# American Institute of Mathematical Sciences

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

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

 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.
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 - A, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779
##### References:
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Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains,, C. R. Acad. Sci. Paris, 342 (2006), 263.   Google Scholar [6] T. Caraballo, P. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays,, J. Differential Equations, 239 (2007), 311.   Google Scholar [7] T. Caraballo, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with delay,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869.   Google Scholar [8] T. Caraballo and J. Real, Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441.  doi: 10.1098/rspa.2001.0807.  Google Scholar [9] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181.  doi: 10.1098/rspa.2003.1166.  Google Scholar [10] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271.   Google Scholar [11] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.   Google Scholar [12] P. Constantin, Near identity transformations for the Navier-Stokes equations,, in, (2003), 117.  doi: 10.1016/S1874-5792(03)80006-X.  Google Scholar [13] F. Flandoli and B. Maslowski, Ergodicity of the $2$-D Navier-Stokes equation under random perturbations,, Commun. Math. Phys., 171 (1995), 119.  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] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains,, Nonlinear Anal., 64 (2006), 1100.  doi: 10.1016/j.na.2005.05.057.  Google Scholar [16] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar [17] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991).   Google Scholar [18] P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three dimensional globally modified Navier-Stokes equations,, in, (2009).   Google Scholar [19] P. E. Kloeden, J. A. Langa and J. Real, Pullback $V$-attractors of a three dimensional globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 6 (2007), 937.  doi: 10.3934/cpaa.2007.6.937.  Google Scholar [20] P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 8 (2009), 785.  doi: 10.3934/cpaa.2009.8.785.  Google Scholar [21] P. E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the $3D$ Navier-Stokes equations,, Proc. Roy. Soc. London Ser. A Math Phys. Eng. Sci., 463 (2007), 1491.   Google Scholar [22] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969).   Google Scholar [23] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655.  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] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,, Nonlinear Anal., 67 (2007), 2784.  doi: 10.1016/j.na.2006.09.035.  Google Scholar [26] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems,, Nonlinear Anal., 71 (2009), 3956.  doi: 10.1016/j.na.2009.02.065.  Google Scholar [27] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, Nonlinear Anal., 74 (2011), 2012.  doi: 10.1016/j.na.2010.11.008.  Google Scholar [28] P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations,, Stoch. Dyn., 3 (2003), 279.  doi: 10.1142/S0219493703000772.  Google Scholar [29] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations,, Adv. Nonlinear Stud., 9 (2009), 425.   Google Scholar [30] R. Temam, "Navier-Stokes Equations,", Theory and Numerical Analysis, 2 (1979).   Google Scholar [31] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Second Edition, 66 (1995).   Google Scholar [32] Z. Yoshida and Y. Giga, A nonlinear semigroup approach to the Navier-Stokes system,, Comm. in Partial Differential Equations, 9 (1984), 215.   Google Scholar

show all references

##### References:
 [1] T. Caraballo, P. E. Kloeden and J. Real, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations,, Advanced Nonlinear Studies, 6 (2006), 411.   Google Scholar [2] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761.  doi: 10.3934/dcdsb.2008.10.761.  Google Scholar [3] T. Caraballo, P. E. Kloeden and J. Real, Addendum to the paper "Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations,", Advanced Nonlinear Studies, 6 (2006), 411.   Google Scholar [4] 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 [5] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains,, C. R. Acad. Sci. Paris, 342 (2006), 263.   Google Scholar [6] T. Caraballo, P. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays,, J. Differential Equations, 239 (2007), 311.   Google Scholar [7] T. Caraballo, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with delay,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869.   Google Scholar [8] T. Caraballo and J. Real, Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441.  doi: 10.1098/rspa.2001.0807.  Google Scholar [9] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181.  doi: 10.1098/rspa.2003.1166.  Google Scholar [10] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271.   Google Scholar [11] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.   Google Scholar [12] P. Constantin, Near identity transformations for the Navier-Stokes equations,, in, (2003), 117.  doi: 10.1016/S1874-5792(03)80006-X.  Google Scholar [13] F. Flandoli and B. Maslowski, Ergodicity of the $2$-D Navier-Stokes equation under random perturbations,, Commun. Math. Phys., 171 (1995), 119.  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] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains,, Nonlinear Anal., 64 (2006), 1100.  doi: 10.1016/j.na.2005.05.057.  Google Scholar [16] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar [17] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991).   Google Scholar [18] P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three dimensional globally modified Navier-Stokes equations,, in, (2009).   Google Scholar [19] P. E. Kloeden, J. A. Langa and J. Real, Pullback $V$-attractors of a three dimensional globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 6 (2007), 937.  doi: 10.3934/cpaa.2007.6.937.  Google Scholar [20] P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 8 (2009), 785.  doi: 10.3934/cpaa.2009.8.785.  Google Scholar [21] P. E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the $3D$ Navier-Stokes equations,, Proc. Roy. Soc. London Ser. A Math Phys. Eng. Sci., 463 (2007), 1491.   Google Scholar [22] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969).   Google Scholar [23] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655.  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] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,, Nonlinear Anal., 67 (2007), 2784.  doi: 10.1016/j.na.2006.09.035.  Google Scholar [26] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems,, Nonlinear Anal., 71 (2009), 3956.  doi: 10.1016/j.na.2009.02.065.  Google Scholar [27] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, Nonlinear Anal., 74 (2011), 2012.  doi: 10.1016/j.na.2010.11.008.  Google Scholar [28] P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations,, Stoch. Dyn., 3 (2003), 279.  doi: 10.1142/S0219493703000772.  Google Scholar [29] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations,, Adv. Nonlinear Stud., 9 (2009), 425.   Google Scholar [30] R. Temam, "Navier-Stokes Equations,", Theory and Numerical Analysis, 2 (1979).   Google Scholar [31] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Second Edition, 66 (1995).   Google Scholar [32] Z. Yoshida and Y. Giga, A nonlinear semigroup approach to the Navier-Stokes system,, Comm. in Partial Differential Equations, 9 (1984), 215.   Google Scholar
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