扭結，彈性桿與磁管的高階與分數階幾何流

The gradient flow of elastic energy of curves could also be treated by a second-order parabolic equation, instead of a fourth-order equation. In this project we focus on the $L^2$-flow of elastic non-closed curves in $\mathbb{R}^2$ with hinged boundary conditions. This $L^2$-flow of planar non-closed curves corresponds to a {\bf second-order} parabolic equation. We prove the existence of $C^\infty$-smooth solutions of the $L^2$-flow in $I\times (0,\infty)$, if the initial curve $f_0: I\rightarrow \mathbb{R}^2$ is $C^\infty$-smooth. Moreover, the asymptotic limit curves are piecewise $C^\infty$-smooth. Our result seems to be extendable to the case of clamped boundary conditions. As one considers certain self-interaction energy of curves, fractional operators often appear. We are still investigating simple cases as the total energy is consisted of elastic energy, stretching energy, and self-interaction energy of curves. We couldn‘t finish problems in this direction, thus leave it to our future project.