For gradient systems depending on a microstructure, it is desirable to derive a macroscopic gradient structure describing the effective behavior of the microscopic scale on the macroscopic evolution. 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 [26], and is now called relaxed EDP-convergence. Both notions are based on De Giorgi’s energy-dissipation principle (EDP), 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 using suitable tiltings we study the kinetic relation directly and, thus, are able to derive a unique macroscopic dissipation potential. The wiggly-energy model of Abeyaratne-Chu-James (1996) serves as a prototypical example where this nontrivial limit passage can be fully analyzed.
Keywords: Variational evolution, energy functional, dissipation potential, gradient flows, Gamma convergence, EDP-convergence, energy-dissipation, balance, homogenization
Dondl, Patrick 1 ; Frenzel, Thomas 1 ; Mielke, Alexander 1
@article{COCV_2019__25__A68_0,
author = {Dondl, Patrick and Frenzel, Thomas and Mielke, Alexander},
title = {A gradient system with a wiggly energy and relaxed {EDP-convergence}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2018058},
mrnumber = {4027693},
zbl = {1444.35101},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2018058/}
}
TY - JOUR AU - Dondl, Patrick AU - Frenzel, Thomas AU - Mielke, Alexander TI - A gradient system with a wiggly energy and relaxed EDP-convergence JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2018058/ DO - 10.1051/cocv/2018058 LA - en ID - COCV_2019__25__A68_0 ER -
%0 Journal Article %A Dondl, Patrick %A Frenzel, Thomas %A Mielke, Alexander %T A gradient system with a wiggly energy and relaxed EDP-convergence %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2018058/ %R 10.1051/cocv/2018058 %G en %F COCV_2019__25__A68_0
Dondl, Patrick; Frenzel, Thomas; Mielke, Alexander. A gradient system with a wiggly energy and relaxed EDP-convergence. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 68. doi: 10.1051/cocv/2018058
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