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June  2015, 35(6): 2679-2700. doi: 10.3934/dcds.2015.35.2679

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

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

Received  January 2014 Revised  March 2014 Published  December 2014

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.
Citation: Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679
##### References:
 [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, (2005). Google Scholar [2] P. Bénilan, Solutions intégrales d'équations d'évolution dans un espace de Banach,, C. R. Acad. Sci. Paris Sér. A-B, 274 (1972). Google Scholar [3] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert,, North-Holland Publishing Co., (1973). Google Scholar [4] P. Collet and J.-P. Eckmann, The time dependent amplitude equation for the Swift-Hohenberg problem,, Comm. Math. Phys., 132 (1990), 139. doi: 10.1007/BF02278004. Google Scholar [5] G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäuser Boston Inc., (1993). doi: 10.1007/978-1-4612-0327-8. Google Scholar [6] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport (Chap. 6),, in Optimal Transportation (eds. Y. Ollivier, (2014), 100. doi: 10.1017/CBO9781107297296.007. Google Scholar [7] W. Eckhaus, Studies in Non-Linear Stability Theory,, Springer-Verlag New York, (1965). Google Scholar [8] W. Eckhaus, The Ginzburg-Landau manifold is an attractor,, J. Nonlinear Sci., 3 (1993), 329. doi: 10.1007/BF02429869. Google Scholar [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. doi: 10.1002/zamm.200700111. Google Scholar [10] H. Hanke, Homogenization in gradient plasticity,, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651. doi: 10.1142/S0218202511005520. Google Scholar [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. doi: 10.1017/S0308210500020989. Google Scholar [12] P. Krejčí and J. Sprekels, Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators,, Math. Methods Appl. Sci. (MMAS), 30 (2007), 2371. doi: 10.1002/mma.892. Google Scholar [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. doi: 10.3934/dcdss.2008.1.283. Google Scholar [14] W. McLean and D. Elliott, On the $p$-norm of the truncated Hilbert transform,, Bull. Austral. Math. Soc., 38 (1988), 413. doi: 10.1017/S0004972700027799. Google Scholar [15] A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation,, in Handbook of Dynamical Systems II (ed. B. Fiedler), 2 (2002), 759. doi: 10.1016/S1874-575X(02)80036-4. Google Scholar [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, 31 (1996), 191. Google Scholar [17] A. Mielke, G. Schneider and A. Ziegra, Comparison of inertial manifolds and application to modulated systems,, Math. Nachr., 214 (2000), 53. doi: 10.1002/1522-2616(200006)214:1<53::AID-MANA53>3.0.CO;2-4. Google Scholar [18] A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems,, WIAS Preprint 1915, (1915). Google Scholar [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. doi: 10.3934/nhm.2014.9.353. Google Scholar [20] A. Mielke, R. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations,, Calc. Var. Part. Diff. Eqns., 46 (2013), 253. doi: 10.1007/s00526-011-0482-z. Google Scholar [21] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Comm. Pure Appl. Math., 57 (2004), 1627. doi: 10.1002/cpa.20046. Google Scholar [22] G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds,, Unpublished extended version (2011, 345 (2007), 151. doi: 10.1016/j.crma.2007.06.018. Google Scholar [23] G. Schneider, Error estimates for the Ginzburg-Landau approximation,, Z. angew. Math. Phys., 45 (1994), 433. doi: 10.1007/BF00945930. Google Scholar [24] G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms,, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69. doi: 10.1007/s000300050034. Google Scholar [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. doi: 10.3934/dcds.2011.31.1427. Google Scholar [26] A. van Harten, On the validity of the Ginzburg-Landau equation,, J. Nonlinear Sci., 1 (1991), 397. doi: 10.1007/BF02429847. Google Scholar

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##### References:
 [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, (2005). Google Scholar [2] P. Bénilan, Solutions intégrales d'équations d'évolution dans un espace de Banach,, C. R. Acad. Sci. Paris Sér. A-B, 274 (1972). Google Scholar [3] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert,, North-Holland Publishing Co., (1973). Google Scholar [4] P. Collet and J.-P. Eckmann, The time dependent amplitude equation for the Swift-Hohenberg problem,, Comm. Math. Phys., 132 (1990), 139. doi: 10.1007/BF02278004. Google Scholar [5] G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäuser Boston Inc., (1993). doi: 10.1007/978-1-4612-0327-8. Google Scholar [6] S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport (Chap. 6),, in Optimal Transportation (eds. Y. Ollivier, (2014), 100. doi: 10.1017/CBO9781107297296.007. Google Scholar [7] W. Eckhaus, Studies in Non-Linear Stability Theory,, Springer-Verlag New York, (1965). Google Scholar [8] W. Eckhaus, The Ginzburg-Landau manifold is an attractor,, J. Nonlinear Sci., 3 (1993), 329. doi: 10.1007/BF02429869. Google Scholar [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. doi: 10.1002/zamm.200700111. Google Scholar [10] H. Hanke, Homogenization in gradient plasticity,, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651. doi: 10.1142/S0218202511005520. Google Scholar [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. doi: 10.1017/S0308210500020989. Google Scholar [12] P. Krejčí and J. Sprekels, Elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators,, Math. Methods Appl. Sci. (MMAS), 30 (2007), 2371. doi: 10.1002/mma.892. Google Scholar [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. doi: 10.3934/dcdss.2008.1.283. Google Scholar [14] W. McLean and D. Elliott, On the $p$-norm of the truncated Hilbert transform,, Bull. Austral. Math. Soc., 38 (1988), 413. doi: 10.1017/S0004972700027799. Google Scholar [15] A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation,, in Handbook of Dynamical Systems II (ed. B. Fiedler), 2 (2002), 759. doi: 10.1016/S1874-575X(02)80036-4. Google Scholar [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, 31 (1996), 191. Google Scholar [17] A. Mielke, G. Schneider and A. Ziegra, Comparison of inertial manifolds and application to modulated systems,, Math. Nachr., 214 (2000), 53. doi: 10.1002/1522-2616(200006)214:1<53::AID-MANA53>3.0.CO;2-4. Google Scholar [18] A. Mielke, On evolutionary $\Gamma$-convergence for gradient systems,, WIAS Preprint 1915, (1915). Google Scholar [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. doi: 10.3934/nhm.2014.9.353. Google Scholar [20] A. Mielke, R. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations,, Calc. Var. Part. Diff. Eqns., 46 (2013), 253. doi: 10.1007/s00526-011-0482-z. Google Scholar [21] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Comm. Pure Appl. Math., 57 (2004), 1627. doi: 10.1002/cpa.20046. Google Scholar [22] G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds,, Unpublished extended version (2011, 345 (2007), 151. doi: 10.1016/j.crma.2007.06.018. Google Scholar [23] G. Schneider, Error estimates for the Ginzburg-Landau approximation,, Z. angew. Math. Phys., 45 (1994), 433. doi: 10.1007/BF00945930. Google Scholar [24] G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms,, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69. doi: 10.1007/s000300050034. Google Scholar [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. doi: 10.3934/dcds.2011.31.1427. Google Scholar [26] A. van Harten, On the validity of the Ginzburg-Landau equation,, J. Nonlinear Sci., 1 (1991), 397. doi: 10.1007/BF02429847. Google Scholar
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