Advertisement

Homogenization of Cahn–Hilliard-type equations via evolutionary \(\varvec{\Gamma }\)-convergence

Article

Abstract

In these notes we discuss two approaches to evolutionary \(\Gamma \)-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ \(\Gamma \)-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004.

We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals \(\Gamma \)-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

Keywords

Evolutionary \(\Gamma \)-convergence Gradient systems Homogenization Cahn–Hilliard equation Evolutionary variational inequality Energy-dissipation principle Two-scale convergence 

Mathematics Subject Classification

35B27 35K55 35K30 35B30 49J40 49J45 

References

  1. 1.
    Abels, H., Wilke, M.: Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67(11), 3176–3193 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(1), 1482–1518 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Allaire, G.: Two-scale convergence and homogenization of periodic structures. In: School on Homogenization, ICTP Trieste (1993)Google Scholar
  4. 4.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Fows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2005)MATHGoogle Scholar
  5. 5.
    Armstrong, S.N., Smart, C.K.: Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. Éc. Norm. Supér. (4) 49(2), 423–481 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Attouch, H.: Variational Convergence of Functions and Operators. Pitman Advanced Publishing Program, Pitman (1984)MATHGoogle Scholar
  7. 7.
    Bellettini, G., Bertini, L., Mariani, M., Novaga, M.: Convergence of the one-dimensional Cahn–Hilliard equation. SIAM J. Math. Anal. 44(5), 3458–3480 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bénilan, P.: Solutions intégrales d’équations d’évolution dans un espace de Banach. C. R. Acad. Sci. Paris Sér. A-B 274, A47–A50 (1972)MATHGoogle Scholar
  9. 9.
    Berlyand, L., Sandier, E., Serfaty, S.: A two scale \(\Gamma \)-convergence approach for random non-convex homogenization. Calc. Var. Partial Differ. Equ. 56(6), 156 (2017)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Brézis, H.: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973)MATHGoogle Scholar
  11. 11.
    Brusch, L., Kühne, H., Thiele, U., Bär, M.: Dewetting of thin films on heterogeneous substrates: pinning versus coarsening. Phys. Rev. E 66, 011602 (2002)CrossRefGoogle Scholar
  12. 12.
    Cahn, J.W., Hilliard, J.E.: Free energy of a non-uniform system I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)CrossRefGoogle Scholar
  13. 13.
    Ciarlet, P.G.: Mathematical elasticity. Vol. II: Theory of Plates, Volume 27 of Studies in Mathematics and Its Applications. North-Holland Publishing Co., Amsterdam (1997)Google Scholar
  14. 14.
    Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris 335(1), 99–104 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40(4), 1585–1620 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Copetti, M.I.M., Elliott, C.M.: Numerical analysis of the Cahn–Hilliard equation with a logarithmic free energy. Numer. Math. 63(1), 39–65 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Daneri, S., Savaré, G.: Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Anal. 40, 1104–1122 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Daneri, S., Savaré, G.: Lecture notes on gradient flows and optimal transport. arXiv:1009.3737v1 (2010)
  19. 19.
    Delgadino, M.G.: Convergence of a one dimensional Cahn–Hilliard equation with degenerate mobility. arXiv: 1510.05021v1 (2016)
  20. 20.
    Elliott, C.M.: The Cahn–Hilliard model for the kinetics of phase separation. In: Mathematical Models for Phase Change Problems (Óbidos, 1988). Volume 88 of International Series of Numerical Mathematics, pp. 35–73. Birkhäuser, Basel (1989)Google Scholar
  21. 21.
    Elliott, C.M., Garcke, H.: On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27(2), 404–423 (1996)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Elliott, C.M., Songmu, Z.: On the Cahn–Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339–357 (1986)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Giannoulis, J., Herrmann, M., Mielke, A.: Continuum description for the dynamics in discrete lattices: derivation and justification. In: Mielke, A. (ed.) Analysis, Modeling and Simulation of Multiscale Problems, pp. 435–466. Springer, Berlin (2006)CrossRefGoogle Scholar
  24. 24.
