Mathematical Sciences: Mathematical Techniques for Unstable Physical Processes

该项目将研究中不稳定的一般区域 使用先进的数学概念的各种物理系统 从微扰理论，动力系统理论， 哈密顿力学 要考虑的议题包括： Ginzburg-Landau色散波解的稳定性 方程和Zakharov方程，流体的平流混合 在粘性流混沌开始时的轨迹和 旋转哈密顿系统粒子轨道的逃逸 来自KAM地区。 湍流现象可能是 流体力学中最重要也是最困难的问题 今天 在这个项目中，首席研究员将研究 湍流的简单模型，使用的技术来自 微扰理论和非线性力学， 更好地理解流动中的不稳定性是如何被放大的， 流动变得完全不稳定， 动荡不安 这也是一个重要的问题， 当水流开始变得湍流时， 希望尽可能地推迟湍流的出现

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.