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Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing
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Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay
1. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain |
2. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla |
3. | 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,, Nonlinear Anal., 64 (2006), 484.
doi: 10.1016/j.na.2005.03.111. |
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.
doi: 10.1007/BF02219225. |
[6] |
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,, Nonlinear Anal., 74 (2011), 4882.
doi: 10.1016/j.na.2011.04.063. |
[7] |
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. |
[8] |
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. Google Scholar |
[9] |
S. Gatti, C. Giorgi and V. Pata, Navier-Stokes limit of Jeffreys type flows,, Phys. D, 203 (2005), 55.
doi: 10.1016/j.physd.2005.03.007. |
[10] |
C. Guillopé and R. Talhouk, Steady flows of slightly compressible viscoelastic fluids of Jeffreys' type around an obstacle,, Differential Integral Equations, 16 (2003), 1293.
|
[11] |
J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.
|
[12] |
Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991).
|
[13] |
E. F. Infante and J. A. Walker, A stability investigation for an incompressible simple fluid with fading memory,, Arch. Rational Mech. Anal., 72 (1980), 203.
doi: 10.1007/BF00281589. |
[14] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod; Gauthier-Villars, (1969).
|
[15] |
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. |
[16] |
P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays,, Discrete Contin. Dyn. Syst., 31 (2011), 779.
doi: 10.3934/dcds.2011.31.779. |
[17] |
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. |
[18] |
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. |
[19] |
S. Nadeem and S. Akram, Peristaltic flow of a Jeffrey fluid in a rectangular duct,, Nonlinear Anal. Real World Appl., 11 (2010), 4238.
doi: 10.1016/j.nonrwa.2010.05.010. |
[20] |
M. Renardy, Local existence theorems for the first and second initial-boundary value problems for a weakly non-Newtonian fluid,, Arch. Rational Mech. Anal., 83 (1983), 229.
doi: 10.1007/BF00251510. |
[21] |
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. |
[22] |
M. Renardy, Initial value problems for viscoelastic liquids,, in, 195 (1984), 333.
doi: 10.1007/3-540-12916-2_65. |
[23] |
M. Renardy, Singularly perturbed hyperbolic evolution problems with infinite delay and an application to polymer rheology,, SIAM J. Math. Anal., 15 (1984), 333.
doi: 10.1137/0515026. |
[24] |
J. C. Robinson, "Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,", Cambridge Texts in Applied Mathematics, (2001).
doi: 10.1007/978-94-010-0732-0. |
[25] |
M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids,, Arch. Rational Mech. Anal., 62 (1976), 303.
|
[26] |
M. Slemrod, Existence, uniqueness, stability for a simple fluid with fading memory,, Bull. Amer. Math. Soc., 82 (1976), 581.
doi: 10.1090/S0002-9904-1976-14113-4. |
[27] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", $2^{nd}$ edition, (1979).
|
[28] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988).
doi: 10.1007/978-1-4684-0313-8. |
show all references
References:
[1] |
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. |
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.
doi: 10.1007/BF02219225. |
[6] |
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,, Nonlinear Anal., 74 (2011), 4882.
doi: 10.1016/j.na.2011.04.063. |
[7] |
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. |
[8] |
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. Google Scholar |
[9] |
S. Gatti, C. Giorgi and V. Pata, Navier-Stokes limit of Jeffreys type flows,, Phys. D, 203 (2005), 55.
doi: 10.1016/j.physd.2005.03.007. |
[10] |
C. Guillopé and R. Talhouk, Steady flows of slightly compressible viscoelastic fluids of Jeffreys' type around an obstacle,, Differential Integral Equations, 16 (2003), 1293.
|
[11] |
J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.
|
[12] |
Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991).
|
[13] |
E. F. Infante and J. A. Walker, A stability investigation for an incompressible simple fluid with fading memory,, Arch. Rational Mech. Anal., 72 (1980), 203.
doi: 10.1007/BF00281589. |
[14] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod; Gauthier-Villars, (1969).
|
[15] |
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. |
[16] |
P. Marín-Rubio, A. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays,, Discrete Contin. Dyn. Syst., 31 (2011), 779.
doi: 10.3934/dcds.2011.31.779. |
[17] |
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. |
[18] |
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. |
[19] |
S. Nadeem and S. Akram, Peristaltic flow of a Jeffrey fluid in a rectangular duct,, Nonlinear Anal. Real World Appl., 11 (2010), 4238.
doi: 10.1016/j.nonrwa.2010.05.010. |
[20] |
M. Renardy, Local existence theorems for the first and second initial-boundary value problems for a weakly non-Newtonian fluid,, Arch. Rational Mech. Anal., 83 (1983), 229.
doi: 10.1007/BF00251510. |
[21] |
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. |
[22] |
M. Renardy, Initial value problems for viscoelastic liquids,, in, 195 (1984), 333.
doi: 10.1007/3-540-12916-2_65. |
[23] |
M. Renardy, Singularly perturbed hyperbolic evolution problems with infinite delay and an application to polymer rheology,, SIAM J. Math. Anal., 15 (1984), 333.
doi: 10.1137/0515026. |
[24] |
J. C. Robinson, "Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,", Cambridge Texts in Applied Mathematics, (2001).
doi: 10.1007/978-94-010-0732-0. |
[25] |
M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids,, Arch. Rational Mech. Anal., 62 (1976), 303.
|
[26] |
M. Slemrod, Existence, uniqueness, stability for a simple fluid with fading memory,, Bull. Amer. Math. Soc., 82 (1976), 581.
doi: 10.1090/S0002-9904-1976-14113-4. |
[27] |
R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", $2^{nd}$ edition, (1979).
|
[28] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988).
doi: 10.1007/978-1-4684-0313-8. |
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