Is it true that no mathematical relation exists between the Navier–Stokes equations and the multifractal model?
Year: 2026
Type: 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 $\eta_{h,pav}$ which is related to the Reynolds number Re by $L\eta _{h,\textit{pa}v}^{-1} = \textit{Re}^{1/(1+h)}$ , and which acts as a mediator between the two theories. This is achieved by considering $L^{2m}$ -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 $1 \leqslant m \leqslant \infty$ is equivalent to $-{{ {2}/{3}}} \leqslant h_{\textit{min}} \leqslant {{{1}/{3}}}$ , 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.
Source: Journal of Fluid Mechanics
Authors: John David Gibbon, Dario Vincenzi
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W7155948958