Deriving amplitude equations via evolutionary $\Gamma$-convergence

Alexander Mielke

doi:10.3934/dcds.2015.35.2679

Article Contents

2015, Volume 35, Issue 6: 2679-2700. Doi: 10.3934/dcds.2015.35.2679

This issue Previous Article A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality Next Article Implicit functions and parametrizations in dimension three: Generalized solutions

Deriving amplitude equations via evolutionary $\Gamma$-convergence

Alexander Mielke^1,

1.
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin

Received: December 31, 2013

Revised: February 28, 2014

Published: May 2017

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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 $L^2$, while for the case of a quadratic nonlinearity we need to impose weak convergence in $H^1$. However, we do not need well-preparedness of the initial conditions.

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Mathematics Subject Classification: Primary: 35Q56, 35K55; Secondary: 76E30, 47H20.

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