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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, \emph{Rend. Sem. Mat. Univ. Padova}, 31 (1961), 308.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[12] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969).
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001).
doi: 10.1007/978-94-010-0732-0. |
[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. |
[24] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988).
doi: 10.1007/978-1-4684-0313-8. |
[25] |
R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979).
|
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, \emph{Rend. Sem. Mat. Univ. Padova}, 31 (1961), 308.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[12] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969).
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001).
doi: 10.1007/978-94-010-0732-0. |
[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. |
[24] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988).
doi: 10.1007/978-1-4684-0313-8. |
[25] |
R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979).
|
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