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

Keywords Łojasiewicz-Simon inequality; gradient flow; Cahn-Hilliard; backward differentiation formula

Citation: Paola F. Antonietti, Benoît Merlet, Morgan Pierre, Marco Verani. Convergence to equilibrium for a second-order time semi-discretization ofthe Cahn-Hilliard equation. AIMS Mathematics, 2016, 1(3): 178-194. doi: 10.3934/Math.2016.3.178


  • 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.


This article has been cited by

  • 1. Yusuf Olatunbosun Tafa, Gang Zhao, Wei Wang, Isogeometric Analysis Prediction of Stress Concentration Factor of Trivariate NURBS-Based Rectangular Plate with Central Elliptical Hole, Applied Mechanics and Materials, 2014, 556-562, 742, 10.4028/www.scientific.net/AMM.556-562.742
  • 2. Anass Bouchriti, Morgan Pierre, Nour Eddine Alaa, Gradient stability of high-order BDF methods and some applications, Journal of Difference Equations and Applications, 2020, 1, 10.1080/10236198.2019.1709062

Reader Comments

your name: *   your email: *  

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

Copyright © AIMS Press All Rights Reserved