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, F m , has a range of dimensions D m = 3 /m (1 ≤ m ≤ ∞ ), and a corresponding range of sub-Kolmogorov dissipation inverse length scales Lη − 1 m ≤ Re 3 / (1+ D m ) spanning Re 3 / 4 to Re 3 . Correspondingly, the multifractal model scaling parameter h , must obey h ≥ h min with − 23 ≤ h min ≤ 13 .