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On dynamical behavior of viscous Cahn-Hilliard equation
1. | Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072, China, China |
References:
[1] |
N. D. Alikakos, P. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential Equations, 90 (1991), 81-135.
doi: 10.1016/0022-0396(91)90163-4. |
[2] |
P. Bates and P. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time-scales for coarsening, Phys. D, 43 (1990), 335-348.
doi: 10.1016/0167-2789(90)90141-B. |
[3] |
F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. I. Computations, Nonliearity, 8 (1995), 131-160.
doi: 10.1088/0951-7715/8/2/002. |
[4] |
A. N. Carvalho and T. Dlotko, Dynamics of the viscous Cahn-Hilliard equation, J. Math. Anal. Appl., 344 (2008), 703-725.
doi: 10.1016/j.jmaa.2008.03.020. |
[5] |
A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514.
doi: 10.1016/0362-546X(94)00205-V. |
[6] |
T. Dlotko, On the Cahn-Hilliard equation with a logarithmic free energy $H^2$ and $H^3$, J. Differential Equations, 113 (1994), 381-393.
doi: 10.1006/jdeq.1994.1129. |
[7] |
C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differeential Equations, 128 (1996), 387-414.
doi: 10.1006/jdeq.1996.0101. |
[8] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.
doi: 10.1002/mana.200310186. |
[9] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. |
[10] |
J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. (4), 154 (1989), 281-326.
doi: 10.1007/BF01790353. |
[11] |
J. K. Hale, X.-B. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123.
doi: 10.1090/S0025-5718-1988-0917820-X. |
[12] |
J. K. Hale, Dynamics of numerical approximations, Appl. Math. Comput., 89 (1998), 5-15.
doi: 10.1016/S0096-3003(97)81644-X. |
[13] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[14] |
D. S. Li and P. E. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasg. Math. J., 46 (2004), 131-141.
doi: 10.1017/S0017089503001605. |
[15] |
D. S. Li and C. K. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations, 149 (1998), 191-210.
doi: 10.1006/jdeq.1998.3429. |
[16] |
D. S. Li and X. X. Zhang, Strongly positively-invariant attractor for periodic processes, J. Math. Anal. Appl., 241 (2000), 10-29.
doi: 10.1006/jmaa.1999.6499. |
[17] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[18] |
B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations, 14 (1989), 245-297. |
[19] |
A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in "Material Instabilities in Continuum Mechanics" (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, (1988), 329-342. |
[20] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. |
[21] |
R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
show all references
References:
[1] |
N. D. Alikakos, P. W. Bates and G. Fusco, Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differential Equations, 90 (1991), 81-135.
doi: 10.1016/0022-0396(91)90163-4. |
[2] |
P. Bates and P. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time-scales for coarsening, Phys. D, 43 (1990), 335-348.
doi: 10.1016/0167-2789(90)90141-B. |
[3] |
F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. I. Computations, Nonliearity, 8 (1995), 131-160.
doi: 10.1088/0951-7715/8/2/002. |
[4] |
A. N. Carvalho and T. Dlotko, Dynamics of the viscous Cahn-Hilliard equation, J. Math. Anal. Appl., 344 (2008), 703-725.
doi: 10.1016/j.jmaa.2008.03.020. |
[5] |
A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy, Nonlinear Anal., 24 (1995), 1491-1514.
doi: 10.1016/0362-546X(94)00205-V. |
[6] |
T. Dlotko, On the Cahn-Hilliard equation with a logarithmic free energy $H^2$ and $H^3$, J. Differential Equations, 113 (1994), 381-393.
doi: 10.1006/jdeq.1994.1129. |
[7] |
C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differeential Equations, 128 (1996), 387-414.
doi: 10.1006/jdeq.1996.0101. |
[8] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.
doi: 10.1002/mana.200310186. |
[9] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. |
[10] |
J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. (4), 154 (1989), 281-326.
doi: 10.1007/BF01790353. |
[11] |
J. K. Hale, X.-B. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123.
doi: 10.1090/S0025-5718-1988-0917820-X. |
[12] |
J. K. Hale, Dynamics of numerical approximations, Appl. Math. Comput., 89 (1998), 5-15.
doi: 10.1016/S0096-3003(97)81644-X. |
[13] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[14] |
D. S. Li and P. E. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasg. Math. J., 46 (2004), 131-141.
doi: 10.1017/S0017089503001605. |
[15] |
D. S. Li and C. K. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations, 149 (1998), 191-210.
doi: 10.1006/jdeq.1998.3429. |
[16] |
D. S. Li and X. X. Zhang, Strongly positively-invariant attractor for periodic processes, J. Math. Anal. Appl., 241 (2000), 10-29.
doi: 10.1006/jmaa.1999.6499. |
[17] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[18] |
B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations, 14 (1989), 245-297. |
[19] |
A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in "Material Instabilities in Continuum Mechanics" (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, (1988), 329-342. |
[20] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. |
[21] |
R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
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