Weak solutions of the curve shortening flow
type: Item
instance of: scholarly article
title: Weak solutions of the curve shortening flow
zbMATH Open document ID: 0990.35076
published in: Calculus of Variations and Partial Differential Equations
publication date: 1997-12-18T00:00:00Z
review text: The author formulates a parametric notion of weak solution for the curve shortening flow in arbitrary codimensions, and he proves the existence of such a solution which is global in time for an arbitrary smooth closed initial curve in \(\mathbb{R}^n\). The idea is to replace the problem by a simpler one which preserves the geometry of the evolving curves, and then to show that the modified problem has a weak solution in the author's sense by proving suitable a priori estimates for the solutions of a family of regularized problems and extracting a convergent subsequence. Alternative notions of generalized solution for mean curvature flow of arbitrary dimension and codimension have been developed by \textit{K. A. Brakke} [The motion of a surface by its mean curvature, Princeton University Press (1978; Zbl 0386.53047)] and by \textit{L. Ambrosio} and \textit{H. M. Soner} [J. Differ. Geom. 43, 693-737 (1996; Zbl 0868.35046)].
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 49Q99
zbMATH DE Number: 1099270
zbMATH Keywords: curve shortening flow in arbitrary codimension
zbMATH Keywords: weak solutions
zbMATH Keywords: existence
zbMATH Keywords: a priori estimates
author: Klaus Deckelnick
reviewed by: John Urbas
MaRDI profile type: Publication
full work available at URL: s005260050076
OpenAlex ID: W2015242782
DOI: 10.1007/S005260050076
description: scientific article; zbMATH DE number 1099270