Existence, uniqueness and regularity of solutions to the parabolic Ambrosio-Tortorelli system

We investigate the existence, uniqueness, and regularity of the gradient flow of the Ambrosio-Tortorelli functional, viewed as an initial-boundary value problem, in arbitrary dimension. For any initial data, using a time-discrete Euler scheme, we establish the existence of a weak gradient flow satisfying a maximum principle. We also identify a functional space in which uniqueness holds. We further show that such gradient flow is smooth in the interior of the space-time domain. Under additional assumptions on the initial data, the regularity of the boundary of the domain, we prove optimal regularity for the solution, up to the space-time boundary.