Publication Cover
Applicable Analysis
An International Journal
Volume 96, 2017 - Issue 2
220
Views
14
CrossRef citations to date
0
Altmetric
Original Articles

Asymptotic behavior of a generalized Cahn–Hilliard equation with a mass source

Pages 324-348 | Received 27 Jan 2015, Accepted 01 Sep 2015, Published online: 11 Jan 2016

References

  • Cahn JW, Hilliard JE. Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 1958;28:258–267.
  • Novick-Cohen A, Segal LA. Nonlinear Cahn--Hiliard equation. Proc. Roy. Soc. London Ser. A. 1989;422:261–278.
  • Cohen D, Murray JM. A generalized diffusion model for growth and dispersion in a population. J. Math. Biol. 1981;12:237–248.
  • Klapper I, Dockery J. Role of cohesion in the material description of biofilms. Phys. Rev. E. 2006;74:0319021.
  • Oron A, Davis SH, Bankoff SG. Long-scale evolution of thin liquid films. Rev. Mod. Phys. 1997;69:931–980.
  • Thiele U, Knobloch E. Thin liquid films on a slightly inclined heated plate. Phys. D. 2004;190:213–248.
  • Chalupecki V. Numerical studies of Cahn--Hilliard equations and applications in image processing. In: Proceedings of Gzech-Japanese Seminar in Applied Mathematics; 2004 Aug 4--7; Czech Technical University in Prague; 2004. p. 1–3.
  • Dolcetta IC, Vita SF, March R. Area-preserving curve-shortening flows: from phase separation to image processing. Interfaces Free Bound. 2002;4:325–343.
  • Tremaine S. On the origin of irregular structure in Saturn’s rings. Astron. J. 2003;125:894–901.
  • Nicolaenko B, Scheurer B. Low-dimensional behavior of the pattern formation Cahn--Hilliard equation. In: Lakshmikantham V, editor. Trends in the theory and practice of nonlinear analysis. Vol. 110, North-Holland mathematics studies. Amsterdam: North-Holland; 1985.
  • Constantin P, Foias C, Nicolaenko B, et al. Integral manifolds and inertial manifolds for dissipative partial differential equations. Vol. 70, Applied mathematical sciences. New York (NY): Springer-Verlag; 1989.
  • Nicolaenko B, Scheurer B, Temam R. Some global dynamical properties of a class of pattern formation equations. Commun. Differ. Equ. 1989;14:245–297.
  • Temam R. Infinite-dimensional dynamical systems in mechanics and physics. 2nd ed. Vol. 68, Applied mathematical sciences. New York (NY): Springer-Verlag; 1997.
  • Oono Y, Puri S. Computationally efficient modeling of ordering of quenched phases. Phys. Rev. Lett. 1987;58:836–839.
  • Villain-Guillot S. Phases modulées et dynamique de Cahn--Hilliard [Habilitation thesis]. , Université Bordeaux; 2010;1.
  • Miranville A. Asymptotic behavior of the Cahn--Hilliard--Oono equation. J. Appl. Anal. Comp. 2011;1:523–536.
  • Cherfils L, Miranville A, Zelik S. On a generalized Cahn--Hilliard equation with biological applications. Discrete Cont. Dyn. Syst. B. 2014;19:2013–2026.
  • Miranville A. Asymptotic behavior of a generalized Cahn--Hilliard equation with a proliferation term. Appl. Anal. 2013;92:1308–1321.
  • Khain E, Sander LM. A generalized Cahn--Hilliard equation for biological applications. Phys. Rev. E. 2008;77:051129.
  • Aristotelous AC, Karakashian OA, Wise SM. Adaptive, second order in time, primitive-variable discontinuous Galerkin schemes for a Cahn--Hilliard equation with a mass source. IMA J. Numer. Anal. 2015;35:1167–1198.
  • Bertozzi A, Esedoglu S, Gillette A. Analysis of a two-scale Cahn--Hilliard model for binary image inpainting. Multiscale Model. Simul. 2007;6:913–936.
  • Bertozzi A, Esedoglu S, Gillette A. Inpainting of binary images using the Cahn--Hilliard equation. IEEE Trans. Image Proc. 2007;16:285–291.
  • Burger M, He L, Schönlieb C. Cahn--Hilliard inpainting and a generalization for grayvalue images. SIAM J. Imag. Sci. 2009;3:1129–1167.
  • Cherfils L, Fakih H, Miranville A. Finite-dimensional attractors for the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation in image inpainting. Inv. Prob. Imag. 2015;9:105–125.
  • Bosch J, Kay D, Stoll M, et al. Fast solvers for Cahn--Hilliard inpainting. SIAM J. Imag. Sci. 2013;7:67–97.
  • Cherfils L, Fakih H, Miranville A. On the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation with logarithmic nonlinear terms. SIAM J. Imag. Sci. 2015;8:1123–1140.
  • Eden A, Foias C, Nicolaenko B, et al. Expenential attractors for dissipative evolution equations. Vol. 37, Research in applied mathematics. New York (NY): Wiley; 1994.
  • Efendiev M, Miranville A, Zelik S. Exponential attractors for a nonlinear reaction-diffusion system in R3. C. R. Acad. Sci. Paris Sér. I. 2000;330:713–718.
  • Efendiev M, Miranville A, Zelik S. Exponential attractors for a singularly pertubed Cahn--Hilliard system. Math. Nachr. 2004;272:11–31.
  • Elliott CM, French DA, Milner FA. A second order splitting method for the Cahn--Hilliard equation. Numer. Math. 1989;54:575–590.
  • Cherfils L, Petcu M, Pierre M. A numerical analysis of the Cahn--Hilliard equation with dynamic boundary conditions. Discrete Cont. Dyn. Syst. 2010;27:1511–1533.
  • Hecht F. New development in FreeFem++. J. Numer. Math. 2012;20:251–265.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.