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On a phase field problem driven by interface area and interface curvature

Published online by Cambridge University Press:  05 November 2009

XIAOFENG REN  and
JUNCHENG WEI
XIAOFENG REN
Affiliation:
Department of Mathematics and Statistics, The George Washington University, Washington, DC 20052, USA email: ren@gwu.edu
JUNCHENG WEI
Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Hong Kong, P.R. China
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Abstract

A two component system driven by both interface area and interface curvature is studied with a new phase field model. We show that if the curvature impact in the system is strong enough, there exist bubble profiles. A bubble profile describes a pattern of an inner core of one component surround by an outer membrane of the other component. It is a radial solution to a fourth-order nonlinear partial differential equation. We show the existence of such profiles in all dimensions, although the profile is unstable if the dimension is greater than 2.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 6 , December 2009 , pp. 531 - 556
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Allen, S. E. & Cahn, J. W. (1979) A microscopic theroy for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metall. 27 (6), 1085–1095.CrossRefGoogle Scholar
[2]Bates, P. W., Dancer, E. N. & Shi, J. (1999) Multi-spike stationary solutions of the Cahn–Hilliard equation in higher-dimension and instability. Adv. Differ. Equ. 4, 1–69.Google Scholar
[3]Casten, R. G. & Holland, C. J. (1978) Instability results for reaction diffusion equations with Neumann boundary conditions. J. Differ. Equ. 27 (2), 266–273.CrossRefGoogle Scholar
[4]Ciarlet, P. G. (1998) Introduction to Linear Shell Theory, Series in Applied Mathematics (Paris), Vol. 1. Édition Scientifiques et Médicales Elsevier, Gauthier-Villars, Paris.Google Scholar
[5]Ciarlet, P. G. (2000) Mathematical Elasticity, III, Studies in Mathematics and its Applications, Vol. 29. North-Holland, Amsterdam.Google Scholar
[6]Dancer, E. N. & Yan, S. (2003) Multi-layer solutions for an elliptic problem. J. Differ. Equ. 194, 382–405.CrossRefGoogle Scholar
[7]Dancer, E. N. & Yan, S. (2004) A minimization problem associated with elliptic systems of FitzHugh–Nagumo type. Annu. Inst. H. Poincaré Anal. Non Linéaire 21 (2), 237–257.CrossRefGoogle Scholar
[8]De Giorgi, E. (1975) Sulla convergenza di alcune successioni d'integrali del tipo dell'area. Rend. Mat. (6), 8, 277–294.Google Scholar
[9]Du, Q., Liu, C., Ryham, R. & Wang, X. (2004) A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198, 450–468.CrossRefGoogle Scholar
[10]Du, Q., Liu, C. & Wang, X. (2006) Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212, 757–777.CrossRefGoogle Scholar
[11]Gui, C. & Wei, J. (1999) Multiple interior peak solutions for some singular perturbation problems. J. Differ. Equ. 158, 1–27.CrossRefGoogle Scholar
[12]Gui, C. & Wei, J. (2000) On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Can. J. Math. 52, 522–538.CrossRefGoogle Scholar
[13]Helfrich, W. (1973) Elastic properties of lipid bilayers: Theory and possible experiments. Z. Naturforsch C 28, 693703.CrossRefGoogle ScholarPubMed
[14]Kohn, R. & Sternberg, P. (1989) Local minimisers and singular perturbations. Proc. R. Soc. Edinburgh Sect. A 111 (1–2), 69–84.CrossRefGoogle Scholar
[15]Matano, H. (1979) Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. Res. Inst. Math. Sci. 15 (2), 401–454.CrossRefGoogle Scholar
[16]Modica, L. (1987) The gradient theory of phase transitions and the minimal interface criterion. Arch. Rat. Mech. Anal. 98 (2), 123–142.CrossRefGoogle Scholar
[17]Modica, L. & Mortola, S. (1977) Un esempio di Γ−-convergenza. Boll. Un. Mat. Ital. B (5) 14 (1), 285–299.Google Scholar
[18]Moser, R. (2005) A higher order asymptotic problem related to phase transitions. SIAM J. Math. Anal. 37 (3), 712–736.CrossRefGoogle Scholar
[19]Ni, W.-M. & Wei, J. (2006) On positive solutions concentrating on spheres for the Gierer–Meinhardt system. J. Differ. Equ. 221 (1), 158–189.CrossRefGoogle Scholar
[20]Ren, X. & Wei, J. (2005) Nucleation in the FitzHugh–Nagumo system: interface-spike solutions. J. Diff. Eqns., 209 (2), 266–301.CrossRefGoogle Scholar
[21]Ren, X. & Wei, J. (2006) Droplet solutions in the diblock copolymer problem with skewed monomer composition. Calc. Var. Partial Differ. Equ. 25 (3), 333–359.CrossRefGoogle Scholar
[22]Seifert, U., Berndl, K. & Lipowsky, R. (1991) Configurations of fluid membranes and vesicles. Phys. Rev. A 44, 1182–1202.CrossRefGoogle Scholar
[23]Simon, L. (1993) Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1 (2), 281–326.CrossRefGoogle Scholar
[24]Tang, H. & Freed, K. F. (1991) Free energy functional expansion for inhomogeneous polymer blends. J. Chem. Phys. 94 (2), 1572–1583.CrossRefGoogle Scholar
[25]Willmore, T. J. (1993) Riemannian Geometry. The Clarendon Press, Oxford University Press, New York.CrossRefGoogle Scholar