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

Journal Article


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.

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


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.

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