Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D

Mathematical Models and Methods in Applied Sciences

Volumn 15, Issue 2, 2005, Pages 165-198

  Gatti, Stefania ^a   Grasselli, Maurizio ^b   Pata, Vittorino ^b   Miranville, Alain ^c  

^a UNIVERSITY OF FERRARA   (Italy)

^b POLITECNICO DI MILANO   (Italy)

^c UNIVERSITÉ DE POITIERS   (France)

Author keywords

Cahn Hilliard equations; Global attractors; Robust exponential attractors

Indexed keywords

EID: 12844276767     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202505000327     Document Type: Article

References (33)

1. A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (North-Holland, 1992).
2. 3, Discr. Cont. Dyn. Systems 7 (2001) 719-735.
3. J. W. Cahn, On spinodal decomposition, Acta Metall. 9 (1961) 795-801.
4. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958) 258-267.
5. A. Debussche, A singular perturbation of the Cahn-Hilliard equation, Asympt. Anal. 4 (1991) 161-185.
6. C. Dupaix, D. Hilhorst and I.N. Kostin, The viscous Cahn-Hilliard equation as a limit of the phase field model: lower semicontinuity of the attractor, J. Dyn. Diff. Eqns. 11 (1999) 333-353.
7. C. Dupaix, D. Hilhorst and Ph. Laurençot, Upper-semicontinuity of the attractor for a singularly perturbed phase field model, Adv. Math. Sci. Appl. 8 (1998) 115-143.
8. A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations (Masson, 1994).
9. M. Efendiev, H. Gajewski and A. Zelik, The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity, Adv. Diff. Eqns. 7 (2002) 1073-1100.
10. M. Efendiev, A. Miranville and A. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., to appear.
11. C. M. Elliott and I. K. Kostin, Lower semicontinuity of a non-hyperbolic attractor for the viscous Cahn-Hilliard equation, Nonlinearity 9 (1996) 687-702.
12. C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation II. Analysis, J. Diff. Eqns. 128 (1996) 387-414.
13. P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Disc. Cont. Dyn. Systems 10 (2004) 211-238.
14. P. C. Fife, Models for phase separation and their mathematics, Electron. J. Diff. Eqns. 48 (2000) 26.
15. P. Galenko, Phase-field models with relaxation of the diffusion flux in nonequilibrium solidification of a binary system, Phys. Lett. A287 (2001) 190-197.
16. H. Garcke, B. Niethammer, M. Rumpf and W. Weikard, Transient coarsening behavior in the Cahn-Hilliard equation, Acta Mat. 51 (2003) 2823-2830.
17. M. Grasselli, A. Miranville, V. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, submitted.
18. M. Grasselli and V. Pata, Upper semicontinuous attractor for a hyperbolic phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J. 50 (2001) 197-214.
19. M. Grasselli and V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure Appl. Anal. 3 (2004) 849-881.
20. M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D92 (1996) 178-192.
21. J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Diff. Eqns. 730 (1988) 1281-1308.
22. I. N. Kostin, Rate of attraction to a non-hyperbolic attractor, Asympt. Anal. 16 (1998) 203-222.
23. L. A. Lyusternik and M. I. Vishik, Regular degeneration and boundary layer for linear differential equations with small parameter, Amer. Math. Soc. Trans. 20 (1962) 239-364.
24. G. B. McFadden, Phase-field models of solidification, Contemp. Math. 306 (2002) 107-145.
25. A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equation with singular potential, Math. Meth. Appl. Sci. 27 (2004) 545-582.
26. A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material Instabilities in Continuum Mechanics (Edinburgh, 1985-1986), ed. J. M. Ball (Oxford University Press, 1988), pp. 329-342.
27. A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl. 8 (1998) 965-985.
28. G. R. Sell and Y. You, Dynamics of Evolutionary Equations (Springer-Verlag, 2002).
29. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Springer-Verlag, 1988).
30. J. D. van der Waals, On the continuity of the gaseous and liquid states, translated from the Dutch edition and with an introduction by J. S. Rowlinson, Studies in Statistical Mechanics, XIV (North-Holland, 1988).
31. J. D. van der Waals, On the continuity of the gaseous and liquid states, translated from the Dutch edition and with an introduction by J. S. Rowlinson, Studies in Statistical Mechanics, XIV (North-Holland, 1988).
32. S. Zheng and A. Milani, Global Attractors for singular perturbations of the Cahn-Hilliard equations, preprint, 2003.
33. S. Zheng and A. Milani, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations, Nonlinear Anal. 57 (2004) 843-877.