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Volumn 17, Issue 3, 2007, Pages 411-437

On the hyperbolic relaxation of the cahn-hilliard equation in 3D: Approximation and long time behaviour

Author keywords

Approximation of global attractors; Cahn Hilliard equation; Damped semilinear wave equation; Generalized semiflows; Global attractor; Hyperbolic relaxation

Indexed keywords


EID: 33947705269     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202507001978     Document Type: Article
Times cited : (43)

References (42)
  • 1
    • 0042887224 scopus 로고
    • Maximal attractors of semigroups corresponding to evolutionary differential equations
    • A. V. Babin and M. I. Vishik, Maximal attractors of semigroups corresponding to evolutionary differential equations, Mat. Sb. (N.S.) 126 (1985) 397-419.
    • (1985) Mat. Sb. (N.S.) , vol.126 , pp. 397-419
    • Babin, A.V.1    Vishik, M.I.2
  • 2
    • 84966232265 scopus 로고
    • Measurability and continuity conditions for nonlinear evolutionary processes
    • J. M. Ball, Measurability and continuity conditions for nonlinear evolutionary processes, Proc. Amer. Math. Soc. 55 (1976) 353-358.
    • (1976) Proc. Amer. Math. Soc , vol.55 , pp. 353-358
    • Ball, J.M.1
  • 3
    • 0000938708 scopus 로고    scopus 로고
    • Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations
    • J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997) 475-502.
    • (1997) J. Nonlinear Sci , vol.7 , pp. 475-502
    • Ball, J.M.1
  • 4
    • 2142768721 scopus 로고    scopus 로고
    • Global attractors for damped semilinear wave equations
    • J. M. Ball, Global attractors for damped semilinear wave equations, Disc. Cont. Dyn. Syst. 10 (2004) 31-52.
    • (2004) Disc. Cont. Dyn. Syst , vol.10 , pp. 31-52
    • Ball, J.M.1
  • 5
    • 16844386846 scopus 로고    scopus 로고
    • The Neumann boundary problem for a nonlocal Cahn-Hilliard equation
    • P. W. Bates and J. Han, The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Diff. Eqs. 212 (2005) 235-277.
    • (2005) J. Diff. Eqs , vol.212 , pp. 235-277
    • Bates, P.W.1    Han, J.2
  • 6
    • 20644432208 scopus 로고    scopus 로고
    • Existence and continuity of uniform exponential attractors for a singular perturbation of a generalized Cahn-Hilliard equation
    • A. Bonfoh, Existence and continuity of uniform exponential attractors for a singular perturbation of a generalized Cahn-Hilliard equation, Asympt. Anal. 43 (2005) 233-247.
    • (2005) Asympt. Anal , vol.43 , pp. 233-247
    • Bonfoh, A.1
  • 7
    • 43449102981 scopus 로고
    • On spinodal decomposition
    • J. W. Calm, On spinodal decomposition, Acta Metall 9 (1961) 795-801.
    • (1961) Acta Metall , vol.9 , pp. 795-801
    • Calm, J.W.1
  • 8
    • 33746012315 scopus 로고
    • Free energy of a nonuniform system. Interfacial free energy
    • J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Interfacial free energy, J. Chem. Phys. 28 (1958) 258-267.
    • (1958) J. Chem. Phys , vol.28 , pp. 258-267
    • Cahn, J.W.1    Hilliard, J.E.2
  • 9
    • 0346448448 scopus 로고    scopus 로고
    • Global attractor and its perturbations for a dissipative hyperbolic equation
    • V. V. Chepyzhov and M. I. Vishik, Global attractor and its perturbations for a dissipative hyperbolic equation, Russ. J. Math. Phys. 8 (2001) 251-266.
    • (2001) Russ. J. Math. Phys , vol.8 , pp. 251-266
    • Chepyzhov, V.V.1    Vishik, M.I.2
  • 11
    • 33947664539 scopus 로고    scopus 로고
    • P. Colli, P. Krejci, E. Rocca and J. Sprekels, Nonlinear evolution inclusion arising from phase change models, preprint IMATI (CNR), PV (2004) 1-28, to appear in Czech. Math. J.
    • P. Colli, P. Krejci, E. Rocca and J. Sprekels, Nonlinear evolution inclusion arising from phase change models, preprint IMATI (CNR), PV (2004) 1-28, to appear in Czech. Math. J.
  • 12
    • 84974767695 scopus 로고
    • A singular perturbation of the Cahn-Hilliard equation
    • A. Debussche, A singular perturbation of the Cahn-Hilliard equation, Asymp. Anal. 4 (1991) 161-185.
    • (1991) Asymp. Anal , vol.4 , pp. 161-185
    • Debussche, A.1
  • 13
    • 4644317944 scopus 로고    scopus 로고
    • Exponential attractors for a singularly perturbed Cahn-Hilliard system
