WIAS Preprint No. 1914, (2014)

Deriving amplitude equations via evolutionary Gamma convergence



Authors

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

2010 Mathematics Subject Classification

  • 35Q56 76E30 35K55 35B35 47H20

Keywords

  • Ginzburg-Landau equation, Swift-Hohenberg equation, gradient systems, Gamma convergence, evolutionary variational inequality

Abstract

We discuss the justification of the Ginzburg-Landau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional Swift-Hohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary Gamma convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show Gamma convergence of the associated energies in suitable function spaces. The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in L2, while for the case of a quadratic nonlinearity we need to impose weak convergence in H1. However, we do not need wellpreparedness of the initial conditions.

Appeared in

  • Discrete Contin. Dyn. Syst., 35 (2015) pp. 2679--2700.

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