Weak Turbulence

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成越来越小的旋涡，结构的尺度越来越小，直到看起来完全混乱。尽管任何人都可以观察到这一现象，尽管它在物理学、航空航天工程等领域非常重要，但在概念层面上人们对它的理解很少。弱湍流是一种相关现象，人们对此也知之甚少，但理论上可能更接近实际。它描述的不是流体流动，而是波相互作用的情况。这一建议的目的是发展理论工具，并将其应用于具体的物理例子，从而加深对弱湍流的理解。更具体地说，焦点将是紧致域上的非线性薛定谔方程，它是描述非线性波或频散现象的最简单和最常用的模型之一。物理学家们认为，观测弱湍流的合适区域包括三个极限：弱非线性(小数据)、大盒子(大区域)、随机相位(统计意义上的傅立叶模的去相关)。这个项目的目的是调查如何使这三个限制程序变得严格。它涉及谱问题(了解区域上拉普拉斯的本征值和本征模)、非线性方面(这些本征模如何相互作用)和统计问题(寻找对定态的正确概率描述)。

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).