Double spiral singularities for a flow of regular planar curves

2021/08/27 by Kokocki, Piotr, Dunst, Kamil
#34M50 #34M55 #41A60 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.12220

Abstract

In this paper we study the singularity formation for the geometric flow of complex curves z_t = -z_xxx + (3)/(2)øz_x z_xx², that was derived [R. E. Goldstein and D. M. Petrich, \em Phys. Rev. Lett., 69 (1992), pp. 555--558] while considering the vortex patch dynamics for the incompressible 2D Euler equation. We prove that arbitrary curve, consisting of two rotating logarithmic spirals, is a finite time singularity developed by a smooth solution of the flow. We provide exact construction of the solution in the terms of appropriate Painlevé II transcendents and furthermore we establish its asymptotic expansion in the vicinity of the singularity.

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