Curve shortening flow of a $C^1$ curve

Question

I am reading something which says that a closed and embedded $C^1$ curve immediately becomes smooth under the curve shortening flow. I am familiar with this result for $C^2$ curves, but was under the impression curve shortening flow isn't necessarily even defined for $C^1$ curves, since the curvature may not be continuous. Could someone perhaps cite this result for $C^1$ curves, or explain what I've misunderstood? Thanks

Answer 1

The earliest reference might be the following paper by S. Angenent. The setting is quite general, and the result is that when the initial curve $\gamma$ is locally Lipschitz (Note that in Theorem C in the paper, one can even weaken the condition to $C^1$ locally graph like, which is a very weak condition) then there is a continuous family $\{ \gamma_t : t\in [0,T)\}$ of immersed curves so that $\gamma_0 = \gamma$ and

$$ \frac{\partial \gamma_t}{\partial t} = \vec k$$

is satisfied on $t>0$. Of course the family is indeed anayltic for $t>0$ (Also proved in the paper).

If you overkill and use Theorem 4.2 of Ecker-Huisken's interior estimates, they do have $1/2$ holder continuity with respect to $t$ at $t=0$. Not completely sure how it is improved (since your case is $C^1$, which is better then locally Lipschitz).