    • M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr. 272 (2004) 11-31.
    • (2004) Math. Nachr , vol.272 , pp. 11-31
    • Efendiev, M.1    Miranville, A.2    Zelik, S.3
  • 14
    • 0000220290 scopus 로고
    • Global attractors for semilinear damped wave equations with supercritical exponent
    • E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Diff. Eqns. 116 (1995) 431-447.
    • (1995) J. Diff. Eqns , vol.116 , pp. 431-447
    • Feireisl, E.1
  • 15
    • 54749104300 scopus 로고    scopus 로고
    • P. C. Fife, Models for phase separation and their mathematics, Electron. J. Diff. Eqns. (electronic), (2000), No. 48.
    • P. C. Fife, Models for phase separation and their mathematics, Electron. J. Diff. Eqns. (electronic), (2000), No. 48.
  • 16
    • 0142119425 scopus 로고    scopus 로고
    • On a nonlocal phase separation model
    • H. Gajewski and K. Zacharias, On a nonlocal phase separation model, J. Math. Anal. Appl. 286 (2003) 11-31.
    • (2003) J. Math. Anal. Appl , vol.286 , pp. 11-31
    • Gajewski, H.1    Zacharias, K.2
  • 17
    • 0035959403 scopus 로고    scopus 로고
    • Phase-field models with relaxation of the diffusion flux in nonequilibrium solidification of a binary system
    • P. Galenko, Phase-field models with relaxation of the diffusion flux in nonequilibrium solidification of a binary system, Phys. Lett. A 287 (2001) 190-197.
    • (2001) Phys. Lett. A , vol.287 , pp. 190-197
    • Galenko, P.1
  • 18
    • 27644595297 scopus 로고    scopus 로고
    • On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation
    • S. Gatti, M. Grasselli, A. Miranville and V. Pata, On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation, J. Math. Anal,. Appl. 312 (2005) 230-247.
    • (2005) J. Math. Anal,. Appl , vol.312 , pp. 230-247
    • Gatti, S.1    Grasselli, M.2    Miranville, A.3    Pata, V.4
  • 20
    • 0032366877 scopus 로고    scopus 로고
    • Attractors for families of processes in weak topologies of Banach spaces
    • F. Gazzola and M. Sardella, Attractors for families of processes in weak topologies of Banach spaces, Disc. Cont. Dyn. Syst. 4 (1998) 455-466.
    • (1998) Disc. Cont. Dyn. Syst , vol.4 , pp. 455-466
    • Gazzola, F.1    Sardella, M.2
  • 21
    • 0001158204 scopus 로고    scopus 로고
    • Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance
    • M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D 92 (1996) 178-192.
    • (1996) Phys. D , vol.92 , pp. 178-192
    • Gurtin, M.E.1
  • 22
    • 33947644227 scopus 로고    scopus 로고
    • J. K. Hale, Asymptotic behaviour and dynamics in infinite dimensions, in Nonlinear Differential Equations (Granada, 1984) of Res. Notes in Math., 132 (Pitman, 1985), pp. 1-42.
    • J. K. Hale, Asymptotic behaviour and dynamics in infinite dimensions, in Nonlinear Differential Equations (Granada, 1984) of Res. Notes in Math., Vol. 132 (Pitman, 1985), pp. 1-42.
  • 23
    • 84948497651 scopus 로고
    • Upper semicontinuity of attractors for approximations of semigroups and partial differential equations
    • J. K. Hale, X. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp. 50 (1988) 89-123.
    • (1988) Math. Comp , vol.50 , pp. 89-123
    • Hale, J.K.1    Lin, X.2    Raugel, G.3
  • 24
    • 0000610074 scopus 로고
    • Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation
    • J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Diff. Egns. 73 (1988) 197-214.
    • (1988) J. Diff. Egns , vol.73 , pp. 197-214
    • Hale, J.K.1    Raugel, G.2
  • 25
    • 0013414545 scopus 로고
    • Compact attractors for weak dynamical systems
    • J. K. Hale and N. Stavrakakis, Compact attractors for weak dynamical systems, Appl. Anal. 26 (1988) 271-287.
    • (1988) Appl. Anal , vol.26 , pp. 271-287
    • Hale, J.K.1    Stavrakakis, N.2
  • 26
    • 0037953627 scopus 로고
    • Minimal compact global attractor for a damped semilinear wave equation
    • L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Diff. Eqns. 20 (1995) 1303-1323.
    • (1995) Comm. Partial Diff. Eqns , vol.20 , pp. 1303-1323
    • Kapitanski, L.1
  • 28
    • 33947678797 scopus 로고    scopus 로고
    • J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, I. (Springer-Verlag, 1972) (translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181).
