This paper treats the existence of pullback attractors for a 2D Navier–Stokes model with finite delay formulated in [Caraballo and Real, J. Differential Equations 205 (2004), 271–297]. Actually, we carry out our study under less restrictive assumptions than in the previous reference. More precisely, we remove a condition on square integrable control of the memory terms, which allows us to consider a bigger class of delay terms. Here we show that the asymptotic compactness of the corresponding processes required to establish the existence of pullback attractors, obtained in [García-Luengo, Marín-Rubio and Real, Adv. Nonlinear Stud. 13 (2013), 331–357] by using an energy method, can be also proved by verifying the flattening property – also known as "Condition (C)". We deal with dynamical systems in suitable phase spaces within two metrics, the $ L^2 $ norm and the $ H^1 $ norm. Moreover, we provide results on the existence of pullback attractors for two possible choices of the attracted universes, namely, the standard one of fixed bounded sets, and secondly, one given by a tempered condition.
Citation: |
[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, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.
doi: 10.1016/j.crma.2005.12.015.![]() ![]() ![]() |
[3] |
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.![]() ![]() ![]() |
[4] |
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.![]() ![]() ![]() |
[5] |
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.![]() ![]() ![]() |
[6] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2012.
doi: 10.1007/978-1-4614-4581-4.![]() ![]() ![]() |
[7] |
L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.
![]() ![]() |
[8] |
S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317.
doi: 10.1016/0022-0396(88)90007-1.![]() ![]() ![]() |
[9] |
S.-N. Chow, K. Lu and G. R. Sell, Smoothness of inertial manifolds, J. Math. Anal. Appl., 169 (1992), 283-312.
doi: 10.1016/0022-247X(92)90115-T.![]() ![]() ![]() |
[10] |
C. Foias, O. Manley and R. Temam, Modelling of the interaction of small and large eddies in two-dimensional turbulent flows, RAIRO Modél. Math. Anal. Numér., 22 (1988), 93-118.
doi: 10.1051/m2an/1988220100931.![]() ![]() ![]() |
[11] |
C. Foias, O. P. Manley, R. Temam and Y. M. Trève, Asymptotic analysis of the Navier–Stokes equations, Phys. D, 9 (1983), 157-188.
doi: 10.1016/0167-2789(83)90297-X.![]() ![]() ![]() |
[12] |
C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353.
doi: 10.1016/0022-0396(88)90110-6.![]() ![]() ![]() |
[13] |
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.![]() ![]() ![]() |
[14] |
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.
![]() ![]() |
[15] |
J. García-Luengo, P. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal., 14 (2015), 1603-1621.
doi: 10.3934/cpaa.2015.14.1603.![]() ![]() ![]() |
[16] |
J. García-Luengo, P. Marín-Rubio, J. Real and J. C. Robinson, Pullback attractors for the non-autonomous 2D Navier–Stokes equations for minimally regular forcing, Discrete Contin. Dyn. Syst., 34 (2014), 203-227.
doi: 10.3934/dcds.2014.34.203.![]() ![]() ![]() |
[17] |
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.![]() ![]() ![]() |
[18] |
S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 225-238.
doi: 10.3934/dcdsb.2011.16.225.![]() ![]() ![]() |
[19] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
![]() ![]() |
[20] |
D. A. Jones and E. S. Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier–Stokes equations, Indiana Univ. Math. J., 42 (1993), 875-887.
doi: 10.1512/iumj.1993.42.42039.![]() ![]() ![]() |
[21] |
P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.
doi: 10.1098/rspa.2006.1753.![]() ![]() ![]() |
[22] |
P. E. Kloeden, J. A. Langa and J. Real, Pullback $V$-attractors of a 3-dimensional system of nonautonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955.
doi: 10.3934/cpaa.2007.6.937.![]() ![]() ![]() |
[23] |
Qi ngfeng Ma, Sh ouhong Wang and Ch enkui Zhong, Necessary and sufficient conditions for the existence of
global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.
doi: 10.1512/iumj.2002.51.2255.![]() ![]() ![]() |
[24] |
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.![]() ![]() ![]() |
[25] |
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.![]() ![]() ![]() |
[26] |
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., 11 (2011), 917-927.
![]() ![]() |
[27] |
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.![]() ![]() ![]() |
[28] |
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.![]() ![]() ![]() |
[29] |
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.![]() ![]() ![]() |
[30] |
P. Marín-Rubio and J. Real, Pullback attractors for $2D$-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006.
doi: 10.3934/dcds.2010.26.989.![]() ![]() ![]() |
[31] |
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.![]() ![]() ![]() |
[32] |
G. Planas and E. Hernández, Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations, Discrete Contin. Dyn. Syst., 21 (2008), 1245-1258.
doi: 10.3934/dcds.2008.21.1245.![]() ![]() ![]() |
[33] |
J. C. Robinson, Infinite-dimensional Dynamical Systems,, Cambridge University Press, Cambridge, 2001.
![]() ![]() |
[34] |
R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, 2$^{nd}$ edition, North Holland, Amsterdam, 1979.
![]() ![]() |
[35] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8.![]() ![]() ![]() |