Multifractality and intermittency in the limit evolution of polygonal vortex filaments

TL;DR

This paper investigates the multifractal and intermittent nature of functions modeling the evolution of polygonal vortex filaments, revealing multifractality for both rational and irrational initial conditions through advanced number theory techniques.

Contribution

It provides a comprehensive analysis of the multifractality and intermittency in vortex filament models, including explicit spectra and norms, using novel Diophantine set constructions and number theory methods.

Findings

Multifractality is established for rational initial conditions.

Multifractal behavior extends to irrational initial conditions.

Explicit spectra and Fourier norms are computed for the functions.

Abstract

With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality and intermittency of the family of generalized Riemann's non-differentiable functions \begin{equation} R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}, \qquad x_0 \in [0,1]. \end{equation} These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow. When $x_0$ is rational, we show that $R_{x_0}$ is multifractal and intermittent by completely determining the spectrum of singularities of $R_{x_0}$ and computing the $L^p$ norms of its Fourier high-pass filters, which are analogues of structure functions. We prove that $R_{x_0}$ has a multifractal behavior also when $x_0$ is irrational. The proofs rely on a careful design of Diophantine sets that depend on $x_0$, which we study by crucially using the Duffin-Schaeffer theorem and the Mass Transference Principle.