Multifractality and polygonal vortex filaments

TL;DR

This paper investigates the multifractal properties of vortex filaments modeled by generalized Riemann functions, revealing their singularity spectrum and connecting turbulence behavior with number theory techniques.

Contribution

It introduces a novel approach to analyze vortex filament dynamics using multifractality and Diophantine approximation, linking fluid dynamics with number theory.

Findings

Determined the spectrum of singularities of the functions

Linked vortex filament behavior to multifractal analysis

Applied Diophantine set techniques to turbulence modeling

Abstract

In this proceedings article we survey the results in [5] and their motivation, as presented at the 50th Journ\'ees EDP 2024. With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality of a family of generalized Riemann's non-differentiable functions. These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow, the classical model for vortex filaments dynamics. We explain how we determined their spectrum of singularities through a careful design of Diophantine sets, which we study by using the Duffin-Schaeffer theorem and the Mass Transference Principle.