Fourier Analysis and Dispersive Equations

傅里叶分析和色散方程

• 批准号: 0330731
• 负责人: Gigliola Staffilani
• 金额: $ 3.5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-02-01 至 2004-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0330731&HistoricalAwards=false
• 关键词: Fourier; Analysis; Dispersive; Equations

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.

提议者在她的研究中提出的问题如下：给定分散方程，对于初始剖面（初始数据），必须假设多少规律性才能确保以后的波浪解决方案的存在和独特性？最初概况的条件是什么，可以保证``浪潮''``长寿''？如果波浪长期``'''的最初属性保留了很长时间？对这些现象的令人满意的分析需要回答有关相关库奇问题解决方案的长时间存在和唯一性的问题，以及解决方案的规律性。它还需要分析有关初始曲线的连续性，在有限的时间内可能对某些能量的爆炸以及爆炸率。对于长期存在和爆炸的问题，一种数学上严格的方法非常困难。当然，数值方法为理论结果提供了指南。但是，人们认为目前可用的分析技术还不够良好，无法恢复数值工作的预测。建议者使用的技术纯粹是分析性的。她使用的工具最近是在傅立叶分析和谐波分析的一般领域中开发的。由于工具是新的，因此调查是可靠地产生真正新颖的结果的。 这些方法可能会将新的见解带入了经过深入研究的理论和经验问题。提议者的主要感兴趣领域是部分微分方程。特别是，她将研究集中在色散非线性PDE上，之所以称呼，是因为它们的解决方案往往是在空间上散布的波浪。两个众所周知的方程属于此类：Schrodinger方程和Korteweg-de-vries方程。这些方程及其与波方程的组合已被提议作为物理学中许多基本波现象的模型。这些现象的示例是：光纤中信号的传播，磁场中等离子体中的非线性离子 - 传感波和血浆中的长波。