On the $H^1(ds)$-gradient flow for the length functional

Citation

• Schrader, P., Wheeler, G., & Wheeler, V. -M. (2021). On the $H^1(ds)$-gradient flow for the length functional. Retrieved from http://arxiv.org/abs/2102.07305v2

Web Of Science Accession Number

Abstract

• In this article we consider the length functional defined on the space of

  immersed planar curves. The $L^2(ds)$ Riemannian metric gives rise to the curve

  shortening flow as the gradient flow of the length functional. Motivated by the

  triviality of the metric topology in this space, we consider the gradient flow

  of the length functional with respect to the $H^1(ds)$-metric. Circles with

  radius $r_0$ shrink with $r(t) = \sqrt{W(e^{c-2t})}$ under the flow, where $W$

  is the Lambert $W$ function and $c = r_0^2 + \log r_0^2$. We conduct a thorough

  study of this flow, giving existence of eternal solutions and convergence for

  general initial data, preservation of regularity in various spaces, qualitative

  properties of the flow after an appropriate rescaling, and numerical

  simulations.

UOW Authors

•   Wheeler, Glen
•   Wheeler, V.-M

Publication Date

• 2021

Citation

• Schrader, P., Wheeler, G., & Wheeler, V. -M. (2021). On the $H^1(ds)$-gradient flow for the length functional. Retrieved from http://arxiv.org/abs/2102.07305v2

Web Of Science Accession Number