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

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 we also derive the Paladin-Vulpiani inverse scale $Lη_{h,pav}^{-1} = Re^{1/(1+h)}$ 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.

Figures (2)

• Figure 1: Scaled $L^{2m}$-norms of the velocity gradient with $F_{m}^{\alpha_{m}}$ defined in (\ref{['nse4']}) for $d=3$ ; courtesy of R. M. Kerr (see DGGKPV2014) : $\alpha_{m}\equiv \alpha_{m,3}$. Evidence of intermittent events appear at $t\approx 110$ for $m\geq 5$.
• Figure 2: Pictorial representation of how the the PaV-scale appears as a mediator between the Euler and NSEs. Following the arrows : (i) Clockwise : application of the PaV-scale to the NSEs implies results consistent with the MFM; (ii) Anti-clockwise : the MFM and NSEs together imply the PaV-scale.