Rigorous results in space-periodic two-dimensional turbulence

Abstract

We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier–Stokes system that is 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, the inviscid limit, and asymptotic results in 3d thin domains. We conclude with some open problems.