Long time existence of solutions to an elastic flow of networks

A Survey of the Elastic Flow of Curves and Networks

“…[40] [ 40] [ 40,41] Open curves, clamped b.c. [52] [ 29] [ 18,41] Non compact curves [40] [ 40] Networks [15,22,23] [ 14,22] [ 14] We refer also to the two recent PhD theses [38,48]. The aim of this expository paper is to arrange (most of) this material in a unitary form, proving in full detail the results for the elastic flow of closed curves and underlying the differences with the other cases.…”

“…One interesting feature following from the parabolic structure of the elastic flow is that solutions are smooth for any (strictly) positive times. We give the idea of two possible lines of proof of this fact and we refer to [15,22] for the complete result.…”

“…It is actually possible not to ask for it by defining the elastic flow of networks in a Sobolev setting. The price that we have to pay is that in such a case a solution will be slightly less regular (see [22,38] for details). On the opposite side, if we want a smooth solution till t = 0 one has to impose many more conditions.…”

“…We can remove the fourth order conditions (iii)-(iv) setting the problem in Sobolev spaces, with the initial datum in W 4−4/p,p ([0, 1]; R 2 ) with p ∈ (5, ∞). Even in this lower regularity class it is possible to prove uniqueness of solutions (see [22]), but we pay in regularity of the solution, that is merely in W 1,p (0, T ); L p (0, 1); R 2 ∩ L p (0, T ); W 4,p (0, 1); R 2 .…”

“…To describe the results contained in [14,22] for networks with junctions subjected to natural boundary conditions we need a preliminary definition. Definition 4.13.…”