I've been looking at the Lemma 1.5 of The Heat Equation Shrinks Embedded Plane Curves to Round Points (here), where Matthew Grayson proved that
If $\kappa(s,t)$ is bounded for $t\in[0,t_0)$. Then for some $\varepsilon>0$, $C(t)$ exists and is smooth for $t\in [0,t_0+\varepsilon)$.
In the proof he uses the formula $${\partial\over\partial t}{\partial\over\partial s}={\partial\over\partial s}{\partial\over\partial t}+\kappa^2{\partial\over\partial s}$$ and also $${\partial\kappa\over\partial t}={\partial^2\kappa\over\partial s^2}+\kappa^3$$ to obtain that $${\partial\over\partial t}\left({\partial\kappa\over\partial s}\right)={\partial\over\partial s^2}\left({\partial\kappa\over\partial s}\right)+4\kappa^2\left({\partial\kappa\over\partial s}\right).$$ He claims that this equation bounds the rate of growth of ${\partial\kappa\over\partial s}$ to exponential.
Repeated applications of the first formula yield $${\partial\over\partial t}\left({\partial^n\kappa\over\partial s^n}\right)={\partial\over\partial s^2}\left({\partial^n\kappa\over\partial s^n}\right)+(n+3)\kappa^2\left({\partial^n\kappa\over\partial s^n}\right)\;+\;\text{previously bounded terms},$$ so he gets the same as before for the $n$-th derivative of $\kappa$.
With this, using again the first formula, it is proved that the curve converges as $t\to t_0$. And similarly, $C(t_0)$ is smooth.
He finally applies the Theorem 1.1 to obtain that $C(t)$ exists and is smooth for some further short time.
I'm struggling to understand the key parts of this proof:
- The boundedness of the derivatives is not that obvious to me from the above formulas. I guess he gets an PDE which solution can be bounded by an exponential, but if so I don't really see how.
- From the boundedness of the derivatives of $\kappa$ with respect to $s$ for a fixed $t$, $C(t)$ converges as $t\to t_0$. This is because with $\kappa$ and its derivatives bounded, the curves under the flow must exist for $t\in[0,t_0)$. And the same argument for the smoothness... Am I right?
As you can see I'm kind of lost here, it would be great if anyone could explain to me a little bit these two ideas. Thanks in advance.