# THE EXISTENCE OF GLOBAL ATTRACTOR FOR CONVECTIVE CAHN-HILLIARD EQUATION

• Zhao, Xiaopeng ;
• Liu, Bo
• Published : 2012.03.01
• 73 18

#### Abstract

In this paper, we consider the convective Cahn-Hilliard equation. Based on the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors, we prove that the convective Cahn-Hilliard equation possesses a global attractor in $H^k$($k\geq0$) space, which attracts any bounded subset of $H^k({\Omega})$ in the $H^k$-norm.

#### Keywords

attractor;convective Cahn-Hilliard equation;absorbing set

#### References

1. J. W. Cholewa and T. Dlotko, Global attractor for the Cahn-Hilliard system, Bull. Austral. Math. Soc. 49 (1994), no. 2, 277-292. https://doi.org/10.1017/S0004972700016348
2. T. Dlotko, Global attractor for the Cahn-Hilliard equation in $H^{2}$ and $H^{3}$, J. Differential Equations 113 (1994), no. 2, 381-393. https://doi.org/10.1006/jdeq.1994.1129
3. A. Eden and V. K. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett. 20 (2007), no. 4, 455-461. https://doi.org/10.1016/j.aml.2006.05.014
4. A. Eden and V. K. Kalantarov, 3D convective Cahn-Hilliard equation, Commun. Pure Appl. Anal. 6 (2007), no. 4, 1075-1086. https://doi.org/10.3934/cpaa.2007.6.1075
5. A. A. Golovin, S. H. Davism, and A. A. Nepomnyashchy, A convective Cahn-Hilliard model for the formation of facets and corners in crystal growth, Phys. D 122 (1998), no. 1-4, 202-230. https://doi.org/10.1016/S0167-2789(98)00181-X
6. K. H. Kwek, On the Cahn-Hilliard type equation, Ph. D. thesis, Georgia Institute of Technology, 1991.
7. D. Li and C. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity, J. Differential Equations 149 (1998), no. 2, 191-210. https://doi.org/10.1006/jdeq.1998.3429
8. C. Liu, On the convective Cahn-Hilliard equation, Bull. Pol. Acad. Sci. Math. 53 (2005), no. 3, 299-314. https://doi.org/10.4064/ba53-3-8
9. C. Liu, On the convective Cahn-Hillirad equation with degenerate mobility, J. Math. Anal. Appl. 344 (2008), no. 1, 124-144. https://doi.org/10.1016/j.jmaa.2008.02.027
10. T. Ma and S. H. Wang, Stability and Bifurcation of Nonlinear Evolution Equations, Science Press, Beijing, 2006.
11. A. Pazy, Semigroups of linear operators and applications to partial differential equations, in: Appl. Math. Sci., vol. 44, Springer-Verlag, 1983.
12. A. Podolny, M. A. Zaks, B. Y. Rubinstein, A. A. Golovin, and A. A. Nepomnyashchy, Dynamics of domain walls governed by the convective Cahn-Hilliard equation, Phys. D 201 (2005), no. 3-4, 291-305. https://doi.org/10.1016/j.physd.2005.01.003
13. L. Song, Y. He, and Y. Zhang, The existence of global attractors for semilinear parabolic equation in $H^{k}$ space, Nonlinear Anal. 68 (2008), no. 11, 3541-3549. https://doi.org/10.1016/j.na.2007.03.045
14. L. Song, Y. Zhang, and T. Ma, Global attractor of the Cahn-Hilliard equation in $H^{k}$ spaces, J. Math. Anal. Appl. 355 (2009), no. 1, 53-62. https://doi.org/10.1016/j.jmaa.2009.01.035
15. L. Song, Y. Zhang, and T. Ma, Global attractor of a modied Swift-Hohenberg equation in $H^{k}$ space, Nonlinear Anal. 72 (2010), no. 1, 183-191. https://doi.org/10.1016/j.na.2009.06.103
16. R. Temam, Innite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
17. S. J. Watson, F. Otto, B. Y. Rubinstein, and S. H. Davis, Coarsening dynamics of the convective Cahn-Hilliard equations, Phys. D 178 (2003), no. 3-4, 127-148. https://doi.org/10.1016/S0167-2789(03)00048-4
18. H. Wu and S. M. Zheng, Global attractor for the 1-D thin lm equation, Asymptot. Anal. 51 (2007), no. 2, 101-111.
19. M. A. Zarks, A. Podolny, A. A. Nepomnyashchy, and A. A. Golovin, Periodic stationary patterns governed by a convective Cahn-Hilliard equation, SIAM J. Appl. Math. 66 (2005), no. 2, 700-720. https://doi.org/10.1137/040615766
20. X. Zhao and C. Liu, Global attractor for the convective Cahn-Hilliard equation, Bull. Pol. Acad. Sci. Math. 58 (2010), no. 2, 117-127. https://doi.org/10.4064/ba58-2-3

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