A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

Michael C. Dallaston, Scott W. McCue

Research output: Contribution to journalArticle

7 Citations (Scopus)
22 Downloads (Pure)

Abstract

Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area; we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for nonconvex initial conditions, distinguishing between pinch-off and coalescence of the curve interior. 2016 The Authors.

Original languageEnglish
Article number20150629
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume472
Issue number2185
DOIs
Publication statusPublished - 1 Jan 2016
Externally publishedYes

Bibliographical note

Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

Keywords

  • Coalescence
  • Curve shortening flow
  • Extinction behaviour
  • Geometric partial differential equation
  • Pinch-off
  • Self-similar solutions

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

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