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

Bjorn Birnir

doi:10.15352/bjma/1272374671

2010 Existence, uniqueness and statistical theory of turbulent solutions of the stochastic Navier--Stokes equation, in three dimensions, an overview

Bjorn Birnir

Banach J. Math. Anal. 4(1): 53-86 (2010). DOI: 10.15352/bjma/1272374671

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.

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Bjorn Birnir "Existence, uniqueness and statistical theory of turbulent solutions of the stochastic Navier--Stokes equation, in three dimensions, an overview," Banach Journal of Mathematical Analysis 4(1), 53-86, (2010). https://doi.org/10.15352/bjma/1272374671

Published: 2010

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