Non-convergence and instability in the asymptotic behaviour of curves evolving by curvature
type: Item
instance of: scholarly article
title: Non-convergence and instability in the asymptotic behaviour of curves evolving by curvature
zbMATH Open document ID: 1029.53079
author: Ben Andrews
published in: Communications in Analysis and Geometry
publication date: 2003-04-07T00:00:00Z
review text: Let \(\gamma_0\) be a convex closed curve in the plane \(C\). Its motion is described by the equation of the form \[ \frac{\partial}{\partial t} x(\xi,t)=-\phi({\mathbf n}(\xi,t))(\kappa(\xi,t))^{\alpha} {\mathbf n}(\xi,t),\quad x(\xi,0)=x_0(\xi), \] for all \(\xi\) in \(C\) and \(t\in[0,T)\), where \(x_t=x(\cdot,t)\) is a smooth embedding for each \(t\in[0,T), {\mathbf n}(\xi,t)\) is the outward unit normal vector to the curve \(\gamma_t=x_t(C)\) at the point \(x(\xi,t),\kappa(\xi,t)\) is the curvature of \(\gamma_t\) at \(x(\xi,t)\) and \(\phi:S^1\to R\) is a smooth positive function. The author proves that the isoparametric ratios of the evolving curves generically approach infinity if the speed of motion is proportional to curvature to a power less than \(1/3\).
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35F05
zbMATH DE Number: 1893852
zbMATH Keywords: convex closed curves
zbMATH Keywords: curvature
zbMATH Keywords: curves motion
zbMATH Keywords: differential equations asymptotics behaviour
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: cag.2002.v10.n2.a8
OpenAlex ID: W3146836907
DOI: 10.4310/CAG.2002.V10.N2.A8
description: scientific article; zbMATH DE number 1893852