Numerical study of a flow of regular planar curves that develop singularities at finite time
Provides numerical validation for a known theoretical result in geometric flow, but is incremental as it focuses on reproducing existing solutions.
This paper numerically reproduces the evolution of regular planar curves under a specific geometric flow that develop corner-shaped singularities at finite time, confirming properties predicted by Perelman and Vega.
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.