American Institute of Mathematical Sciences

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September  2011, 31(3): 779-796. doi: 10.3934/dcds.2011.31.779

Pullback attractors for globally modified Navier-Stokes equations with infinite delays

Pedro Marín-Rubio 1, , Antonio M. Márquez-Durán 2, and  José Real 3,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain
3. 

Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla

Received  December 2009 Revised  June 2011 Published  August 2011

We establish the existence of pullback attractors for the dynamical system associated to a globally modified model of the Navier-Stokes equations containing delay operators with infinite delay in a suitable weighted space. Actually, we are able to prove the existence of attractors in different classes of universes, one is the classical of fixed bounded sets, and the other is given by a tempered condition. Relationship between these two kind of objects is also analyzed.
Keywords: Globally Modified Navier-Stokes Equations; pullback attractors; infinite delays..
Mathematics Subject Classification: Primary: 35K55, 35Q30, 34D4.
Citation: Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779
References:
[1]

Advanced Nonlinear Studies, 6 (2006), 411-436.  Google Scholar

[2]

Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761.  Google Scholar

[3]

Advanced Nonlinear Studies, 6 (2006), 411-436, Adv. Nonlinear Stud., 10 (2010), 245-247.  Google Scholar

[4]

Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.  Google Scholar

[5]

C. R. Acad. Sci. Paris, 342 (2006), 263-268.  Google Scholar

[6]

J. Differential Equations, 239 (2007), 311-342.  Google Scholar

[7]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883.  Google Scholar

[8]

R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807.  Google Scholar

[9]

R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166.  Google Scholar

[10]

J. Differential Equations, 205 (2004), 271-297.  Google Scholar

[11]

J. Dynam. Differential Equations, 9 (1997), 307-341.  Google Scholar

[12]

in "Handbook of Mathematical Fluid Dynamics," Vol. II, 117-141, North-Holland, Amsterdam, 2003. doi: 10.1016/S1874-5792(03)80006-X.  Google Scholar

[13]

Commun. Math. Phys., 171 (1995), 119-141. doi: 10.1007/BF02104513.  Google Scholar

[14]

J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D Navier-Stokes equations and their tempered behaviour,, submitted., ().   Google Scholar

[15]

Nonlinear Anal., 64 (2006), 1100-1118. doi: 10.1016/j.na.2005.05.057.  Google Scholar

[16]

Funkcial. Ekvac., 21 (1978), 11-41.  Google Scholar

[17]

Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991.  Google Scholar

[18]

in "Mathematical Problems in Engineering Aerospace and Sciences" (eds. S. Sivasundaram, J. Vasundhara Devi, Zahia Drici and Farzana Mcrae), Vol. 3, Chapter 2, Cambridge Scientific Publishers, 2009. Google Scholar

[19]

Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937.  Google Scholar

[20]

Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785.  Google Scholar

[21]

Proc. Roy. Soc. London Ser. A Math Phys. Eng. Sci., 463 (2007), 1491-1508.  Google Scholar

[22]

Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[23]

Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655-673. doi: 10.3934/dcdsb.2010.14.655.  Google Scholar

[24]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, On the convergence of solutions of Globally Modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays,, Adv. Nonlinear Stud., ().   Google Scholar

[25]

Nonlinear Anal., 67 (2007), 2784-2799. doi: 10.1016/j.na.2006.09.035.  Google Scholar

[26]

Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065.  Google Scholar

[27]

Nonlinear Anal., 74 (2011), 2012-2030. doi: 10.1016/j.na.2010.11.008.  Google Scholar

[28]

Stoch. Dyn., 3 (2003), 279-297. doi: 10.1142/S0219493703000772.  Google Scholar

[29]

Adv. Nonlinear Stud., 9 (2009), 425-427.  Google Scholar

[30]

Theory and Numerical Analysis, Revised edition, Studies in Mathematics and its Applications, 2, North Holland Publishig Co., Amsterdam-New York, 1979.  Google Scholar

[31]

Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, SIAM, Philadelphia, PA, 1995.  Google Scholar

[32]

Comm. in Partial Differential Equations, 9 (1984), 215-230.  Google Scholar

show all references

References:
[1]

Advanced Nonlinear Studies, 6 (2006), 411-436.  Google Scholar

[2]

Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761.  Google Scholar

[3]

Advanced Nonlinear Studies, 6 (2006), 411-436, Adv. Nonlinear Stud., 10 (2010), 245-247.  Google Scholar

[4]

Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.  Google Scholar

[5]

C. R. Acad. Sci. Paris, 342 (2006), 263-268.  Google Scholar

[6]

J. Differential Equations, 239 (2007), 311-342.  Google Scholar

[7]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883.  Google Scholar

[8]

R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807.  Google Scholar

[9]

R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181-3194. doi: 10.1098/rspa.2003.1166.  Google Scholar

[10]

J. Differential Equations, 205 (2004), 271-297.  Google Scholar

[11]

J. Dynam. Differential Equations, 9 (1997), 307-341.  Google Scholar

[12]

in "Handbook of Mathematical Fluid Dynamics," Vol. II, 117-141, North-Holland, Amsterdam, 2003. doi: 10.1016/S1874-5792(03)80006-X.  Google Scholar

[13]

Commun. Math. Phys., 171 (1995), 119-141. doi: 10.1007/BF02104513.  Google Scholar

[14]

J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D Navier-Stokes equations and their tempered behaviour,, submitted., ().   Google Scholar

[15]

Nonlinear Anal., 64 (2006), 1100-1118. doi: 10.1016/j.na.2005.05.057.  Google Scholar

[16]

Funkcial. Ekvac., 21 (1978), 11-41.  Google Scholar

[17]

Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991.  Google Scholar

[18]

in "Mathematical Problems in Engineering Aerospace and Sciences" (eds. S. Sivasundaram, J. Vasundhara Devi, Zahia Drici and Farzana Mcrae), Vol. 3, Chapter 2, Cambridge Scientific Publishers, 2009. Google Scholar

[19]

Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937.  Google Scholar

[20]

Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785.  Google Scholar

[21]

Proc. Roy. Soc. London Ser. A Math Phys. Eng. Sci., 463 (2007), 1491-1508.  Google Scholar

[22]

Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[23]

Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655-673. doi: 10.3934/dcdsb.2010.14.655.  Google Scholar

[24]

P. Marín-Rubio, A. M. Márquez-Durán and J. Real, On the convergence of solutions of Globally Modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays,, Adv. Nonlinear Stud., ().   Google Scholar

[25]

Nonlinear Anal., 67 (2007), 2784-2799. doi: 10.1016/j.na.2006.09.035.  Google Scholar

[26]

Nonlinear Anal., 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065.  Google Scholar

[27]

Nonlinear Anal., 74 (2011), 2012-2030. doi: 10.1016/j.na.2010.11.008.  Google Scholar

[28]

Stoch. Dyn., 3 (2003), 279-297. doi: 10.1142/S0219493703000772.  Google Scholar

[29]

Adv. Nonlinear Stud., 9 (2009), 425-427.  Google Scholar

[30]

Theory and Numerical Analysis, Revised edition, Studies in Mathematics and its Applications, 2, North Holland Publishig Co., Amsterdam-New York, 1979.  Google Scholar

[31]

Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, SIAM, Philadelphia, PA, 1995.  Google Scholar

[32]

Comm. in Partial Differential Equations, 9 (1984), 215-230.  Google Scholar

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