  We consider the gradient flow of the Ambrosio-Tortorelli functional at fixed
$\epsilon>0$, proving existence, uniqueness and $L^2 _t (H_x ^2) \cap L^\infty
_t (H^1 _x) \cap H^1 _t (L^2 _x) $ regularity in dimension 2. In particular we
improve a previous result where such regularity was known only up to a finite
number of space time points, which diverged as $\epsilon \to 0$. By employing a
different technique for the crucial $L^2 _t (H^2 _x)$ estimates we can see how
in fact the desired regularity holds everywhere.
