Long-Time Behavior and Stability of Infinite-Dimensional Dynamical Systems

在这个项目中，PI将应用谱和变分方法，以及傅立叶分析技术，研究哈密顿和色散偏微分方程理论中的一些突出的开放问题。该项目的第一部分重点关注Kuramoto-Sivashinsky和Burgers-Sivashinsky方程解的长期行为。要解决的问题涉及的整体适定性，吸引集的存在性，和正则性的解决方案。PI将继续她的工作，发展一个系统的方法来解决问题的存在性和规律性的吸引力耗散偏微分方程的情况下，无界域。特别令人感兴趣的是二维Navier-Stokes方程和流体动力学的“alpha模型”。一种创新的方法，使用调制方程和谱分解技术将被用来研究孤立波，驻波，和类似的特殊的解决方案在中心流形附近的一类抛物问题的存在性和稳定性。在解决这些问题时，PI将使用各种技术，包括演化半群，频谱和频率分析。本文的目标是研究Green-Naghdi系统，即非线性光学中的耦合模系统，建立一类抽象的Hamilton偏微分方程的非线性稳定性和完整的不变流形描述，解决了数学物理中的一大类偏微分方程的解的各种问题。此类方程被视为无限维空间上的动力系统。特别感兴趣的是Kuramoto-Sivashinsky和Burgers-Sivashinsky方程，二维Navier-Stokes方程，和？阿尔法模型流体动力学。PI将使用各种技术，包括谱分析，变分方法，傅立叶分析技术和演化半群。在这个项目中解决的哈密顿偏微分方程出现在许多物理情况下-一些起源于量子力学系统和非线性光学的研究，其他的在流体动力学和燃烧。了解一个特殊的解决方案附近的解决方案的局部行为是一个问题的应用程序中至关重要的，因为只有稳定的波可以预期是物理上可实现的。另一方面，确定不稳定性或不稳定性的来源具有重要的实际意义。在这些问题上的任何进展不仅在数学上很重要，而且将在物理科学中找到直接的应用。PI的另一个目标是让一些本科生参与这个迷人的研究领域。PI还计划与数学和物理系的其他教师共同举办一个关于非线性动力系统的主题课程。

In this project, the PI will apply spectral and variational methods, as well as the techniques of Fourier analysis, to study some outstanding open problems in the theory of Hamiltonian and dispersive partial differential equations. The first part of the project focuses on the long-time behavior of the solutions of the Kuramoto-Sivashinsky and Burgers-Sivashinsky equations. Questions to be addressed concern the global well-posedness, the existence of attracting sets, and the regularity of the solutions. The PI will continue her work on developing a systematic approach to the questions of existence and regularity of attractors for dissipative partial differential equations in the case of unbounded domains. Of particular interest are the two-dimensional Navier-Stokes equation and the "alpha models" from fluid dynamics. An innovative approach, using modulation equations and spectral decomposition techniques will be used to study the existence and stability of solitary waves, standing waves, and similar particular solutions in the vicinity of the central manifold for a class of parabolic problems. In solving these problems the PI will use a variety of techniques including evolution semigroups, spectral and frequency analysis. The goal here is to investigate the Green-Naghdi system, the coupled-mode system from nonlinear optics as well as to establish nonlinear stability and complete invariant manifolds description for a class of abstract Hamiltonian partial differential equations.This proposal deals with a variety of problems concerning solutions of a large class of partial differential equations of mathematical physics. Such equations are viewed as dynamical systems on an infinite-dimensional space. Of particular interest are the Kuramoto-Sivashinsky and Burgers-Sivashinsky equations, the two-dimensional Navier-Stokes equation, and the ?alpha models? of fluid dynamics. The PI will use a variety of techniques, including spectral analysis, variational methods, the techniques of Fourier analysis, and evolution semigroups. The Hamiltonian partial differential equations addressed in this project arise in numerous physical situations - some have their origins in the study of quantum mechanical systems and nonlinearoptics, others in fluid dynamics and combustion. Understanding the local behavior of solutions nearby a special solution is an issue of paramount importance for the applications, since only waves that are stable can be expected to be physically realizable. On the other hand, identifying instability or the source of it is of great practical importance. Any progress on these questions will not only be important mathematically, but will find immediate applications in physical sciences. Another goal of the PI is to involve some undergraduate students in this fascinating area of research. The PI also plans a topics class on nonlinear dynamical systems jointly with other faculty of the Mathematics and Physics Departments.