Abstract
The unique global solvability of the initial-boundary value problem (1)–(3) is proved for the system of phase field equations (1), (2). It is shown that the problem (1)–(3) generates a continuous compact semigroup Vt, t>0, for which there exists a minimal global B-attractor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Caginalp, “An analysis of a phase field model of a free boundary,” Arch. Rational Mech. Anal.92, 205–245 (1986).
G. Caginalp and S. Hastings, “Properties of some ordinary differential equations related to free boundary problems,” Proc. Roy. Soc. Edinburgh,104A, 217–234 (1986).
C. M. Elliott and Song Mu Zheng, “Global existence and stability of solutions to the phase field equations,” in: Free Boundary Value Problems (Oberwolfach, 1989), Internat. Ser. Numer. Math., 95, BirkhÄuser, Basel (1990), pp. 46–58.
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence (1968).
O. A. Ladyzhenskaya, “On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations,” Usp. Mat. Nauk,42, No. 6, 25–60 (1987).
O. A. Ladyzhenskaya, “Some complements and corrections to my papers on the theory of attractors for abstracts semigroups,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,182, 102–112 (1990).
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations [in Russian], Nauka, Moscow (1989).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 188, pp. 70–86, 1991.
Rights and permissions
About this article
Cite this article
Kalantarov, V.K. On the minimal global attractor of a system of phase field equations. J Math Sci 70, 1767–1777 (1994). https://doi.org/10.1007/BF02149147
Issue Date:
DOI: https://doi.org/10.1007/BF02149147