The second-order $L^2$-flow of inextensible elastic curves with hinged   ends in the plane
  In the paper published in Duke Math. J. 1993, Y. Wen studied a second-order parabolic equation for inextensible elastic \emph{closed} curves in $\mathbb{R}^{2}$ toward inextensible elasticae. In this article, we extend Wen's result to the case of open inextensible planar curves with hinged ends. We obtain the long time existence of smooth solutions when the initial curves fulfill certain regular conditions. 
Chun-Chi Lin, Yang-Kai Lue, Hartmut R. Schwetlick
math.AP