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

被称为动力系统和偏微分方程的数学方程被用来模拟各种各样的过程，包括生态，生物和物理应用。数学预测解决方案的行为提供了一种机制，用于预测现实世界的系统行为，也用于预测这样的行为将如何改变系统参数的变化。稳定状态是吸引的，因此稳定状态是最有可能被观察到的状态。与稳定性相关的拓扑性质是那些在足够好的变形下具有鲁棒性的性质，因此可以用于对大类系统进行预测，可能不需要参考所研究方程的更精细的细节。虽然在数学上对只有一个空间维度的模型有很多理解，但对于更高的空间维度，还有许多有趣的开放性问题，这与应用特别相关。该项目的重点是开发理论工具，用于预测这些类型方程的解的长期行为。该项目还将支持培训女博士生和开展外联工作的努力，例如为高中生举办为期一周的数学夏令营。该项目的重点是开发理论工具，用于分析与一大类偏微分方程（PDE）相关的动力学，包括反应扩散方程和四阶系统，如Swift-Hohenberg方程，在一维和更高的空间维度。理解这样的系统不仅需要预测存在什么类型的解，还需要预测它们的稳定性，这对它们在现实世界中的可观测性有直接的影响。有两个主要目标：（i）研究高阶偏微分方程和偏微分方程系统稳定性的拓扑含义;（ii）分析空间维度大于1的偏微分方程的解。第一个目标是与经典Sturm-Liouville理论的标量，二阶特征值问题的结果。然而，由于基本原因，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。第二个目标将通过发展一个有用的空间动力学来实现，这是指在多个空间维度的系统中，将一个独特的空间变量作为一个类时演化变量来处理。这将使人们能够应用动力系统理论的工具来理解相关的解决方案。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的支持。

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.