Phase Space Analysis of Evolution Equations

本研究的主要目的是分析色散偏微分方程解的全局时空性质。这种分析有几个方面。首先，线性方程的全局分析在非线性发展方程的局部和全局问题中都是至关重要的。其次，在整体问题中，人们可以发现偏微分方程组(PDE)中的问题与基本几何之间的许多重要关系。所考虑的方程包括双曲型方程、具有或不具有多重性的双曲型方程组、单耦合薛定谔方程、相对论方程、Klein-Gordon方程、KdV方程和许多其他方程。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)的实例。几十年来，人们一直在研究线性方程的局部定性性质，有许多重要而有趣的发现。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，而这里几乎没有普遍可用的结果。该项目提出了一种新的统一的方法来解决这些问题，该方法基于“全局微局部分析”这一新领域，它处理所谓的傅立叶积分算子(FIOS)的全局性质，并允许远远超越已知的谱和其他方法。这些算子(FIOS)已经在局部理论中使用了超过35年，并且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算子等，都可以归结为傅里叶积分算子的形式或其相应的推广。这个项目的第一个目标是分析所需的傅立叶积分型算子的全局(空间和时间)性质。到目前为止，仅在对操作者非常严格的条件下，才在许多特殊情况下成功地研究了这些性质(部分原因是它们没有以FIOS的形式实现)。然而，最近的研究表明，通过结合局部正则性理论的最新发展和建立全局估计的新方法，应该可以处理非退化傅立叶积分算子的一般情况。对这些算子的全局估计对于非线性问题是至关重要的，但在过去很大程度上是不可接近的。预计本建议中描述的新方法将使我能够处理具有变系数的方程，这是目前整个领域的主要挑战之一。现有的基于谱理论或基于调和分析的方法在处理变系数时一般都是失效的。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大多数可以用其他方法获得的结果，并且远远超过！本项目的另一部分是使用所有这些以及其他最近发现的思想和技术来研究具有变系数和低阶项的色散方程的色散估计、Strichartz估计和光顺估计，以及它们之间的关系。所得结果将应用于非线性双曲方程、薛定谔方程和其他色散方程的局部和整体适定性问题，在线性和非线性色散方程的理论以及它们与几何和其他领域的关系方面具有重要而具有挑战性的研究。这项研究将在帝国理工学院数学系进行，同时预计将在该项目的某些方面与其他数学家合作。

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.