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

TL;DR

This paper numerically investigates a geometric flow of planar curves, originally derived from vortex dynamics, that develops finite-time singularities, providing insights into the formation of corners in such evolutions.

Contribution

It introduces a numerical method to simulate the flow of planar curves with singularity formation, expanding understanding of geometric flows related to vortex dynamics.

Findings

Numerical reproduction of singularity formation in curve flow

Characterization of corner development at finite time

Properties of solutions near singularities

Abstract

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.