Journal of the
Korean Mathematical Society JKMS

In this paper, we are devoted to study a forth order curve flow for a smooth closed curve in centro-affine geometry. Firstly, a new evolutionary equation about this curve flow is proposed. Then the related geometric quantities and some meaningful conclusions are obtained through the equation. Next, we obtain finite order differential inequalities for energy by applying interpolation inequalities, Cauchy-Schwartz inequalities, etc. After using a completely new symbolic expression, the $n$-order differential inequality for energy is considered. Finally, by the means of energy estimation, we prove that the forth order curve flow has a smooth solution all the time for any closed smooth initial curve.

Keywords: Centro-affine geometry, energy estimation, smooth solution, forth order curve flow

MSC numbers: Primary 51N10, 53A15, 53A55, 53E40