# 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.

• 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

### 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.

• 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