Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Convergence to equilibrium for a second-order time semi-discretization ofthe Cahn-Hilliard equation

1 MOX– Laboratory for Modeling and Scientific Computing, Dipartimento di Matematica,Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
2 Laboratoire Paul Painlevé, U.M.R. CNRS 8524, Université Lille 1, Cité Scientifique, F-59655 Villeneuve d’Ascq Cedex, France
3 Team RAPSODI, Inria Lille - Nord Europe, 40 av. Halley, F-59650 Villeneuve d’Ascq, France
4 Laboratoire de Mathématiques et Applications, Université de Poitiers, CNRS, F-86962 Chasseneuil, France

Special Issues: Nonlinear Evolution PDEs, Interfaces and Applications

We consider a second-order two-step time semi-discretization of the Cahn-Hilliard equation with an analytic nonlinearity. The time-step is chosen small enough so that the pseudo-energy associated withdiscretization is nonincreasing at every time iteration. We prove that the sequence generated by the scheme converges to a steady state as time tends to infinity. We also obtain convergence rates in the energy norm. The proof is based on the Łojasiewicz-Simon inequality.
  Article Metrics


1. H. Abels and M. Wilke, Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 67 (2007), 3176-3193.

2. P.-A. Absil and K. Kurdyka, On the stable equilibrium points of gradient systems, Systems Control Lett., 55 (2006), 573-577.

3. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math.,12 (1959), 623-727.

4. N. E. Alaa and M. Pierre, Convergence to equilibrium for discretized gradient-like systems with analytic features, IMA J. Numer. Anal., 33 (2013), 1291-1321.

5. P. F. Antonietti, L. Beir˜ao da Veiga, S. Scacchi, and M. Verani, A C1 virtual element method for the Cahn-Hilliard equation with polygonal meshes, SIAM J. Numer. Anal., 54 (2016), 34-56.

6. H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program., 116 (2009), 5-16.

7. H. Attouch, J. Bolte, and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Math.Program., 137 (2013), 91-129.

8. T. B´arta, R. Chill, and E. Faˇsangov´a, Every ordinary di erential equation with a strict Lyapunov function is a gradient system, Monatsh. Math., 166 (2012), 57-72.

9. J. Bolte, A. Daniilidis, O. Ley, and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity, Trans. Amer. Math. Soc., 362 (2010), 3319-3363.

10. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.

11. L. Cherfils, A. Miranville, and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.

12. L. Cherfils and M. Petcu, A numerical analysis of the Cahn-Hilliard equation with non-permeable walls, Numer. Math., 128 (2014), 517-549.

13. L. Cherfils, M. Petcu, and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 27 (2010), 1511-1533.

14. R. Chill, E. Faˇsangov´a, and J. Pr¨uss, Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.

15. R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2003), 1017-1039.

16. Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAMJ. Numer. Anal., 28 (1991), 1310-1322.

17. C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in Mathematical models for phase change problems (O´ bidos, 1988), vol. 88 of Internat. Ser. Numer. Math., Birkha¨user,Basel, 1989, 35-73.

18. C. G. Gal and A. Miranville, Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions, Discrete Contin. Dyn.Syst. Ser. S, 2(2009), 113-147.

19. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

20. H. Gomez, V. M. Calo, Y. Bazilevs, and T. J. R. Hughes, Isogeometric analysis of the Cahn-Hilliard phase-field model, Comput. Methods Appl. Mech. Engrg., 197 (2008), 4333-4352.

21. H. Gomez and T. J. R. Hughes, Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys., 230 (2011), 5310-5327.

22. L. Gouden`ege, D. Martin, and G. Vial, High order finite element calculations for the Cahn-Hilliard equation, J. Sci. Comput., 52 (2012), 294-321.

23. M. Grasselli and M. Pierre, Energy stable and convergent finite element schemes for the modified phase field crystal equation, ESAIM Math. Model. Numer. Anal., (in press).

