# American Institute of Mathematical Sciences

December  2015, 8(6): 1079-1101. doi: 10.3934/dcdss.2015.8.1079

## A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 2 221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849

Received  June 2015 Revised  September 2015 Published  December 2015

In this survey paper we review several aspects related to Navier-Stokes models when some hereditary characteristics (constant, distributed or variable delay, memory, etc) appear in the formulation. First some results concerning existence and/or uniqueness of solutions are established. Next the local stability analysis of steady-state solutions is studied by using the theory of Lyapunov functions, the Razumikhin-Lyapunov technique and also by constructing appropriate Lyapunov functionals. A Gronwall-like lemma for delay equations is also exploited to provide some stability results. In the end we also include some comments concerning the global asymptotic analysis of the model, as well as some open questions and future lines for research.
Citation: Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079
##### References:
 [1] M. Anguiano, T. Caraballo, J. Real and J. Valero, Pullback attractors for nonautonomous dynamical systems,, Differential and Difference Eqns. with Apps., 47 (2013), 217.  doi: 10.1007/978-1-4614-7333-6_15.  Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Amsterdam, (1992).   Google Scholar [3] V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity,, Z. angew. Math. Phys., 54 (2003), 449.  doi: 10.1007/s00033-003-1087-y.  Google Scholar [4] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, Vol. I, (1992).   Google Scholar [5] T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays,, Dyn. Partial Differ. Equ., 11 (2014), 345.  doi: 10.4310/DPDE.2014.v11.n4.a3.  Google Scholar [6] T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays,, J. Math. Anal. Appl., 260 (2001), 421.  doi: 10.1006/jmaa.2000.7464.  Google Scholar [7] T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: Existence, uniqueness and asymptotic decay property,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 1775.  doi: 10.1098/rspa.2000.0586.  Google Scholar [8] T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays,, J. Differential Equations, 208 (2005), 9.  doi: 10.1016/j.jde.2003.09.008.  Google Scholar [9] T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems,, Discrete Contin. Dyn. Syst., 34 (2014), 51.  doi: 10.3934/dcds.2014.34.51.  Google Scholar [10] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity,, J. Difference Equ. Appl., 17 (2011), 161.  doi: 10.1080/10236198.2010.549010.  Google Scholar [11] T. Caraballo and J. Real, Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441.  doi: 10.1098/rspa.2001.0807.  Google Scholar [12] T. Caraballo and J. Real, Asymptotic behaviour of Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181.  doi: 10.1098/rspa.2003.1166.  Google Scholar [13] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar [14] T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations,, J. Math. Anal. Appl., 334 (2007), 1130.  doi: 10.1016/j.jmaa.2007.01.038.  Google Scholar [15] D. Cheban, P. E. Kloeden and B. Schmalfuss, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,, Nonlinear Dynamics and Systems Theory, 2 (2002), 125.   Google Scholar [16] H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays,, Proc. Indian Acad. Sci. (Math. Sci.), 122 (2012), 283.  doi: 10.1007/s12044-012-0071-x.  Google Scholar [17] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002).   Google Scholar [18] P. Constantin and C. Foias, Navier Stokes Equations,, The University of Chicago Press, (1988).   Google Scholar [19] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probability Theory and Related Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar [20] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eq., 9 (1995), 307.  doi: 10.1007/BF02219225.  Google Scholar [21] J. 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Planas, On a class of three dimensional Navier-Stokes equations with bounded delay,, Discret Cont. Dyn. Syst. Series B, 16 (2011), 225.  doi: 10.3934/dcdsb.2011.16.225.  Google Scholar [26] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988).   Google Scholar [27] X. Han, Exponential attractors for lattice dynamical systems in weighted spaces,, Discrete Contin. Dyn. Syst., 31 (2011), 445.  doi: 10.3934/dcds.2011.31.445.  Google Scholar [28] X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbbZ^k$ in weighted spaces,, J. Math. Anal. Appl., 397 (2013), 242.  doi: 10.1016/j.jmaa.2012.07.015.  Google Scholar [29] P. E. Kloeden and M. Rasmussem, Nonautonomous Dynamical Systems,, American Mathematical Society, (2011).  doi: 10.1090/surv/176.  Google Scholar [30] P. E. Kloeden, Pullback attractors in nonautonomous difference equations,, J. Difference Eqns. Appl., 6 (2000), 33.  doi: 10.1080/10236190008808212.  Google Scholar [31] P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics Continuous Discrete and Impulsive Systems, 4 (1998), 211.   Google Scholar [32] P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141.  doi: 10.1023/A:1019156812251.  Google Scholar [33] V. B. Kolmanovskii and L. E. Shaikhet, General method of Lyapunov functionals construction for stability investigations of stochastic difference equations,, in Dynamical Systems and Applications, 4 (1995), 397.  doi: 10.1142/9789812796417_0026.  Google Scholar [34] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge, (1991).  doi: 10.1017/CBO9780511569418.  Google Scholar [35] J. L. Lions, Quelques Méthodes de Résolutions des Probèmes aux Limites non Linéaires,, Paris; Dunod, (1969).   Google Scholar [36] J. Málek and D. Pražák, Large time behavior via the Method of l-trajectories,, J. Diferential Equations, 181 (2002), 243.  doi: 10.1006/jdeq.2001.4087.  Google Scholar [37] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, Nonlinear Analysis, 74 (2011), 2012.  doi: 10.1016/j.na.2010.11.008.  Google Scholar [38] B. S. Razumikhin, Application of Liapunov's method to problems in the stability of systems with a delay,, Automat. i Telemeh., 21 (1960), 740.   Google Scholar [39] B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations,, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, (1992), 185.   Google Scholar [40] G. Sell, Non-autonomous differential equations and topological dynamics I,, Trans. Amer. Math. Soc., 127 (1967), 241.   Google Scholar [41] T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force,, Discrete Contin. Dyn. Syst., 12 (2005), 997.  doi: 10.3934/dcds.2005.12.997.  Google Scholar [42] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979).   Google Scholar [43] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [44] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis,, 2nd Ed., (1995).  doi: 10.1137/1.9781611970050.  Google Scholar [45] L. Wan and Q. Zhou, Asymptotic behaviors of stochastic two-dimensional Navier-Stokes equations with finite memory,, Journal of Mathematical Physics, 52 (2011).  doi: 10.1063/1.3574630.  Google Scholar [46] S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces,, Nonlinear Anal., 78 (2013), 141.  doi: 10.1016/j.na.2012.10.001.  Google Scholar

