arXiv subjects

Francisco de la Hoz

Publications and source records attributed to Francisco de la Hoz.

22 records · Page 2Linked to original sources

arXiv2015

Vortex Filament Equation for a Regular Polygon

In this paper, we study the evolution of the vortex filament equation (VFE), $$\mathbf X_t = \mathbf X_s \wedge \mathbf X_{ss},$$ with $\mathbf X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $\mathbf X(s, t)$ is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gauß sum. We also study the fractal behavior of $\mathbf X(0, t)$, relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal.

math.AP↗

arXiv2014

Doubly connected V-states for the planar Euler equations

We prove existence of doubly connected V-states for the planar Euler equations which are not annuli. The proof proceeds by bifurcation from annuli at simple "eigenvalues". The bifurcated $V$-states we obtain enjoy a $m$-fold symmetry for some $m\ge 3.$ The existence of doubly connected $V$-states of strict $2$-fold symmetry remains open.

math.AP↗

arXiv2008

Numerical study of a flow of regular planar curves that develop singularities at finite time

In this paper, we will study the following geometric flow, obtained by Goldstein and Petrich while considering the evolution of a vortex patch in the plane under Euler's equations, X_t = -k_s n - (1/2) k^2 T, with s being the arc-length parameter and k the curvature. Perelman and Vega proved that this flow has a one-parameter family of regular solutions that develop a corner-shaped singularity at finite time. We will give a method to reproduce numerically the evolution of those solutions, as well as the formation of the corner, showing several properties associated to them.

math.NA↗

arXiv2008

A numerical study of the self-similar solutions of the Schroedinger Map

We present a numerical study of the self-similar solutions of the Localized Induction Approximation of a vortex filament. These self-similar solutions, which constitute a one-parameter family, develop a singularity at finite time. We study a number of boundary conditions that allow us reproduce the mechanism of singularity formation. Some related questions are also considered.

math.NA↗