Pullback \mathcal{D}-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain

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March 2012, 32(3): 991-1009. doi: 10.3934/dcds.2012.32.991

Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain

Xue-Li Song ^1, and Yan-Ren Hou ^1,

1.	College of science, Xi’an Jiaotong University, Xi’an, 710049, China

Received September 2010 Revised May 2011 Published October 2011

Abstract
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We investigate the asymptotic behavior of solutions of a class of non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. The existence of pullback global attractors is proved in $L^2(\Omega)\times L^2(\Omega)$ and $H^1(\Omega)\times H^1(\Omega)$, respectively.

Keywords: Pullback attractor, Asymptotic compactness, Newton-Boussinesq equation..

Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L3.

Citation: Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991

References:

[1]	B. Guo, Spectral method for solving the two-dimensional Newton-Boussinesq equations, Acta. Math. Appl. Sinica (English Ser.), 5 (1989), 208-218.
[2]	B. Guo, Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chinese Ann. Math. Ser. B, 16 (1995), 379-390.
[3]	B. Guo and B. Wang, Gevrey class regularity and approximate inertial manifolds for the Newton-Boussinesq equations, Chinese Ann. Math. Ser. B, 19 (1998), 179-188.
[4]	B. Guo and B. Wang, Approximate inertial manifolds to the Newton-Boussinesq equations, J. Partial Differential Equations, 9 (1996), 237-250.
[5]	G. Fucci, B. Wang and P. Singh, Asymptotic behavior of the Newton-Boussinesq equation in a two-dimensional channel, Nonlinear Anal., 70 (2009), 2000-2013. doi: 10.1016/j.na.2008.02.098.
[6]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[7]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Acad. Sci. Paris, 342 (2006), 263-268.
[8]	B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal., 70 (2009), 3799-3815. doi: 10.1016/j.na.2008.07.011.
[9]	B. Wang and R. Jones, Asymptotic behavior of a class of non-autonomous degenerate parabolic equations, Nonlinear Anal., 72 (2010), 3887-3902. doi: 10.1016/j.na.2010.01.026.

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References:

[1]	B. Guo, Spectral method for solving the two-dimensional Newton-Boussinesq equations, Acta. Math. Appl. Sinica (English Ser.), 5 (1989), 208-218.
[2]	B. Guo, Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chinese Ann. Math. Ser. B, 16 (1995), 379-390.
[3]	B. Guo and B. Wang, Gevrey class regularity and approximate inertial manifolds for the Newton-Boussinesq equations, Chinese Ann. Math. Ser. B, 19 (1998), 179-188.
[4]	B. Guo and B. Wang, Approximate inertial manifolds to the Newton-Boussinesq equations, J. Partial Differential Equations, 9 (1996), 237-250.
[5]	G. Fucci, B. Wang and P. Singh, Asymptotic behavior of the Newton-Boussinesq equation in a two-dimensional channel, Nonlinear Anal., 70 (2009), 2000-2013. doi: 10.1016/j.na.2008.02.098.
[6]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.
[7]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Acad. Sci. Paris, 342 (2006), 263-268.
[8]	B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal., 70 (2009), 3799-3815. doi: 10.1016/j.na.2008.07.011.
[9]	B. Wang and R. Jones, Asymptotic behavior of a class of non-autonomous degenerate parabolic equations, Nonlinear Anal., 72 (2010), 3887-3902. doi: 10.1016/j.na.2010.01.026.

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