On the H¹(ds^γ) -Gradient Flow for the Length Functional

In this article, we consider the length functional defined on the space of immersed planar curves. The L2(dsγ) Riemannian metric gives rise to the curve shortening flow as the gradient flow of the length functional. Motivated by the vanishing of the L2(dsγ) Riemannian distance, we consider the gradient flow of the length functional with respect to the H1(dsγ) -metric. Circles with radius r shrink with r(t)=W(ec-2t) under the flow, where W is the Lambert W function and c=r02+logr02 . 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.