WIAS Preprint No. 2174, (2015)

Deriving effective models for multiscale systems via evolutionary $Gamma$-convergence



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 34E13 35R15 35K57 47J35 74Qxx

Keywords

  • Reaction-diffusion systems, homogenization, gradient systems, evolutionary variational inequality, energy-dissipation principle, amplitude equations

Abstract

We discuss possible extensions of the recently established theory of evolutionary Gamma convergence for gradient systems to nonlinear dynamical systems obtained by perturbation of a gradient systems. Thus, it is possible to derive effective equations for pattern forming systems with multiple scales. Our applications include homogenization of reaction-diffusion systems, the justification of amplitude equations for Turing instabilities, and the limit from pure diffusion to reaction-diffusion. This is achieved by generalizing the Gamma-limit approaches based on the energy-dissipation principle or the evolutionary variational estimate.

Appeared in

  • Control of Self-Organizing Nonlinear Systems, E. Schöll, S. Klapp, P. Hövel, eds., Understanding Complex Systems, Springer, 2016, pp. 235--251

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