Curve straightening flow deforms closed plane curves with nonzero rotation number to circles
The curve straightening flow of closed plane curves describes the problem of minimizing the potential energy of an elastic beam. This energy is proportional to the total squared curvature. The curve straightening flow has been considered by several authors with different representations for the curves and different choices for the metric; references are given in the paper. In the present article, the author considers the curve straightening flow as the \(L^2\)-gradient flow of \(\int_\gamma k^2 ds\) where \(s\) is the arclength parameter of the closed plane curve \(\gamma\) with given length. First, he shows the existence of a solution for a short time and then he derives a priori curvature bounds using integral inequalities to prove longtime existence. The third section is devoted to the longtime behaviour of the flow: if the curve \(\gamma\) has rotation number \(\eta\neq 0\), then it converges exponentially to an \(\eta\)-fold circle. A slightly different approach to the same problem has been undertaken in the meantime. In his Ph.D. thesis, Alexander Polden considers the energy functional \(\int k^2+\lambda^2\) for arbitrary \(\lambda\neq 0\) instead of fixing the length of the closed plane curve \(\gamma\) and he examines the \(L^2\)-gradient flow of this functional. He was able to prove short time existence in a much more general setting, namely for quasilinear parabolic equations of order \(2p\). Concerning the longtime behaviour of the flow, Polden could handle the case of \(\gamma\) having rotation number 0 as well, showing in this case convergence to a figure eight curve.
scientific article; zbMATH DE number 790884