Free Access
Issue
ESAIM: M2AN
Volume 55, 2021
Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
Page(s) S225 - S250
DOI https://doi.org/10.1051/m2an/2020028
Published online 26 February 2021
  1. N. Abukhdeir, D. Vlachos, M. Katsoulakis and M. Plexousakis, Long-time integration methods for mesoscopic models of pattern-forming systems. J. Comput. Phys. 18 (2013) 2211–2238. [Google Scholar]
  2. A.C. Aristotelous, O. Karakashian and S.M. Wise, A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonliniear multigrid solver. Disc. Cont. Dyn. Syst. B 20 (2015) 1529–1553. [Google Scholar]
  3. P.W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation. J. Diff. Equ. 212 (2005) 235–277. [Google Scholar]
  4. P.W. Bates and J. Han, The Dirichlet boundary problem for a nonlocal Cahn-Hilliard equation. J. Math. Anal. App. 311 (2005) 289–312. [Google Scholar]
  5. P.W. Bates, S. Brown and J. Han, Numerical analysis for a nonlocal Allen-Cahn equation. Int. J. Numer. Anal. Model. 6 (2009) 33–49. [Google Scholar]
  6. A. Bertozzi, S. Esedoglu and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for binary image inpainting. Multiscale Model. Simul. 6 (2007) 913–936. [Google Scholar]
  7. L.A. Caffarelli and N.E. Muler, An L bound for solutions of the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 135 (1995) 129–144. [Google Scholar]
  8. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. [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. L. Cherfils, A. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications. Disc. Cont. Dyn. Sys. B 19 (2014) 2013–2026. [Google Scholar]
  11. L. Cherfils, H. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation in image inpainting. Inv. Prob. Imag. 9 (2015) 105–125. [Google Scholar]
  12. L. Cherfils, H. Fakih and A. Miranville, On the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation with logarithmic nonlinear terms. SIAM J. Imag. Sci. 8 (2015) 1123–1140. [Google Scholar]
  13. L. Cherfils, H. Fakih and A. Miranville, A Cahn-Hilliard system with a fidelity term for color image inpainting. J. Math. Imag. Vision 54 (2016) 117–131. [Google Scholar]
  14. L. Cherfils, H. Fakih and A. Miranville, A complex version of the Cahn-Hilliard equation for grayscale image inpainting. Multiscale Model. Simul. 15 (2017) 575–605. [Google Scholar]
  15. D. Cohen and J.M. Murray, A generalized diffusion model for growth and dispersion in a population. J. Math. Biol. 12 (1981) 237–248. [Google Scholar]
  16. E. Davoli, L. Scarpa and L. Trussardi, Local asymptotics for nonlocal convective Cahn-Hilliard equations with W1,1 kernel and singular potential. Preprint arXiv:1911.12770v1 [math.AP] (2019). [Google Scholar]
  17. E. Davoli, H. Ranetbauer, L. Scarpa and L. Trussardi, Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptotics. Ann. Inst. Henri Poincaré C, Anal. non Lin. 37 (2020) 627–651. [Google Scholar]
  18. F. Della Porta and M. Grasselli, Convective nonlocal Cahn-Hilliard equations with reaction terms. Disc. Cont. Dyn. Syst. B 20 (2015) 1529–1553. [Google Scholar]
  19. I.C. Dolcetta, S.F. Vita and R. March, Area-preserving curve-shortening flows: from phase separation to image processing. Interfaces Free Bound. 4 (2002) 325–343. [Google Scholar]
  20. Q. Du, M. Gunzburger, R. LeHoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54 (2012) 667–696. [Google Scholar]
  21. D.J. Eyre, An unconditionally stable one-step scheme for gradient systems. Available at: https://www.math.utah.edu/eyre/research/methods/stable.ps (2020). [Google Scholar]
  22. H. Fakih, A Cahn-Hilliard equation with a proliferation term for biological and chemical applications. Asympt. Anal. 94 (2015) 71–104. [Google Scholar]
  23. P.C. Fife, Models for phase separation and their mathematics. Electron. J. Diff. Equ. 13 (2002) 353–370. [Google Scholar]
  24. H. Gajewski and K. Gärtner, On a nonlocal model of image segmentation. Z. Angew. Math. Phys. 56 (2005) 572–591. [Google Scholar]
  25. C.G. Gal, A. Giorgini and M. Grasselli, The nonlocal Cahn-Hilliard equation with singular potential: well-posedness, regularity and strict separation property. J. Differ. Equ. 263 (2017) 5253–5297. [Google Scholar]
