Phase mixing in the fluid mechanics and kinetic theory

非线性混合是在流体和等离子体中观察到的普遍存在的行为，在理解大气和海洋科学、湍流、天体物理学和受控聚变的某些方面起着重要作用。例如，它被认为是大气动力学中的一个重要组织机制，有助于解释为什么飓风在很长一段时间内保持其特有的涡旋形状。尽管是一个基本的物理机制，非线性相位混合的理论和数学分析仍然难以捉摸。此外，许多这些影响是难以隔离的计算或物理实验，在这里数学分析可以提供重要的见解。本项目旨在通过研究流体力学和等离子体物理学中各种基本“案例研究”中的效应来扩展我们的数学理解。本项目的目的是进一步对非线性相混合进行数学分析。由于高度非正常的线性演化和与微妙的非线性共振交织在一起的不寻常的规律性问题的出现，非线性相混合通常被视为困难。PI将致力于开发解决这些问题所需的新工具，重点关注四个特定的设置：2D Couette流附近不可压缩Navier-Stokes方程（NSE）的粘性极限消失，3D剪切流的非线性不稳定性，2D Euler中的无粘“涡流轴对称化”和Vlasov-Poisson（或Vlasov-Maxwell）中存在外部磁场的朗道阻尼。许多相关的物理相关的问题存在，并可能作为中间步骤或额外的研究线。流体力学方面的工作将进一步扩展PI和合作者Nader Masmoudi在最近关于2D Couette流稳定性的工作中所采用的思想[arXiv：1306.5028]，特别是自适应坐标系的使用和考虑弱非线性共振和非正常瞬态线性行为的规范的构建。动力学理论方面的工作将进一步发展PI及其合作者Nader Masmoudi和Clement Mouhot在朗道阻尼[arXiv：1311.2870]方面的工作中使用的技术，特别是等离子体回波的有效处理。最有可能的是，为了扩大动力学理论的应用范围，流体力学工作中的一些想法可能需要适应动力学设置。

Nonlinear mixing is a ubiquitous behavior observed in fluids and plasmas which plays an important role in understanding certain aspects of atmosphere and ocean sciences, turbulence, astrophysics and controlled fusion. For example, it is thought to be an important organizing mechanism in atmospheric dynamics that helps to explain why hurricanes retain their characteristic vortex shape for extended periods of time. Despite being a fundamental physical mechanism, the theoretical and mathematical analysis of nonlinear phase mixing has remained elusive. Moreover, many of these effects are difficult to isolate in computational or physical experiments and here mathematical analysis can provide important insights. This project specifically aims to expand our mathematical understanding by studying the effect in a variety of fundamental 'case studies' found in fluid mechanics and plasma physics.The purpose of this program is to further the mathematical analysis of nonlinear phase mixing, generally regarded as difficult due to the appearance of highly non-normal linear evolutions and unusual regularity issues intertwined with subtle nonlinear resonances. The PI will work to develop new tools necessary to approach these problems with a focus on working towards four specific settings: the vanishing viscosity limit of the incompressible Navier-Stokes equations (NSE) near the 2D Couette flow, the nonlinear instability of 3D shear flows, inviscid 'vortex axisymmetrization' in 2D Euler and Landau damping in Vlasov-Poisson (or Vlasov-Maxwell) in presence of external magnetic fields. Many related physically relevant problems exist, and may serve as intermediate steps or additional lines of research. The work in fluid mechanics will expand further on ideas employed by the PI and collaborator Nader Masmoudi in the recent work on the stability of 2D Couette flow [arXiv:1306.5028], especially the use of adaptive coordinate systems and the construction of norms which account for weakly nonlinear resonances and the non-normal transient linear behavior. The work in kinetic theory will further develop techniques used in the work of the PI and collaborators Nader Masmoudi and Clement Mouhot on Landau damping [arXiv:1311.2870], especially the efficient treatment of the plasma echoes. Most likely, in order to expand the range of kinetic theory applications, some ideas from the work in fluid mechanics may need to be adapted to the kinetic setting.