Multifractality and polygonal vortex filaments

In this proceedings article we survey the results in [5] and their motivation, as presented at the 50th Journées Équations aux dérivées partielles 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.