On dynamical behavior of viscous Cahn-Hilliard equation

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

On dynamical behavior of viscous Cahn-Hilliard equation

Desheng Li ^1, and Xuewei Ju ^1,

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

Received February 2011 Revised October 2011 Published February 2012

Abstract
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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 $A_\alpha$.

Keywords: attractors, The $\alpha$-uniform dissipativeness, global solutions, the $\alpha$-uniform attractiveness of attractors..

Mathematics Subject Classification: 35Q99, 35B40, 35B4.

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," 2^nd 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," 2^nd 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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