Is it true that no mathematical relation exists between the Navier–Stokes equations and the multifractal model?
Author(s)
Gibbon, John David
Vincenzi, Dario
Type
Journal Article
Abstract
Contrary to accepted turbulence folklore, which holds that no mathematical relation exists between the Navier–Stokes equations (NSEs) and the multifractal model (MFM) of Parisi and Frisch, we develop a theory that reconciles the MFM with Leray’s weak solutions of Navier–Stokes analysis. From a combination of Euler invariant scaling and the NSEs set in a three-dimensional box of side L, we also derive the Paladin–Vulpiani scale ηh,pav which is related to the Reynolds number Re by Lη−1 h,pav = Re1/(1+h), and which acts as a mediator between the two theories. This is achieved by considering L2m-norms of the velocity gradient to find a correspondence between m and the local scaling exponent h in the multifractal model. The parameter m acts as if it were the sliding focus control
on a telescope which allows us to zoom in and out on different structures. The range 1⩽m⩽∞isequivalent to −2/3⩽hmin ⩽1/3, which lies precisely in the region where
Bandak et al. (Phys.Rev.E, 2022, vol. 105, p. 065113; Phys.Rev.Lett., 2024, vol. 132, p. 104002) have suggested that thermal noise makes the NSEs inadequate and generates spontaneous stochasticity. The implications of this are discussed.
on a telescope which allows us to zoom in and out on different structures. The range 1⩽m⩽∞isequivalent to −2/3⩽hmin ⩽1/3, which lies precisely in the region where
Bandak et al. (Phys.Rev.E, 2022, vol. 105, p. 065113; Phys.Rev.Lett., 2024, vol. 132, p. 104002) have suggested that thermal noise makes the NSEs inadequate and generates spontaneous stochasticity. The implications of this are discussed.
Date Issued
2026-05-10
Date Acceptance
2026-03-15
Citation
Journal of Fluid Mechanics, 2026, 1034
ISSN
0022-1120
Publisher
Cambridge University Press (CUP)
Journal / Book Title
Journal of Fluid Mechanics
Volume
1034
Copyright Statement
© The Author(s), 2026. Published by Cambridge University Press This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
License URL
Publication Status
Published
Article Number
A2
Date Publish Online
2026-04-27