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

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 $\text{Dim}=3\mathrm{/}m$ ($1\le m\le \mathrm{\infty }$), and a corresponding range of sub-Kolmogorov dissipation inverse length scales $L{\eta }_m^{-1}\le R{e}^{3\mathrm{/}(1+\text{Dim})}$ spanning $R{e}^{3\mathrm{/}4}$ to $R{e}^3$. Correspondingly, the multifractal model scaling parameter $h$, must obey $h\ge h_{min}$ with $-\text{twothirds}\le h_{min}\le \text{third}$.