    Gloria, A., Neukamm, S., Otto, F.: An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. ESAIM Math. Model. Numer. Anal. 48(2), 325–346 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Grasselli, M., Miranville, A., Rossi, R., Schimperna, G.: Analysis of the Cahn–Hilliard equation with a chemical potential dependent mobility. Commun. Partial Differ. Equ. 36(7), 1193–1238 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Heida, M.: On systems of Cahn–Hilliard and Allen–Cahn equations considered as gradient flows in Hilbert spaces. J. Math. Anal. Appl. 423(1), 410–455 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ioffe, A.D.: On lower semicontinuity of integral functionals. I. SIAM J. Control Optim. 15(4), 521–538 (1977)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Le, N.Q.: A gamma-convergence approach to the Cahn–Hilliard equation. Calc. Var. Partial Differ. Equ. 32(4), 499–522 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Liero, M.: Passing from bulk to bulk-surface evolution in the Allen–Cahn equation. NoDEA Nonlinear Differ. Equ. Appl. 20(3), 919–942 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Liero, M., Mielke, A.: An evolutionary elastoplastic plate model derived via \(\Gamma \)-convergence. Math. Models Methods Appl. Sci. (M3AS) 21(9), 1961–1986 (2011)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    López-Gómez, J.: Linear Second Order Elliptic Operators. World Scientific Publishing Co., Pte. Ltd., Hackensack (2013)CrossRefMATHGoogle Scholar
  32. 32.
    Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence. Int. J. Pure Appl. Math. 2, 35–86 (2002)MathSciNetMATHGoogle Scholar
  33. 33.
    Metafune, G., Spina, C.: An integration by parts formula in Sobolev spaces. Mediterr. J. Math. 5(3), 357–369 (2008)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Mielke, A., Timofte, A.: Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal. 39(2), 642–668 (2007)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Mielke, A.: Weak-convergence methods for Hamiltonian multiscale problems. Discrete Contin. Dyn. Syst. 20(1), 53–79 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Mielke, A.: On evolutionary \(\Gamma \)-convergence for gradient systems. WIAS Preprint 1915 (2014)Google Scholar
  37. 37.
    Mielke, A.: Deriving amplitude equations via evolutionary \(\Gamma \)-convergence. Discrete Contin. Dyn. Syst. Ser. A 35(6), 2679–2700 (2015)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Mielke, A., Reichelt, S., Thomas, M.: Two-scale homogenization of nonlinear reaction–diffusion systems with slow diffusion. Netw. Heterog. Media 9(2), 353–382 (2014)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Mielke, A., Roubíček, T., Stefanelli, U.: \(\Gamma \)-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31(3), 387–416 (2008)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Modica, L., Mortola, S.: Un esempio di \(\Gamma ^{-}\)-convergenza. Boll. Un. Mat. Ital. B (5) 14(1), 285–299 (1977)MathSciNetMATHGoogle Scholar
  41. 41.
    Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Niethammer, B., Oshita, Y.: A rigorous derivation of mean-field models for diblock copolymer melts. Calc. Var. Partial Differ. Equ. 39(3–4), 273–305 (2010)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Niethammer, B., Otto, F.: Ostwald ripening: the screening length revisited. Calc. Var. Partial Differ. Equ. 13(1), 33–68 (2001)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Novick-Cohen, A.: The Cahn–Hilliard equation. In: Dafermos, C.M., Pokomy, M. (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. IV, pp. 201–228. Elsevier/North-Holland, Amsterdam (2008)Google Scholar
  45. 45.
    Rossi, R., Savaré, G.: Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM Control Optim. Calc. Var. 12, 564–614 (2006)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Rossi, R., Segatti, A., Stefanelli, U.: Attractors for gradient flows of nonconvex functionals and applications. Arch. Ration. Mech. Anal. 187(1), 91–135 (2008)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Commun. Pure Appl. Math. LVII, 1627–1672 (2004)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Schmuck, M., Pradas, M., Pavliotis, G.A., Kalliadasis, S.: Derivation of effective macroscopic Stokes-Cahn-Hilliard equations for periodic immiscible flows in porous media. Nonlinearity 26(12), 3259–3277 (2013)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Serfaty, S.: Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31(4), 1427–1451 (2011)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (Mathematical Surveys and Monographs), vol. 49. American Mathematical Society, Providence (1997)MATHGoogle Scholar
  51. 51.
    Thiele, U., Brusch, L., Bestehorn, M., Bär, M.: Modelling thin-film dewetting on structured substrates and templates: bifurcation analysis and numerical simulation. Eur. Phys. J. E 11, 255–271 (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

Personalised recommendations