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Some special solutions to curve shortening flow

December 2, 2019 – 2:01 pm

Consider ${\boldsymbol{\gamma}(p,t)}$ is a closed curve in ${\mathbb{R}^2}$ which moves by its curvature. More preciesly, let

$\displaystyle \partial_t\boldsymbol{\gamma}(p,t)=\kappa \mathbf{N}$

where ${\mathbf{N}}$ is the unit normal pointing to the inside of the region inscribed by the curve. Such a flow is curve shortening flow. We are interested in some particular (convex, graphical) solutions of the flow.

Suppose ${\boldsymbol{\gamma}}$ locally can written as a graph, say ${\boldsymbol{\gamma}=(x,u(x,t))}$ . Then the flow equation means

$\displaystyle u_t=\frac{u_{xx}}{1+u_x^2}$

Consider ${u(x,t)=g(\lambda t+f(x))}$ in particular, where ${\lambda}$ is a constant. Then we can decouple the equation into two differential equations.

$\displaystyle f''+\mu f'^2-\lambda=0,$

$\displaystyle g''-\lambda g'^3-\mu g'=0.$

where ${\mu}$ is a constant. Depending on the values of ${\lambda}$ and ${\mu}$ , one can have the following four cases

(1) ${\lambda=0}$ , ${\mu=0}$ . Then ${u(x,t)=ax}$ and ${\boldsymbol{\gamma}}$ is actually a static line.

(2) ${\lambda\neq 0}$ , ${\mu=0}$ . Then one can choose ${f=\frac{1}{2}\lambda x^2}$ and ${g(x)=\frac{1}{2}\sqrt{\frac{x}{-2\lambda}}}$ . In this case

$\displaystyle u^2+\frac{1}{4}x^2=-\frac{1}{2}t$

${\boldsymbol{\gamma}}$ is a circle.

(3) ${\lambda\mu<0}$ . Say ${\lambda=-1}$ and ${\mu=1}$ . Then one can solve the ODE to get ${f(x)=\log\cos x}$ and ${g(x)=const}$ , ${\sinh^{-1}(e^x)}$ , ${\pm \cosh^{-1}(e^x)}$ . They corresponds to grim reaper, hair clip and paper clip (Angenent oval) respectively.