Evolution of elastic curves in \(\mathbb R^n\): Existence and computation (Q2782979)

The authors investigate the flow of curves \(f:{\mathbb R}/{\mathbb Z}\to{\mathbb R}^n\) by the gradient of their elastic energy; in particular, long-time existence of the flow and the limit behaviour of the flow is studied: NEWLINENEWLINENEWLINEfor the (curve diffusion) flow \(\partial_tf=-\nabla_s^2\kappa\) (where \(\kappa\) is the curvature vector field of the curve) the authors provide a lower bound for the existence of the flow; NEWLINENEWLINENEWLINEfor the flow \(\partial_tf=-\nabla_s^2\kappa+(\lambda-{|\kappa|^2\over 2})\kappa\) (where \(\lambda\geq 0\) is constant or is determined by the fixed length constraint, respectively) the authors establish the global existence of the flow and subconvergence to an elastic curve (for \(\lambda\neq 0\)). NEWLINENEWLINENEWLINEIn the last section of the paper the authors propose numerical algorithms to compute these flows. The algorithm is tested against the analytic solution in a case where it is known, and various examples are computed and visualized.