Identifying the multifractal set on which energy dissipates in a turbulent Navier-Stokes fluid

Abstract

The rich multifractal properties of fluid turbulence illustrated by the work of Parisi and Frisch are related explicitly to Leray's weak solutions of the three-dimensional Navier-Stokes equations. Directly from this correspondence it is found that the set on which energy dissipates, ${\mathbb{F}}_m$, has a range of dimensions $\Dim=3/m$ ($1\le m\le \infty$), and a corresponding range of sub-Kolmogorov dissipation inverse length scales $Lη_{m}^{-1} \leq Re^{3/(1+\Dim)}$ spanning $R{e}^{3/4}$ to $R{e}^3$. Correspondingly, the multifractal model scaling parameter $h$, must obey $h\ge h_{min}$ with $-\twothirds \leq h_{min} \leq \third$.

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