Homogeneous Sobolev gradient flow of the length functional

Abstract

The well-known curve shortening flow can be formulated as the gradient flow of the length functional on the space of immersed closed planar curves, where the gradient is taken with respect to a reparametrisation-invariant $L^2$ Riemannian metric. This metric is degenerate, giving a geodesic distance of zero between any two curves. We instead consider a family of Sobolev $H^1$ metrics depending on two parameters $\lambda>0$ and $a\in \mathbb R$, where $\lambda$ sets the weight of the first-derivative term, and $a$ indexes a length normalisation which ensures that the metric is scale-homogeneous. For each such metric, the gradient of length can be written explicitly in terms of a convolution with respect to normalised arc length against the periodic Green's function of $(\lambda^2 \partial_x^2-1)$. The associated evolution is a reparametrisation invariant nonlocal ODE whose right-hand side is well-defined even on curves that are not immersed. Working in the optimal low-regularity setting $W^{1,1}(\mathbb S,\mathbb R^2)$, we prove local well-posedness using the Picard--Lindel\"of theorem and convergence to constant maps in finite time when $a<2$, and as $t\to\infty$ when $a\geq 2$. This behaviour is exhibited by round circles, which evolve self-similarly and collapse at an explicit time. We further prove that if the initial curve is an immersion, $C^1$, $C^2$, or bounds a strictly convex set, then each of these properties is preserved along the flow.