  26. G. Giacomin and J.L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions. Phys. Rev. Lett. 76 (1996) 1094–1097. [PubMed] [Google Scholar]
  27. A. Giorgini, M. Grasselli and A. Miranville, The Cahn–Hilliard–Oono equation with singular potential. Math. Models Methods Appl. Sci. 27 (2017) 2485–2510. [Google Scholar]
  28. M. Grasselli and H. Wu, Well-posedness and longtime behavior for the modified phase-field crystal equation. Math. Models Methods Appl. Sci. 24 (2014) 2743–2783. [Google Scholar]
  29. Z. Guan, C. Wang and S.M. Wise, A Convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation. Numer. Math. 128 (2014) 377–406. [Google Scholar]
  30. Z. Guan, J.S. Lowengrub, C. Wang and S.M. Wise, Second-order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations. J. Comput. Phys. 277 (2014) 48–71. [Google Scholar]
  31. Z. Guan, J.S. Lowengrub and C. Wang, Convergence analysis for second order accurate convex splitting schemes for the periodic nonlocal Allen-Cahn and Cahn-Hilliard equations. Math. Methods Appl. Sci. 40 (2017) 6836–6863. [Google Scholar]
  32. T. Hartley and T. Wanner, A semi-implicit spectral method for stochastic nonlocal phase-field models. Disc. Cont. Dyn. Syst. 25 (2009) 399–429. [Google Scholar]
  33. F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. [Google Scholar]
  34. D. Hornthrop, M. Katsoulakis and D. Vlachos, Spectral methods for mesoscopic models of pattern formation. J. Comput. Phys. 173 (2001) 364–390. [Google Scholar]
  35. E. Khain and L.M. Sander, A generalized Cahn-Hilliard equation for biological applications. Phys. Rev. E 77 (2008) 051129. [Google Scholar]
  36. I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms. Phys. Rev. E 74 (2006) 0319021. [Google Scholar]
  37. J.S. Langer, Theory of spinodal decomposition in alloys. Ann. Phys. 65 (1975) 53–86. [Google Scholar]
  38. Q.-X. Liu, A. Doelman, V. Rottschäfer, M. de Jager, P.M.J. Herman, M. Rietkerk and, J. van de Koppel, Phase separation explains a new class of self-organized spatial patterns in ecological systems. Proc. Nat. Acad. Sci. 110 (2013) 11905–11910. [Google Scholar]
  39. S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: nonlinear dynamics. Arch. Ration. Mech. Anal. 151 (2000) 187–219. [Google Scholar]
  40. S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term. Adv. Math. Sci. App. 24 (2014) 461–497. [Google Scholar]
  41. A. Miranville, Asymptotic behavior of the Cahn–Hilliard–Oono equation. J. Appl. Anal. Comput. 1 (2011) 523–536. [Google Scholar]
  42. A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Disc. Cont. Dyn. Syst. 28 (2010) 275–310. [Google Scholar]
  43. C.B. Muratov and Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions. Commun. Math. Phys. 299 (2010) 45–87. [Google Scholar]
  44. R.H. Nochetto, A.J. Salgado and S.W. Walker, A diffuse interface model for electrowetting with moving contact lines. Math. Models Methods Appl. Sci. 24 (2014) 67–111. [Google Scholar]
  45. Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases. Phys. Rev. Lett. 58 (1987) 836–839. [CrossRef] [PubMed] [Google Scholar]
  46. A. Oron, S.H. Davis and S.G. Bankoff, Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (1997) 931–980. [Google Scholar]
  47. E.W. Sachs and A.K. Strauss, Efficient solution of a partial integro-differential equation in finance. Appl. Numer. Math. 58 (2008) 1687–1703. [Google Scholar]
  48. C.-B. Schönlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting. Commun. Math. Sci. 9 (2011) 413–457. [Google Scholar]
  49. U. Thiele and E. Knobloch, Thin liquid films on a slightly inclined heated plate. Phys. D 190 (2004) 213–248. [Google Scholar]
  50. S. Tremaine, On the origin of irregular structure in Saturn’s rings. Astron. J. 125 (2003) 894–901. [Google Scholar]
  51. S. Villain-Guillot, Phases modulées et dynamique de Cahn–Hilliard, Habilitation thesis. Université Bordeaux I (2010). [Google Scholar]
  52. K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48 (2010) 1759–1780. [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