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##### References:
 [1] M. Anguiano, T. Caraballo, J. Real and J. Valero, Pullback attractors for nonautonomous dynamical systems,, Differential and Difference Eqns. with Apps., 47 (2013), 217.  doi: 10.1007/978-1-4614-7333-6_15.  Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Amsterdam, (1992).   Google Scholar [3] V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity,, Z. angew. Math. Phys., 54 (2003), 449.  doi: 10.1007/s00033-003-1087-y.  Google Scholar [4] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems,, Vol. I, (1992).   Google Scholar [5] T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays,, Dyn. Partial Differ. Equ., 11 (2014), 345.  doi: 10.4310/DPDE.2014.v11.n4.a3.  Google Scholar [6] T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays,, J. Math. Anal. Appl., 260 (2001), 421.  doi: 10.1006/jmaa.2000.7464.  Google Scholar [7] T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: Existence, uniqueness and asymptotic decay property,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 1775.  doi: 10.1098/rspa.2000.0586.  Google Scholar [8] T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays,, J. Differential Equations, 208 (2005), 9.  doi: 10.1016/j.jde.2003.09.008.  Google Scholar [9] T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems,, Discrete Contin. Dyn. Syst., 34 (2014), 51.  doi: 10.3934/dcds.2014.34.51.  Google Scholar [10] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity,, J. Difference Equ. Appl., 17 (2011), 161.  doi: 10.1080/10236198.2010.549010.  Google Scholar [11] T. Caraballo and J. Real, Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441.  doi: 10.1098/rspa.2001.0807.  Google Scholar [12] T. Caraballo and J. Real, Asymptotic behaviour of Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181.  doi: 10.1098/rspa.2003.1166.  Google Scholar [13] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271.  doi: 10.1016/j.jde.2004.04.012.  Google Scholar [14] T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations,, J. Math. Anal. Appl., 334 (2007), 1130.  doi: 10.1016/j.jmaa.2007.01.038.  Google Scholar [15] D. Cheban, P. E. Kloeden and B. Schmalfuss, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,, Nonlinear Dynamics and Systems Theory, 2 (2002), 125.   Google Scholar [16] H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays,, Proc. Indian Acad. Sci. (Math. Sci.), 122 (2012), 283.  doi: 10.1007/s12044-012-0071-x.  Google Scholar [17] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002).   Google Scholar [18] P. Constantin and C. Foias, Navier Stokes Equations,, The University of Chicago Press, (1988).   Google Scholar [19] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probability Theory and Related Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar [20] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eq., 9 (1995), 307.  doi: 10.1007/BF02219225.  Google Scholar [21] J. García-Luengo, P. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D-Navier-Stokes model,, Discret Cont. Dyn. Syst., 34 (2014), 4085.  doi: 10.3934/dcds.2014.34.4085.  Google Scholar [22] J. García-Luengo, P. Marín-Rubio and J. Real, Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes with infinite delay,, Discret Cont. Dyn. Syst., 34 (2014), 181.  doi: 10.3934/dcds.2014.34.181.  Google Scholar [23] J. García-Luengo, P. Marín-Rubio, J. Real and J. Robinson, Pullback attractors for the non-autonomous 2D Navier-Stokes equations for minimally regular forcing,, Discret Cont. Dyn. Syst., 34 (2014), 203.  