Pullback attractors for globally modified Navier-Stokes equations with infinite delays

Previous Article
On the birth of minimal sets for perturbed reversible vector fields
DCDS Home
This Issue
Next Article
Coherent lists and chaotic sets

September 2011, 31(3): 779-796. doi: 10.3934/dcds.2011.31.779

Pullback attractors for globally modified Navier-Stokes equations with infinite delays

Pedro Marín-Rubio ^1, , Antonio M. Márquez-Durán ^2, and José Real ^3,

1.	Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2.	Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain
3.	Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla

Received December 2009 Revised June 2011 Published August 2011

Abstract
Related Papers

We establish the existence of pullback attractors for the dynamical system associated to a globally modified model of the Navier-Stokes equations containing delay operators with infinite delay in a suitable weighted space. Actually, we are able to prove the existence of attractors in different classes of universes, one is the classical of fixed bounded sets, and the other is given by a tempered condition. Relationship between these two kind of objects is also analyzed.

Keywords: Globally Modified Navier-Stokes Equations; pullback attractors; infinite delays..

Mathematics Subject Classification: Primary: 35K55, 35Q30, 34D4.

Citation: Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779

References:

[1]	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,, Advanced Nonlinear Studies, 6 (2006), 411. Google Scholar
[2]	T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761. doi: 10.3934/dcdsb.2008.10.761. Google Scholar
[3]	T. Caraballo, P. E. Kloeden and J. Real, Addendum to the paper "Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations,", Advanced Nonlinear Studies, 6 (2006), 411. Google Scholar
[4]	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. Google Scholar
[5]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains,, C. R. Acad. Sci. Paris, 342 (2006), 263. Google Scholar
[6]	T. Caraballo, P. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays,, J. Differential Equations, 239 (2007), 311. Google Scholar
[7]	T. Caraballo, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with delay,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869. Google Scholar
[8]	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. Google Scholar
[9]	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. Google Scholar
[10]	T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. Google Scholar
[11]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307. Google Scholar
[12]	P. Constantin, Near identity transformations for the Navier-Stokes equations,, in, (2003), 117. doi: 10.1016/S1874-5792(03)80006-X. Google Scholar
[13]	F. Flandoli and B. Maslowski, Ergodicity of the $2$-D Navier-Stokes equation under random perturbations,, Commun. Math. Phys., 171 (1995), 119. doi: 10.1007/BF02104513. Google Scholar
[14]	J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D Navier-Stokes equations and their tempered behaviour,, submitted., (). Google Scholar
[15]	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. Google Scholar
[16]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. Google Scholar
[17]	Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991). Google Scholar
[18]	P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three dimensional globally modified Navier-Stokes equations,, in, (2009). Google Scholar
[19]	P. E. Kloeden, J. A. Langa and J. Real, Pullback $V$-attractors of a three dimensional globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 6 (2007), 937. doi: 10.3934/cpaa.2007.6.937. Google Scholar
[20]	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. Google Scholar
[21]	P. E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the $3D$ Navier-Stokes equations,, Proc. Roy. Soc. London Ser. A Math Phys. Eng. Sci., 463 (2007), 1491. Google Scholar
[22]	J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969). Google Scholar
[23]	P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655. doi: 10.3934/dcdsb.2010.14.655. Google Scholar
[24]	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., (). Google Scholar
[25]	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. Google Scholar
[26]	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. Google Scholar
[27]	P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, Nonlinear Anal., 74 (2011), 2012. doi: 10.1016/j.na.2010.11.008. Google Scholar
[28]	P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations,, Stoch. Dyn., 3 (2003), 279. doi: 10.1142/S0219493703000772. Google Scholar
[29]	M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations,, Adv. Nonlinear Stud., 9 (2009), 425. Google Scholar
[30]	R. Temam, "Navier-Stokes Equations,", Theory and Numerical Analysis, 2 (1979). Google Scholar
[31]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Second Edition, 66 (1995). Google Scholar
[32]	Z. Yoshida and Y. Giga, A nonlinear semigroup approach to the Navier-Stokes system,, Comm. in Partial Differential Equations, 9 (1984), 215. Google Scholar

show all references

References:

