Abstract
An important role in studying the classical Cahn–Hilliard problem [5] is played by its singular-limit problem, the so-called Melin–Sikerk free boundary problem, which, at present allows one to only numerically describe the instability of the crystallization process. The purpose of this work is to prepare the material for deducing the singular-limit problem for the essentially asymmetric model [8, 21].
Similar content being viewed by others
References
R. Akhmerov, “On structure of a set of solutions of Dirichlet boundary-value problem for stationary one-dimensional forward-backward parabolic equation,” Nonlinear Anal. Theory Meth. Appl., 11, No. 11, 1303–1316 (1987).
N. Alikakos, P. Bates, and G. Fusco “Slow motion for Cahn–Hilliard equation,” SIAM J. Appl. Math., 90, 81–135 (1991).
G. I. Barenblat, V. M. Entov, and V. M. Rizhik, Behavior of Fluids and Gases in Porous Media [in Russian], Nedra, Moscow (1984).
P. Bates and P. Fife “The dynamics of nucleation for Cahn–Hilliard equation,” SIAM J. Appl. Math. 53, 990–1008 (1993).
J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system, Part I: Interfacial free energy,” J. Chemical Physics, 28, No. 1, 258–267 (1958).
V. G. Danilov, G. A. Omel’yanov, and E. V. Radkevich, “Hogoniot-type conditions and weak solutions to the phase field system,” Eur. J. Appl. Math., 10, 55–77 (1999).
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).
W. Dreyer and W. H. Muller, “A study of the coarsening in tin/lead solders,” Int. J. Solids Structures, 37, 3841–3871 (2000).
Ch. Elliott, “The Stefan problem with non-monotone constitutive relations,” IMA J. Appl. Math., 35, 257–264 (1985).
Ch. Elliot and S. Zheng, “On the Chan–Hilliard equation,” Arch. Ration. Mech. Anal., 96, No. 4, 339–357 (1986).
A. Fridman, Variational Prnciples and Free Boundary Problem, Wiley, New York-Chichester, Brisbane, Toronto, Singapore (1982).
C. Grant, “Spinodal decomposition for the Cahn–Hilliard equation,” Commu. Part. Differ. Equat., 18, Nos. 3-4, 453–490 (1985).
D. Hilhorst, R.Kersner, E. Logak, and M. Mimura, “On some asymptotic limits of the Fisher equation with degenerate diffusion” (in press).
K. Hollig, “Existence of infinity many solutions for a forward-backward parabolic equation,” Trans. Am. Math. Soc., 278, No. 1, 299–316 (1983).
D. Kinderlehrer and P. Pedregal, “Weak convergence of integrands and the Young measure representation,” SIAM J. Math. Anal., 23, 1–19 (1992).
B. Nicolaenko, B. Scheurer, and R. Temam, “Some global properties of a class of pattern formation equations,” Commun. Part. Differ. Equ., 14, No. 2, 245–297 (1989).
L. Nirenberg, Topics on Nonlinear Functional Analysis, Courant Inst. Math. Sciences, New York (1974).
P. Plotnikov, “Singular limits of solutions to Cahn–Hilliard equation” (in press).
E. V. Radkevich, “Existence conditions of a classical solution of modified Stefan problem (Gibbs–Thomson law),” Mat. Sb., 183, No. 2, 77–101 (1982).
E. V. Radkevich, “On asymptotic solution of phase field system,” Differ. Uravn., 29, No. 3, 487–500 (1993).
E. V. Radkevich and M. Zakharchenko, “Asymptotic solution of extended Cahn–Hilliard model,” In: Contemporary Mathematics and Its Applications [in Russian], 2, Institute of Cybernetics, Tbilisi (2003), pp. 121–138.
P. G. Saffman and G. I. Taylor, “The penetration of a fluid into porous medium or Hele-Shaw cell containing a more viscous liquid,” Proc. Roy. Soc. London, A. 245, 312–329 (1958).
M. Slemrod, “Dynamics of measure-valued solutions to a backward-forward parabolic equation,” J. Dyn. Differ. Equ., 2, 1–28 (1991).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 78, Partial Differential Equations and Optimal Control, 2012.
Rights and permissions
About this article
Cite this article
Selivanova, N.Y., Shamolin, M.V. Studying the interphase zone in a certain singular-limit problem. J Math Sci 189, 284–293 (2013). https://doi.org/10.1007/s10958-013-1185-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-013-1185-0