Asymptotic properties of solutions to hyperbolic equations

本文将集中研究标量双曲型方程和耦合双曲型方程组的渐近性质。在所提出的分析中，有许多重要的例子激发了巨大的需求：波动方程、耗散波动方程、Klein-Gordon方程、Kirchhoff方程、Maxwell系统、弹性方程等。也有许多大系统和高阶方程的激励例子。例如，气体动力学中的Grad系统依赖于一些矩，并导致13、20和更高阶的系统。同时，福克-普朗克方程的所谓Hermite-Grad方法导致系数的无限方程组。考虑该系统的Galerkin逼近，产生了一组规模增加到无穷大的双曲型方程组，该项目的主要目的是分析这些方程的线性化形式解的渐近性质。这些性质在分析相应的非线性方程的局部和全局时间适定性中起着重要作用。事实上，这些渐近性质将被用来建立所谓的解的Strichartz估计，这是处理非线性问题的最有效的现代工具。事实上，事实证明，要追踪原始方程系数的渐近性质是极其困难的。这就是为什么目前关于变系数双曲型方程的结果非常有限的主要原因。在这类问题的分析中，以前从未尝试过几何方法，而这正是我们期望作出重大贡献的地方。这个问题将被一分为二。首先，关于系数和手头方程结构的信息将被转化为其特征和相应的哈密顿流的几何性质。其次，这些几何量将被用来对传播子进行渐近估计。最近，一种类似的方法被成功地用于分析薛定谔方程。然而，对于双曲型方程，我们有几个我们计划使用的大优点。这些方程的传播子以及用于它们的约化或共轭的变换算子基本上具有相同的形式。这将使我们能够充分利用这些算子的演算，将一类非常广泛的方程的渐近分析问题归结为基本上单个标量一阶方程的问题。这样的模型方程将是一般形式的，但其不同形式的全局传播子已从几个角度进行了部分分析。我们将用时间全局渐近分析来极大地发展和补充已有的结果，从而理解大类方程的色散性质。这将使我们能够在对不同数学理论(微局部分析、辛几何、调和分析、规范形等)进行广泛分析的基础上建立新的方法。旨在介绍双曲型方程的渐近分析的一个重要进展。在线性和非线性双曲型方程的理论以及它们与几何和其他领域的关系方面，这是一项重要的、具有挑战性的、及时的、具有深刻意义的研究。

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.