Stability and metastability of coherent structures in dissipative PDE

该项目的重点是开发理论工具，用于预测生物学，化学和流体动力学等各种应用中出现的数学模型的解决方案的行为。一个关键的属性被称为稳定性。粗略地说，一个解是稳定的，如果当它被扰动时，它仍然随着时间的推移返回到原始行为。在真实的世界中，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，只有稳定的解决方案，人们期望从长远来看是可观察的。数学模型可以帮助我们了解给定系统的哪些解是稳定的，以及这种稳定性如何取决于系统参数。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，以观察某些行为，从而建议在实验环境中测试什么。此外，模型可以提供信息，以确定稳定性的物理机制是首要的。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程序。主要研究者专注于两种类型的稳定性：（1）渐近稳定性，即随着时间的无限演化，解吸引附近的数据;（2）亚稳定性，即解吸引附近的数据，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进行极限分析，这一事实大大简化了相关的技术。然而，重要的悬而未决的问题仍然存在，如稳定的 被称为缺陷的时间周期性模式和马斯洛夫指数在理解大于1的空间维度的稳定性方面的潜在用途，它们将在本项目中得到解决。亚稳性是一种更微妙的现象，其分析方法要少得多，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测行为时，渐近解或亚稳态解是否更重要取决于相关的时间尺度。例如，如果渐近稳定状态仅在指数长时间尺度上接近，那么人们不会期望等待足够长的时间来观察它们。在这种情况下，亚稳态，然后出现在长期，中间时间，变得更加相关。例如，这发生在Navier-Stokes方程中，这是流体动力学的一个重要模型，本项目将开发在这方面分析亚稳性的方法。

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.