Mathematical Sciences: Dynamics of Partial Differential Equations

主要研究人员打算继续发展源于科学和工程的无穷维动力系统的结构稳定性理论，这些系统是由抛物方程产生的，并研究相关问题，如抛物方程的Floquet理论，发展正常双曲不变流形的持久性理论，正常双曲不变流形的稳定和不稳定流形的存在性，以及由偏微分方程产生的无限维动力系统的不变叶理论。然而，由抛物方程产生的无限维动力系统是不可逆的，相空间也不是局部紧的。这一特点给有限维动力系统的研究带来了前所未有的困难，需要发展新的方法来理解这些系统的本质。最近得到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。本研究将采用他在前作中发现的技巧。预计这项工作将有助于更好地理解由偏微分方程组描述的物理系统模型的动力学。演化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(比如微分方程)近似地描述了物理现实。要了解物理系统的定性性质，不仅需要研究数学模型，还需要研究模型的扰动。我们还需要研究扰动模型的定性属性如何与原始模型的定性属性相关。这对于模型的数值计算尤其重要。由于舍入误差和数值格式，数值计算所研究的模型实际上是原始模型的摄动。从Poincare，Liapunov和Birkhoff开始，许多数学家和科学家都考虑过动力系统理论。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。对于结构稳定的系统，定性性质在系统的小扰动下保持不变。为了理解系统的动力学，人们需要研究不变集的存在，特别是平衡点、周期轨道、不变环和吸引子，来研究它们的结构，并知道它们附近发生了什么(附近的解是接近不变集，还是留在附近，或者离开邻域)。一个基本的问题是研究不变流形的持久性和不变流形附近流动的定性性质。不变流形和不变叶理论已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.