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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, 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, Birkhäuser Verlag, Basel, 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), A47-A50. Google Scholar

[3]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 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), 139-153. doi: 10.1007/BF02278004.  Google Scholar

[5]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser Boston Inc., Boston, MA, 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, H. Pajot and C. Villani), Cambridge University Press (2014), 100-144. arXiv:1009.3737v1. doi: 10.1017/CBO9781107297296.007.  Google Scholar

[7]

W. Eckhaus, Studies in Non-Linear Stability Theory, Springer-Verlag New York, New York, Inc., 1965.  Google Scholar

[8]

W. Eckhaus, The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci., 3 (1993), 329-348. 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), 199-217. doi: 10.1002/zamm.200700111.  Google Scholar

[10]

H. Hanke, Homogenization in gradient plasticity, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651-1684. 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), 85-91. 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), 2371-2393. 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), 283-292. 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), 413-420. 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), Elsevier Science B.V., 2 (2002), 759-834. 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, CA, 1994) (eds. P. Deift, C. Levermore and C. Wayne), Amer. Math. Soc., Providence, RI, 31 (1996), 191-216.  Google Scholar

[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.  Google Scholar

[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. 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), 353-382. 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), 253-310. 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), 1627-1672. 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, 47 pp.) of C. R. Acad. Sci. Paris 345 (2007), 151-154. 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), 433-457. 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), 69-82. 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), 1427-1451. 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), 397-422. doi: 10.1007/BF02429847.  Google Scholar

show all references

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, Birkhäuser Verlag, Basel, 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), A47-A50. Google Scholar

[3]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 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), 139-153. doi: 10.1007/BF02278004.  Google Scholar

[5]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser Boston Inc., Boston, MA, 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, H. Pajot and C. Villani), Cambridge University Press (2014), 100-144. arXiv:1009.3737v1. doi: 10.1017/CBO9781107297296.007.  Google Scholar

[7]

W. Eckhaus, Studies in Non-Linear Stability Theory, Springer-Verlag New York, New York, Inc., 1965.  Google Scholar

[8]

W. Eckhaus, The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci., 3 (1993), 329-348. 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), 199-217. doi: 10.1002/zamm.200700111.  Google Scholar

[10]

H. Hanke, Homogenization in gradient plasticity, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651-1684. 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), 85-91. 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), 2371-2393. 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), 283-292. 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), 413-420. 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), Elsevier Science B.V., 2 (2002), 759-834. 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, CA, 1994) (eds. P. Deift, C. Levermore and C. Wayne), Amer. Math. Soc., Providence, RI, 31 (1996), 191-216.  Google Scholar

[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.  Google Scholar

[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. 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), 353-382. 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), 253-310. 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), 1627-1672. 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, 47 pp.) of C. R. Acad. Sci. Paris 345 (2007), 151-154. 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), 433-457. 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), 69-82. 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), 1427-1451. 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), 397-422. doi: 10.1007/BF02429847.  Google Scholar

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