A survey of the elastic flow of curves and networks

Abstract

We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In particular, we give a complete proof of global existence and smooth convergence to critical points of the solution of the elastic flow of closed curves in R 2 . In the last section of the paper we also discuss a list of open problems.

Li-Yau type inequality for curves in any codimension

“…Concerning these flows, long-time existence and smooth convergence to an elastica are valid in general at least from smooth closed initial curves. The results follow by combining the fundamental result by Dziuk-Kuwert-Schätzle [8] with recent developments on the Lojasiewicz-Simon inequality as is demonstrated in [23] (see also [18]); we note that the argument in [23] directly works for higher codimensions as so do the key ingredients [8,24,19,25]. Our Li-Yau type inequality can be used for ensuring all-time embeddedness of solutions starting from initial curves below certain energy thresholds.…”

“…For elastic flows of open curves (cf. references in [18]) we may also apply Theorem 2.5 to obtaining similar thresholds to the above theorems (with 4̟ * replaced by ̟ * ) that ensure all-time embeddedness. Such thresholds are in particular effective for the Navier boundary condition with vanishing curvature, since in this case there always exist admissible curves below ̟ * .…”

“…At least from Polden's 1996 thesis, elastic flows are studied by many authors, see e.g. a recent nice survey [18] and references therein. Concerning these flows, long-time existence and smooth convergence to an elastica are valid in general at least from smooth closed initial curves.…”