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

Abstract

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.