曲线扩散流短时存在性的初等证明

An Elementary Proof of the Short Time Existence of Curve Diffusion Flow

利用初等方法证明了曲线扩散流的短时存在性问题.首先,从曲线的曲率演化方程入手,将研究的问题等价转化为探究四阶抛物型拟线性偏微分方程解的存在性问题;然后,将拟线性方程线性化,证明四阶线性偏微分方程有解;最后,根据曲线论基本定理构造出曲线,再结合参数变换,验证曲线满足曲线扩散流的演化方程.

In this paper,we present an elementary proof of the short time existence of curve diffusion flow.Starting from the curvature evolution equation of the curve,we transform the problem into the equivalent problem of the existence of the solution of a fourth order parabolic quasi-linear partial differential equation.Then we linearize the quasi-linear equation to prove that the fourth-order linear partial differential equation has a solution.Finally,we construct the curve by the fundamental theorem of curve theory,and we verify that the curve satisfies the original evolution equation of curve diffusion flow.