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

在这个项目中，PI将应用谱方法和变分方法，以及傅立叶分析技术，来研究哈密顿和色散偏微分方程组理论中一些突出的公开问题。该项目的第一部分集中于Kuramoto-Sivashinsky和Burgers-Sivashinsky方程的解的长期行为。需要解决的问题涉及全局适定性、吸引集的存在性以及解的正则性。PI将继续她的工作，发展一种系统的方法来研究无界区域情况下耗散偏微分方程吸引子的存在性和正则性问题。特别令人感兴趣的是二维N-S方程和流体力学中的“阿尔法模型”。采用一种新颖的方法，利用调制方程和谱分解技术，研究一类抛物型问题在中心流形附近的孤立波、驻波和类似特解的存在性和稳定性。在解决这些问题时，PI将使用各种技术，包括进化半群、谱和频谱分析。本文的目的是从非线性光学的角度研究格林-纳格迪系统，即耦合模系统，并建立一类抽象的哈密顿偏微分方程解的非线性稳定性和完全不变流形描述。这类方程被视为无限维空间上的动力系统。特别令人感兴趣的是Kuramoto-Sivashinsky和Burgers-Sivashinsky方程，二维Navier-Stokes方程，以及Alpha模型？流体力学的研究。PI将使用各种技术，包括频谱分析、变分方法、傅立叶分析技术和演化半群。本项目中讨论的哈密顿偏微分方程式在许多物理情况下出现--一些起源于量子力学系统和非线性光学的研究，另一些起源于流体力学和燃烧的研究。了解特殊解附近解的局部行为对于应用来说是一个至关重要的问题，因为只有稳定的波才有望在物理上实现。另一方面，确定不稳定或不稳定的根源具有重要的现实意义。在这些问题上的任何进展不仅在数学上是重要的，而且将立即在物理科学中得到应用。PI的另一个目标是让一些本科生参与这个令人着迷的研究领域。PI还计划与数学系和物理系的其他系联合举办一门关于非线性动力系统的专题课程。

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.