Regularity Properties of Dispersive PDE

色散偏微分方程的正则性质

• 批准号: 0602792
• 负责人: Markus Keel
• 金额: --
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0602792&HistoricalAwards=false
• 关键词: Regularity; Properties; Dispersive; PDE

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.

色散偏微分方程解的正则性研究摘要研究非线性色散偏微分方程组的长期演化，包括KdV方程、某些非线性薛定谔方程和非线性二阶双曲型方程。目标是更好地了解解决方案的长期规律性，它们的定性属性，如解决方案如何(或是否)散射，以及能量如何随着时间的推移被传输到或不被传输到更高的频率。我们将探索几种新的方法来解决这些问题，包括傅立叶空间和物理空间方法。这项研究是出于一些考虑。首先，这里要研究的模型方程是公认理论的众所周知的近似或对称性约化。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，以及描述弱非线性介质中一般色散波的包络动力学。波图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分量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提供有用的数学工具，用其他可能完全不同的非线性来研究物理理论。