Regularity of pullback attractors and attraction in H^1 in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay

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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

Julia García-Luengo ^1, , Pedro Marín-Rubio ^2, and José Real ^3,

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

Abstract
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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.

Keywords: infinite delay, pullback attractors, regularity., 2D Navier--Stokes equations.

Mathematics Subject Classification: 35B41, 35B65, 35Q30, 35R1.

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 and 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.
[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.
[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.
[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.
[5]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. 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), 4882-4887. 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), 4333-4356. 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), 331-357.
[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.
[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.
[11]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
[12]	Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 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), 203-218. doi: 10.1007/BF00281589.
[14]	J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod; Gauthier-Villars, Paris, 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), 1058-1068. 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), 779-796. 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), 3956-3963. 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), 2012-2030. 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), 4238-4247. 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), 229-244. 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), 280-292. doi: 10.1016/0022-0396(83)90053-0.
[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.
[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.
[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.
[25]	M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal., 62 (1976), 303-321.
[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.
[27]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," $2^{nd}$ edition, North Holland, Amsterdam, 1979.
[28]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 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), 484-498. 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), 2441-2453. 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), 3181-3194. 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), 271-297. doi: 10.1016/j.jde.2004.04.012.
[5]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. 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), 4882-4887. 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), 4333-4356. 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), 331-357.
[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.
[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.
[11]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
[12]	Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 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), 203-218. doi: 10.1007/BF00281589.
[14]	J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod; Gauthier-Villars, Paris, 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), 1058-1068. 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), 779-796. 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), 3956-3963. 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), 2012-2030. 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), 4238-4247. 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), 229-244. 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), 280-292. doi: 10.1016/0022-0396(83)90053-0.
[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.
[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.
[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.
[25]	M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal., 62 (1976), 303-321.
[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.
[27]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," $2^{nd}$ edition, North Holland, Amsterdam, 1979.
[28]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

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