On the Evolution of the Vortex Filament Equation for regular M-polygons with nonzero torsion

In this paper, we consider the evolution of the Vortex Filament equa- tion (VFE): Xt = Xs ∧ Xss, taking M-sided regular polygons with nonzero torsion as initial data. Us- ing 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; more- over, 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 the one already established for circles. This is accomplished by showing the existence of variants of the so-called Rie- mann’s non-differentiable 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 selfsimilar solutions of VFE have finite renormalized energy

Vortex Filament Equation for some Regular Polygonal Curves

One of the most interesting phenomena in fluid literature is the occurrence and evolution of vortex filaments. Some of their examples in the real world are smoke rings, whirlpools, and tornadoes. For an ideal fluid, there have been several models and governing equations to describe this evolution; however, due to its simplicity and geometric properties, the vortex filament equation (VFE) has recently gained attention. As an approximation of the dynamics of a vortex filament, the equation first appeared in the work of Da Rios at the beginning of the twentieth century. This model is usually known as the local induction approximation (LIA). In this work, we examine the evolution of VFE for regular polygonal curves both from a numerical and theoretical point of view in the Euclidean as well as hyperbolic geometry. In the first part of the thesis, we observe the evolution of the Vortex Filament equation taking $M$-sided regular polygons with nonzero torsion as initial data in the Euclidean space. 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 the one already established for circles. This is accomplished by showing the existence of variants of the so-called Riemann’s non-differentiable 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. In the rest of the work, we delve into the hyperbolic setting and examine the evolution of VFE for a planar $l$-polygon, i.e., a regular planar polygon in the Minkowski 3-space $\mathbb{R}^{1,2}$. Unlike in the Euclidean case, a planar $l$-polygon is open which makes the problem more challenging from a numerical point of view. After trying several numerical methods, we conclude that a finite-difference discretization in space combined with an explicit Runge--Kutta method in time, gives the best numerical results both in terms of efficiency and accuracy. On the other hand, using theoretical arguments, we recover the evolution algebraically, and thus, we show the agreement between the two approaches. During the numerical evolution, it has been observed that the trajectory of a corner is multifractal and as the parameter $l$ goes to zero, it converges to the Riemann’s non-differentiable function. Furthermore, as in the Euclidean case, we provide strong numerical evidence to show that at infinitesimal times, the evolution of VFE for a planar $l$-polygon as an initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only we can compute the speed of the center of mass of the planar $l$-polygon theoretically, the relationship also reveals important properties of the trajectory of its corners which we compare with its equivalent in the Euclidean case. Finally, a nonzero torsion in the hyperbolic case, yields two different kinds of helical polygonal curves, however, with the numerical and theoretical techniques developed so far, we are able to address them as well. This remains part of the future work of the thesis

On the Schrödinger map for regular helical polygons in the hyperbolic space

The main purpose of this article is to understand the evolution of X t = X s ∧− X ss , with X(s, 0) a regular polygonal curve with a nonzero torsion in the three-dimensional Minkowski space. Unlike in the case of the Euclidean space, a nonzero torsion now implies two different helical curves. This generalizes recent works by the author with de la Hoz and Vega on helical polygons in the Euclidean space as well as planar polygons in the Minkowski space. Numerical experiments in this article show that the trajectory of the point X(0, t) exhibits new variants of Riemann’s non-differentiable function whose structure depends on the initial torsion in the problem. As a result, we observe that the smooth solutions (helices, straight line) in the Minkowski space show the same instability as displayed by their Euclidean counterparts and curves with zero-torsion. These numerical observations are in agreement with some recent theoretical results obtained by Banica and Vega

Vortex Filament Equation for a regular polygon in the hyperbolic plane

The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case

Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

The aim of this paper is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow up exponentially, which makes the problem more challenging from a numerical point of view. However, using a finite difference scheme in space combined with a fourth-order Runge-Kutta method in time and fixed boundary conditions, we show that the numerical solution is in complete agreement with the one obtained by means of algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with the evolution in the Euclidean case.Sandeep Kumar would like to thank Carlos J. Garcia-Cervera for the discussions on Sect. 3 which took place during his visit to the University of California, Santa Barbara (UCSB), USA. The authors would like to thank the anonymous referees for their valuable comments and suggestions that have largely improved the presentation of this paper. This paperwas partially supported by the ERCEA Advanced Grant 2014 669689-HADE, by the MICINN Projects PGC2018-094522-B-I00 and SEV-2017-0718, by the Basque Government Grant IT1247-19, and by the Basque Government BERC Program 2018-2021

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

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