Multifractality and intermittency in the limit evolution of polygonal vortex filaments

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^{2πi ( 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.

44 pages. v2: Introduction rewritten. Overview rewritten in Section 2. Appendix B added. v3: Small corrections. v4: Accepted manuscript

References in corpus (6)

• Vortex Filament Equation for a Regular Polygon
• Fractal solutions of linear and nonlinear dispersive partial differential equations
• Riemann's non-differentiable function and the binormal curvature flow
• The pointwise behavior of Riemann's function
• The phenomenon of revivals on complex potential Schrödinger's equation
• An analytical study of flatness and intermittency through Riemann's non-differentiable functions