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 ₇, ₀ₕ which is related to the Reynolds number Re by L ₇, ₀ₕ^-1 = 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 m is equivalent to -{ {2/3}} h₌₈₍ {{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.
Gibbon et al. (Mon,) studied this question.