Coherent Structures and Nonlinear Partial Differential Equations

相干结构，如宇宙中的椭圆星系或螺旋星系，大气和海洋中的大尺度涡旋结构，或表面水波，在许多自然现象中被观察到。这项研究的目的是了解这种相干结构的形成和演变（在某些情况下可以进行数学分析）。第一个问题是它们的稳定性，也就是说，即使它们经历扰动，它们是否持续存在。了解不稳定性和稳定性的机制是许多应用的基础。一个例子是在等离子体物理学和工程学中理解不稳定性机制的重要性，以便能够设计出能够容纳足够长时间的等离子体以用于实际聚变能量生产的设备。另一个主题是理解相干结构的动力学作用。例如，我们如何解释在大气和海洋中观察到的大尺度结构的出现？一个初始的非结构化态接近最终的相干态的机制是什么？这些问题还远未被彻底理解。数学分析方法是本研究的主要工具。严格的数学使得进行稳定的数值计算成为可能，并更好地理解数值和实验研究中发现的现象。该项目的重点之一是具有无限莫尔斯指数的能量泛函的Hamilton偏微分方程的稳定性问题。它们包括重力水波（和许多长波模型），非线性狄拉克方程，离子声波方程和无碰撞等离子体的弗拉索夫模型。与能量泛函具有有限莫尔斯指标的情形相比，迄今为止，对于具有无限莫尔斯指标的稳定性问题的通用工具还很少。本文将研究正则化的Boussinesq方程和二维Euler方程等几种模型，目的是为解决无穷大莫尔斯指数的稳定性问题开发一些通用工具。该项目的另一个重点是相干结构的亚稳性和小耗散下不变结构的持久性。其中包括小粘性二维Navier-Stokes方程准定常流的亚稳定性，Navier-Stokes方程不变流形的无粘极限。在这些研究中，将探讨流体方程的哈密顿量和几何结构。

Coherent structures, such as the elliptical or spiral galaxies in the universe, or large scale vortex structures in the atmosphere and ocean, or surface water waves, are observed in many natural phenomena. This research is aimed at understanding the formation and evolution of such coherent structures (that in some cases are amenable to mathematical analysis). A first question is their stability, that is, whether they persist even if they experience perturbations. Understanding the mechanisms of instability and stability is fundamental for many applications. An example is the importance in plasma physics and engineering of understanding mechanisms of instability, to enable design of devices able to contain plasma long enough for practical fusion energy production. Another topic is to understand the dynamical roles of coherent structures. For example, how do we explain the appearance of the large-scale structures observed in the atmosphere and oceans? What is the mechanism for an initially unstructured state to approach a final coherent state? Such problems are far from being thoroughly understood. Methods of mathematical analysis are the primary tools employed in this investigation. The rigorous mathematics makes it feasible to do stable numerical computations and to better understand the phenomena found in numerical and experimental studies.One focus of the project is the stability problem for Hamiltonian partial differential equations with energy functional of infinite Morse index. They include gravity water waves (and many long wave models), nonlinear Dirac equations, ion acoustic wave equations, and Vlasov models for collisionless plasmas. In contrast to the cases with energy functional of finite Morse index, to date there are very few general tools for stability problems with infinite Morse index. Several models, including the regularized Boussinesq equation and two-dimensional Euler equation, will be studied, with the goal to develop some general tools for stability problems of infinite Morse index. Another focus of the project is the metastability of coherent structures and the persistence of invariant structures under small dissipation. They include the metastability of quasi-stationary flows of the two-dimensional Navier-Stokes equation with small viscosity, and the inviscid limit of invariant manifolds of Navier-Stokes equations. Hamiltonian and geometric structures of the fluid equations will be explored in these investigations.