Elastic flow of curves with partial free boundary

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 this 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 Hölder 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 \(T= + \infty \).