The Łojasiewicz–Simon inequality for the elastic flow

We define the elastic energy of smooth immersed closed curves in $${\mathbb {R}}^n$$ R n as the sum of the length and the $$L^2$$ L 2 -norm of the curvature, with respect to the length measure. We prove that the $$L^2$$ L 2 -gradient flow of this energy smoothly converges asymptotically to a critical point. One of our aims was to the present the application of a Łojasiewicz–Simon inequality, which is at the core of the proof, in a quite concise and versatile way.