Existence, uniqueness and $L_t^2(H_x^2)\cap L_t^{\mathrm{\infty }}(H_x^1)\cap H_t^1(L_x^2)$ regularity of the gradient flow of the Ambrosio-Tortorelli functional

We consider the gradient flow of the Ambrosio-Tortorelli functional at fixed $\epsilon >0$, proving existence, uniqueness and $L_t^2(H_x^2)\cap L_t^{\mathrm{\infty }}(H_x^1)\cap H_t^1(L_x^2)$ 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 $L_t^2(H_x^2)$ estimates we can see how in fact the desired regularity holds everywhere.