# American Institute of Mathematical Sciences

January  2014, 34(1): 181-201. doi: 10.3934/dcds.2014.34.181

## 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

Received  October 2012 Revised  February 2013 Published  June 2013

In this paper we strengthen some results on the existence and properties of pullback attractors for a non-autonomous 2D Navier-Stokes model with infinite delay. Actually we prove that under suitable assumptions, and thanks to regularity results, the attraction also happens in the $H^1$ norm for arbitrarily large finite intervals of time. Indeed, from comparison results of attractors we establish that all these families of attractors are in fact the same object. The tempered character of these families in $H^1$ is also analyzed.
Citation: 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
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [11] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar [12] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991).   Google Scholar [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.  Google Scholar [14] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod; Gauthier-Villars, (1969).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] M. Renardy, Initial value problems for viscoelastic liquids,, in, 195 (1984), 333.  doi: 10.1007/3-540-12916-2_65.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids,, Arch. Rational Mech. Anal., 62 (1976), 303.   Google Scholar [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.  Google Scholar [27] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", $2^{nd}$ edition, (1979).   Google Scholar [28] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [11] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar [12] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lecture Notes in Mathematics, 1473 (1991).   Google Scholar [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.  Google Scholar [14] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod; Gauthier-Villars, (1969).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] M. Renardy, Initial value problems for viscoelastic liquids,, in, 195 (1984), 333.  doi: 10.1007/3-540-12916-2_65.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids,, Arch. Rational Mech. Anal., 62 (1976), 303.   Google Scholar [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.  Google Scholar [27] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", $2^{nd}$ edition, (1979).   Google Scholar [28] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar
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