June  2012, 32(6): 2207-2221. doi: 10.3934/dcds.2012.32.2207

On dynamical behavior of viscous Cahn-Hilliard equation

1. 

Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072, China, China

Received  February 2011 Revised  October 2011 Published  February 2012

In this paper, we consider the initial and Dirichlet boundary value problem of the viscous Cahn-Hilliard equation with a general nonlinearity $f$, that is $$ d((1-\alpha)u-\alpha\Delta u)+(\Delta^2u-\Delta f(u))dt= 0, $$where $\alpha\in[0,1]$. Firstly, we establish the existence and continuity results on weak solutions and attractors to this problem. Secondly, we show the $\alpha$-uniform attractiveness of the attractors $A_\alpha$.
Citation: Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
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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