Relaxation of the curve shortening flow on the plane via the parabolic Ginzburg-Landau equation

I will discuss a method to represent curves evolving under curve shortening flow as nodal sets of the limit of solutions to the parabolic Ginzburg-Landau equation. For any given compact curve $\Gamma$ that satisfies the equation \be \frac{\partial \Gamma}{\partial t}=k_\Gamma\hat{n},\label{csf1}\ee I construct a family of solutions to the parabolic Ginzburg-Landau: \be \frac{\partial u_\epsilon}{\partial t}-\Delta u_\epsilon +\frac{(\nabla_u W)(u_\epsilon)}{2\epsilon ^2}=0 \label{gl11}\ee such that for $\bar{t}<T$ $$\lim_{\epsilon\to 0}u_\epsilon(x,\bar{t})=\left\{\begin{array}{cc} 1& \hbox{for $x$ outside $\Gamma(\lambda,\bar{t})$}\\ 0& \hbox{for $x$ on } \Gamma(\lambda,\bar{t}) \\ -1& \hbox{for $x$ inside }\Gamma(\lambda,\bar{t}). \end{array} \right.$$ It holds for every $\bar{t}\geq T$ $$\lim_{\epsilon\to 0} $$\lim_{\epsilon\to 0} |u_\epsilon(\cdot,\bar{t})-1|_0=0.$$ Moreover there is a continuous function $\tilde{v}^*_\epsilon$, with characteristics discussed above, that satisfies: $$\lim_{\epsilon \to 0}\sup_{(x,t)\in \rr2\times[0,\infty)}|u_\epsilon(x,t)-\tilde{v}^*_\epsilon(x,t)|=0.$$ <br />
This result is proven by constructing an approximate solution $\tilde{v}^*_\epsilon$ to the equation \equ{gl11}. The difference $u_\epsilon(x,t)-\tilde{v}^*_\epsilon(x,t)$ is estimated by regarding the functions $u_\epsilon$ as fixed points of some given functional.