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Theoretical and numerical results for some bi-objective optimal control problems
Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property
1. | Departamento de Matemática Aplicada a las TIC, Universidad Politécnica de Madrid, C/ Nikola Tesla s/n, 28031, Madrid, Spain |
2. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/ Tarfia s/n, 41012, Sevilla, Spain |
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.
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, 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. |
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, 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. |
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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 and Applied Analysis, 2015, 14 (5) : 1603-1621. doi: 10.3934/cpaa.2015.14.1603 |
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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 and Continuous Dynamical Systems, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203 |
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