Stability and Spatial Dynamics

稳定性和空间动力学

基本信息

批准号：
1907923
负责人：
Margaret Beck
金额：
$ 33.37万
依托单位：
Trustees of Boston University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-15 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907923&HistoricalAwards=false
关键词：
Stability Spatial Dynamics

项目摘要

Whether or not a specific state of a system, such as a striped pattern in desert vegetation or a vortex in a fluid flow, is observed depends on how robust it is to perturbations. If a small perturbation, or fluctuation, in the system would drive it away from that state, then the state is unlikely to persist over long times and hence unlikely to be observed. Such a state is unstable. Conversely, a stable state is one that would persist even in the presence of small perturbations, thus rendering it physically observable. Mathematical models are often used to help predict the evolution of real-world systems. This project is focused on the development of mathematical methods for analyzing the stability of solutions of such models. Detecting whether or not a given state in a mathematical model is stable is a key step in predicting its observability, and hence in predicting real-world behavior. In this project, stability in models described by partial differential equations is analyzed using tools from topology, geometry, and analysis, with a focus on systems having more than one spatial dimension. Because many of the existing mathematical tools are valid in only one spatial dimension, while many physical systems, such as those mentioned above, have two or more spatial dimensions, this focus is of particular importance. Graduate students participate in the research of the project.Understanding the long-time behavior of solutions is important when using partial differential equations to model physical systems. A key aspect of this is identifying certain solutions or coherent structures of the model and determining if they are stable. This means that small perturbations of them remain small as time increases, thus rendering them physically observable. In one space dimension, spatial dynamics has allowed many problems related to nonlinear waves, and in particular their stability, to be cast in a dynamical systems framework by viewing the spatial variable as a time-like evolution variable. In higher dimensions, the notion of spatial dynamics is not well-defined, because in general there is no distinguished spatial variable to view as the time-like variable. Recently, multi-dimensional stability problems have been recast using a family of shrinking domains and by relating the Morse index of the linearized operator to a Maslov index connected with this family. This indicates both that the Maslov index could be a powerful tool for determining the spectral stability of multi-dimensional nonlinear waves and that a generalization of spatial dynamics to the multi-dimensional setting could be developed through the one-dimensional variable that indexes the domain. This project is focused on developing both of these possibilities. Graduate students participate 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.

观察到系统的特定状态，例如沙漠植被中的条纹图案还是流体流中的涡流状态取决于它对扰动的稳健性。如果系统中的小扰动或波动会使它远离该状态，那么状态不太可能在很长一段时间内持续，因此不太可能观察到。这样的状态是不稳定的。相反，即使存在小扰动，稳定的状态也将持续存在，从而使其在物理上可观察到。数学模型通常用于帮助预测现实世界系统的演变。该项目的重点是开发数学方法，用于分析此类模型解决方案的稳定性。检测数学模型中的给定状态是否稳定是预测其可观察性的关键步骤，因此在预测现实世界行为方面。在该项目中，使用拓扑，几何和分析的工具分析了由部分微分方程描述的模型中的稳定性，重点是具有多个空间维度的系统。由于许多现有的数学工具仅在一个空间维度上有效，而许多物理系统（例如上述）具有两个或多个空间维度，因此这种重点特别重要。研究生参与项目的研究。理解解决方案的长期行为在使用部分微分方程对物理系统建模时很重要。其中的一个关键方面是确定模型的某些解决方案或连贯的结构，并确定它们是否稳定。这意味着随着时间的增加，它们的少量扰动保持很小，从而使它们在物理上可观察到。在一个空间维度中，空间动力学使许多与非线性波相关的问题，尤其是它们的稳定性，可以通过将空间变量视为时间样变量，以在动态系统框架中施放。在较高的维度中，空间动力学的概念没有明确定义，因为通常没有明显的空间变量可以将其视为时间样变量。最近，使用缩小域的家族并将线性化操作员的摩尔斯索引与与该家族相关的Maslov指数联系起来，多维稳定性问题已被重施加。这表明MASLOV指数可能是确定多维非线性波的光谱稳定性的强大工具，并且可以通过索引域的一维变量来开发空间动力学对多维设置的概括。该项目的重点是开发这两种可能性。研究生参加了该项目的研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。

项目成果

期刊论文数量（6）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Rigorous Justification of Taylor Dispersion via Center Manifolds and Hypocoercivity

通过中心流形和低矫顽力对泰勒色散的严格论证

DOI：
10.1007/s00205-019-01440-2
发表时间：
2020
期刊：
Archive for Rational Mechanics and Analysis
影响因子：
2.5
作者：
Beck, Margaret;Chaudhary, Osman;Wayne, C. Eugene
通讯作者：
Wayne, C. Eugene

Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus

环面上随机受力纳维-斯托克斯方程中准稳态的选择

DOI：
10.1007/s00332-020-09621-0
发表时间：
2020
期刊：
Journal of nonlinear science
影响因子：
3
作者：
Beck, M;Cooper, E.;Spiliopoulos, K.
通讯作者：
Spiliopoulos, K.

Validated Spectral Stability via Conjugate Points

通过共轭点验证光谱稳定性

DOI：
10.1137/21m1420095
发表时间：
2022
期刊：
SIAM Journal on Applied Dynamical Systems
影响因子：
2.1
作者：
Beck, Margaret;Jaquette, Jonathan
通讯作者：
Jaquette, Jonathan

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
影响因子：
0
作者：
M. Beck, G. Cox
通讯作者：
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
影响因子：
0
作者：
Beck, Margaret;Cox, Graham;Jones, Christopher;Latushkin, Yuri;Sukhtayev, Alim
通讯作者：
Sukhtayev, Alim