Rigorous results in space-periodic two-dimensional turbulence

TL;DR

This paper reviews recent rigorous mathematical advances in understanding two-dimensional turbulence through stochastic Navier-Stokes equations, focusing on existence, uniqueness, statistical properties, and large deviations.

Contribution

It provides a comprehensive survey of recent results on the qualitative theory of 2D stochastic Navier-Stokes equations relevant to turbulence modeling.

Findings

Existence and uniqueness of stationary measures.

Establishment of mixing properties and statistical laws.

Analysis of large deviations and inviscid limits.

Abstract

We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier-Stokes system that are relevant to the description of turbulence in two-dimensional fluids. After discussing briefly the initial-boundary value problem and the associated Markov process, we formulate results on the existence, uniqueness and mixing of a stationary measure. We next turn to various consequences of these properties: strong law of large numbers, central limit theorem, and random attractors related to a unique stationary measure. We also discuss the Donsker-Varadhan and Freidlin-Wentzell type large deviations, as well as the inviscid limit and asymptotic results in 3d thin domains. We conclude with some open problems.