The Eighth-Order Evolution Model for Closed Planar Curves

We consider the $L^2(ds)$ gradient flow of $\frac{1}{2}\int k_{ss}^{2}ds$ for smooth closed immersed planar curves. The stationary curves are precisely the multiply covered circles. We give an explicit sufficient condition under which the elastic energy is non-increasing and for each nonzero turning number, explicit small initial scale-invariant energy yields a global smooth solution with bounded positive length and exponential convergence to a fixed multiply covered circle, after reparametrisation. We note that that the qualitative rigidity and convergence results are already contained in the generalised ideal flow theory of McCoy and Wheeler. The present paper gives quantitative measure of small initial data, corrected evolution identities, and a complete argument in the case of the curvature energy $\frac{1}{2}\int k_{ss}^{2}ds$.