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arXiv · 2304.14470

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

Martina Hofmanová·Umberto Pappalettera·Rongchan Zhu·Xiangchan Zhu
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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.

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Martina Hofmanová, Umberto Pappalettera, Rongchan Zhu, Xiangchan Zhu. 2023-04-27. Kolmogorov $4/5$ law for the forced 3D Navier-Stokes equations. https://doi.org/10.1007/s40072-025-00380-1

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