A gradient system with a wiggly energy and relaxed EDP-convergence
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 68.

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.

DOI : 10.1051/cocv/2018058
Classification : 35K55, 35B27, 35A15, 49S05, 49J40, 49J45
Mots-clés : 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

1 
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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. http://www.numdam.org/articles/10.1051/cocv/2018058/

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