Fourier Analysis and Dispersive Equations

提出者在她的研究中解决的问题如下：给定一个色散方程，人们必须对初始剖面(初始数据)假定多大的规律性，才能确保稍后波解的存在和唯一性？初始剖面上的条件是什么，可以保证海浪的“长寿”？如果海浪“活”了很长一段时间，它的哪些初始属性被保留了下来？要对这些现象进行满意的分析，就需要回答相关柯西问题解的长期存在唯一性以及解的正则性问题。它还需要分析关于初始轮廓的连续性、某些能量在有限时间内可能发生的爆炸以及爆炸的速率。对于长期存在和爆炸的问题，严格的数学方法是非常困难的。当然，数值方法为理论结果提供了指导。但人们认为，目前可用的分析技术还不足以恢复数值功的预测。提出者使用的技术纯粹是分析性的。她使用的工具是最近在傅立叶分析和调和分析的一般领域开发的。由于这些工具是新的，调查更多地是产生真正的新奇结果。这些方法可能会给已经研究得很好的理论和经验问题带来新的见解。特别是，她专注于色散非线性偏微分方程组的研究，之所以被称为色散非线性偏微分方程组，是因为它们的解往往是在空间上传播的波。有两个著名的方程属于这一类：薛定谔方程和Korteweg-de-Vries方程。这些方程及其与波动方程的组合，已被提出作为物理学中许多基本波动现象的模型。这些现象的例子有：信号在光纤中的传播，磁场中等离子体中的非线性离子声波和等离子体中的长波。

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 morelikely 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.