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Peer-reviewedCC BY 4.0Calculus of Variations and Partial Differential Equations

Delayed parabolic regularity for curve shortening flow

  • Arjun Sobnack1
  • Peter M. Topping2
  1. 1Department of Mathematics, The University of Texas at Austin, Austin, TX 78712 USA
  2. 2Mathematics Institute, University of Warwick, Coventry, CV4 7AL UK
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Abstract

Given two curves bounding a region of area A that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time A/π A/π. We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L1\end{document}L1 function.

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Cite

BibTeXRIS
BibTeX
@article{sobnack2026delayed,
  title = {{Delayed parabolic regularity for curve shortening flow}},
  author = {Arjun Sobnack and Peter M. Topping},
  journal = {Calculus of Variations and Partial Differential Equations},
  year = {2026},
  month = nov,
  doi = {10.1007/s00526-026-03437-8},
  url = {https://doi.org/10.1007/s00526-026-03437-8},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Delayed parabolic regularity for curve shortening flow
AU  - Arjun Sobnack
AU  - Peter M. Topping
PY  - 2026
DA  - 2026/11/04
JO  - Calculus of Variations and Partial Differential Equations
DO  - 10.1007/s00526-026-03437-8
UR  - https://doi.org/10.1007/s00526-026-03437-8
AB  - Given two curves bounding a region of area A that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time A/π A/π. We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L^1\end{document}L1 function.
SN  - 0944-2669
SN  - 1432-0835
ER  -