On the planar free elastic flow with small oscillation of curvature

The free elastic flow that begins at any curve exists for all time. If the initial curve is an $ω$-fold covered circle (``$ω$-circle'') the solution expands self-similarly. Very recently, Miura and the second author showed that (topological) $ω$-circles that are close to multiply-covered round circles are asymptotically stable under the planar free elastic flow, which means that upon rescaling the rescaled flow converges smoothly to the stationary (in the rescaled setting) $ω$-circle. Closeness in that work was measured via the derivative of the curvature scalar. In the present paper, we improve this by requiring closeness in terms of the curvature scalar itself. The convergence rate we obtain is sharp.