×

Pullback attractor of 2D non-autonomous \(g\)-Navier-Stokes equations on some bounded domains. (English) Zbl 1209.35023

The existence of the pullback attractor for the 2D non-autonomous \(g\)-Navier-Stokes equations on some bounded domains is investigated under the general assumptions of pullback asymptotic compactness. A new method to prove the existence of the pullback attractor for the 2D \(g\)-Navier-Stokes equations is given.

MSC:

35B41 Attractors
37B55 Topological dynamics of nonautonomous systems
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
Full Text: DOI

References:

[1] Abergel, F. Attractor for a Navier-Stokes flow in an unbounded domain. Math. Model. Numer. Anal. 23(3), 359–370 (1989) · Zbl 0676.76028
[2] Babin, A. V. The attractor of a Navier-Stokes system in an unbounded channel-like domain. J. Dyn. Diff. Equat. 4(4), 555–584 (1992) · Zbl 0762.35082 · doi:10.1007/BF01048260
[3] Rosa, R. The global attractor for the 2D-Navier-Stokes flow in some unbounded domain. Nonlinear Anal. 32(1), 71–85 (1998) · Zbl 0901.35070 · doi:10.1016/S0362-546X(97)00453-7
[4] Cheban, D. N. and Duan, J. Almost periodic solutions and global attractors of non-autonomous Navier-Stokes equation. J. Dyn. Diff. Equat. 16(1), 1–34 (2004) · Zbl 1088.35043 · doi:10.1023/B:JODY.0000041279.25095.8a
[5] Raugel, G. and Sell, G. R. Navier-Stokes equations on thin 3D domains (I), global attractors and global regularity of solutions. J. Amer. Math. Soc. 6(3), 503–568 (1993) · Zbl 0787.34039
[6] Temam, R. Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence (2001) · Zbl 0981.35001
[7] Temam, R. Infinite-dimensional dynamical system in mechanics and physics. Applied Mathematical Sciences 68, Springer-Verlag, New York (1988) · Zbl 0662.35001
[8] Babin, A. V. and Vishik, M. I. Attractors of partial differential evolution equations in an unbounded domain. Proc. Roy. Soc. Edinburgh Sect. A 116(A), 221–243 (1990) · Zbl 0721.35029
[9] Constantin, P., Foias, C., and Temam, R. Attractor representing turbulent flows. Mem. Amer. Math. Soc. 53(314), 1–67 (1985) · Zbl 0567.35070
[10] Cheban, D. N. Global Attractors of Non-autonomous Dissipative Dynamical Systems, World Scientific, Singapore (2004) · Zbl 1098.37002
[11] Caraballo, T., Kloeden, P. E., and Marin-Rubio, P. Global and pullback attractor of set-valued skew product flows. Ann. Mat. 185(2), S23–S45 (2006) · Zbl 1115.37065 · doi:10.1007/s10231-004-0135-3
[12] Caraballo, T., Lukaszewicz, G., and Real, J. Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Anal. 64(3), 484–498 (2006) · Zbl 1128.37019 · doi:10.1016/j.na.2005.03.111
[13] Caraballo, T., Kloeden, P. E., and Real, J. Pullback and forward attractors for a damped wave equation with delays. Stochastics and Dynamics 4(3), 405–423 (2004) · Zbl 1061.35153 · doi:10.1142/S0219493704001139
[14] Caraballo, T., Real, J., and Chueshov, I. D. Pullback attractors for stochastic heat equations in materials with memory. Discrete and Continuous Dynamical Systems Series B 9(3), 525–539 (2008) · Zbl 1152.60044 · doi:10.3934/dcdsb.2008.9.525
[15] Kloeden, P. E. Pullback attractors in non-autonomous difference equations. Journal of Difference Equations and Applications 6(1), 33–52 (2000) · Zbl 0961.39007 · doi:10.1080/10236190008808212
[16] Kloeden, P. E. Pullback attractors of non-autonomous semi-dynamical systems. Stochastics and Dynamics 3(1), 101–112 (2003) · Zbl 1029.37010 · doi:10.1142/S0219493703000632
[17] Wang, Y. J., Zhong, C. K., and Zhou, S. F. Pullback attractors of non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems 16(3), 587–614 (2006) · Zbl 1121.34027 · doi:10.3934/dcds.2006.16.705
[18] Song, H. T. and Wu, H. Q. Pullback attractor of non-autonomous reaction-diffusion equations. J. Math. Anal. Appl. 325(2), 1200–1215 (2007) · Zbl 1104.37013 · doi:10.1016/j.jmaa.2006.02.041
[19] Li, Y. J. and Zhong, C. K. Pullback attractors for the norm-to-weak continuous process and application to the non-autonomous reaction-diffusion equations. Appl. Math. Comput. 190(2), 1020–1029 (2007) · Zbl 1126.37049 · doi:10.1016/j.amc.2006.11.187
[20] Roh, J. g-Navier-Stokes Equations, Ph. D. dissertation, University of Minnesota (2001) · Zbl 1179.35224
[21] Kwak, M., Kwean, H., and Roh, J. The dimension of attractor of the 2D g-Navier-Stokes equations. J. Math. Anal. Appl. 315(2), 436–461 (2006) · Zbl 1099.35084 · doi:10.1016/j.jmaa.2005.04.050
[22] Jiang, J. P. and Hou, Y. R. The global attractor of g-Navier-Stokes equations with linear dampness on \(\mathbb{R}\)2. Appl. Math. Comput. 215(3), 1068–1076 (2009) · Zbl 1172.76009 · doi:10.1016/j.amc.2009.06.035
[23] Zhong, C. K., Yang, M. H., and Sun, C. Y. The existence of global attractors for the norm-to-weak continuous semi-group and application to the non-linear reaction-diffusion equations. Journal of Differential Equations 223(2), 367–399 (2006) · Zbl 1101.35022 · doi:10.1016/j.jde.2005.06.008
[24] Foias, C. and Teman, R. Finite parameter approximative structures of actual flows. Nonlinear Problems: Present and Future (eds. Biship, A. R., Campbell, D. K., and Nicolaenco, B.), North Holland, Amsterdam (1982)
[25] Sell, G. R. and You, Y. Dynamics of evolutionary equations. Applied Mathematical Sciences 143, Springer-Verlag, New York (2002) · Zbl 1254.37002
[26] Bae, H. and Roh, J. Existence of solutions of the g-Navier-Stokes equations. Taiwanese J. Math. 8(1), 85–102 (2004) · Zbl 1060.35103
[27] Hale, J. K. Asymptotic Behaviour of Dissipative Dynamicals Systems, Amer. Math. Soc., Providence (1988)
[28] Hou, Y. R. and Li, K. T. The uniform attractor for the 2D non-autonomous Navier-Stokes flow in some unbounded domain. Nonlinear Anal. 58(5–6), 609–630 (2004) · Zbl 1057.35031 · doi:10.1016/j.na.2004.02.031
[29] Roh, J. Dynamics of the g-Navier-stokes equations. Journal of Differential Equations 211(2), 452–484 (2005) · Zbl 1072.35145 · doi:10.1016/j.jde.2004.08.016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.