Existence, uniqueness and statistical theory of turbulent solutions of the stochastic Navier--Stokes equation, in three dimensions, an overview

Abstract

We discuss the proofs of the existence and uniqueness of solutions of the Navier--Stokes equation driven with additive noise in three dimensions, in the presence of a strong uni-directional mean flow with some rotation. We also discuss how the existence of a unique invariant measure is established and the properties of this measure are described. The invariant measure is used to prove Kolmogorov's scaling in 3-dimensional turbulence including the celebrated $-5/3$ power law for the decay of the power spectrum of a turbulent 3-dimensional flow. Then we briefly describe the mathematical proof of Kolmogorov's statistical theory of turbulence.