A gradient system with a wiggly energy and relaxed EDP-convergence

TL;DR

This paper introduces a new concept called relaxed EDP-convergence to derive macroscopic gradient structures from microscopic models with microstructure effects, generalizing previous EDP-convergence and analyzing a wiggly-energy model.

Contribution

It develops a generalized notion of Gamma-convergence for gradient systems that accounts for microstructure effects and non-preservation of dissipation structure, with a detailed analysis of a prototypical example.

Findings

Introduced relaxed EDP-convergence as a generalization of EDP-convergence.

Proved the uniqueness of the macroscopic dissipation potential.

Analyzed the wiggly-energy model to demonstrate the limit passage.

Abstract

If gradient systems depend on a microstructure, we want to derive a macroscopic gradient structure describing the effective behavior of the microscopic effects. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. This new notion generalizes the concept of EDP-convergence, which was introduced in arXiv:1507.06322, and is called "relaxed EDP-convergence". Both notions are based on De Giorgi's energy-dissipation principle, however the special structure of the dissipation functional in terms of the primal and dual dissipation potential is, in general, not preserved under Gamma-convergence. By investigating the kinetic relation directly and using general forcings we still derive a unique macroscopic dissipation potential. The wiggly-energy model of James et al serves as a prototypical example where this nontrivial limit passage can be fully analyzed.