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

2023/04/27 by Hofmanová, Martina, Pappalettera, Umberto, Zhu, Rongchan +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.14470

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 Bruè, 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.

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