Weak solutions for unidirectional gradient flows: existence, uniqueness, and convergence of time discretization schemes

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraints. First, we provide a notion of weak solution, inspired by the theory of curves of maximal slope, and then we prove existence (employing time-discrete schemes with differ- ent implementations of the constraint), uniqueness, power and energy identity, comparison principle and continuous dependence. As a by- product, we show that the energy identity gives a selection criterion for the (non-unique) evolutions obtained by other notions of solutions. Finally, we show that for autonomous energies the evolution obtained with the monotonicity constraint actually coincides with the evolution obtained by replacing the constraint with a fixed obstacle, given by the initial datum.