A phase field system with memory: Global existence

Journal of Integral Equations and Applications

Volumn 14, Issue 1, 2002, Pages 73-107

  Novick Cohen, Amy ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Galerkin methods; Integro differential equations; Memory; Phase field equations; Phase transitions

Indexed keywords

EID: 21244446998     PISSN: 08973962     EISSN: None     Source Type: Journal    
DOI: 10.1216/jiea/1031315435     Document Type: Article

References (32)

1. A.B. Abramovitz and A. Novick-Cohen, Motion by mean curvature with memory, in preparation.
2. S. Aizicovici and V. Barbu, Existence and asymptotic results for a system of integro-partial differential equations, NoDea 3 (1996), 1 18.
3. P.W. Bates and S.M. Zheng, Inertial manifolds and inertial sets for the phasefield equations, J. Dynamical Differential Equations 4 (1992), 375 398.
4. D. Brochet, D. Hilhorst and X. Chen, Finite-dimensional exponential attractor for the phase field model, Appl. Anal. 49 (1993), 197 212.
5. D. Brochet, D. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase-field model, Advances in Differential Equations 1 (1996), 547 578.
6. M. Brokate and J. Sprekels, Hysteresis and phase transitions, Springer-Verlag, Berlin, 1996.
7. G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational. Mech. Anal. 92 (1986), 205 245.
8. G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, Hele-Shaw, and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math. 44 (1990), 77 94.
9. G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math. 9 (1998), 417 445.
10. G. Caginalp and P.C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math. 48 (1988), 506 518.
11. P. Colli and Ph. Laurençot, Existence and stabilization of solutions to the phase field model with memory, J. Integral Equations Appl. 10 (1998), 169 194.
12. P. Colli and Ph. Laurençot, Uniqueness of weak solutions to the phase-field model with memory, J. Math. Sci. Tokyo 5 (1998), 459 476.
13. C.M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations, in Free boundary value problems, (K.H. Hoffman and J. Sprekels, eds.), Internat. Ser. Numer. Math., 95, Birkhauser, Basel, 1990, 46 58.
14. G.J. Fix, Phase field methods for free boundaries, in Free boundary problems theory and applications, Pitman Res. Notes Math. Ser., 77, Longman, London, 1983, 580 589.
15. C. Giorgi, M. Grasselli and V. Pata, Well-posedness and longtime behavior of the phase-field model with memory in a history space setting, Quart. Appl. Math., in press.
16. C. Giorgi, M. Grasselli and V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J. 48 (1999), 1395 1445.
17. M. Grasselli, On a phase-field model with memory, in Proceedings of the international conference on free boundary problems: theory and application, (Chiba, Japan 1999), ed.: N. Kenmochi, I. Tokyo: Gakkotosho, GAKUTO Int. Ser., Math. Sci. Appl. 13 (2000), 89 102.
18. M. Grasselli and H.G. Rotstein, Hyperbolic phase-field dynamics with memory, J. Math. Anal. Appl. 261 (2001), 205 230.
19. M. Grinfeld and A. Novick-Cohen, Some remarks on thermodynamic consistency and stability for a phase field model with memory, in preparation.
20. G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations, Encyclopedia ofMathematics and Its Applications, Cambridge University Press, 1990.
21. M.E. Gurtin and A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), 113 126.
22. B.H. Halperin, P.C. Hohenberg and S.-K. Ma, Renormalization group methods for critical dynamics, I. Recursion relations and effects of energy of condensation, Phys. Rev. B. 10 (1974), 139 153.
23. J.S. Langer, Theory of the condensation point, Ann. Phys. 41 (1967), 108 157.
24. R. Miller, Nonlinear Volterra integral equations, W.A. Benjamin, Inc., Menlo Park, CA, 1971.
25. A. Novick-Cohen, Conserved phase field equations with memory, in Curvature Flows and Related Topics, Levico, 1994, eds. A. Damlamian, J. Spruck, A. Visintin, Tokyo: Gakkotosho, Gakuto Int. Ser., Math. Sci. Appl. 5 (1995), 179 197.
26. A. Novick-Cohen, A phase-field system with memory: global existence, in Proceedings of the RIMS Meeting on Free Boundary Problems, (Kyoto, 2000), ed. H. Ishii, S-urikaisekikenky-usho Koky-uroku, 1210 (2001), 129 141.
27. H.G. Rotstein, Phase transition dynamics with memory, Ph.D. thesis, Department of Mathematics, Technion-IIT, 1998.
28. H.G. Rotstein, S. Brandon and A. Novick-Cohen, Hyperbolic flow by mean curvature, J. Crystal Growth 198/199 (1999), 1256 1261.
29. H.G. Rotstein, S. Brandon, A. Novick-Cohen and A. Nepomnyashchy, Phase field equations with memory: The hyperbolic case, SIAM J. Appl. Math., 62, (2001), 264 282.
30. H.G. Rotstein, A.I. Domoshnitsky and A.A. Nepomnyashchy, Front motion for phase transitions in systems with memory, Physica D 146 (2000), 137 149.
31. p(0, T;B), Ann. Mat. Pura Appl. 146 (1987), 65 97.
32. R. Temam, Navier-Stokes equations, North-Holland, Amsterdam, 1977.