On the Evolution of the Vortex Filament Equation for Regular $M$-Polygons with Nonzero Torsion

Abstract

In this paper, we consider the evolution of the vortex filament equation (VFE): $X_t = X_s \wedge X_{ss},$ taking $M$-sided regular polygons with nonzero torsion as initial data. Using algebraic techniques backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; moreover, the multifractal trajectory of the point $X(0,t)$ is not planar and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as those already established for circles. This is accomplished by showing the existence of variants of the so-called Riemann's nondifferentiable function that are as close to smooth curves as desired when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the self-similar solutions of VFE have finite renormalized energy.

Locations

• arXiv (Cornell University) - View - PDF
• BIRD (Basque Center for Applied Mathematics) - View - PDF
• SIAM Journal on Applied Mathematics - View