# 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, 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), 484-498. 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), 2441-2453. 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), 3181-3194. 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), 271-297. 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), 307-341. 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), 4882-4887. 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), 4333-4356. 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), 331-357. Google Scholar [9] S. Gatti, C. Giorgi and V. Pata, Navier-Stokes limit of Jeffreys type flows, Phys. D, 203 (2005), 55-79. 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), 1293-1320.  Google Scholar [11] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.  Google Scholar [12] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 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), 203-218. doi: 10.1007/BF00281589.  Google Scholar [14] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod; Gauthier-Villars, Paris, 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), 1058-1068. 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), 779-796. 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), 3956-3963. 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), 2012-2030. 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), 4238-4247. 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), 229-244. 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), 280-292. doi: 10.1016/0022-0396(83)90053-0.  Google Scholar [22] M. Renardy, Initial value problems for viscoelastic liquids, in "Trends and Applications of Pure Mathematics to Mechanics" (Palaiseau, 1983), Lecture Notes in Phys. 195, Springer, Berlin, (1984), 333-345. 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), 333-349. 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, Cambridge University Press, Cambridge, 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), 303-321.  Google Scholar [26] M. Slemrod, Existence, uniqueness, stability for a simple fluid with fading memory, Bull. Amer. Math. Soc., 82 (1976), 581-583. doi: 10.1090/S0002-9904-1976-14113-4.  Google Scholar [27] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," $2^{nd}$ edition, North Holland, Amsterdam, 1979.  Google Scholar [28] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 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), 484-498. 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), 2441-2453. 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), 3181-3194. 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), 271-297. 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), 307-341. 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), 4882-4887. 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), 4333-4356. 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), 331-357. Google Scholar [9] S. Gatti, C. Giorgi and V. Pata, Navier-Stokes limit of Jeffreys type flows, Phys. D, 203 (2005), 55-79. 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), 1293-1320.  Google Scholar [11] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.  Google Scholar [12] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 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), 203-218. doi: 10.1007/BF00281589.  Google Scholar [14] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod; Gauthier-Villars, Paris, 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), 1058-1068. 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), 779-796. 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), 3956-3963. 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), 2012-2030. 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), 4238-4247. 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), 229-244. 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), 280-292. doi: 10.1016/0022-0396(83)90053-0.  Google Scholar [22] M. Renardy, Initial value problems for viscoelastic liquids, in "Trends and Applications of Pure Mathematics to Mechanics" (Palaiseau, 1983), Lecture Notes in Phys. 195, Springer, Berlin, (1984), 333-345. 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), 333-349. 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, Cambridge University Press, Cambridge, 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), 303-321.  Google Scholar [26] M. Slemrod, Existence, uniqueness, stability for a simple fluid with fading memory, Bull. Amer. Math. Soc., 82 (1976), 581-583. doi: 10.1090/S0002-9904-1976-14113-4.  Google Scholar [27] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," $2^{nd}$ edition, North Holland, Amsterdam, 1979.  Google Scholar [28] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar
 [1] 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, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203 [2] 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 [3] Julia García-Luengo, Pedro Marín-Rubio. Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2127-2146. doi: 10.3934/cpaa.2020094 [4] Yutaka Tsuzuki. Solvability of generalized nonlinear heat equations with constraints coupled with Navier--Stokes equations in 2D domains. Conference Publications, 2015, 2015 (special) : 1079-1088. doi: 10.3934/proc.2015.1079 [5] C. Foias, M. S Jolly, O. P. Manley. Recurrence in the 2-$D$ Navier--Stokes equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 253-268. doi: 10.3934/dcds.2004.10.253 [6] Pedro Marín-Rubio, José Real. Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 989-1006. doi: 10.3934/dcds.2010.26.989 [7] 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 [8] Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779 [9] Songsong Lu, Hongqing Wu, Chengkui Zhong. Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 701-719. doi: 10.3934/dcds.2005.13.701 [10] Luca Bisconti, Davide Catania. Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 59-75. doi: 10.3934/dcdsb.2015.20.59 [11] Wenlong Sun. The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay. Electronic Research Archive, 2020, 28 (3) : 1343-1356. doi: 10.3934/era.2020071 [12] Patrick Penel, Milan Pokorný. Improvement of some anisotropic regularity criteria for the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1401-1407. doi: 10.3934/dcdss.2013.6.1401 [13] Daomin Cao, Xiaoya Song, Chunyou Sun. Pullback attractors for 2D MHD equations on time-varying domains. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021132 [14] Milan Pokorný, Piotr B. Mucha. 3D steady compressible Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 151-163. doi: 10.3934/dcdss.2008.1.151 [15] Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5421-5448. doi: 10.3934/dcdsb.2020352 [16] Yutaka Tsuzuki. Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains. Evolution Equations & Control Theory, 2014, 3 (1) : 191-206. doi: 10.3934/eect.2014.3.191 [17] J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 [18] Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080 [19] Ruihong Ji, Yongfu Wang. Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1117-1133. doi: 10.3934/dcds.2019047 [20] Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408

2020 Impact Factor: 1.392