权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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.

一个系统的特定状态，例如沙漠植被中的条纹图案或流体流动中的漩涡，是否被观察到，取决于它对扰动的鲁棒性。如果系统中的一个小扰动或波动会使它远离该状态，那么该状态不太可能持续很长时间，因此不太可能被观察到。这种状态是不稳定的。相反，稳定状态是一种即使在存在小扰动的情况下也会持续存在的状态，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。该项目的重点是发展数学方法来分析这种模型的解决方案的稳定性。检测数学模型中的给定状态是否稳定是预测其可观测性的关键步骤，因此也是预测现实世界行为的关键步骤。在这个项目中，使用拓扑学，几何学和分析工具分析由偏微分方程描述的模型的稳定性，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只在一维空间有效，而许多物理系统，如上面提到的那些，有两个或更多的空间维度，这个重点是特别重要的。研究生参与本项目的研究。当使用偏微分方程模拟物理系统时，了解解的长期行为是很重要的。其中一个关键方面是识别模型的某些解或相干结构，并确定它们是否稳定。这意味着它们的小扰动随着时间的增加而保持很小，从而使它们在物理上可观察到。在一维空间中，空间动力学允许许多与非线性波相关的问题，特别是它们的稳定性，通过将空间变量视为类时演化变量，可以将其置于动力系统框架中。在更高的维度中，空间动力学的概念并没有很好的定义，因为一般来说，没有特殊的空间变量可以被视为类时变量。最近，多维稳定性问题已被重铸使用家庭的收缩域和相关的莫尔斯指数的线性化运营商的Maslov指数与这个家庭。这表明，马斯洛夫指数可以是一个强大的工具，用于确定多维非线性波的谱稳定性和空间动力学的多维设置的推广可以通过一维变量，索引域开发。该项目的重点是开发这两种可能性。研究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知识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

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