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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 weimprove a previous result where such regularity was known only up to a finitenumber of space time points, which diverged as $\epsilon \to 0$. By employing adifferent technique for the crucial $L^2 _t (H^2 _x)$ estimates we can see howin fact the desired regularity holds everywhere.

Existence, uniqueness and $L^2 _t (H_x ^2) \cap L^\infty _t (H^1 _x)  \cap H^1 _t (L^2 _x) $ regularity of the gradient flow of the  Ambrosio-Tortorelli functional