doi: 10.3934/dcds.2014.34.203.  Google Scholar [24] 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.   Google Scholar [25] S. M. Guzzo and G. Planas, On a class of three dimensional Navier-Stokes equations with bounded delay,, Discret Cont. Dyn. Syst. Series B, 16 (2011), 225.  doi: 10.3934/dcdsb.2011.16.225.  Google Scholar [26] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988).   Google Scholar [27] X. Han, Exponential attractors for lattice dynamical systems in weighted spaces,, Discrete Contin. Dyn. Syst., 31 (2011), 445.  doi: 10.3934/dcds.2011.31.445.  Google Scholar [28] X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbbZ^k$ in weighted spaces,, J. Math. Anal. Appl., 397 (2013), 242.  doi: 10.1016/j.jmaa.2012.07.015.  Google Scholar [29] P. E. Kloeden and M. Rasmussem, Nonautonomous Dynamical Systems,, American Mathematical Society, (2011).  doi: 10.1090/surv/176.  Google Scholar [30] P. E. Kloeden, Pullback attractors in nonautonomous difference equations,, J. Difference Eqns. Appl., 6 (2000), 33.  doi: 10.1080/10236190008808212.  Google Scholar [31] P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics Continuous Discrete and Impulsive Systems, 4 (1998), 211.   Google Scholar [32] P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141.  doi: 10.1023/A:1019156812251.  Google Scholar [33] V. B. Kolmanovskii and L. E. Shaikhet, General method of Lyapunov functionals construction for stability investigations of stochastic difference equations,, in Dynamical Systems and Applications, 4 (1995), 397.  doi: 10.1142/9789812796417_0026.  Google Scholar [34] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge, (1991).  doi: 10.1017/CBO9780511569418.  Google Scholar [35] J. L. Lions, Quelques Méthodes de Résolutions des Probèmes aux Limites non Linéaires,, Paris; Dunod, (1969).   Google Scholar [36] J. Málek and D. Pražák, Large time behavior via the Method of l-trajectories,, J. Diferential Equations, 181 (2002), 243.  doi: 10.1006/jdeq.2001.4087.  Google Scholar [37] P. Marín-Rubio, J. Real and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,, Nonlinear Analysis, 74 (2011), 2012.  doi: 10.1016/j.na.2010.11.008.  Google Scholar [38] B. S. Razumikhin, Application of Liapunov's method to problems in the stability of systems with a delay,, Automat. i Telemeh., 21 (1960), 740.   Google Scholar [39] B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations,, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, (1992), 185.   Google Scholar [40] G. Sell, Non-autonomous differential equations and topological dynamics I,, Trans. Amer. Math. Soc., 127 (1967), 241.   Google Scholar [41] T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force,, Discrete Contin. Dyn. Syst., 12 (2005), 997.  doi: 10.3934/dcds.2005.12.997.  Google Scholar [42] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979).   Google Scholar [43] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar [44] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis,, 2nd Ed., (1995).  doi: 10.1137/1.9781611970050.  Google Scholar [45] L. Wan and Q. Zhou, Asymptotic behaviors of stochastic two-dimensional Navier-Stokes equations with finite memory,, Journal of Mathematical Physics, 52 (2011).  doi: 10.1063/1.3574630.  Google Scholar [46] S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces,, Nonlinear Anal., 78 (2013), 141.  doi: 10.1016/j.na.2012.10.001.  Google Scholar
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