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

2008/12/05 by de la Hoz, Francisco
Mathematics · Physics and Astronomy · #65D10 #65D30 #65N35 #65T50 #76B47 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.0812.1153

openalex publication_date 2008/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) k² 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.

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