Skip to main content
Log in

Asymptotic Solution of the Extended Cahn—Hilliard Model

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The paper is devoted to asymptotic analysis of the mathematical model of two-composite materials. The main result is the deduction of the extended Stefan problem being a singular limit of the initial problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. R. Akhmerov, “On the structure of a set of solutions of the Dirichlet boundary value problem for a stationary one-dimensional forward-backward parabolic equation,” Nonlinear Anal. Theory Meth. Appl., 11 No. 11, 1303–1316 (1987).

    Google Scholar 

  2. N. Alikakos, P. Bates, and G. Fusco, “Slow motion for the Cahn-Hilliard equation, ” SIAM J. Appl. Math., 90 81–135 (1991).

    Google Scholar 

  3. G. I. Barenblat, V. M. Entov, and V. M. Rizhik, Gas and Liquid Behavior in Porous Layers [in Russian ], Nedra, Moscow (1984).

    Google Scholar 

  4. P. Bates and P. Fife, “The dynamics of nucleation for the Cahn-Hilliard equation, ” SIAM J. Appl. Math., 53 990–1008 (1993).

    Google Scholar 

  5. J. W. Cahn and J. E. Hilliard, “Free energy of a non-uniform system. Part I Interfacial free energy,” J. Chem. Phys., 28 No. 1, 258–267 (1958).

    Google Scholar 

  6. V. G. Danilov, G. A. Omel'yanov, and E. V. Radkevich, “Hugoniot-type conditions and weak solutions to the phase eld system,” Eur. J. Appl. Math., 10 55–77 (1999).

    Google Scholar 

  7. V. G. Danilov, G. A. Omel'yanov, and E. V. Radkevich, “Asymptotic solution of the conserved phase field system in the fast relaxation case,” Eur. J. Appl. Math., 9 1–21 (1998).

    Google Scholar 

  8. W. Dreyer and W. H. Muller, “A study of the coarsening in tin/lead solders, ”Int. J. Solids Structures, 37 3841–3871 (2000).

    Google Scholar 

  9. C. Elliot, “The Stefan problem with non-monotone constitutive relations, ”IMA J. Appl. Math., 35 257–264 (1985).

    Google Scholar 

  10. C. Elliot and S. Zheng, “On the Cahn-Hilliard equation,” Arch. Rat. Mech. Anal., 96 No. 4, 339–357 (1986).

    Google Scholar 

  11. A. Friedman, Variational Principles and Free Boundary Problems, Wiley, New York (1982).

    Google Scholar 

  12. C. Grant, “Spinodal decomposition for the Cahn-Hilliard equation,” Comm. Part. Differ. Equat., 18 Nos. 3–4, 453–490 (1985).

    Google Scholar 

  13. D. Hilhorst, R. Kersner, E. Logak, and M. Mimura, “On some asymptotic limits of the Fisher equation with degenerate diffusion,” to appear.

  14. K. Hollig, “Existence of in nitely many solutions for a forward-backward parabolic equation, ” Trans. Amer. Math. Soc., 278 No. 1, 299–316 (1983).

    Google Scholar 

  15. D. Kinderlehrer and P. Pedregal, “Weak convergence of integrands and the Young measure representation,” SIAM J. Math. Anal., 23 1–19 (1992).

    Google Scholar 

  16. V. P. Maslov, Propagation of Shock Waves in an Isocentric Nonviscous Gas, In Progress in Science and Technology, Series on Contemporary Problems in Mathematics [in Russian ], Vol. 8, All-Union Institute for Scienti c and Technical Information, Akad. Nauk SSSR, Moscow (1977).

    Google Scholar 

  17. B. Nicolaenko, B. Scheurer, and R. Temam, “Some global properties of a class of pattern formation equations,”Comm. Part. Differ. Equat., 14 No. 2, 245–297 (1989).

    Google Scholar 

  18. L. Nirenberg, Topics on Nonlinear Functional Analysis, New York Courant Inst. Math. Sciences (1974).

  19. P. Plotnikov, “Singular limits of solutions to the Cahn-Hilliard equation, ”in press.

  20. E. V. Radkevich, “The existence conditions for the classical solution to the modi ed Stefan problem (Gibbs-Thompson law),” Mat. Sb., 183 No. 2, 77–101 (1992).

    Google Scholar 

  21. E. V. Radkevich, “On the asymptotic solution of the system of the phase field, ” Differ. Uravn., 29 No. 3, 487–500 (1993).

    Google Scholar 

  22. P. G. Saffman and G. I. Taylor “The penetration of a fluid into a porous medium or a Hele-Shaw cell containing a more viscous liquid,” Proc. Roy. Soc. London., A. 245, 312–329 (1958).

    Google Scholar 

  23. M. Slemrod, “Dynamics of measure valued solutions to a backward-forward parabolic equation, ” J. Dyn. Diff. Eq., 2 1–28 (1991).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Radkevich, E.V., Zakharchenko, M.V. Asymptotic Solution of the Extended Cahn—Hilliard Model. Journal of Mathematical Sciences 123, 4456–4474 (2004). https://doi.org/10.1023/B:JOTH.0000040304.23521.8b

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOTH.0000040304.23521.8b

Keywords

Navigation