1. A gradient system with a wiggly energy and relaxed EDP-convergence
- Author
-
Dondl, Patrick, Frenzel, Thomas, and Mielke, Alexander
- Subjects
Mathematics - Analysis of PDEs ,35K55 35B27 35A15 49S05 49J40 49J45 - 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., Comment: 43 pages, 8 figures
- Published
- 2018