Inertial manifolds and inertial sets for the phase-field equations

  • Peter W. Bates
  • Songmu Zheng


The phase-field system is a mathematical model of phase transition, coupling temperature with a continuous order parameter which describes degree of solidification. The flow induced by this system is shown to be smoothing in H1×L2 and a global attractor is shown to exist. Furthermore, in low-dimensional space, the flow is essentially finite dimensional in the sense that a strongly attracting finite-dimensional manifold (or set) exists.

Key words

Parabolic attractor infinite-dimensional dynamical system global existence and regularity 


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  1. [AB]
    N. Alikakos and P. Bates, On the singular limit in a phase field model.Ann. I.H.P. Anal. Nonlin. 6, 141–178 (1988).Google Scholar
  2. [BF]
    P. Bates and P. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening.Physica D 43, 335–348 (1990).Google Scholar
  3. [C1]
    G. Caginalp, An analysis of a phase field model of a free boundary.Arch. Rat. Mech. Anal. 92, 205–245 (1986).Google Scholar
  4. [C2]
    G. Caginalp, Solidification problems as systems of nonlinear differential equations.Lect. Appl. Math. 23, Am. Math. Soc., Providence, R.I., 1986, pp. 247–269.Google Scholar
  5. [C3]
    G. Caginalp, Phase field models: Some conjectures and theorems for their sharp initerface limits.Proceedings, Conference on Free Boundary Problems, Irsee, 1987.Google Scholar
  6. [C4]
    G. Caginalp, Mathematical models of phase boundaries.Material Instabilities in Continuium Mechanics and Related Mathematical Problems, Clarendon Press, Oxford, 1988, pp. 35–52.Google Scholar
  7. [CF]
    G. Caginalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries.SIAM J. Appl. Math. 48, 506–518 (1988).Google Scholar
  8. [CH]
    R. Courant and D. Hilbert,Methods of Mathematical Physics, Intersciences, New York, 1953.Google Scholar
  9. [CL]
    J. B. Collins and H. Levine, Diffuse interface model of diffusion-limited crystal growth.Phys. Rev. B 31, 6119–6122; 33, 2020E (1985).Google Scholar
  10. [EFNT1]
    A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Ensembles inertiels pour des équations d'évolution dissipatives.C.R. Acad. Sci. Paris 310, Ser. 1, 559–562 (1990).Google Scholar
  11. [EFNT2]
    A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Inertial sets for dissipative evolution equations.Appl. Math. Lett, (in press).Google Scholar
  12. [EMN]
    A. Eden, A. J. Milani, and B. Nicolaenko, Finite dimensional exponential attrac-tors for semilinear wave equations with damping. IMA preprint Series No. 693, 1990.Google Scholar
  13. [EZ]
    C. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations. InFree Boundary Problems, K. H. Hoffmann and J. Sprekels (eds.), International Series of Numerical Mathematics, Vol. 95, Birkhauser Verlag, Basel, 1990, pp. 46–58.Google Scholar
  14. [F]
    P. C. Fife, Pattern dynamics for parabolic PDEs. Preprint, University of Utah, Salt Lake City, 1990.Google Scholar
  15. [FG]
    P. C. Fife and G. S. Gill, The phase-field description of mushy zones.Physica D 35, 267–275 (1989).Google Scholar
  16. [Fx]
    G. J. Fix, Phase field methods for free boundary problems. InFree Boundary Problems Theory and Applications, Pittman, London, 1983, pp. 580–589.Google Scholar
  17. [FST]
    C. Foias, G. Sell, and R. Teman, Inertial manifolds for nonlinear evolution equations.J. Diff. Eg. 73, 309–353 (1988).Google Scholar
  18. [H]
    D. Henry, Geometric theory of semilinear parabolic equations.Lect. Notes Math. 840, Springer-Verlag, New York, 1981.Google Scholar
  19. [HHM]
    B. I. Halperin, P. C. Hohenberg, and S.-K. Ma, Renormalization group methods for critical dynamics. I. Recursion relations and effects of energy conservation.Phys. Rev. B 10, 139–153 (1974).Google Scholar
  20. [K]
    R. Kobayashiet al., Videotape of solidification fronts and their instabilities for the phase field equation, University of Hiroshima, Hiroshima, 1990.Google Scholar
  21. [Kw]
    M. Kwak, Finite dimensional description of convective reaction-diffusion equations, Univ. Minn. AHPCRC Preprint 91-29, 1991.Google Scholar
  22. [L1]
    J. S. Langer, Theory of the condensation point.Ann. Phys. 41, 108–157 (1967).Google Scholar
  23. [L2]
    J. S. Langer, Models of pattern formation in first-order phase transitions. InDirections in Condensed Matter Physics, World Scientific, Singapore, 1986, pp. 164–186.Google Scholar
  24. [M]
    M. Miklavcic, A sharp condition for existence of an inertial manifold. Univ. Minn. IMA Preprint 604, 1990.Google Scholar
  25. [P]
    A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.Google Scholar
  26. [PF]
    O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions.Physica D 43, 44–62 (1990).Google Scholar
  27. [R]
    J. Richards, On the gap between numbers which are the sum of two squares.Adv. Math. 46, 1–2 (1982).Google Scholar
  28. [SY]
    G. R. Sell and Y. You, Inertial manifolds: The nonselfadjoint case. Univ. Minn. Preprint, 1990.Google Scholar
  29. [T]
    R. Temam,Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer-Verlag, New York, 1988.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Peter W. Bates
    • 1
    • 2
  • Songmu Zheng
    • 3
  1. 1.Department of MathematicsBrigham Young UniversityProvo
  2. 2.Currently visiting the Department of MathematicsUniversity of UtahSalt Lake City
  3. 3.Institute of MathematicsFudan UniversityShanghaiPRC

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