On Evolutionary $$\varGamma $$ Γ -Convergence for Gradient Systems

Abstract

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 \({\mathscr {E}}_\varepsilon \) and the dissipation potential \({\mathscr {R}}_\varepsilon \) or the associated dissipation distance. We assume that the functionals depend on a small parameter and that the associated gradient systems have solutions \(u_\varepsilon \). We investigate the question under which conditions the limits u of (subsequences of) the solutions \(u_\varepsilon \) are solutions of the gradient system generated by the \(\varGamma \)-limits \({\mathscr {E}}_0\) and \({\mathscr {R}}_0\). Here the choice of the right topology will be crucial as well as additional structural conditions. We cover classical gradient systems, where \({\mathscr {R}}_\varepsilon \) 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.