Intermittency and Dissipation Regularity in Turbulence

TL;DR

This paper develops a geometric-analytic framework to analyze energy dissipation in weak solutions of the incompressible Euler equations, linking regularity, fractal dissipation sets, and intermittency phenomena in turbulence.

Contribution

It introduces a novel framework connecting Besov regularity, dissipation measures, and fractal dimensions, extending Onsager's theory in turbulence analysis.

Findings

Duchon-Robert distribution has improved regularity in negative Besov spaces.

Dissipative sets can have fractal structures with constrained dimensions.

Many Onsager singularity results are recovered as special cases.

Abstract

We lay down a geometric-analytic framework to capture properties of energy dissipation within weak solutions to the incompressible Euler equations. For solutions with spatial Besov regularity, it is proved that the Duchon-Robert distribution has improved regularity in a negative Besov space and, in the case it is a Radon measure, it is absolutely continuous with respect to a suitable Hausdorff measure. This imposes quantitative constraints on the dimension of the, possibly fractal, dissipative set and the admissible structure functions exponents, relating to the phenomenon of "intermittency" in turbulence. As a by-product of the approach, we also recover many known "Onsager singularity" type results.