Evolution of Elastic Curves in $\Rn$: Existence and Computation

We consider curves in ${\mathbb R}^n$ moving by the gradient flow for elastic energy, i.e., the L ² integral of curvature. Long-time existence is proved in the two cases when a multiple of length is added to the energy or the length is fixed as a constraint. Along these lines, a lower bound for the lifespan of solutions to the curve diffusion flow is observed. We derive algorithms for both the elastic flows and the curve diffusion equation. After a numerical test we compute several examples, including cases of curve diffusion in which a singularity develops.