Inertial manifolds and inertial sets for the phase-field equations
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- N. Alikakos and P. Bates, On the singular limit in a phase field model.Ann. I.H.P. Anal. Nonlin. 6, 141–178 (1988).
- 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).
- G. Caginalp, An analysis of a phase field model of a free boundary.Arch. Rat. Mech. Anal. 92, 205–245 (1986).
- G. Caginalp, Solidification problems as systems of nonlinear differential equations.Lect. Appl. Math. 23, Am. Math. Soc., Providence, R.I., 1986, pp. 247–269.
- G. Caginalp, Phase field models: Some conjectures and theorems for their sharp initerface limits.Proceedings, Conference on Free Boundary Problems, Irsee, 1987.
- G. Caginalp, Mathematical models of phase boundaries.Material Instabilities in Continuium Mechanics and Related Mathematical Problems, Clarendon Press, Oxford, 1988, pp. 35–52.
- G. Caginalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries.SIAM J. Appl. Math. 48, 506–518 (1988).
- R. Courant and D. Hilbert,Methods of Mathematical Physics, Intersciences, New York, 1953.
- J. B. Collins and H. Levine, Diffuse interface model of diffusion-limited crystal growth.Phys. Rev. B 31, 6119–6122; 33, 2020E (1985).
- 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).
- A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Inertial sets for dissipative evolution equations.Appl. Math. Lett, (in press).
- 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.
- 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.
- P. C. Fife, Pattern dynamics for parabolic PDEs. Preprint, University of Utah, Salt Lake City, 1990.
- P. C. Fife and G. S. Gill, The phase-field description of mushy zones.Physica D 35, 267–275 (1989).
- G. J. Fix, Phase field methods for free boundary problems. InFree Boundary Problems Theory and Applications, Pittman, London, 1983, pp. 580–589.
- C. Foias, G. Sell, and R. Teman, Inertial manifolds for nonlinear evolution equations.J. Diff. Eg. 73, 309–353 (1988).
- D. Henry, Geometric theory of semilinear parabolic equations.Lect. Notes Math. 840, Springer-Verlag, New York, 1981.
- 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).
- R. Kobayashiet al., Videotape of solidification fronts and their instabilities for the phase field equation, University of Hiroshima, Hiroshima, 1990.
- M. Kwak, Finite dimensional description of convective reaction-diffusion equations, Univ. Minn. AHPCRC Preprint 91-29, 1991.
- J. S. Langer, Theory of the condensation point.Ann. Phys. 41, 108–157 (1967).
- J. S. Langer, Models of pattern formation in first-order phase transitions. InDirections in Condensed Matter Physics, World Scientific, Singapore, 1986, pp. 164–186.
- M. Miklavcic, A sharp condition for existence of an inertial manifold. Univ. Minn. IMA Preprint 604, 1990.
- A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
- 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).
- J. Richards, On the gap between numbers which are the sum of two squares.Adv. Math. 46, 1–2 (1982).
- G. R. Sell and Y. You, Inertial manifolds: The nonselfadjoint case. Univ. Minn. Preprint, 1990.
- R. Temam,Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer-Verlag, New York, 1988.
- Inertial manifolds and inertial sets for the phase-field equations
Journal of Dynamics and Differential Equations
Volume 4, Issue 2 , pp 375-398
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- infinite-dimensional dynamical system
- global existence and regularity
- Author Affiliations
- 1. Department of Mathematics, Brigham Young University, 84602, Provo, Utah
- 2. Currently visiting the Department of Mathematics, University of Utah, 84112, Salt Lake City, Utah
- 3. Institute of Mathematics, Fudan University, Shanghai, PRC