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Asymptotic properties of solutions to hyperbolic equations

双曲方程解的渐近性质

基本信息

批准号：
EP/E062873/1
负责人：
Michael Ruzhansky
金额：
$ 39.46万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FE062873%2F1
关键词：
Asymptotic properties solutions hyperbolic equations

项目摘要

The proposed research will concentrate on the asymptotic properties of scalar and coupled hyperbolic equations and systems. There are many important examples motivating the great need in the proposed analysis: wave equations, dissipative wave equations, Klein-Gordon equations, Kirchhoff equations, Maxwell systems, elastic equations, and others. There are also many motivating examples of large systems and higher order equations. For example, Grad systems in gas dynamics depend on a number of moments, and lead to systems of order 13, 20, and higher.At the same time the so-called Hermite-Grad approach to the Fokker-Planck equation leads to an infinite system of equations for coefficients. Considering Galerkin approximations of this system produces a sequence of hyperbolic systems of the size increasing to infinity.The main aim of the project is to analyse the asymptotic properties of solutions to the linearised versions of these equations. These properties play a major role in the analysis of the local and global time well-posedness of the corresponding nonlinear equations. In fact, these asymptotic properties will be used to establish the so-called Strichartz estimates for solutions, which are the most effective modern tool to tackle the nonlinear problems.The proposed approach will be based on the geometric interpretation of the asymptotic profiles. Indeed, it turns out to be extremely difficult to trace asymptotic properties to coefficients of the original equation. This is the main reason why only very limited results are currently available on hyperbolic equations with variable coefficients. No geometric approach has been attempted before in the analysis of such problems and that is where we expect to make a major contribution. The problem will be split in two parts. First, information on coefficients and on the structure of the equation at hand will be translated into geometric properties of its characteristics and the corresponding Hamiltonian flow. Second, these geometric quantities will be used to carry out asymptotic estimation of the propagators.A similar approach was recently successfully carried out for the analysis of Schrodinger equations. However, for hyperbolic equations we have several big advantages that we plan to use. Propagators for these equations as well as transformation operators used for their reduction or conjugation are essentially of the same form. This will allow us to fully use the calculus of these operators to be able to reduce the problem of asymptotic analysis for a very wide class of equations to essentially a single scalar first order equation. Such model equation will be of the general form, but its global propagators in different form have been partly analysed from several points of view. We will considerably develop and complement the existing results with time global asymptotic analysis leading to the understanding of the dispersive properties of wide classes of equations.This will allow us to build the new approach on the available extensive analysis of different mathematical theories (microlocal analysis, symplectic geometry, harmonic analysis, normal forms, etc.) to aim at a major development of the asymptotic analysis of hyperbolic equations. It is important, challenging and timely research with deep implications in theories of linear and nonlinear hyperbolic equations and their relation to geometry and other areas.

拟议的研究将集中于标量和耦合双曲方程和系统的渐近性质。有许多重要的例子激发了对所提出的分析的巨大需求：波动方程、耗散波动方程、克莱因-戈登方程、基尔霍夫方程、麦克斯韦系统、弹性方程等等。还有许多大型系统和高阶方程的激动人心的例子。例如，气体动力学中的 Grad 系统取决于多个矩，并导致 13、20 阶和更高阶的系统。同时，福克-普朗克方程的所谓 Hermite-Grad 方法导致了系数方程的无限系统。考虑到该系统的伽辽金近似，会产生一系列尺寸增加到无穷大的双曲系统。该项目的主要目的是分析这些方程的线性化版本的解的渐近性质。这些性质在相应非线性方程的局部和全局时间适定性分析中发挥着重要作用。事实上，这些渐近性质将用于建立所谓的 Strichartz 解估计，这是解决非线性问题的最有效的现代工具。所提出的方法将基于渐近轮廓的几何解释。事实上，追踪原始方程系数的渐近性质是极其困难的。这就是为什么目前对于变系数双曲方程只有非常有限的结果的主要原因。以前没有尝试过几何方法来分析此类问题，这正是我们期望做出重大贡献的地方。问题将分为两部分。首先，有关系数和手头方程结构的信息将转化为其特征的几何属性和相应的哈密顿流。其次，这些几何量将用于进行传播子的渐近估计。最近成功地对薛定谔方程进行了分析。然而，对于双曲方程，我们计划利用几个重大优势。这些方程的传播器以及用于它们的约简或共轭的变换算子本质上具有相同的形式。这将使我们能够充分利用这些运算符的微积分，能够将非常广泛的一类方程的渐近分析问题减少到本质上单个标量一阶方程。这种模型方程将具有一般形式，但其不同形式的全局传播子已经从几个角度进行了部分分析。我们将通过时间全局渐近分析来大大发展和补充现有结果，从而理解各种方程的色散特性。这将使我们能够在不同数学理论（微局域分析、辛几何、调和分析、范式等）的现有广泛分析的基础上建立新方法，以实现双曲方程渐近分析的重大发展。这是一项重要的、具有挑战性的、及时的研究，对线性和非线性双曲方程及其与几何和其他领域的关系的理论具有深远的影响。