# American Institute of Mathematical Sciences

September  2015, 14(5): 1603-1621. doi: 10.3934/cpaa.2015.14.1603

## Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays

 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

Received  April 2012 Revised  July 2012 Published  June 2015

In this paper we strengthen some results on the existence and properties of pullback attractors for a 2D Navier-Stokes model with finite delay formulated in [Caraballo and Real, J. Differential Equations 205 (2004), 271--297]. Actually, we prove that under suitable assumptions, pullback attractors not only of fixed bounded sets but also of a set of tempered universes do exist. Moreover, thanks to regularity results, the attraction from different phase spaces also happens in $C([-h,0];V)$. Finally, from comparison results of attractors, and under an additional hypothesis, we establish that all these families of attractors are in fact the same object.
Citation: Julia García-Luengo, Pedro Marín-Rubio, José Real. Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1603-1621. doi: 10.3934/cpaa.2015.14.1603
##### References:
 [1] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, \emph{Nonlinear Anal.}, 64 (2006), 484. doi: 10.1016/j.na.2005.03.111. Google Scholar [2] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,, \emph{C. R. Math. Acad. Sci. Paris}, 342 (2006), 263. doi: 10.1016/j.crma.2005.12.015. Google Scholar [3] T. Caraballo and J. Real, Navier-Stokes equations with delays,, \emph{R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.}, 457 (2001), 2441. doi: 10.1098/rspa.2001.0807. Google Scholar [4] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays,, \emph{R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.}, 459 (2003), 3181. doi: 10.1098/rspa.2003.1166. Google Scholar [5] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, \emph{J. Differential Equations}, 205 (2004), 271. doi: 10.1016/j.jde.2004.04.012. Google Scholar [6] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, \emph{Rend. Sem. Mat. Univ. Padova}, 31 (1961), 308. Google Scholar [7] J. García-Luengo, P. Marín-Rubio and J. Real, $H^2$-boundedness of the pullback attractors for non-autonomous 2D Navier-Stokes equations in bounded domains,, \emph{Nonlinear Anal.}, 74 (2011), 4882. doi: 10.1016/j.na.2011.04.063. Google Scholar [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,, \emph{J. Differential Equations}, 252 (2012), 4333. doi: 10.1016/j.jde.2012.01.010. Google Scholar [9] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 331. Google Scholar [10] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains,, \emph{Nonlinear Anal.}, 64 (2006), 1100. doi: 10.1016/j.na.2005.05.057. Google Scholar [11] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 16 (2011), 225. doi: 10.3934/dcdsb.2011.16.225. Google Scholar [12] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969). Google Scholar [13] A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulation,, \emph{IEEE Trans. Automat. Control}, 29 (1984), 1058. doi: 10.1109/TAC.1984.1103436. Google Scholar [14] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 14 (2010), 655. doi: 10.3934/dcdsb.2010.14.655. Google Scholar [15] 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,, \emph{Adv. Nonlinear Stud.}, 11 (2011), 917. Google Scholar [16] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays,, \emph{Discrete Contin. Dyn. Syst.}, 31 (2011), 779. doi: 10.3934/dcds.2011.31.779. Google Scholar [17] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,, \emph{Nonlinear Anal.}, 67 (2007), 2784. doi: 10.1016/j.na.2006.09.035. Google Scholar [18] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems,, \emph{Nonlinear Anal.}, 71 (2009), 3956. doi: 10.1016/j.na.2009.02.065. Google Scholar [19] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators,, \emph{Discrete Contin. Dyn. Syst.}, 26 (2010), 989. doi: 10.3934/dcds.2010.26.989. Google Scholar [20] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, \emph{Nonlinear Anal.}, 74 (2011), 2012. doi: 10.1016/j.na.2010.11.008. Google Scholar [21] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations,, \emph{Discrete Contin. Dyn. Syst.}, 21 (2008), 1245. doi: 10.3934/dcds.2008.21.1245. Google Scholar [22] J. C. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar [23] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains,, \emph{Nonlinear Anal.}, 32 (1998), 71. doi: 10.1016/S0362-546X(97)00453-7. Google Scholar [24] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [25] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979). Google Scholar

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##### References:
 [1] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, \emph{Nonlinear Anal.}, 64 (2006), 484. doi: 10.1016/j.na.2005.03.111. Google Scholar [2] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,, \emph{C. R. Math. Acad. Sci. Paris}, 342 (2006), 263. doi: 10.1016/j.crma.2005.12.015. Google Scholar [3] T. Caraballo and J. Real, Navier-Stokes equations with delays,, \emph{R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.}, 457 (2001), 2441. doi: 10.1098/rspa.2001.0807. Google Scholar [4] T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays,, \emph{R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.}, 459 (2003), 3181. doi: 10.1098/rspa.2003.1166. Google Scholar [5] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, \emph{J. Differential Equations}, 205 (2004), 271. doi: 10.1016/j.jde.2004.04.012. Google Scholar [6] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, \emph{Rend. Sem. Mat. Univ. Padova}, 31 (1961), 308. Google Scholar [7] J. García-Luengo, P. Marín-Rubio and J. Real, $H^2$-boundedness of the pullback attractors for non-autonomous 2D Navier-Stokes equations in bounded domains,, \emph{Nonlinear Anal.}, 74 (2011), 4882. doi: 10.1016/j.na.2011.04.063. Google Scholar [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,, \emph{J. Differential Equations}, 252 (2012), 4333. doi: 10.1016/j.jde.2012.01.010. Google Scholar [9] J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity,, \emph{Adv. Nonlinear Stud.}, 13 (2013), 331. Google Scholar [10] M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains,, \emph{Nonlinear Anal.}, 64 (2006), 1100. doi: 10.1016/j.na.2005.05.057. Google Scholar [11] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 16 (2011), 225. doi: 10.3934/dcdsb.2011.16.225. Google Scholar [12] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969). Google Scholar [13] A. Z. Manitius, Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulation,, \emph{IEEE Trans. Automat. Control}, 29 (1984), 1058. doi: 10.1109/TAC.1984.1103436. Google Scholar [14] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 14 (2010), 655. doi: 10.3934/dcdsb.2010.14.655. Google Scholar [15] 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,, \emph{Adv. Nonlinear Stud.}, 11 (2011), 917. Google Scholar [16] P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays,, \emph{Discrete Contin. Dyn. Syst.}, 31 (2011), 779. doi: 10.3934/dcds.2011.31.779. Google Scholar [17] P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,, \emph{Nonlinear Anal.}, 67 (2007), 2784. doi: 10.1016/j.na.2006.09.035. Google Scholar [18] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems,, \emph{Nonlinear Anal.}, 71 (2009), 3956. doi: 10.1016/j.na.2009.02.065. Google Scholar [19] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators,, \emph{Discrete Contin. Dyn. Syst.}, 26 (2010), 989. doi: 10.3934/dcds.2010.26.989. Google Scholar [20] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, \emph{Nonlinear Anal.}, 74 (2011), 2012. doi: 10.1016/j.na.2010.11.008. Google Scholar [21] G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations,, \emph{Discrete Contin. Dyn. Syst.}, 21 (2008), 1245. doi: 10.3934/dcds.2008.21.1245. Google Scholar [22] J. C. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar [23] R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains,, \emph{Nonlinear Anal.}, 32 (1998), 71. doi: 10.1016/S0362-546X(97)00453-7. Google Scholar [24] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar [25] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979). Google Scholar
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