Abstract
In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, translators, and rotators; and (3) an explicit parametrisation of a shrinking figure eight curve.
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Notes
We would like to thank one of the anonymous referees for pointing out this improved bound.
The authors would like to thank Hojoo Lee for pointing this out.
It may be helpful for the reader to recall the biharmonic heat flow (∂ t +Δ2)v=0 for a function v.
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Acknowledgements
The research of the second author was supported by a summer vacation scholarship of the Australian Mathematical Sciences Institute. The research of the third, fourth and fifth authors was supported by Discovery Project grant DP120100097 of the Australian Research Council.
The authors would like to thank the anonymous referees for their careful reading and comments that have led to improvements in the article. The authors would also like to thank Hojoo Lee for enlightening discussions related to this work.
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The research of the second author was supported by a summer vacation scholarship of the Australian Mathematical Sciences Institute. The research of the third, fourth and fifth authors was supported by Discovery Project grant DP120100097 of the Australian Research Council.
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Edwards, M., Gerhardt-Bourke, A., McCoy, J. et al. The Shrinking Figure Eight and Other Solitons for the Curve Diffusion Flow. J Elast 119, 191–211 (2015). https://doi.org/10.1007/s10659-014-9502-5
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DOI: https://doi.org/10.1007/s10659-014-9502-5