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

Patrick Dondl; Thomas Frenzel; Alexander Mielke

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arXiv · 1801.07144

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

Patrick Dondl·Thomas Frenzel·Alexander Mielke

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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.

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Patrick Dondl, Thomas Frenzel, Alexander Mielke. 2018-01-13. A gradient system with a wiggly energy and relaxed EDP-convergence. https://arxiv.org/abs/1801.07144

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