April  2020, 19(4): 2127-2146. doi: 10.3934/cpaa.2020094

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

Dedicated to Professor Tomás Caraballo on occasion of his Sixtieth Birthday

Received  May 2019 Revised  September 2019 Published  January 2020

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: 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
References:
[1]

T. CaraballoG. 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. CaraballoG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.   Google Scholar

[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.  Google Scholar

[9]

S.-N. ChowK. 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.  Google Scholar

[10]

C. FoiasO. 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.  Google Scholar

[11]

C. FoiasO. P. ManleyR. 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.  Google Scholar

[12]

C. FoiasG. 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.  Google Scholar

[13]

J. García-LuengoP. 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

[14]

J. García-LuengoP. 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

[15]

J. García-LuengoP. 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.  Google Scholar

[16]

J. García-LuengoP. Marín-RubioJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

P. E. KloedenJ. 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.  Google Scholar

[23]

Qi ngfeng MaSh 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.  Google Scholar

[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.  Google Scholar

[25]

P. Marín-RubioA. 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.  Google Scholar

[26]

P. Marín-RubioA. 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.   Google Scholar

[27]

P. Marín-RubioA. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

P. Marín-RubioJ. 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

[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.  Google Scholar

[33] J. C. Robinson, Infinite-dimensional Dynamical Systems,, Cambridge University Press, Cambridge, 2001.   Google Scholar
[34]

R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, 2$^{nd}$ edition, North Holland, Amsterdam, 1979.  Google Scholar

[35]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

show all references

References:
[1]

T. CaraballoG. 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. CaraballoG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.   Google Scholar

[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.  Google Scholar

[9]

S.-N. ChowK. 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.  Google Scholar

[10]

C. FoiasO. 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.  Google Scholar

[11]

C. FoiasO. P. ManleyR. 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.  Google Scholar

[12]

C. FoiasG. 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.  Google Scholar

[13]

J. García-LuengoP. 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

[14]

J. García-LuengoP. 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

[15]

J. García-LuengoP. 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.  Google Scholar

[16]

J. García-LuengoP. Marín-RubioJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

P. E. KloedenJ. 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.  Google Scholar

[23]

Qi ngfeng MaSh 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.  Google Scholar

[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.  Google Scholar

[25]

P. Marín-RubioA. 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.  Google Scholar

[26]

P. Marín-RubioA. 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.   Google Scholar

[27]

P. Marín-RubioA. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

P. Marín-RubioJ. 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

[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.  Google Scholar

[33] J. C. Robinson, Infinite-dimensional Dynamical Systems,, Cambridge University Press, Cambridge, 2001.   Google Scholar
[34]

R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, 2$^{nd}$ edition, North Holland, Amsterdam, 1979.  Google Scholar

[35]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[1]

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

[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, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203

[4]

Pedro Marín-Rubio, José Real. Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 989-1006. doi: 10.3934/dcds.2010.26.989

[5]

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

[6]

Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779

[7]

Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068

[8]

Songsong Lu, Hongqing Wu, Chengkui Zhong. Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 701-719. doi: 10.3934/dcds.2005.13.701

[9]

P.E. Kloeden, José A. Langa, José Real. Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 937-955. doi: 10.3934/cpaa.2007.6.937

[10]

Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172

[11]

Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559

[12]

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

[13]

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

[14]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647

[15]

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

[16]

Ruihong Ji, Yongfu Wang. Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1117-1133. doi: 10.3934/dcds.2019047

[17]

Hongyong Cui, Mirelson M. Freitas, José A. Langa. Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1297-1324. doi: 10.3934/dcdsb.2018152

[18]

Igor Kukavica. Interior gradient bounds for the 2D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 873-882. doi: 10.3934/dcds.2001.7.873

[19]

Pierre Fabrie, C. Galusinski. Exponential attractors for the slightly compressible 2D-Navier-Stokes. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 315-348. doi: 10.3934/dcds.1996.2.315

[20]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (53)
  • HTML views (74)
  • Cited by (0)

Other articles
by authors

[Back to Top]