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.