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

1. 

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

Received  September 2010 Revised  May 2011 Published  October 2011

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.
Citation: Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

B. Guo, Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations,, Chinese Ann. Math. Ser. B, 16 (1995), 379.   Google Scholar

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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.   Google Scholar

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B. Guo and B. Wang, Approximate inertial manifolds to the Newton-Boussinesq equations,, J. Partial Differential Equations, 9 (1996), 237.   Google Scholar

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G. Fucci, B. Wang and P. Singh, Asymptotic behavior of the Newton-Boussinesq equation in a two-dimensional channel,, Nonlinear Anal., 70 (2009), 2000.  doi: 10.1016/j.na.2008.02.098.  Google Scholar

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T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

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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.   Google Scholar

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B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains,, Nonlinear Anal., 70 (2009), 3799.  doi: 10.1016/j.na.2008.07.011.  Google Scholar

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B. Wang and R. Jones, Asymptotic behavior of a class of non-autonomous degenerate parabolic equations,, Nonlinear Anal., 72 (2010), 3887.  doi: 10.1016/j.na.2010.01.026.  Google Scholar

show all references

References:
[1]

B. Guo, Spectral method for solving the two-dimensional Newton-Boussinesq equations,, Acta. Math. Appl. Sinica (English Ser.), 5 (1989), 208.   Google Scholar

[2]

B. Guo, Nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations,, Chinese Ann. Math. Ser. B, 16 (1995), 379.   Google Scholar

[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.   Google Scholar

[4]

B. Guo and B. Wang, Approximate inertial manifolds to the Newton-Boussinesq equations,, J. Partial Differential Equations, 9 (1996), 237.   Google Scholar

[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.  doi: 10.1016/j.na.2008.02.098.  Google Scholar

[6]

T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[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.   Google Scholar

[8]

B. Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains,, Nonlinear Anal., 70 (2009), 3799.  doi: 10.1016/j.na.2008.07.011.  Google Scholar

[9]

B. Wang and R. Jones, Asymptotic behavior of a class of non-autonomous degenerate parabolic equations,, Nonlinear Anal., 72 (2010), 3887.  doi: 10.1016/j.na.2010.01.026.  Google Scholar

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