Free Access
Issue
ESAIM: M2AN
Volume 55, Number 1, January-February 2021
Page(s) 229 - 282
DOI https://doi.org/10.1051/m2an/2020090
Published online 18 February 2021
  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. [Google Scholar]
  2. H.W. Alt, Linear Functional Analysis – An Application-Oriented Introduction. Springer, London (2016). [Google Scholar]
  3. L. Ambrosio, N. Gigli and G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser Basel (2008). [Google Scholar]
  4. P. Bates and P. Fife, The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53 (1993) 990–1008. [Google Scholar]
  5. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer (2002). [Google Scholar]
  6. A. Buffa, M. Costabel and D. Sheen, On traces for H(curl, Ω) in Lipschitz domains. J. Math. Anal. Appl. 276 (2002) 845–867. [Google Scholar]
  7. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 2 (1958) 205–245. [Google Scholar]
  8. E. Campillo-Funollet, G. Grün and F. Klingbeil, On modeling and simulation of electrokinetic phenomena in two-phase flow with general mass densities. SIAM J. Appl. Math. 72 (2012) 1899–1925. [Google Scholar]
  9. L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials. Milan J. Math. 79 (2011) 561–596. [Google Scholar]
  10. P. Colli and T. Fukao, Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary. J. Math. Anal. Appl. 429 (2015) 1190–1213. [Google Scholar]
  11. P. Colli and T. Fukao, Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials. Nonlinear Anal. 127 (2015) 413–433. [Google Scholar]
  12. P. Colli, G. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential. J. Math. Anal. Appl. 419 (2014) 972–994. [Google Scholar]
  13. P. Colli, G. Gilardi, R. Nakayashiki and K. Shirakawa, A class of quasi-linear Allen-Cahn type equations with dynamic boundary conditions. Nonlinear Anal. 158 (2017) 32–59. [Google Scholar]
  14. C. Cowan, The Cahn–Hilliard equation as a gradient flow. Master’s thesis, Simon Fraser University (2005). [Google Scholar]
  15. G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces. in: Lecture Notes in Mathematics, Springer, Berlin Heidelberg (1988) 142–155. [Google Scholar]
  16. G. Dziuk and C.M. Elliott, Finite element methods for surface PDEs. Acta Numer. 22 (2013) 289–396. [CrossRef] [MathSciNet] [Google Scholar]
  17. C.M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (1996) 404–424. [CrossRef] [MathSciNet] [Google Scholar]
  18. C.M. Elliott and T. Ranner, Finite element analysis for a coupled bulk-surface partial differential equation. IMA J. Numer. Anal. 33 (2012) 377–402. [Google Scholar]
  19. C.M. Elliott and S. Zheng, On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96 (1986) 339–357. [Google Scholar]
  20. H.P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition. Phys. Rev. Lett. 79 (1997) 893–896. [Google Scholar]
  21. H.P. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall. J. Chem. Phys. 108 (1998) 3028–3037. [Google Scholar]
  22. T. Fukao and H. Wu, Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn-Hilliard type with singular potential. To appear. In: Asymptot. Anal. DOI: 10.3233/ASY-201646 (2020). [Google Scholar]
  23. C.G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls. Math. Methods App. Sci. 29 (2006) 2009–2036. [Google Scholar]
  24. C.G. Gal and H. Wu, Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete Contin. Dyn. Syst. 22 (2008) 1041–1063. [Google Scholar]
  25. H. Garcke, On Cahn-Hilliard systems with elasticity. Proc. Roy. Soc. Edinburgh 133A (2003) 307–331. [Google Scholar]
  26. H. Garcke and P. Knopf, Weak solutions of the Cahn-Hilliard equation with dynamic boundary conditions: A gradient flow approach. SIAM J. Math. Anal. 52 (2020) 340–369. [Google Scholar]
  27. G.R. Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non permeable walls. Phys. D 240 (2011) 754–766. [Google Scholar]
