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，以及傅里叶分析技术，研究哈密顿和色散偏微分方程理论中的一些突出的开放性问题。项目的第一部分着重于Kuramoto-Sivashinsky和Burgers-Sivashinsky方程解的长期行为。要处理的问题涉及全局适定性、吸引集的存在性和解的正则性。PI将继续她的工作，开发一个系统的方法来解决无界域情况下耗散偏微分方程的吸引子的存在性和规律性问题。特别有趣的是二维Navier-Stokes方程和流体动力学中的“alpha模型”。利用调制方程和谱分解技术，研究一类抛物型问题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。为了解决这些问题，PI将使用各种技术，包括演化半群、频谱和频率分析。本文的目标是从非线性光学角度研究Green-Naghdi系统和耦合模系统，并建立一类抽象哈密顿偏微分方程的非线性稳定性和完全不变流形描述。本文讨论了数学物理中一类偏微分方程的解的各种问题。这样的方程被看作是无限维空间上的动力系统。特别有趣的是Kuramoto-Sivashinsky方程和Burgers-Sivashinsky方程，二维Navier-Stokes方程，以及？α模型?流体动力学。PI将使用多种技术，包括光谱分析、变分方法、傅立叶分析技术和进化半群。本项目中涉及的哈密顿偏微分方程出现在许多物理情况中——有些源于量子力学系统和非线性光学的研究，有些源于流体动力学和燃烧。了解特殊解附近解的局部行为对应用来说是一个至关重要的问题，因为只有稳定的波才能在物理上实现。另一方面，确定不稳定或不稳定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是重要的，而且在物理科学上也会有直接的应用。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还计划与数学系和物理系的其他教员联合开设非线性动力系统专题课程。

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.