  We develop the theory of fractional gradient flows: an evolution aimed at the
minimization of a convex, l.s.c.~energy, with memory effects. This memory is
characterized by the fact that the negative of the (sub)gradient of the energy
equals the so-called Caputo derivative of the state. We introduce the notion of
energy solutions, for which we provide existence, uniqueness and certain
regularizing effects. We also consider Lipschitz perturbations of this energy.
For these problems we provide an a posteriori error estimate and show its
reliability. This estimate depends only on the problem data, and imposes no
constraints between consecutive time-steps. On the basis of this estimate we
provide an a priori error analysis that makes no assumptions on the smoothness
of the solution.