  28. G. Grün and F. Klingbeil, Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model. J. Comput. Phys. 257 (2014) 708–725. [Google Scholar]
  29. G. Grün, F. Guillén-González and S. Metzger, On fully decoupled, convergent schemes for diffuse interface models for two-phase flow with general mass densities. Commun. Comput. Phys. 19 (2016) 1473–1502. [Google Scholar]
  30. P. Harder and B. Kovács, Error estimates for the Cahn-Hilliard equation with dynamic boundary conditions. Preprint arxiv:2005.03349 [math.NA] (2020). [Google Scholar]
  31. R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl and W. Dietrich, Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions. Comp. Phys. Comm. 133 (2001) 139–157. [Google Scholar]
  32. P. Knopf and K.F. Lam, Convergence of a Robin boundary approximation for a Cahn-Hilliard system with dynamic boundary conditions. Nonlinearity 33 (2020) 4191–4235. [Google Scholar]
  33. P. Knopf and C. Liu, On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems. Preprint arxiv:2008.00895 [math.AP] (2020). [Google Scholar]
  34. M. Liero, Passing from bulk to bulk-surface evolution in the Allen-Cahn equation. Nonlinear Differ. Equ. Appl. 20 (2013) 919–942. [Google Scholar]
  35. C. Liu and H. Wu, An energetic variational approach for the Cahn-Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis. Arch. Ration. Mech. Anal. 233 (2019) 167–247. [Google Scholar]
  36. S. Metzger, On convergent schemes for two-phase flow of dilute polymeric solutions. ESAIM:M2AN 52 (2018) 2357–2408. [EDP Sciences] [Google Scholar]
  37. S. Metzger, On stable, dissipation reducing splitting schemes for two-phase flow of electrolyte solutions. Numer. Algorithms 80 (2018) 1361–1390. [Google Scholar]
  38. S. Metzger, An efficient and convergent finite element scheme for Cahn-Hilliard equations with dynamic boundary conditions. Preprint arxiv:1908.04910 [math.NA], accepted for publication in SIAM Journal on Numerical Analysis (2020). [Google Scholar]
  39. R.M. Mininni, A. Miranville and S. Romanelli, Higher-order Cahn-Hilliard equations with dynamic boundary conditions. J. Math. Anal. Appl. 449 (2017) 1321–1339. [Google Scholar]
  40. A. Miranville, The Cahn-Hilliard Equation: Recent Advances and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2019). [CrossRef] [Google Scholar]
  41. A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete Contin. Dyn. Syst. 28 (2010) 275–310. [Google Scholar]
  42. T. Motoda, Time periodic solutions of Cahn-Hilliard systems with dynamic boundary conditions. AIMS Math. 3 (2018) 263–287. [Google Scholar]
  43. R. Pego, Front migration in the nonlinear Cahn-Hilliard equation. Proc. R. Soc. Lond. A 422 (1989) 261–278. [Google Scholar]
  44. R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamic boundary conditions. Adv. Differ. Equ. 8 (2003) 83–110. [Google Scholar]
  45. P. Rybka and K.H. Hoffmann, Convergence of solutions to Cahn-Hilliard equation. Commun. Part. Differ. Equ. 24 (1999) 1055–1077. [Google Scholar]
  46. J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura App. 146 (1986) 65–96. [Google Scholar]
  47. M.E. Taylor, Partial differential equations I. . Basic theory, 2nd edition. In: Vol. 115 of Applied Mathematical Sciences, Springer, New York (2011). [Google Scholar]
  48. P.A. Thompson and M.O. Robbins, Simulations of contact-line motion: slip and the dynamic contact angle. Phys. Rev. Lett. 63 (1989) 766–769. [CrossRef] [PubMed] [Google Scholar]
  49. H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition. Asymptot. Anal. 54 (2007) 71–92. [Google Scholar]
  50. H. Wu and S. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions. J. Differ. Equ. 204 (2004) 511–531. [Google Scholar]
  51. S. Zheng, Asymptotic behavior of solution to the Cahn-Hilliard equation. Appl. Anal. 23 (1986) 165–184. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you