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

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.

References in corpus (6)

• On the vortex filament conjecture for Euler flows
• Vortex Filament Equation for a Regular Polygon
• Stability of the selfsimilar dynamics of a vortex filament
• Fractal solutions of linear and nonlinear dispersive partial differential equations
• On the Relationship between the One-Corner Problem and the $M$-Corner Problem for the Vortex Filament Equation
• Geometric differentiability of Riemann's non-differentiable function

Cited by in corpus (3)

• Convergence over fractals for the periodic Schrödinger equation
• The pointwise behavior of Riemann's function
• Multifractality and intermittency in the limit evolution of polygonal vortex filaments