A fractal version of the Onsager's conjecture: the $\beta-$model

Abstract

Intermittency phenomena are known to be among the main reasons why Kolmogorov's theory of fully developed Turbulence is not in accordance with several experimental results. This is why some \emph{fractal} statistical models have been proposed in order to realign the theoretical physical predictions with the empirical experiments. They indicate that energy dissipation, and thus singularities, are not space filling for high Reynolds numbers. This note aims to give a precise mathematical statement on the energy conservation of such fractal models of Turbulence. We prove that for $\theta-$H\"older continuous weak solutions of the incompressible Euler equations energy conservation holds if the upper Minkowski dimension of the spatial singular set $S \subseteq \T^3$ (possibly also time-dependent) is small, or more precisely if $\overline{\dim}_{\mathcal{M}}(S)<2+3\theta\,.$ In particular, the spatial singularities of \emph{non-conservative} $\theta-$H\"older continuous weak solutions of Euler are concentrated on a set with dimension lower bound $2+3\theta$. This result can be viewed as the fractal counterpart of the celebrated Onsager conjecture and it matches both with the prediction given by the $\beta-$model introduced by Frisch, Sulem and Nelkin in \cite{FSN78} and with other mathematical results in the endpoint cases.