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Counting stationary solutions of the Cahn–Hilliard equation by transversality arguments

Published online by Cambridge University Press:  14 November 2011

M. Grinfeld  and
A. Novick-Cohen
M. Grinfeld
Affiliation:
Department of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Bristol BS8 1TR, U.K.
A. Novick-Cohen
Affiliation:
Department of Mathematics, Technion-ITT, Haifa, Israel 32000
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Abstract

In this paper we use arguments based on Picard-Fuchs equations and transversality of intersections of level curves to obtain an exact count of the number of stationary solutions of the one-dimensional Cahn-Hilliard equation with a cubic nonlinearity.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 2 , 1995 , pp. 351 - 370
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Alikakos, N., Bates, P. W. and Fusco, G.. Slow motion for Cahn-Hilliard equation in one space dimension. J. Differential Equations 90 (1991), 81–134.Google Scholar
2Alikakos, N. and Fusco, G.. Equilibrium and dynamics for the Cahn-Hilliard equation (Preprint).Google Scholar
3Alikakos, N. and Fusco, G.. Slow dynamics for the Cahn-Hilliard equation in higher space dimensions. Part I: Spectral estimates. Comm. P.D.E. 19 (1994), 1397–1447.Google Scholar
4Bates, P. and Fife, P.. The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53(1993), 990–1008.Google Scholar
5Bronsard, L. and Hilhorst, D.. Slow evolution for the one-dimensional Cahn-Hilliard equation. Proc. Roy. Soc. London Series A 439 (1992), 669–682.Google Scholar
6Cahn, J. W. and Hilliard, J. E.. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28 (1958), 258–267.Google Scholar
7Carr, J., Gurtin, M. E. and Slemrod, M.. Structured phase transitions on a finite interval. Arch. Rational Meek Anal 86 (1984), 317–351.Google Scholar
8Carr, J. and Pego, R.. Metastable patterns in solutions ut, = ε2Uxx – f(u). Comm. Pure Appl. Math. 42 (1989), 523–576.CrossRefGoogle Scholar
9B. Char, W., Geddes, K. O., Gonnet, G. H., Monagan, M. B. and Watt, S. M.. MAPLE Reference Manual, 5th edn (Waterloo, Ontario: University of Waterloo, 1988).Google Scholar
10Cushman, R. and Sanders, J. A.. A codimension two bifurcation with a third order Picard-Fuchs equation. J. Differential Equations 59 (1985), 243–256.Google Scholar
11Eden, A., Foias, C., Nicolaenko, B. and Temam, R.. Inertial sets for dissipative evolution equations. Part I: Construction and application (IMA Preprint 812, 1991).Google Scholar
12Eilbeck, J. C., Fife, P. C., Furter, J. E., Grinfeld, M. and Mimura, M.. Stationary states associated with phase separation in a pure material. I. The large latent heat case. Phys. Lett. A 139 (1989), 42–46.Google Scholar
13Eilbeck, J. C., Furter, J. E. and Grinfeld, M.. On a stationary state characterization of transition from spinodal decomposition to nucleation behaviour in the Cahn-Hilliard model of phase separation. Phys. Lett. A 135 (1989), 272–275.Google Scholar
14Elliott, C. M. and Zheng, S.. On the Cahn-Hilliard equation. Arch. Rational Mech. Anal. 96 (1986), 339–357.Google Scholar
15Fife, P. C.. Models for phase separation and their mathematics. In Nonlinear Partial Differential Equations and Applications, eds Mimura, M. and Nishida, T. (Tokyo: Kinokuniya) (1992), 183–212.Google Scholar
16Freitas, P.. A nonlocal Sturm-Liouville problem. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 169–188.Google Scholar
17Fusco, G. and Hale, J. K.. Slow motion manifolds, dormant instability and singular perturbations. Dynamics Differential Equations 1 (1989), 75–94.CrossRefGoogle Scholar
18Grant, C. P.. Slow motion in one-dimensional Cahn-Morral systems (Preprint).Google Scholar
19Grinfeld, M., Furter, J. E. and Eilbeck, J. C.. A monotonicity theorem and its applications to stationary solutions of the phase field model. IMA J. Appl. Math. 49 (1992), 61–72; 50 (1993), 203–204.CrossRefGoogle Scholar
20Gunton, J. D. and Droz, M.. Introduction to the Theory of Metastable and Unstable States, Lecture Notes in Physics 183 (Berlin: Springer, 1983).Google Scholar
21Gurtin, M. E. and Matano, H.. On the structure of equilibrium phase transitions within the gradient theory of fluids. Quart. Appl. Math. 46 (1988), 301–317.Google Scholar
22Hale, J. K.. Asymptotic Behaviour of Dissipative Systems, Mathematical Surveys and Monographs 25 (Providence, R.I.: American Mathematical Society, 1988).Google Scholar
23Hearn, A. C.. Reduce User Manual, Version 3.3 (Santa Monica, CA: Rand Corporation, 1987).Google Scholar
24Mischaikow, K.. Global asymptotic dynamics of gradient-like bistable equations (Preprint).Google Scholar
25Modica, L.. The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987), 123–142.Google Scholar
26Modica, L. and Luckhaus, S.. The Gibbs-Thompson relation within the gradient theory of phase transitions. Arch. Rational Mech. Anal. 107 (1989), 71–83.Google Scholar
27Nicolaenko, B., Scheurer, B. and Temam, R.. Some global dynamical properties of a class of pattern formation equations. Comm. Partial Differential Equations 14 (1989), 245–297.CrossRefGoogle Scholar
28Novick-Cohen, A.. On Cahn-Hilliard type equations. Nonlinear Anal. 15 (1990), 797–814.Google Scholar
29Novick-Cohen, A. and Peletier, L. A.. Steady states of the one-dimensional Cahn-Hilliard equation. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 1071–1098.Google Scholar
30Novick-Cohen, A. and Segel, L. A.. Nonlinear aspects of the Cahn-Hilliard equation. Physica D 10 (1984), 277–298.Google Scholar
31Novick-Cohen, A. and Zheng, S.. Steady states for a non-local system of ODEs (in preparation).Google Scholar
32Penrose, O. and Fife, P. C.. Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43 (1990), 44–62.Google Scholar
33Rubinstein, J. and Sternberg, P.. Non-local reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48(1991), 249–264.Google Scholar
34Smoller, J. and Wasserman, A.. Global bifurcation of steady state solutions. J. Differential Equations 39 (1981), 269–290.Google Scholar
35Sternberg, P.. The effect of a singular perturbation on nonconvex variational problems. Arch. Rational Mech. Anal. 101 (1988), 209–260.Google Scholar
36Temam, R.. Infinite-dimensional dynamical systems in mechanics and physics, Applied Math. Sciences 68 (New York: Springer, 1988).Google Scholar
37Zheng, S.. Asymptotic behaviour of solutions to the Cahn-Hilliard equation. Applicable Anal. 23 (1986), 165–184.Google Scholar