[1]	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,, Advanced Nonlinear Studies, 6 (2006), 411. Google Scholar
[2]	T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761. doi: 10.3934/dcdsb.2008.10.761. Google Scholar
[3]	T. Caraballo, P. E. Kloeden and J. Real, Addendum to the paper "Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations,", Advanced Nonlinear Studies, 6 (2006), 411. Google Scholar
[4]	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. Google Scholar
[5]	T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains,, C. R. Acad. Sci. Paris, 342 (2006), 263. Google Scholar
[6]	T. Caraballo, P. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays,, J. Differential Equations, 239 (2007), 311. Google Scholar
[7]	T. Caraballo, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with delay,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869. Google Scholar
[8]	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. Google Scholar
[9]	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. Google Scholar
[10]	T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. Google Scholar
[11]	H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307. Google Scholar
[12]	P. Constantin, Near identity transformations for the Navier-Stokes equations,, in, (2003), 117. doi: 10.1016/S1874-5792(03)80006-X. Google Scholar
[13]	F. Flandoli and B. Maslowski, Ergodicity of the $2$-D Navier-Stokes equation under random perturbations,, Commun. Math. Phys., 171 (1995), 119. doi: 10.1007/BF02104513. Google Scholar
[14]	J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in $V$ for non-autonomous 2D Navier-Stokes equations and their tempered behaviour,, submitted., (). Google Scholar
[15]	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. Google Scholar
[16]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11. Google Scholar
[17]	Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991). Google Scholar
[18]	P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three dimensional globally modified Navier-Stokes equations,, in, (2009). Google Scholar
[19]	P. E. Kloeden, J. A. Langa and J. Real, Pullback $V$-attractors of a three dimensional globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 6 (2007), 937. doi: 10.3934/cpaa.2007.6.937. Google Scholar
[20]	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. Google Scholar
[21]	P. E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the $3D$ Navier-Stokes equations,, Proc. Roy. Soc. London Ser. A Math Phys. Eng. Sci., 463 (2007), 1491. Google Scholar
[22]	J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969). Google Scholar
[23]	P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655. doi: 10.3934/dcdsb.2010.14.655. Google Scholar
[24]	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., (). Google Scholar
[25]	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. Google Scholar
[26]	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. Google Scholar
[27]	P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, Nonlinear Anal., 74 (2011), 2012. doi: 10.1016/j.na.2010.11.008. Google Scholar
[28]	P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations,, Stoch. Dyn., 3 (2003), 279. doi: 10.1142/S0219493703000772. Google Scholar
[29]	M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations,, Adv. Nonlinear Stud., 9 (2009), 425. Google Scholar
[30]	R. Temam, "Navier-Stokes Equations,", Theory and Numerical Analysis, 2 (1979). Google Scholar
[31]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Second Edition, 66 (1995). Google Scholar
[32]	Z. Yoshida and Y. Giga, A nonlinear semigroup approach to the Navier-Stokes system,, Comm. in Partial Differential Equations, 9 (1984), 215. Google Scholar

[1]	P.E. Kloeden, José A. Langa, José Real. Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 937-955. doi: 10.3934/cpaa.2007.6.937
[2]	Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Three dimensional system of globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 655-673. doi: 10.3934/dcdsb.2010.14.655
[3]	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
[4]	Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761
[5]	Julia García-Luengo, Pedro Marín-Rubio, José Real. Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 181-201. doi: 10.3934/dcds.2014.34.181
[6]	P.E. Kloeden, Pedro Marín-Rubio, José Real. Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 785-802. doi: 10.3934/cpaa.2009.8.785
[7]	G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[8]	Pedro Marín-Rubio, José Real. Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 989-1006. doi: 10.3934/dcds.2010.26.989
[9]	Grzegorz Łukaszewicz. Pullback attractors and statistical solutions for 2-D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 643-659. doi: 10.3934/dcdsb.2008.9.643
[10]	Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-26. doi: 10.3934/dcdsb.2019172
[11]	Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269
[12]	Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361
[13]	Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239
[14]	Hammamia Mohamed Ali, Lassaad Mchiria, Sana Netchaoui, Stefanie Sonner. Pullback exponential attractors for differential equations with variable delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2019183
[15]	Julia García-Luengo, Pedro Marín-Rubio, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203
[16]	Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57
[17]	Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[18]	Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[19]	Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697
[20]	Songsong Lu, Hongqing Wu, Chengkui Zhong. Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 701-719. doi: 10.3934/dcds.2005.13.701

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences