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Author: Mielke, Alexander × End date: 2019 × Start date: 2010 × Type: Text × Subject: Gamma convergence × Subject: evolutionary variational inequality ×

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    Deriving amplitude equations via evolutionary [Gamma]-convergence
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, Alexander
    We discuss the justification of the GinzburgLandau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional SwiftHohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary [Gamma]-convergence by reformulating both equation 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 Savare 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.