Homogeneous Sobolev gradient flow of the length functional

We study the gradient flow of the length functional on the space of planar immersed closed curves, where the gradient is taken with respect to a family of homogeneous Sobolev $H^1$-type Riemannian metrics depending on parameters $\lambda>0$ and $a\in\mathbb{R}$. The gradient can be written explicitly in terms of arc-length convolution with the periodic Green's function for the second-order operator associated with the $H^1$ metric, and then the gradient flow is a reparametrisation-invariant nonlocal ODE. Working in the optimal low-regularity setting $W^{1,1}(\mathbb{S},\mathbb{R}^2)$, we show that the gradient is locally Lipschitz to obtain local well-posedness via the Picard--Lindel\"of theorem in Banach spaces. A time-reparametrisation reduces the analysis for general $a$ to the model case ${a=2}$, for which we obtain exponential decay of the length and global existence with uniform convergence in $W^{1,1}$ to a constant map. For $C^1$ immersed initial data we show that immersion is preserved for all time, and we further prove that if the initial curve bounds a convex set then convexity is also preserved by the flow.