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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

Julia García-Luengo 1, , Pedro Marín-Rubio 2, and  José Real 2,

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.
Keywords: 2D Navier-Stokes equations, pullback attractors., delay terms.
Mathematics Subject Classification: Primary: 35B41, 35Q30, 37L3.
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

show all references

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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