On Evolutionary Γ-Convergence for Gradient Systems

In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional E-epsilon and the dissipation potential R-epsilon or the associated dissipation distance. We assume that the functionals depend on a small parameter and that the associated gradient systems have solutions u(epsilon). We investigate the question under which conditions the limits u of ( subsequences of) the solutions u(epsilon) are solutions of the gradient system generated by the Gamma-limits E-0 and R-0. Here the choice of the right topology will be crucial as well as additional structural conditions. We cover classical gradient systems, where R-epsilon is quadratic, and rate-independent systems as well as the passage from classical gradient to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.