Our aim in this article is to study the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system. In particular, we prove the existence of an exponential attractor and, as a consequence, the existence of the global attractor with finite fractal dimension.
A.V.Babin and M.I.Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
2.
D.Brochet, D.Hilhorst and A.Novick-Cohen, Finite-dimensional exponential attractor for a model for order–disorder and phase separation, Appl. Math. Lett.7 (1994), 83–87. doi:10.1016/0893-9659(94)90118-X.
3.
J.W.Cahn and A.Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, Statistical Physics76 (1994), 877–909. doi:10.1007/BF02188691.
4.
R.Dal Passo, L.Giacomelli and A.Novick-Cohen, Existence for an Allen–Cahn/Cahn–Hilliard system with degenerate mobility, Interfaces and Free Boundaries1 (1999), 199–226.
5.
V.Derkach, A.Novick-Cohen and A.Vilenkin, Geometric interfacial motion: Coupling surface diffusion and mean curvature motion, submitted.
6.
A.Eden, C.Foias, B.Nicolaenko and R.Temam, Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Vol. 37, Wiley, New York, 1994.
7.
M.Efendiev, A.Miranville and S.Zelik, Exponential attractors for a nonlinear reaction-diffusion system in , C. R. Acad. Sci., Paris Sér. I330 (2000), 713–718. doi:10.1016/S0764-4442(00)00259-7.
8.
P.C.Millett, S.Rokkam, A.El-Azab, M.Tonks and D.Wolf, Void nucleation and growth in irradiated polycrystalline metals: A phase-field model, Modelling Simul. Mater. Sci. Eng.17 (2009), 0064003. doi:10.1088/0965-0393/17/6/064003.
9.
A.Miranville and S.Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in: Handbook of Differential Equations, Evolutionary Partial Differential Equations, C.M.Dafermos and M.Pokorny, eds, Vol. 4, Elsevier, Amsterdam, 2008, pp. 103–200.
10.
A.Novick-Cohen, Triple-junction motion for an Allen–Cahn/Cahn–Hilliard system, Physica D137 (2000), 1–24. doi:10.1016/S0167-2789(99)00162-1.
11.
A.Novick-Cohen and L.Peres Hari, Geometric motion for a degenerate Allen–Cahn/Cahn–Hilliard system: The partial wetting case, Physica D209 (2005), 205–235. doi:10.1016/j.physd.2005.06.028.
12.
S.Rokkam, A.El-Azab, P.Millett and D.Wolf, Phase field modeling of void nucleation and growth in irradiated metals, Modelling Simul. Mater. Sci. Eng.17 (2009), 0064002. doi:10.1088/0965-0393/17/6/064002.
13.
R.Temam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, 2nd edn, Springer, New York, 1997. doi:10.1007/978-1-4612-0645-3.
14.
M.R.Tonks, D.Gaston, P.C.Millett, D.Andrs and P.Talbot, An object-oriented finite element framework for multiphysics phase field simulations, Comput. Mater. Sci.51 (2012), 20–29. doi:10.1016/j.commatsci.2011.07.028.
15.
L.Wang, J.Lee, M.Anitescu, A.E.Azab, L.C.Mcinnes, T.Munson and B.Smith, A differential variational inequality approach for the simulation of heterogeneous materials, in: Proc. SciDAC 2011, 2011.
16.
Y.Xia, Y.Xu and C.W.Shu, Application of the local discontinuous Galerkin method for the Allen–Cahn/Cahn–Hilliard system, Commun. Comput. Phys.5 (2009), 821–835.
17.
C.Yang, X.C.Cai, D.E.Keyes and M.Pernice, NKS method for the implicit solution of a coupled Allen–Cahn/Cahn–Hilliard system, in: Proceedings of the 21th International Conference on Domain Decomposition Methods, 2012.
