The dynamics of elastic closed curves under uniform high pressure

We consider the dynamics of an inextensible elastic closed wire in the plane under uniform high pressure. In 1967, Tadjbakhsh and Odeh (J. Math. Anal. Appl. 18:59-74, 1967) posed a variational problem to determine the shape of a buckled elastic ring under uniform pressure. In order to comprehend a dynamics of the wire, we consider the following two mathematical questions: (i) can we construct a gradient flow for the Tadjbakhsh-Odeh functional under the inextensibility condition?; (ii) what is a behavior of the wire governed by the gradient flow near every critical point of the Tadjbakhsh-Odeh variational problem? For (i), first we derive a system of equations which governs the gradient flow, and then, give an affirmative answer to (i) by solving the system involving fourth order parabolic equations. For (ii), we first prove a stability and instability of each critical point by considering the second variation formula of the Tadjbakhsh-Odeh functional. Moreover, we give a lower bound of its Morse index. Finally we prove a dynamical aspects of the wire near each equilibrium state.