶空间和物理空间方法。这项研究的动机是出于一些考虑。首先，这里要研究的模型方程是众所周知的近似，或对称约简，接受的理论。例如，KdV给出了某些流体流动的近似描述；而NLS则出现在对各种物理现象的描述中，包括玻色-爱因斯坦凝聚，以及对弱非线性介质中一般色散波的包络动力学的描述。波映射方程描述了爱因斯坦真空方程的某些对称解。这项研究的第二个动机是，所考虑的问题需要对非线性波的不同组成部分相互作用的方式进行仔细的数学分析。这一点目前还没有得到很好的理解。我们期望在理解这个问题上取得的进展将为研究其他可能完全不同的非线性的物理理论提供有用的数学工具。

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.