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

TL;DR

This paper proves the existence, uniqueness, and comprehensive regularity of the gradient flow of the Ambrosio-Tortorelli functional in two dimensions, improving previous results by establishing regularity everywhere.

Contribution

It introduces a new technique for $L^2_t(H^2_x)$ estimates, ensuring regularity holds globally rather than at diverging space-time points.

Findings

Established existence and uniqueness of the gradient flow.

Proved full regularity in the specified function spaces.

Improved previous results by removing divergence at finite points.

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.