# American Institute of Mathematical Sciences

October  2014, 34(10): 4085-4105. doi: 10.3934/dcds.2014.34.4085

## Attractors for a double time-delayed 2D-Navier-Stokes model

 1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla 2 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla 3 Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, 13083-859 Campinas, SP, Brazil

Received  December 2012 Revised  July 2013 Published  April 2014

In this paper, a double time-delayed 2D-Navier-Stokes model is considered. It includes delays in the convective and the forcing terms. Existence and uniqueness results and suitable dynamical systems are established. We also analyze the existence of pullback attractors for the model in several phase-spaces and the relationship among them.
Citation: Julia García-Luengo, Pedro Marín-Rubio, Gabriela Planas. Attractors for a double time-delayed 2D-Navier-Stokes model. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4085-4105. doi: 10.3934/dcds.2014.34.4085
##### References:
 [1] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037. [2] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [3] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268. doi: 10.1016/j.crma.2005.12.015. [4] 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, Adv. Nonlinear Stud., 6 (2006), 411-436. [5] T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807. [6] 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), 3181-3194. doi: 10.1098/rspa.2003.1166. [7] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012. [8] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010. [9] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357. [10] J. García-Luengo, P. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, to appear in Commun. Pure Appl. Anal. [11] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118. doi: 10.1016/j.na.2005.05.057. [12] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238. doi: 10.3934/dcdsb.2011.16.225. [13] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system, J. Math. Anal. Appl., 373 (2011), 535-547. doi: 10.1016/j.jmaa.2010.07.040. [14] A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278. doi: 10.1016/j.jde.2007.06.008. [15] P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three-dimensional globally modified Navier-Stokes equations, in Advances in Nonlinear Analysis: Theory Methods and Applications, Math. Probl. Eng. Aerosp. Sci., 3, Cambridge Scientific Publishers, Cambridge, 2009, 11-22. [16] 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), 785-802. doi: 10.3934/cpaa.2009.8.785. [17] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [18] W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47-56. doi: 10.3934/dcdsb.2002.2.47. [19] A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulation, IEEE Trans. Automat. Control, 29 (1984), 1058-1068. doi: 10.1109/TAC.1984.1103436. [20] 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., 11 (2011), 917-927. [21] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799. doi: 10.1016/j.na.2006.09.035. [22] 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), 3956-3963. doi: 10.1016/j.na.2009.02.065. [23] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006. doi: 10.3934/dcds.2010.26.989. [24] P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations, Stoch. Dyn., 3 (2003), 279-297. doi: 10.1142/S0219493703000772. [25] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258. doi: 10.3934/dcds.2008.21.1245. [26] M. Renardy, A class of quasilinear parabolic equations with infinite delay and application to a problem of viscoelasticity, J. Differential Equations, 48 (1983), 280-292. doi: 10.1016/0022-0396(83)90053-0. [27] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427. [28] T. Taniguchi, The exponencial behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997. [29] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

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##### References:
 [1] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037. [2] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [3] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268. doi: 10.1016/j.crma.2005.12.015. [4] 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, Adv. Nonlinear Stud., 6 (2006), 411-436. [5] T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453. doi: 10.1098/rspa.2001.0807. [6] 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), 3181-3194. doi: 10.1098/rspa.2003.1166. [7] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. doi: 10.1016/j.jde.2004.04.012. [8] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356. doi: 10.1016/j.jde.2012.01.010. [9] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357. [10] J. García-Luengo, P. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, to appear in Commun. Pure Appl. Anal. [11] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118. doi: 10.1016/j.na.2005.05.057. [12] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238. doi: 10.3934/dcdsb.2011.16.225. [13] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system, J. Math. Anal. Appl., 373 (2011), 535-547. doi: 10.1016/j.jmaa.2010.07.040. [14] A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278. doi: 10.1016/j.jde.2007.06.008. [15] P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three-dimensional globally modified Navier-Stokes equations, in Advances in Nonlinear Analysis: Theory Methods and Applications, Math. Probl. Eng. Aerosp. Sci., 3, Cambridge Scientific Publishers, Cambridge, 2009, 11-22. [16] 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), 785-802. doi: 10.3934/cpaa.2009.8.785. [17] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [18] W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47-56. doi: 10.3934/dcdsb.2002.2.47. [19] A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulation, IEEE Trans. Automat. Control, 29 (1984), 1058-1068. doi: 10.1109/TAC.1984.1103436. [20] 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., 11 (2011), 917-927. [21] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799. doi: 10.1016/j.na.2006.09.035. [22] 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), 3956-3963. doi: 10.1016/j.na.2009.02.065. [23] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006. doi: 10.3934/dcds.2010.26.989. [24] P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations, Stoch. Dyn., 3 (2003), 279-297. doi: 10.1142/S0219493703000772. [25] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258. doi: 10.3934/dcds.2008.21.1245. [26] M. Renardy, A class of quasilinear parabolic equations with infinite delay and application to a problem of viscoelasticity, J. Differential Equations, 48 (1983), 280-292. doi: 10.1016/0022-0396(83)90053-0. [27] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427. [28] T. Taniguchi, The exponencial behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997. [29] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.
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