24. M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 351-370.

25. F. Guill´en-Gonz´alez and M. Samsidy Goudiaby, Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows, Discrete Contin. Dyn.Syst., 32 (2012), 4229-4246.

26. J. Guo, C. Wang, S. M. Wise, and X. Yue, An H2 convergence of a second-order convex-splitting,finite difference scheme for the three-dimensional Cahn-Hilliard equation, Commun. Math. Sci.,14 (2016), 489-515.

27. A. Haraux and M. A. Jendoubi, The convergence problem for dissipative autonomous systems,SpringerBriefs in Mathematics, Springer, Cham; BCAM Basque Center for Applied Mathematics,Bilbao, 2015. Classical methods and recent advances, BCAM SpringerBriefs.

28. S.-Z. Huang, Gradient inequalities, vol. 126 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2006.

29. S. Injrou and M. Pierre, Error estimates for a finite element discretization of the Cahn-Hilliard-Gurtin equations, Adv. Di erential Equations, 15 (2010), 1161-1192.

30. J. Jiang, H.Wu, and S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth, J. Di erential Equations, 259(2015), 3032-3077.

31. O. Kavian, Introduction `a la th´eorie des points critiques et applications aux probl`emes elliptiques,vol. 13 of Math´ematiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag,Paris, 1993.

32. S. Kosugi, Y. Morita, and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: proof by the complete elliptic integrals, Discrete Contin. Dyn. Syst., 19 (2007), 609-629.

33. S. Łojasiewicz, Une propri´et´e topologique des sous-ensembles analytiques r´eels, in Les ´ Equations aux D´eriv´ees Partielles (Paris, 1962), ´ Editions du Centre National de la Recherche Scientifique, Paris, 1963, 87-89.

34. B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications,Commun. Pure Appl. Anal., 9 (2010), 685-702.

35. A. Miranville and A. Rougirel, Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equations, Z. Angew. Math. Phys., 57 (2006), 244-268.

36. F. Nabet, Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions, IMA J. Numer. Anal., (in press).

37. A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of di erential equations: evolutionary equations. Vol. IV, Handb. Di er. Equ., Elsevier/North-Holland, Amsterdam, 2008, 201-228.

38. M. Pierre and P. Rogeon, Convergence to equilibrium for a time semi-discrete damped wave equation,J. Appl. Anal. Comput., 6 (2016), 1041-1048.

39. P. Pol´aˇcik and F. Simondon, Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains, J. Di erential Equations, 186 (2002), 586-610.

40. J. Pr¨uss, R. Racke, and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions, Ann. Mat. Pura Appl. (4), 185 (2006),627-648.

41. J. Pr¨uss and M. Wilke, Maximal Lp-regularity and long-time behaviour of the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions, in Partial di erential equations and functional analysis, vol. 168 of Oper. Theory Adv. Appl., Birkh¨auser, Basel, 2006, 209-236.

42. P. Rybka and K.-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation, Comm. Partial Differential Equations, 24 (1999), 1055-1077.

43. J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669-1691.

44. L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2), 118 (1983), 525-571.

45. A. M. Stuart and A. R. Humphries, Dynamical systems and numerical analysis, vol. 2 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 1996.

46. F. Tone, On the long-time stability of the Crank-Nicolson scheme for the 2D Navier-Stokes equations,Numer. Methods Partial Di erential Equations, 23 (2007), 1235-1248.

47. J. Wei and M. Winter, On the stationary Cahn-Hilliard equation: interior spike solutions, J. Differential Equations, 148 (1998), 231-267.

48. H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition, Asymptot. Anal., 54 (2007), 71-92.

49. H. Wu and S. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, J. Di erential Equations, 204 (2004), 511-531.

50. X.Wu, G. J. van Zwieten, and K. G. van der Zee, Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models, Int. J. Numer. MethodsBiomed. Eng., 30 (2014), 180-203.

51. S. Zheng, Asymptotic behaviour to the solution of the Cahn-Hilliard equation, Applic. Anal., 23 (1986), 165-184.

Copyright Info: © 2016, Morgan Pierre, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved