×

zbMATH — the first resource for mathematics

Slow-motion manifolds, dormant instability, and singular perturbations. (English) Zbl 0684.34055
Summary: If the coexistence of two phases at the transition temperature is kept under observation for a long time, then one observes that the system is not exactly in equilibrium and a very slow evolution driven by surface tension is taking place. Theoretically, one should eventually see a spatially homogeneous state, but the time for settling down is so long that what one actually observes is “motion towards a stable state”. The complexity of the spatial distribution of the two phases keeps decreasing but appears to be stable for very long periods of time with intermittent periods of fast motion when there are small inclusions of one of the two regions embedded in the other phase. For a simple reaction diffusion model, it is shown that this phenomenon can be explained by investigating the flow on the attractor and the unstable manifolds of equilibria.

MSC:
34D15 Singular perturbations of ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Angenent, S. (1988). The zeroset of a solution of a parabolic equation (preprint). · Zbl 0644.35050
[2] Benassi, A., and Fouque, J. P. (1988). Hydrodynamic limit for asymmetric simple exclusion processes.Ann. Probability 16 (to appear). · Zbl 0646.60038
[3] Brunovsky, P., and Fiedler, B. (1988a). Connecting orbits in scalar reaction diffusion equations. I. InDynamics Reported, Vol. 1, John Wiley, New York, pp. 57-89.
[4] Brunovsky, P., and Fiedler, B. (1988)b. Connecting orbits in scalar reaction diffusion equations. II.J. Differential Equations (to appear).
[5] Bongiorno, V., Scriven, L. E., and Davis, H. T. (1976). Molecular theory of fluid interfaces.J. Colloid Interface Sci. 57, 462-475.
[6] Carr, J., and Pego, R. L. (1988). Metastable patterns in solutions ofut, =? 2 uxx-f(u).Comm. Pure Appl. Math., (to appear).
[7] Casten, R. C., and Holland, C. J. (1978). Instability results for reaction diffusion equations with Neumann boundary conditions.J. Differential Equations 27, 266-273. · Zbl 0359.35039
[8] Fusco, G., and Oliva, W. M. (1988). Jacobi matrices and transversality.Proc. R. Soc. Edinburgh (to appear). · Zbl 0692.58019
[9] Gantmacher, F. R. (1959).The Theory of Matrices, Vol. 2, Chelsea, New York. · Zbl 0085.01001
[10] Gurtin, M. E. (1986a). On the two-phase Stefan problem with interfacial energy and entropy.Arch. Rational Mech. Anal. 96, 199-241. · Zbl 0654.73008
[11] Gurtin, M. E. (1986b). On phase transitions with bulk, interfacial, and boundary energy.Arch. Rational Mech. Anal. 96, 243-264.
[12] Hale, J. K. (1988). Asymptotic behavior of dissipative systems.Math. Surv. Monogr. 25. · Zbl 0642.58013
[13] Henry, D. (1983). Geometric theory of semilinear parabolic equations.Lect. Notes Math., Am. Math. Soc. 840, Springer-Verlag, New York.
[14] Henry, D. (1985). Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations.J. Differential Equations 53, 165-205. · Zbl 0572.58012
[15] Matano, H. (1979). Asymptotic behavior and stability of solutions of semilinear diffusion equations.Publ. Res. Inst. Math. Sci. 15, 401-458. · Zbl 0445.35063
[16] Matano, H. (1982). Nonincrease of lap-number of a solution for a one-dimensional semilinear parabolic equations.J. Fac. Sci. Univ. Tokyo [Sect. IA] 23, 401-441. · Zbl 0496.35011
[17] Novick-Cohen, A., and Segal, L. A. (1984). Nonlinear aspects of the Cahn-Hilliard equation.Physica D 10, 278-298.
[18] Presutti, E. (1987). Collective behavior of interacting particle systems. InProceedings of the First World Congress of the Bernoully Society, Tashkent, USSR, September 1986, Vol. 1, VNU Scientific Press, Utrecht, The Netherlands, pp. 295-413.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.