THE EXISTENCE OF GLOBAL ATTRACTOR FOR CONVECTIVE CAHN-HILLIARD EQUATION

Title & Authors
THE EXISTENCE OF GLOBAL ATTRACTOR FOR CONVECTIVE CAHN-HILLIARD EQUATION
Zhao, Xiaopeng; Liu, Bo;

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 $\small{H^k}$($\small{k\geq0}$) space, which attracts any bounded subset of $\small{H^k({\Omega})}$ in the $\small{H^k}$-norm.
Keywords
attractor;convective Cahn-Hilliard equation;absorbing set;
Language
English
Cited by
1.
GLOBAL ATTRACTOR FOR COUPLED TWO-COMPARTMENT GRAY-SCOTT EQUATIONS,;;

대한수학회보, 2013. vol.50. 1, pp.143-159
2.
SPECTRAL APPROXIMATIONS OF ATTRACTORS FOR CONVECTIVE CAHN-HILLIARD EQUATION IN TWO DIMENSIONS,;

대한수학회보, 2015. vol.52. 5, pp.1445-1465
1.
The convective viscous Cahn–Hilliard equation: Exact solutions, European Journal of Applied Mathematics, 2016, 27, 01, 42
2.
Solutions for a fourth-order nonlinear parabolic equation describing Marangoni convection, Applicable Analysis, 2016, 95, 9, 2081
3.
GLOBAL ATTRACTOR FOR COUPLED TWO-COMPARTMENT GRAY-SCOTT EQUATIONS, Bulletin of the Korean Mathematical Society, 2013, 50, 1, 143
4.
SPECTRAL APPROXIMATIONS OF ATTRACTORS FOR CONVECTIVE CAHN-HILLIARD EQUATION IN TWO DIMENSIONS, Bulletin of the Korean Mathematical Society, 2015, 52, 5, 1445
5.
Fourier Spectral Approximation to Global Attractor for 2D Convective Cahn–Hilliard Equation, Bulletin of the Malaysian Mathematical Sciences Society, 2016
6.
Optimal Control for the Convective Cahn–Hilliard Equation in 2D Case, Applied Mathematics & Optimization, 2014, 70, 1, 61
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.

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.

3.
A. Eden and V. K. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett. 20 (2007), no. 4, 455-461.

4.
A. Eden and V. K. Kalantarov, 3D convective Cahn-Hilliard equation, Commun. Pure Appl. Anal. 6 (2007), no. 4, 1075-1086.

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.

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.

8.
C. Liu, On the convective Cahn-Hilliard equation, Bull. Pol. Acad. Sci. Math. 53 (2005), no. 3, 299-314.

9.
C. Liu, On the convective Cahn-Hillirad equation with degenerate mobility, J. Math. Anal. Appl. 344 (2008), no. 1, 124-144.

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.

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.

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.

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.

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.

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.

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.