Existence and approximation results for gradient flows

This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space governed by a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional.

The interest for this kind of equations is motivated by a number of examples, which show that several mathematical models describing phase transitions phenomena and leading to systems of evolutionary PDE’s have a common gradient flow structure. In particular, when quasi-stationary models are considered, highly non-convex functionals naturally arise.

We will present some existence results for the solution of the gradient flow equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements.