The free elastic flow for closed planar curves

Abstract

The free elastic flow is the $L^2$-gradient flow for Euler's elastic energy, or equivalently the Willmore flow with translation invariant initial data. In contrast to elastic flows under length penalisation or preservation, it is more challenging to study the free elastic flow's asymptotic behavior, and convergence for closed curves is lost. In this paper, we nevertheless determine the asymptotic shape of the flow for initial curves that are geometrically close to circles, possibly multiply-covered, proving that an appropriate rescaling smoothly converges to a unique round circle.