Curve-shortening flow

Remarks

Equivalently, $\partial_t \gamma = \partial_s^2 \gamma$, where $s$ is the arc length parameter of $\gamma_t$; this is a nonlinear partial differential equation of heat type. The normal velocity is what matters: adding a tangential velocity only changes the parametrization of the curves. The length $L(t)$ of $\gamma_t$ satisfies $L'(t) = -\int_{\gamma_t} \kappa^2 \, ds$, and the enclosed area $A(t)$ of an embedded counterclockwise curve satisfies $A'(t) = -2\pi$.

Example: the circles $\gamma(u, t) = \sqrt{r_0^2 - 2t}\,(\cos 2\pi u, \sin 2\pi u)$ for $0 \le t < r_0^2 / 2$ form a solution. For a counterclockwise circle of radius $r$, $\kappa = 1/r$ and $N$ points to the center, and indeed $\partial_t r = -1/r$.

Non-example: the shrinking circles $\gamma(u, t) = (r_0 - t)(\cos 2\pi u, \sin 2\pi u)$ move with normal speed $1$ instead of $1/r$, so they do not solve the equation.