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，即最初非常光滑的流体流可以产生越来越小的涡流，越来越小的尺度的结构，直到看起来完全混乱为止。尽管任何人都可以观察到这种现象，尽管它在物理，航空航天工程等中至关重要，但它在概念层面上的理解很差。微弱的湍流是一种相关现象，它也被鲜为人知，但理论上可能更平易近人。它不是流体流动，而是描述了波相互作用的情况。该建议的目的是开发理论工具，并将其应用于具体的物理示例，从而更深入地了解弱湍流。要更具体，重点的主要对象将是在紧凑型域上设置的非线性Schroedinger方程，这是描述非线性波或分散现象的最简单，最常用的模型之一。物理学家认为，观察弱湍流的正确制度涉及三个限制：弱非线性（小数据），大框（大域），随机阶段（统计意义上的傅立叶模式的去相关）。该项目的目的是调查如何使这三个限制程序变得严格。它涉及光谱问题（了解域上拉普拉斯式的特征值和特征模），非线性方面（这些特征模式如何相互作用）和统计问题（找到对固定状态的正确概率描述）。