Riemann's non-differentiable function and the binormal curvature flow

TL;DR

This paper links Riemann's non-differentiable function to the binormal curvature flow, revealing a nonlinear geometric interpretation and constructing solutions with multifractal behavior akin to turbulence.

Contribution

It establishes a novel connection between Riemann's function and a nonlinear PDE, and constructs solutions exhibiting multifractal properties.

Findings

Solutions with smooth trajectories close to multifractal curves

Multifractal behavior aligns with turbulence models

Analytical object has a geometric interpretation

Abstract

We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious nonlinear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids.