Weak Turbulence

点击翻译按钮获取中文摘要

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).