Mathematical Sciences: Partial Differential Equations and Harmonic Analysis for the Sublaplacians

项目负责人拟继续发展由抛物方程等产生的无限维动力系统的结构稳定性理论，并研究抛物方程的Floquet理论等相关问题，发展通常双曲不变流形的持续性理论。一般双曲不变流形的稳定流形与不稳定流形的存在性，以及由偏微分方程生成的无限维动力系统的不变叶理理论。然而，由抛物方程等产生的无限维动力系统是不可逆的，相空间也不是局部紧致的。这一特点造成了在有限维动力系统的研究中没有遇到的困难，需要开发新的方法来理解这些系统的本质。最近获得了标量反应扩散方程、Cahn-Hilliard方程和相场系统的结构稳定性理论。PI在他以前的作品中发现的技术将在当前的研究中使用。期望这项工作将有助于更好地理解由偏微分方程描述的物理系统模型的动力学。演化的物理系统状态的数学模型是动力系统研究的主题。动力系统研究的主要目标是了解系统中状态的长期行为。在应用中，数学模型（如微分方程）近似地描述物理现实。要了解一个物理系统的定性性质，不仅要研究数学模型，还要研究模型的扰动。我们还需要研究摄动模型的定性性质与原始模型的定性性质之间的关系。这对模型的数值计算尤其重要。由于四舍五入误差和数值格式的存在，数值计算所研究的模型实际上是对原模型的扰动。从庞加莱、李亚普诺夫和伯克霍夫开始，许多数学家和科学家都在考虑动力系统的理论。动力系统理论中的一个基本问题是动力系统的结构稳定性问题。对于结构稳定的系统，在系统的小扰动下，其定性性质保持不变。为了理解系统的动力学，人们需要研究不变量集的存在性，特别是平衡点、周期轨道、不变量环面和吸引子，研究它们的结构并知道在它们附近发生了什么（附近的解是接近不变量集，还是留在附近，还是离开附近）。一个基本问题是研究不变流形的持续性和研究不变流形附近流动的定性性质。不变流形和不变叶形理论已经成为研究动力系统的基本工具。

Abstract Lu The principal investigator intends to continue to develop the theory of structural stability for infinite dimensional dynamical systems originating in science and engineering, which are generated by, for example, parabolic equations and to study the related problems such as Floquet theory for parabolic equations, to develop the theory of the persistence of normally hyperbolic invariant manifolds, the existence of stable and unstable manifolds of the normally hyperbolic invariant manifolds, and the theory of invariant foliations for infinite dimensional dynamical systems generated by partial differential equations. However, the infinite dimensional dynamical systems generated by, for example, parabolic equations are not reversible and the phase spaces are not locally compact. This characteristic creates difficulties not encountered in the study of finite dimensional dynamical systems and new methods need to be developed to understand the nature of these systems. A theory of structural stability for scalar reaction-diffusion equations, the Cahn-Hilliard Equation and Phase-Field System has recently been obtained. Techniques found by the PI in his previous works will be employed in the current studies. It is expected that this work will contribute to a better understanding of the dynamics of the models of physical systems described by partial differential equations. Mathematical models for the state of an evolving physical system are the subject of investigation of dynamical systems. The main goal of the study of dynamical systems is to understand the long term behavior of states in the systems. In applications, the mathematical models (say differential equations) approximately describe physical reality. To understand the qualitative properties of a physical system, one needs to investigate not only the mathematical model but also the perturbations of the model. One also needs to study how the qualitative properties of the perturbed models are related to the qualitative properties of the original model. This is especially important to the numerical computations for the models. Because of round off error and numerical schemes, the model studied by the numerical computations actually is a perturbation of the original model.The theory for dynamical systems has been considered by many mathematicians and scientists starting with Poincare, Liapunov, and Birkhoff. One of the fundamental problems in the theory of dynamical systems is the structural stability of dynamical systems. For a structurally stable system, the qualitative properties are preserved under small perturbations of the system. To understand the dynamics of a system, one needs to investigate the existence of invariant sets, in particular, such as equilibrium points, periodic orbits, invariant tori, and attractors, to study their structures and to know what happens in their vicinity (do the nearby solutions approach the invariant set, or stay nearby, or leave the neighborhood). A fundamental problem is to study the persistence of invariant manifolds and to study the qualitative properties of the flow nearby invariant manifolds. The theory of invariant manifolds and invariant foliations has become a fundamental tool for the study of dynamical systems.