Some Dynamical Problems in Fluids and Plasmas

本计画研究无碰撞电浆、不可压缩流体及表面水波之动力结构。各种相干结构的稳定性，包括稳定的星系，周期性和孤立的水波，和电磁等离子体平衡将进一步探讨。其中一些结构是不稳定的，这意味着一个小的扰动会产生一个大的扰动。目的是确定哪些结构是稳定的，哪些是不稳定的。另一个重点是了解这些稳定的结构在长时间动态中起什么作用。不稳定定态附近的不变结构，包括不稳定（稳定）流形和异宿轨道，将被研究。特别强调的是这些动力学结构的无粘极限，这是很重要的理解过渡到湍流和观察到的结构。非耗散流体和等离子体在某些稳定平衡点附近的非线性阻尼也将被研究。这项研究的总目标是发展新的和有效的数学方法来分析流体和等离子体的动力学行为。这里的数学进步将提高我们对基本物理机制的理解，也可以导致更好的数值方法来模拟这些复杂的现象。等离子体是由粒子组成的气体，每个粒子都带有正电荷或负电荷。宇宙的大部分都是等离子体;例子包括太阳风、电离层、银河系星云和彗星尾巴。等离子体也出现在物理学和工程学中，例如核聚变。流体、等离子体和水波的稳定性和动力学结构与聚变能研究、空间卫星和无线电通信、大气和海洋中的波动以及流体的湍流行为等各种应用有关。非耗散模型中的非线性阻尼机制对于理解流体和等离子体湍流中的大尺度相干结构（如星系）的发展和演化具有重要意义。

This project studies dynamical structures in collisionless plasmas, incompressible fluids, and surface water waves. Stability of a variety of coherent structures including steady galaxies, periodic and solitary water waves, and electromagnetic plasma equilibria will be further explored. Some of these structures are unstable, which means that a small perturbation creates a large disturbance. The goal is to determine which structures are stable and which are unstable. Another focus is to understand what role these steady structures play in the long time dynamics. The invariant structures near unstable steady states, including unstable (stable) manifolds and heteroclinic orbits, will be investigated. Particular emphasis will be on the inviscid limit of these dynamical structures, which is important for understanding the transition to turbulence and the observed structures. Nonlinear damping near certain stable equilibria of non-dissipative fluids and plasmas will also be studied.The general goal of this research is to develop new and effective mathematical methods for the analysis of dynamic behavior in fluids and plasmas. Mathematical advances here will improve our understanding of the basic physical mechanisms, and also can lead to better numerical methods for simulations of these complex phenomena. Plasma is a gas composed of particles, each carrying a positive or negative electrical charge. Most of the universe is plasma; examples include the solar wind, the ionosphere, galactic nebulae, and comet tails. Plasmas also arise in physics and engineering, for instance in nuclear fusion. The stability and dynamical structures of fluids, plasmas and water waves are relevant for such diverse applications as fusion energy research, satellite and radio communications in space, wave motion in the atmosphere and oceans, and turbulent behavior of fluids. The mechanisms of nonlinear damping in the non-dissipative models are important for understanding how the large scale coherent structures (such as galaxies) in the fluid and plasma turbulence develop and evolve.