# American Institute of Mathematical Sciences

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:
 [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. [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. [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. [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. [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. [6] T. Caraballo, P. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays,, J. Differential Equations, 239 (2007), 311. [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. [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. [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. [10] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. [11] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307. [12] P. Constantin, Near identity transformations for the Navier-Stokes equations,, in, (2003), 117. doi: 10.1016/S1874-5792(03)80006-X. [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. [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., (). [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. [16] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. [17] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991). [18] P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three dimensional globally modified Navier-Stokes equations,, in, (2009). [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. [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. [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. [22] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969). [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. [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., (). [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. [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. [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. [28] P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations,, Stoch. Dyn., 3 (2003), 279. doi: 10.1142/S0219493703000772. [29] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations,, Adv. Nonlinear Stud., 9 (2009), 425. [30] R. Temam, "Navier-Stokes Equations,", Theory and Numerical Analysis, 2 (1979). [31] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Second Edition, 66 (1995). [32] Z. Yoshida and Y. Giga, A nonlinear semigroup approach to the Navier-Stokes system,, Comm. in Partial Differential Equations, 9 (1984), 215.

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##### 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. [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. [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. [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. [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. [6] T. Caraballo, P. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays,, J. Differential Equations, 239 (2007), 311. [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. [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. [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. [10] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. [11] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307. [12] P. Constantin, Near identity transformations for the Navier-Stokes equations,, in, (2003), 117. doi: 10.1016/S1874-5792(03)80006-X. [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. [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., (). [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. [16] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. [17] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991). [18] P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three dimensional globally modified Navier-Stokes equations,, in, (2009). [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. [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. [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. [22] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969). [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. [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., (). [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. [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. [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. [28] P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations,, Stoch. Dyn., 3 (2003), 279. doi: 10.1142/S0219493703000772. [29] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations,, Adv. Nonlinear Stud., 9 (2009), 425. [30] R. Temam, "Navier-Stokes Equations,", Theory and Numerical Analysis, 2 (1979). [31] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Second Edition, 66 (1995). [32] Z. Yoshida and Y. Giga, A nonlinear semigroup approach to the Navier-Stokes system,, Comm. in Partial Differential Equations, 9 (1984), 215.
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