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 $${\mathbb {R}}^2$$ R 2 . In the last section of the paper we also discuss a list of open problems.

A dynamic approach to heterogeneous elastic wires

“…Local and global existence of solutions for the classical elastic flow, given by a parabolic fourth order equation in R n , has been shown in several works, for instance [10,16,8,7,15,21,6,19,17,18]. For a more detailed overview, see the recent survey [14]. A parabolic system related to (1.5) is studied in [1] and numerically elaborated in [9].…”

Complete classification of planar p-elasticae

“…The optimal regularity results would be informative for studying gradient flows, of second order [33,34] and of fourth order [7,8,35] (called p-elastic flow). In the classical case p = 2 many results about smooth convergence to elasticae as time goes to infinity are already known, in particular for elastic flows of fourth order; see the survey [27] and references therein (and also [45,35] for second-order flows). In contrast, for p = 2, long-time behavior is less understood, and the only known results seem to be recent sub-convergence results [33,7]; the former [33] is about (at least) W 2,pweak convergence of a second-order gradient flow for Θ-networks; the latter [7] is about W 2,p -strong convergence of a fourth-order p-elastic flow for closed curves with p ≥ 2.…”

On an elastic flow for parametrized curves in $\mathbb{R}^n$ suitable for numerical purposes

“…The study of evolution equations associated to the bending energy E has attracted a lot of attention in recent years: for motivation and extended references we refer here simply to [6], which has inspired a lot of the work presented here, and to a recent survey [9], where several recent results are discussed.…”