### 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 language | English |
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Article number | 20150629 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 472 |

Issue number | 2185 |

DOIs | |

Publication status | Published - 1 Jan 2016 |

Externally published | Yes |

### 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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## Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*472*(2185), [20150629]. https://doi.org/10.1098/rspa.2015.0629