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

We prove that the solutions to the 3D forced Navier–Stokes equations constructed by Bruè, Colombo, Crippa, De Lellis, Sorella in [2] satisfy an Lp-in-time version of the celebrated Kolmogorov 4/5 law for 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. This is then applied to derive a bound for the exponent of the third order absolute structure function in accordance with the Kolmogorov turbulence theory. Furthermore, we also derive analogous results for the stochastic 3D Navier–Stokes equations under a regularity assumption.