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On the birth of minimal sets for perturbed reversible vector fields
Pullback attractors for globally modified Navier-Stokes equations with infinite delays
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 |
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), 411-436. |
[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), 761-781.
doi: 10.3934/dcdsb.2008.10.761. |
[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), 411-436, Adv. Nonlinear Stud., 10 (2010), 245-247. |
[4] |
T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[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), 263-268. |
[6] |
T. Caraballo, P. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays, J. Differential Equations, 239 (2007), 311-342. |
[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), 2869-2883. |
[8] |
T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.
doi: 10.1098/rspa.2001.0807. |
[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), 3181-3194.
doi: 10.1098/rspa.2003.1166. |
[10] |
T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. |
[11] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. |
[12] |
P. Constantin, Near identity transformations for the Navier-Stokes equations, in "Handbook of Mathematical Fluid Dynamics," Vol. II, 117-141, North-Holland, Amsterdam, 2003.
doi: 10.1016/S1874-5792(03)80006-X. |
[13] |
F. Flandoli and B. Maslowski, Ergodicity of the $2$-D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 171 (1995), 119-141.
doi: 10.1007/BF02104513. |
[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., ().
|
[15] |
M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118.
doi: 10.1016/j.na.2005.05.057. |
[16] |
J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. |
[17] |
Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. |
[18] |
P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three dimensional globally modified Navier-Stokes equations, in "Mathematical Problems in Engineering Aerospace and Sciences" (eds. S. Sivasundaram, J. Vasundhara Devi, Zahia Drici and Farzana Mcrae), Vol. 3, Chapter 2, Cambridge Scientific Publishers, 2009. |
[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), 937-955.
doi: 10.3934/cpaa.2007.6.937. |
[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), 785-802.
doi: 10.3934/cpaa.2009.8.785. |
[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), 1491-1508. |
[22] |
J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. |
[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), 655-673.
doi: 10.3934/dcdsb.2010.14.655. |
[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., ().
|
[25] |
P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799.
doi: 10.1016/j.na.2006.09.035. |
[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), 3956-3963.
doi: 10.1016/j.na.2009.02.065. |
[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), 2012-2030.
doi: 10.1016/j.na.2010.11.008. |
[28] |
P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations, Stoch. Dyn., 3 (2003), 279-297.
doi: 10.1142/S0219493703000772. |
[29] |
M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427. |
[30] |
R. Temam, "Navier-Stokes Equations," Theory and Numerical Analysis, Revised edition, Studies in Mathematics and its Applications, 2, North Holland Publishig Co., Amsterdam-New York, 1979. |
[31] |
R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, SIAM, Philadelphia, PA, 1995. |
[32] |
Z. Yoshida and Y. Giga, A nonlinear semigroup approach to the Navier-Stokes system, Comm. in Partial Differential Equations, 9 (1984), 215-230. |
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), 411-436. |
[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), 761-781.
doi: 10.3934/dcdsb.2008.10.761. |
[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), 411-436, Adv. Nonlinear Stud., 10 (2010), 245-247. |
[4] |
T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[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), 263-268. |
[6] |
T. Caraballo, P. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays, J. Differential Equations, 239 (2007), 311-342. |
[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), 2869-2883. |
[8] |
T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.
doi: 10.1098/rspa.2001.0807. |
[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), 3181-3194.
doi: 10.1098/rspa.2003.1166. |
[10] |
T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297. |
[11] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. |
[12] |
P. Constantin, Near identity transformations for the Navier-Stokes equations, in "Handbook of Mathematical Fluid Dynamics," Vol. II, 117-141, North-Holland, Amsterdam, 2003.
doi: 10.1016/S1874-5792(03)80006-X. |
[13] |
F. Flandoli and B. Maslowski, Ergodicity of the $2$-D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 171 (1995), 119-141.
doi: 10.1007/BF02104513. |
[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., ().
|
[15] |
M. J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118.
doi: 10.1016/j.na.2005.05.057. |
[16] |
J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. |
[17] |
Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. |
[18] |
P. E. Kloeden, T. Caraballo, J. A. Langa, J. Real and J. Valero, The three dimensional globally modified Navier-Stokes equations, in "Mathematical Problems in Engineering Aerospace and Sciences" (eds. S. Sivasundaram, J. Vasundhara Devi, Zahia Drici and Farzana Mcrae), Vol. 3, Chapter 2, Cambridge Scientific Publishers, 2009. |
[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), 937-955.
doi: 10.3934/cpaa.2007.6.937. |
[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), 785-802.
doi: 10.3934/cpaa.2009.8.785. |
[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), 1491-1508. |
[22] |
J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod, Gauthier-Villars, Paris, 1969. |
[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), 655-673.
doi: 10.3934/dcdsb.2010.14.655. |
[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., ().
|
[25] |
P. Marín-Rubio and J. Real, Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains, Nonlinear Anal., 67 (2007), 2784-2799.
doi: 10.1016/j.na.2006.09.035. |
[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), 3956-3963.
doi: 10.1016/j.na.2009.02.065. |
[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), 2012-2030.
doi: 10.1016/j.na.2010.11.008. |
[28] |
P. Marín-Rubio and J. Robinson, Attractors for the stochastic 3D Navier-Stokes equations, Stoch. Dyn., 3 (2003), 279-297.
doi: 10.1142/S0219493703000772. |
[29] |
M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427. |
[30] |
R. Temam, "Navier-Stokes Equations," Theory and Numerical Analysis, Revised edition, Studies in Mathematics and its Applications, 2, North Holland Publishig Co., Amsterdam-New York, 1979. |
[31] |
R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66, SIAM, Philadelphia, PA, 1995. |
[32] |
Z. Yoshida and Y. Giga, A nonlinear semigroup approach to the Navier-Stokes system, Comm. in Partial Differential Equations, 9 (1984), 215-230. |
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