  We consider a curve with boundary points free to move on a line in $\mathbb
R^2$, which evolves by the $L^2$--gradient flow of the elastic energy, that is
a linear combination of the Willmore and the length functional. For such planar
evolution problem we study the short and long--time existence.
  Once we establish under which boundary conditions the PDE's system is
well--posed (in our case the Navier boundary conditions), employing the
Solonnikov theory for linear parabolic systems in H\"older space, we show that
there exists a unique flow in a maximal time interval $[0,T)$. Then, using
energy methods we prove that the maximal time is actually $T= + \infty$.
