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Existence, uniqueness and statistical theory of turbulent solutions of the stochastic Navier-Stokes equation, in three dimensions – an overview. (English) Zbl 1206.35266
This paper is devoted to 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. The authors discusses 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 the author briefly describes the mathematical proof of Kolmogorov’s statistical theory of turbulence.
MSC:
 35R60 PDEs with randomness, stochastic PDE 35Q30 Stokes and Navier-Stokes equations 76F02 Fundamentals of turbulence 76F55 Statistical turbulence modeling 60H15 Stochastic partial differential equations