Gradient flows and rate independent problems

Solutions of rate-independent evolution problems, as recently proposed by A. Mielke and his collaborators, can be obtained by solving a recursive minimization scheme which involves a functional governing the evolution perturbed by a suitable convex dissipation term. Rate-independence is guaranteed by the 1-homogeneity of the dissipation, which therefore has a linear growth.

The same variational scheme, with quadratic (or at least superlinear) dissipation, plays a crucial role in the variational approach to Gradient Flows.

It is then natural to investigate the relationships between these two theories, in particular when viscous approximations of rate- independent problems are considered: they are simply obtained by adding a (asymptotically small) quadratic perturbation to the dissipation term.

In this talk we address this kind of problems and we discuss some characterizations of the limit solutions obtained by general viscous approximations.

(Joint work in collaboration with A. Mielke and R. Rossi)