Article Contents
Article Contents

# Deriving amplitude equations via evolutionary $\Gamma$-convergence

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

 Citation:

•  [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. [2] P. Bénilan, Solutions intégrales d'équations d'évolution dans un espace de Banach, C. R. Acad. Sci. Paris Sér. A-B, 274 (1972), A47-A50. [3] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973. [4] P. Collet and J.-P. Eckmann, The time dependent amplitude equation for the Swift-Hohenberg problem, Comm. Math. Phys., 132 (1990), 139-153.doi: 10.1007/BF02278004. [5] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser Boston Inc., Boston, MA, 1993.doi: 10.1007/978-1-4612-0327-8. [6] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport (Chap. 6), in Optimal Transportation (eds. Y. Ollivier, H. Pajot and C. Villani), Cambridge University Press (2014), 100-144. arXiv:1009.3737v1.doi: 10.1017/CBO9781107297296.007. [7] W. Eckhaus, Studies in Non-Linear Stability Theory, Springer-Verlag New York, New York, Inc., 1965. [8] W. Eckhaus, The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci., 3 (1993), 329-348.doi: 10.1007/BF02429869. [9] R. B. Guenther, P. Krejčí and J. Sprekels, Small strain oscillations of an elastoplastic Kirchhoff plate, Z. angew. Math. Mech. (ZAMM), 88 (2008), 199-217.doi: 10.1002/zamm.200700111. [10] H. Hanke, Homogenization in gradient plasticity, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651-1684.doi: 10.1142/S0218202511005520. [11] P. Kirrmann, G. Schneider and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 85-91.doi: 10.1017/S0308210500020989. [12] P. Krejčí and J. Sprekels, Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Math. Methods Appl. Sci. (MMAS), 30 (2007), 2371-2393.doi: 10.1002/mma.892. [13] P. Krejčí and J. Sprekels, Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators, Discr. Cont. Dynam. Systems Ser. S, 1 (2008), 283-292.doi: 10.3934/dcdss.2008.1.283. [14] W. McLean and D. Elliott, On the $p$-norm of the truncated Hilbert transform, Bull. Austral. Math. Soc., 38 (1988), 413-420.doi: 10.1017/S0004972700027799. [15] A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation, in Handbook of Dynamical Systems II (ed. B. Fiedler), Elsevier Science B.V., 2 (2002), 759-834.doi: 10.1016/S1874-575X(02)80036-4. [16] A. Mielke and G. Schneider, Derivation and justification of the complex Ginzburg-Landau equation as a modulation equation, in Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994) (eds. P. Deift, C. Levermore and C. Wayne), Amer. Math. Soc., Providence, RI, 31 (1996), 191-216. [17] A. Mielke, G. Schneider and A. Ziegra, Comparison of inertial manifolds and application to modulated systems, Math. Nachr., 214 (2000), 53-69.doi: 10.1002/1522-2616(200006)214:1<53::AID-MANA53>3.0.CO;2-4. [18] A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems, WIAS Preprint 1915, URL http://www.wias-berlin.de/preprint/1915/wias_preprints_1915.pdf, To appear in Proc. Summer School in Twente University June 2012. [19] A. Mielke, S. Reichelt and M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Networks Heterg. Materials, 9 (2014), 353-382.doi: 10.3934/nhm.2014.9.353. [20] A. Mielke, R. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calc. Var. Part. Diff. Eqns., 46 (2013), 253-310.doi: 10.1007/s00526-011-0482-z. [21] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.doi: 10.1002/cpa.20046. [22] G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, Unpublished extended version (2011, 47 pp.) of C. R. Acad. Sci. Paris 345 (2007), 151-154.doi: 10.1016/j.crma.2007.06.018. [23] G. Schneider, Error estimates for the Ginzburg-Landau approximation, Z. angew. Math. Phys., 45 (1994), 433-457.doi: 10.1007/BF00945930. [24] G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69-82.doi: 10.1007/s000300050034. [25] S. Serfaty, Gamma-convergence of gradient flows on Hilbert spaces and metric spaces and applications, Discr. Cont. Dynam. Systems Ser. A, 31 (2011), 1427-1451.doi: 10.3934/dcds.2011.31.1427. [26] A. van Harten, On the validity of the Ginzburg-Landau equation, J. Nonlinear Sci., 1 (1991), 397-422.doi: 10.1007/BF02429847.