On the H^1(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 L^2(ds^γ ) Riemannian metric gives rise to the curve shortening flow as the gradient flow of the length functional. Motivated by the vanishing of the L^2(ds^γ ) Riemannian distance, we consider the gradient flow of the length functional with respect to the H^1(ds^γ ) -metric. Circles with radius r_0 shrink with r(t) = √(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.