Mathematical Sciences: Mathematical Techniques for Unstable Physical Processes

这个项目将使用微扰理论、动力系统理论和哈密顿力学中的高级数学概念来研究各种物理系统中不稳定的一般领域。其中要考虑的问题包括Ginzburg-Landau方程和Zakharov方程的色散波解的稳定性，粘性流动中混沌开始时流体轨迹的平流混合，以及旋转哈密顿系统的粒子轨迹从KAM区域逃逸。湍流现象可能是当今流体力学中最重要、最困难的问题。在这个项目中，主要的研究人员将利用微扰理论和非线性力学的技术来研究简单的湍流模型，以便更好地了解流动中的不稳定性是如何被放大到流动变得完全不稳定并变成湍流的。这也是一个重要的问题，要预测一种流动在何时何地首先变成湍流，因为人们往往想要尽可能地推迟湍流的开始。

This project will study the general area of instability in various physical systems using advanced mathematical concepts from perturbation theory, dynamical systems theory and Hamiltonian mechanics. Among the topics to be considered are the stability of dispersive wave solutions of the Ginzburg-Landau equation and the Zakharov equation, the advective mixing of fluid trajectories at the onset of chaos in viscous flows and the escape of particle trajectories of rotating Hamiltonian systems from KAM regions. The phenomenon of turbulence is perhaps the most important and the most difficult problem in fluid dynamics today. In this project the principal investigator will study simple models of turbulent flows using techniques from perturbation theory and nonlinear mechanics in order to understand better how instabilities in flows are magnified to the point where the flows go completely unstable and become turbulent. This too is an important problem, predicting where and when a flow first becomes turbulent, since very often one wants to delay the onset of turbulence for as long as possible.