Stability and metastability of coherent structures in dissipative PDE

该项目的重点是发展理论工具，用于预测各种应用中出现的数学模型的解的行为，如生物学，化学和流体动力学。一个关键的特性叫做稳定性。粗略地说，一个解是稳定的，如果当它受到扰动时，随着时间的推移，它仍然返回到原来的行为。在现实世界中，人们期望系统经历频繁的小扰动，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，这种波动将驱使系统远离它，走向稳定状态。因此，从长远来看，只有稳定的解才能被观察到。数学模型可以深入了解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最重要的信息。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，而不是适用于任何一个特定的应用程序。主要研究者关注的稳定性有两种类型：(1)渐近稳定性，这意味着随着时间向无穷大发展，解会吸引附近的数据；(2)亚稳态（Metastability），即该解在大而有限的时间内吸引附近的数据。渐近稳定性的概念是相对标准的，它允许在时间趋于无穷时的极限进行分析，这一事实大大简化了相关的技术。尽管如此，重要的开放性问题仍然存在，比如被称为缺陷的时间周期模式的稳定性，以及马斯洛夫指数在理解大于1的空间维度稳定性方面的潜在用途，这些问题将在本项目中得到解决。亚稳态是一种更为微妙的现象，其分析方法要少得多，因此在这一领域取得进展具有根本性的重要性。在确定给定系统的观测行为时，渐近解或亚稳解是否更重要取决于相关的时间尺度。例如，如果渐近稳定状态只在指数长的时间尺度上接近，那么人们就不会期望等待足够长的时间来观察它们。在这种情况下，在漫长的中间时间里出现的亚稳态就变得更加相关了。例如，这发生在流体动力学的重要模型Navier-Stokes方程中，本项目将开发在这种情况下分析亚稳态的方法。

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.