    • J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I. (Springer-Verlag, 1972) (translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181).
  • 29
    • 84974178842 scopus 로고
    • Compact attractor for a nonlinear wave equation with critical exponent
    • O. Lopes, Compact attractor for a nonlinear wave equation with critical exponent, Proc. Roy. Soc. Edinburgh Sec. A 115 (1990) 61-64.
    • (1990) Proc. Roy. Soc. Edinburgh Sec. A , vol.115 , pp. 61-64
    • Lopes, O.1
  • 30
    • 0032474671 scopus 로고    scopus 로고
    • Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. I. Probability and wavelength estimate
    • S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. I. Probability and wavelength estimate, Comm. Math. Phys. 195 (1998) 435-464.
    • (1998) Comm. Math. Phys , vol.195 , pp. 435-464
    • Maier-Paape, S.1    Wanner, T.2
  • 31
    • 0040191753 scopus 로고    scopus 로고
    • Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics
    • S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Arch. Rat. Mech. Anal. 151 (2000) 187-219.
    • (2000) Arch. Rat. Mech. Anal , vol.151 , pp. 187-219
    • Maier-Paape, S.1    Wanner, T.2
  • 32
    • 0034562774 scopus 로고    scopus 로고
    • On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions
    • V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal. 8 (2000) 375-403.
    • (2000) Set-Valued Anal , vol.8 , pp. 375-403
    • Melnik, V.S.1    Valero, J.2
  • 33
    • 2542500465 scopus 로고    scopus 로고
    • Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations
    • A. Milani and S. Zheng, Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations, Nonlinear Anal. 57 (2004) 843-877.
    • (2004) Nonlinear Anal , vol.57 , pp. 843-877
    • Milani, A.1    Zheng, S.2
  • 34
    • 10944240024 scopus 로고    scopus 로고
    • Global attractors for singular perturbations of the Cahn-Hilliard equations
    • A. Milani and S. Zheng, Global attractors for singular perturbations of the Cahn-Hilliard equations, J. Diff. Eqns. 209 (2005) 101-139.
    • (2005) J. Diff. Eqns , vol.209 , pp. 101-139
    • Milani, A.1    Zheng, S.2
  • 35
    • 1642337137 scopus 로고    scopus 로고
    • Robust exponential attractors for Cahn-Hilliard type equations with singular potentials
    • A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Meth. Appl. Sci. 27 (2004) 545-582.
    • (2004) Math. Meth. Appl. Sci , vol.27 , pp. 545-582
    • Miranville, A.1    Zelik, S.2
  • 36
    • 15744377181 scopus 로고    scopus 로고
    • Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions
    • A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci. 28 (2005) 709-735.
    • (2005) Math. Methods Appl. Sci , vol.28 , pp. 709-735
    • Miranville, A.1    Zelik, S.2
  • 37
    • 2142698531 scopus 로고    scopus 로고
    • Attractors for noncompact nonautonomous systems via energy equations
    • I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discr. Cont. Dyn. Syst. 10 (2004) 473-496.
    • (2004) Discr. Cont. Dyn. Syst , vol.10 , pp. 473-496
    • Moise, I.1    Rosa, R.2    Wang, X.3
  • 38
    • 0001192116 scopus 로고    scopus 로고
    • The Cahn-Hilliard equation: Mathematical and modeling perspectives
    • A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl. 8 (1998) 965-985.
    • (1998) Adv. Math. Sci. Appl , vol.8 , pp. 965-985
    • Novick-Cohen, A.1
  • 39
    • 0141519753 scopus 로고    scopus 로고
    • Attracteurs compacts pour des problèmes d'évolution sans unicité
    • A. Ould Elmounir and F. Simondon, Attracteurs compacts pour des problèmes d'évolution sans unicité, Ann. Fac. Sci. Toulouse Math. (6), 9 (2000) 631-654.
    • (2000) Ann. Fac. Sci. Toulouse Math. (6) , vol.9 , pp. 631-654
    • Ould Elmounir, A.1    Simondon, F.2
  • 40
    • 33947711307 scopus 로고    scopus 로고
    • R. Rossi, A. Segatti and U. Stefanelli, Attractors for gradient flows of non-convex functionals and applications, Preprint IMATI (CNR), No. 5 PV, 2006.
    • R. Rossi, A. Segatti and U. Stefanelli, Attractors for gradient flows of non-convex functionals and applications, Preprint IMATI (CNR), No. 5 PV, 2006.


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