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Phase Space Analysis of Evolution Equations

演化方程的相空间分析

基本信息

批准号：
EP/G007233/1
负责人：
Michael Ruzhansky
金额：
$ 71.02万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG007233%2F1
关键词：
Phase Space Analysis Evolution Equations

项目摘要

The main purpose of the proposed research is to analyse global space time properties of dispersive partial differential equations. There are several aspects of such analysis. First, the global analysis of linear equations is crucial in both local and global problems for nonlinear evolution equations. Second, in global problems one finds many important relations between problems in partial differential equations (PDEs) and the underlying geometry. Equations under consideration include hyperbolic equations, hyperbolic systems with or without multiplicities, single and coupled Schrodinger type equations, relativistic equations, Klein-Gordon, KdV and many others. Such equations are all called dispersive equations because there are many similarities in the behaviour of their solutions exhibiting instances of the so-called dispersion (of energy, moments, singularities, or of other information).Local qualitative properties of linear equations have been studied for decades with many important and fascinating discoveries. However, for their nonlinear versions one needs global quantitative information on the behaviour of linearised equations, and here almost no results are available in general. The proposed project suggests a new unified approach to these problems based on the new area of ``global microlocal analysis'' which deals with global properties of so-called Fourier integral operators (FIOs) and which allows to go far beyond the known spectral and other methods.These operators (FIOs) have been used in the local theories for over 35 years and proved to be very efficient since they encode many analytic and geometric properties of equations. For example, solutions to Cauchy problems for hyperbolic equations, transformations operators between different types of dispersive equations, etc., can all be reduced to the form of Fourier integral operators or their relevant extensions. The first aim of this project is to analyse required global (space and time) properties of Fourier integral type operators. These properties have been successfully studied so far in a number of special cases only under very restrictive conditions on the operator (partly because they were not realised in the form of FIOs). However, recent research indicates that it should be possible to treat the general case of nondegenerate Fourier integral operators by combining recent developments in the local regularity theory with new approaches for establishing global estimates. Global estimates for these operators are of crucial importance for nonlinear problems but were largely unapproachable in the past.It is expected that the new approach described in this proposal will allow me to deal with equations with variable coefficients which is nowadays one of the main challenges of the whole area. Present methods coming from spectral theory or from harmonic analysis generally fail when dealing with variable coefficients. At the same time the approach that I propose here is very well suited for it. In fact, already for some classes of equations it allowed to recover and improve most of the results that can be obtained with other approaches, and go far beyond!Another part of the project is to use all this as well as other recently discovered ideas and techniques to investigate dispersive, Strichartz, and smoothing estimates for dispersive equations with variable coefficients and lower order terms, and relations between them. The obtained results will be applied to local and global well-posedness questions of nonlinear hyperbolic, Schrodinger and other dispersive equations.It is important and challenging research with deep implications in theories of linear and nonlinear dispersive equations and their relation to geometry and other areas. The research will be undertaken at the Mathematics Department of Imperial College, while collaboration with other mathematicians on some aspects of this project is expected.

本文的主要目的是研究色散偏微分方程的全局时空性质。这种分析有几个方面。首先，线性方程组的整体分析在非线性发展方程的局部和整体问题中都是至关重要的。第二，在整体问题中，人们发现偏微分方程（PDE）问题与基本几何之间存在许多重要关系。考虑的方程包括双曲方程，双曲系统有或没有多重性，单一和耦合薛定谔型方程，相对论方程，克莱因-戈登，KdV和许多其他。这些方程都被称为色散方程，因为它们的解的行为有许多相似之处，表现出所谓的色散（能量，矩，奇点或其他信息）。线性方程的局部定性性质已经研究了几十年，有许多重要和迷人的发现。然而，对于它们的非线性版本，人们需要关于线性化方程行为的全局定量信息，并且这里几乎没有结果。拟议的项目提出了一种新的统一办法来解决这些问题，其基础是“全球微局部分析”这一新领域，该领域处理所谓傅立叶积分算子的全球性质，并允许远远超出已知的谱和其他方法。已经被用于局部理论超过35年，并被证明是非常有效的，因为它们编码了方程的许多解析和几何性质。例如，双曲型方程柯西问题的解、不同类型色散方程之间的变换运算符等，都可以归结为傅里叶积分算子或其相关的扩展形式。该项目的第一个目的是分析所需的整体（空间和时间）的傅立叶积分型运营商的属性。这些属性已经成功地研究到目前为止，在一些特殊情况下，只有在非常严格的条件下，对运营商（部分原因是他们没有实现的形式FIO）。然而，最近的研究表明，它应该是可能的治疗一般情况下的非退化傅立叶积分算子相结合的局部正则性理论的最新发展与新的方法建立全球估计。这些运营商的全球估计是至关重要的非线性问题，但在很大程度上是不可接近的在过去，预计在这个建议中所描述的新方法将允许我处理的变系数方程，这是当今整个地区的主要挑战之一。目前的方法来自频谱理论或谐波分析通常失败时，处理变系数。与此同时，我在这里提出的方法非常适合它。事实上，对于某些类型的方程，它已经允许恢复和改进可以用其他方法获得的大部分结果，并且远远超出！该项目的另一部分是使用所有这些以及其他最近发现的想法和技术来研究具有变系数和低阶项的色散方程的色散，Eschhartz和平滑估计，以及它们之间的关系。所得结果将应用于非线性双曲型、Schrodinger等色散方程的局部和整体适定性问题，对线性和非线性色散方程的理论及其与几何和其他领域的关系具有重要意义和挑战性.这项研究将在帝国理工学院数学系进行，预计将与其他数学家就该项目的某些方面进行合作。