An invariant forth-order curve flow in centro-affine geometry (Q6619748)

Let \(C:S^1\times I\to\mathbb R^2\) be a family of embedded smooth closed curves. The authors investigate a fourth order curve flow in centro-affine geometry defined by \N\[\N\frac{\partial C}{\partial t}=-\frac{1}{2}\varphi_{\xi\xi}C_{\xi}-\varphi_{\xi}C,\N\]\Nwhere \(t\in I\) a parameter for time, \(\xi\) is the centro-affine arc length element and \(\varphi\) is the centro-affine curvature. They obtain finite order differential inequalities for the energy by applying interpolation inequalities, Cauchy-Schwarz inequalities, etc, and they consider a \(n\)-order differential inequality for the energy obtained by using a completely new symbolic expression. Finally, they use energy estimates to prove that the fourth order curve flow has a smooth solution for all time for any closed smooth initial curve.