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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 |
References:
[1] |
J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,, J. Nonlinear Sci., 7 (1997), 475.
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.
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.
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.
|
[5] |
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. |
[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.
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.
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.
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.
|
[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., (). Google Scholar |
[11] |
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. |
[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.
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.
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.
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, (2009), 11.
|
[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.
doi: 10.3934/cpaa.2009.8.785. |
[17] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969).
|
[18] |
W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation,, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47.
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.
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.
|
[21] |
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. |
[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.
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.
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.
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.
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.
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.
|
[28] |
T. Taniguchi, The exponencial behavior of Navier-Stokes equations with time delay external force,, Discrete Contin. Dyn. Syst., 12 (2005), 997.
doi: 10.3934/dcds.2005.12.997. |
[29] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).
doi: 10.1007/978-1-4684-0313-8. |
show all references
References:
[1] |
J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations,, J. Nonlinear Sci., 7 (1997), 475.
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.
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.
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.
|
[5] |
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. |
[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.
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.
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.
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.
|
[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., (). Google Scholar |
[11] |
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. |
[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.
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.
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.
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, (2009), 11.
|
[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.
doi: 10.3934/cpaa.2009.8.785. |
[17] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969).
|
[18] |
W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation,, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47.
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.
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.
|
[21] |
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. |
[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.
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.
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.
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.
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.
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.
|
[28] |
T. Taniguchi, The exponencial behavior of Navier-Stokes equations with time delay external force,, Discrete Contin. Dyn. Syst., 12 (2005), 997.
doi: 10.3934/dcds.2005.12.997. |
[29] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).
doi: 10.1007/978-1-4684-0313-8. |
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