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Homogenization of Cahn–Hilliard-type equations via evolutionary Γ-convergence

Please always quote using this URN:urn:nbn:de:0296-matheon-13700
  • In these notes we discuss two approaches to evolutionary Γ- convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ- convergence methods. In the first approach we consider families of uni- formly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savar ́e 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

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Metadaten
Author:Matthias Liero, Sina Reichelt
URN:urn:nbn:de:0296-matheon-13700
Referee:Alexander Mielke
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2016/02/14
Release Date:2016/02/14
Tag:Evolutionary Γ-convergence, gradient systems, homogeniza- tion, Cahn–Hilliard equation, evolutionary variational inequality, energy- dissipation principle, two-scale convergence.
Institute:Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
Project:D Optics and Electronics (Electronic and photonic devices) / D-SE2 Electrothermal modeling of large-area OLEDs
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B27 Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B30 Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K30 Initial value problems for higher-order parabolic equations
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K55 Nonlinear parabolic equations
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Jxx Existence theories / 49J40 Variational methods including variational inequalities [See also 47J20]
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Jxx Existence theories / 49J45 Methods involving semicontinuity and convergence; relaxation
Preprint Number:1102
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