Dynamics of Partial Differential Equations: Topological Implications for Stability and Analysis in Higher Spatial Dimensions

偏微分方程的动力学：更高空间维度稳定性和分析的拓扑含义

• 批准号: 2205434
• 负责人: Margaret Beck
• 金额: $ 42.98万
• 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: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205434&HistoricalAwards=false
• 关键词: Dynamics; Partial; Differential; Equations; Topological

Mathematical equations known as dynamical systems and partial differential equations are used to model a wide variety of processes, including ecological, biological, and physical applications. Mathematically predicting solution behavior provides a mechanism for predicting real-world system behavior, and also for predicting how such behavior will change as system parameters are varied. Stable states are attracting, and thus stable states are those that are mostly likely to be observed. Topological properties related to stability are those that are robust under sufficiently nice deformations, and can therefore be used to make predictions about large classes of systems, possibly without reference to the finer details of the equation under study. Although much is mathematically understood about models with only one space dimension, there are many interesting open questions about higher spatial dimensions, which is particularly relevant for applications. This project is focused on developing theoretical tools for predicting the long-time behavior of solutions to these types of equations. This project will also support efforts to train women PhD students and conduct outreach work, such as a week-long summer math camp for high school students. This project is focused on developing theoretical tools for analyzing the dynamics associated with a large class of partial differential equations (PDEs), including reaction-diffusion equations and fourth-order systems such as the Swift-Hohenberg equation, in both one and higher spatial dimensions. Understanding such systems requires predicting not only what types of solutions will exist, but also their stability, which has direct consequences for their observability in the real-world. There are two primary goals: (i) Investigating topological implications for stability in higher-order PDEs and systems of PDEs; and (ii) Analyzing solutions to PDEs in spatial dimensions greater than one. The first goal is connected with the results of classical Sturm-Liouville theory for scalar, second order eigenvalues problems. However, for fundamental reasons that theory cannot be directly extended to higher-order systems, and so truly new methods must be developed. The second goal will be achieved by developing a useful spatial dynamics, which refers to treating a distinguished spatial variable as a time-like evolution variable, for systems in multiple spatial dimensions. This will allow one to apply the tools of dynamical systems theory to understand the associated solutions.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.

数学方程称为动力学系统和部分微分方程用于建模各种过程，包括生态，生物学和物理应用。数学上预测解决方案行为提供了一种预测现实世界系统行为的机制，还可以预测随着系统参数的变化，这种行为将如何变化。稳定的状态正在吸引，因此稳定的状态是大多数观察到的状态。与稳定性相关的拓扑特性是在足够好的变形下具有鲁棒性的拓扑特性，因此可以用于对大型系统进行预测，可能无需参考所研究方程的更细节。尽管关于只有一个空间维度的模型的数学了解很多，但关于较高空间维度有许多有趣的开放问题，这与应用程序特别相关。该项目的重点是开发理论工具，以预测解决这些类型方程的解决方案的长期行为。该项目还将支持培训女性博士生并进行外展工作的努力，例如为高中生的为期一周的夏季数学训练营。该项目的重点是开发理论工具，用于分析与大型部分微分方程（PDE）相关的动力学，包括反应扩散方程和四阶系统，例如Swift-Hohenberg方程，包括一个和更高的空间维度。理解这样的系统不仅需要预测存在哪种类型的解决方案，而且还需要其稳定性，这对它们在现实世界中的可观察性有直接的影响。有两个主要目标：（i）研究对PDE的高阶PDE和系统中稳定性的拓扑意义； （ii）在空间维度中分析PDE的溶液大于一个。第一个目标是与标量二阶特征值问题的经典Sturm-Liouville理论的结果有关。但是，由于根本原因，理论不能直接扩展到高阶系统，因此必须开发真正的新方法。第二个目标将通过开发有用的空间动力学来实现，该动力学是指在多个空间维度中的系统将杰出的空间变量视为时间样变量。这将使人们能够应用动力系统理论的工具来了解相关的解决方案。该奖项反映了NSF的法定使命，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审查标准来评估值得支持的。