Is it true that no mathematical relation exists between the Navier-Stokes equations and the multifractal model?

TL;DR

This paper develops a theory linking the Navier-Stokes equations with the multifractal model of turbulence, challenging the belief that no relation exists, and introduces a mediator scale related to Reynolds number.

Contribution

The authors reconcile the multifractal model with Leray's weak solutions of Navier-Stokes equations using invariant scaling and derive a new inverse scale connecting both theories.

Findings

Derived the Paladin-Vulpiani inverse scale related to Reynolds number.

Established a correspondence between velocity gradient norms and local scaling exponents.

Identified the parameter range where thermal noise affects the Navier-Stokes equations.

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 size $L$, we also derive the Paladin-Vulpiani inverse scale $\eta_{h,pav}$, which is related to the Reynolds number $\mathit{Re}$ by $L\eta_{h,pav}^{-1} = \mathit{Re}^{1/(1+h)}$, 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 $-2/3 \leqslant h_{min} \leqslant 1/3$, which lies precisely in the region where Bandak et al. (2022, 2024) have suggested that thermal noise makes the NSEs inadequate and generates spontaneous stochasticity. The implications of this are discussed.