Weak Turbulence

弱湍流

• 批准号: 1501019
• 负责人: Pierre Germain
• 金额: $ 30万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501019&HistoricalAwards=false
• 关键词: Weak; Turbulence

Turbulence is the phenomenon by which a fluid flow, initially very smooth, can develop smaller and smaller eddies, structures at smaller and smaller scales, until it looks completely chaotic. Though anybody can observe this phenomenon, and though it is of paramount importance in physics, aerospace engineering, etc., it is very poorly understood at a conceptual level. Weak turbulence is a related phenomenon, which is also poorly understood, but which might be theoretically more approachable. Rather than fluid flows, it describes situations where waves interact. The aim of this proposal is to develop theoretical tools, and apply them to concrete physical examples, leading to a deeper understanding of weak turbulence.To be more specific, the main object of focus will be the nonlinear Schroedinger equation set on a compact domain, which is one of the simplest and most used models to describe nonlinear wave, or dispersive phenomena. It is believed by physicists that the right regime to observe weak turbulence involves three limits: weakly nonlinear (small data), big box (large domain), random phases (decorrelation of the Fourier modes in a statistical sense). The aim of this project is to investigate how these three limiting procedures can be made rigorous. It involves spectral questions (understanding the eigenvalues and eigenmodes of the Laplacian on domains), nonlinear aspects (how these eigenmodes interact) and statistical questions (finding the right probabilistic description of the stationary state).

湍流是一种现象，通过这种现象，最初非常平稳的流体流动可以发展越来越小的漩涡，越来越小的尺度结构，直到它看起来完全混乱。尽管任何人都可以观察到这种现象，尽管它在物理学、航空航天工程等方面至关重要， 在概念层面上对它的理解非常少。弱湍流是一个相关的现象，也是知之甚少，但在理论上可能更接近。它描述的不是流体流动，而是波相互作用的情况。本计划的目的是发展理论工具，并将其应用于具体的物理实例，从而加深对弱湍流的理解，更具体地说，主要关注的对象将是紧致区域上的非线性Schroedinger方程，它是描述非线性波或色散现象的最简单和最常用的模型之一。物理学家认为，观察弱湍流的正确机制涉及三个极限：弱非线性（小数据），大盒子（大域），随机相位（统计意义上的傅立叶模式去相关）。本项目的目的是研究如何使这三个限制程序严格。它涉及谱问题（理解拉普拉斯算子的本征值和本征模），非线性方面（这些本征模如何相互作用）和统计问题（找到稳态的正确概率描述）。