Spectral Stability and Oscillations of Dynamical Systems, Boltzmann-Like Models

动力系统的谱稳定性和振荡，类玻尔兹曼模型

• 批准号: 1910820
• 负责人: Alim Sukhtayev
• 金额: $ 11.67万
• 依托单位: Miami University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1910820&HistoricalAwards=false
• 关键词: Spectral; Stability; Oscillations; Dynamical; Systems

This project aims to develop the mathematical tools that can help investigate stability and instability mechanisms affecting the solutions of partial differential equations. In particular, these tools can be used to assess stability of distinguished states of a nonlinear partial differential equation, which is a key step in understanding the behavior of the physical system modeled by the equation. Stability here means the robustness of the dynamics to perturbations in initial conditions from a particular state. The distinguished state may be a nonlinear wave, pattern, or coherent structure arising in applications such as optics, fluids, neuroscience, ecology, chemical reactions, or shallow water dynamics. The stability of the given state indicates its physical realizability, while any instability suggests more complex dynamics. Understanding the nature of such instabilities can be used as a starting point for understanding the organization of the nonlinear dynamics away from the unstable state. The project includes research activities that will train undergraduate studentThe investigator's goal is to generalize the oscillation-type results for eigenvalue problems that are associated with partial differential equation models. These range from nonlinear eigenvalue problems arising in spectral stability of shock profiles of hyperbolic systems of balance laws to problems in multi-dimensional spatial domains. In particular, he exploits a new set of ideas that cast general multi-dimensional problems in a dynamical systems framework and thus offer a new characterization of the underlying issues of existence and stability of solutions. One of the key directions of the project is the introduction of the Spatial Evolutionary System, which can be viewed as a far-reaching extension of the spatial dynamics framework to general multi-dimensional domains. Undergraduate students are engaged in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目旨在开发数学工具，帮助研究影响偏微分方程解的稳定性和不稳定机制。 特别是，这些工具可用于评估非线性偏微分方程的不同状态的稳定性，这是理解由方程建模的物理系统行为的关键步骤。 这里的稳定性是指动力学对特定状态的初始条件下的扰动的鲁棒性。 特殊状态可能是光学、流体、神经科学、生态学、化学反应或浅水动力学等应用中出现的非线性波、图案或相干结构。 给定状态的稳定性表明其物理可实现性，而任何不稳定性都表明更复杂的动力学。 了解这种不稳定性的本质可以作为理解远离不稳定状态的非线性动力学组织的起点。 该项目包括培训本科生的研究活动。研究者的目标是推广与偏微分方程模型相关的特征值问题的振荡型结果。 这些范围从平衡定律双曲系统冲击曲线谱稳定性中出现的非线性特征值问题到多维空间域中的问题。 特别是，他利用了一组新的想法，将一般的多维问题放入动态系统框架中，从而为解决方案的存在和稳定性的根本问题提供了新的表征。 该项目的关键方向之一是引入空间进化系统，它可以被视为空间动力学框架对一般多维领域的深远延伸。 本科生参与该项目的研究。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Fredholm determinants, Evans functions and Maslov indices for partial differential equations

• DOI: 10.1007/s00208-023-02696-6
• 发表时间: 2022-06
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: G. Cox;Y. Latushkin;A. Sukhtayev

A Sturm–Liouville theorem for quadratic operator pencils

二次算子铅笔的 Sturm-Liouville 定理

• DOI: 10.1016/j.jde.2019.10.010
• 发表时间: 2020
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Sukhtayev, Alim;Zumbrun, Kevin
• 通讯作者: Zumbrun, Kevin

Renormalized Oscillation Theory for Singular Linear Hamiltonian Systems

• DOI: 10.1016/j.jfa.2022.109525
• 发表时间: 2020-09
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: P. Howard;A. Sukhtayev

Exponential dichotomies for elliptic PDE on radial domains

径向域上椭圆偏微分方程的指数二分法

• DOI: 10.1007/978-3-030-47174-3
• 发表时间: 2020
• 期刊: Mathematics of Wave Phenomenon
• 作者: M. Beck, G. Cox

A dynamical approach to semilinear elliptic equations

半线性椭圆方程的动力学方法

• DOI: 10.1016/j.anihpc.2020.08.001
• 发表时间: 2021
• 期刊: Analyse non linéaire
• 作者: Beck, Margaret;Cox, Graham;Jones, Christopher;Latushkin, Yuri;Sukhtayev, Alim
• 通讯作者: Sukhtayev, Alim