Kolmogorov $4/5$ law for the forced 3D Navier-Stokes equations

TL;DR

This paper establishes a sufficient condition under which solutions to the 3D forced Navier-Stokes equations exhibit the Kolmogorov 4/5 law in an averaged sense, linking probabilistic behavior with turbulence theory.

Contribution

It introduces a new sufficient condition for the Kolmogorov 4/5 law to hold for solutions of the 3D forced Navier-Stokes equations, connecting probabilistic interpretation and turbulence theory.

Findings

The 4/5 law holds on average after a random waiting time.

The condition applies to solutions constructed by Bruè et al.

A bound for the third order structure function exponent is derived.

Abstract

We identify a sufficient condition under which solutions to the 3D forced Navier--Stokes equations satisfy an $L^p$-in-time version of the Kolmogorov 4/5 law for the behavior of the averaged third order longitudinal structure function along the vanishing viscosity limit. The result has a natural probabilistic interpretation: the predicted behavior is observed on average after waiting for some sufficiently generic random time. The sufficient condition is satisfied e.g. by the solutions constructed by Bru\`e, Colombo, Crippa, De~Lellis, and Sorella. In this particular case, our results can be applied to derive a bound for the exponent of the third order absolute structure function in accordance with the Kolmogorov turbulence theory.