Stability and metastability of coherent structures in dissipative PDE

该项目专注于开发理论工具，用于预测在各种应用中出现的数学模型的解的行为，如生物、化学和流体动力学。其中一个关键属性称为稳定性。粗略地说，一个解决方案是稳定的，如果当它被扰动时，它仍然随着时间的推移回到原来的行为。在现实世界中，人们预计系统会经历频繁的小扰动，例如由于不可预测的外部输入或噪声。如果一个特定的状态是不稳定的，这样的波动将驱使系统远离它，走向稳定状态。因此，只有稳定的解决方案才能长期观察到。数学模型可以洞察给定系统的哪些解是稳定的，以及这种稳定性如何依赖于系统参数。这可以帮助其他学科的科学家预测哪些参数范围可能是感兴趣的，以便观察某些行为，从而建议在实验环境中测试什么。此外，这些模型还可以提供关于哪种物理机制在决定稳定性方面起主要作用的信息。该项目的主要目标是开发适用于各种特定模型的通用数学技术，而不是任何一个特定的应用。主要研究人员关注两种类型的稳定性：(1)渐近稳定性，即随着时间的推移，解吸引附近的数据；(2)亚稳定，意味着解吸引附近的数据的时间较长，但时间有限。渐近稳定性的概念是相对标准的，它允许在时间趋于无穷大时在极限内进行分析，这一事实极大地简化了相关技术。然而，重要的未决问题仍然存在，例如被称为缺陷的时间周期模式的稳定性，以及Maslov指数在理解大于1的空间维度的稳定性方面的潜在用途，这些问题将在本项目中得到解决。亚稳性是一种更微妙的现象，分析它的方法要少得多，因此这一领域的进展具有根本的重要性。在确定给定系统的观察行为时，渐近解或亚稳态解哪个更重要取决于相关的时间尺度。例如，如果渐近稳定的状态只能在指数级长的时间尺度上接近，那么人们就不会期待等待足够长的时间来观察它们。在这种情况下，亚稳态，然后出现在漫长的中间时间，变得更加相关。例如，这发生在流体动力学的重要模型--纳维斯托克斯方程中，本项目将在此背景下开发分析亚稳性的方法。

This project is focused on the development of theoretical tools for predicting the behavior of solutions to mathematical models arising in a variety of applications, such as biology, chemistry, and fluid dynamics. One key property is called stability. Roughly speaking, a solution is stable if when it is perturbed it still returns back to the original behavior as time evolves. In the real world, one expects a system to experience frequent small perturbations, for example due to unpredictable external inputs or noise. If a particular state is unstable, such fluctuations will drive the system away from it, towards a stable state. Thus, it is only the stable solutions that one expects to be observable in the long run. Mathematical models can provide insight into which solutions of a given system are stable, and how that stability depends on system parameters. This can help scientists in other disciplines predict which parameter ranges may be of interest in order to observe certain behaviors, thus suggesting what to test in experimental settings. Moreover, the models can provide information as to which physical mechanism is of primary importance in determining stability. The main goal of the project is to develop general mathematical techniques that are applicable to a variety of specific models, rather than to any one particular application. There are two types of stability that the principal investigator has focused on: (1) Asymptotic stability, meaning that the solution attracts nearby data as time evolves towards infinity; (2) Metastability, meaning that the solution attracts nearby data for large, but finite, times. The notion of asymptotic stability is relatively standard, and the fact that it allows for analysis in the limit as time tends towards infinity greatly simplifies the associated techniques. Nevertheless important open questions remain, such the stability of time-periodic patterns known as defects and the potential use of the Maslov index in understanding stability in spatial dimensions greater than one, and they will be addressed in this project. Metastability is a more subtle phenomenon, with far fewer methods for its analysis, and so advances in this area are of fundamental importance. Whether asymptotic or metastable solutions are more important in determining the observed behavior of a given system is dependent on the associated timescales. For example, if the asymptotically stable states are approached only on exponentially long time scales, then one would not expect to wait long enough to observe them in practice. In that case, the metastable states, which then appear during the long, intermediate times, become more relevant. This occurs, for example, in the Navier-Stokes equations, an important model of fluid dynamics, and this project will develop methods for analyzing metastability in this context.