# 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 and 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), 217-225. doi: 10.1007/978-1-4614-7333-6_15. [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Amsterdam, North Holland, 1992. [3] V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Z. angew. Math. Phys., 54 (2003), 449-461. doi: 10.1007/s00033-003-1087-y. [4] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. I, Birkhäuser, Boston-Basel-Berlin, 1992. [5] T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays, Dyn. Partial Differ. Equ., 11 (2014), 345-359. doi: 10.4310/DPDE.2014.v11.n4.a3. [6] T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438. doi: 10.1006/jmaa.2000.7464. [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), 1775-1802. doi: 10.1098/rspa.2000.0586. [8] T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41. doi: 10.1016/j.jde.2003.09.008. [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), 51-77. doi: 10.3934/dcds.2014.34.51. [10] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010. [11] 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. [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), 3181-3194. doi: 10.1098/rspa.2003.1166. [13] 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. [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), 1130-1145. doi: 10.1016/j.jmaa.2007.01.038. [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), 125-144. [16] H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays, Proc. Indian Acad. Sci. (Math. Sci.), 122 (2012), 283-295. doi: 10.1007/s12044-012-0071-x. [17] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. [18] P. Constantin and C. Foias, Navier Stokes Equations, The University of Chicago Press, Chicago, 1988. [19] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [20] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eq., 9 (1995), 307-341. doi: 10.1007/BF02219225. [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), 4085-4105. doi: 10.3934/dcds.2014.34.4085. [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), 181-201. doi: 10.3934/dcds.2014.34.181. [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), 203-227. doi: 10.3934/dcds.2014.34.203. [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), 331-357. [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), 225-238. doi: 10.3934/dcdsb.2011.16.225. [26] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, AMS, Providence, 1988. [27] X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467. doi: 10.3934/dcds.2011.31.445. [28] X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbbZ^k$ in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254. doi: 10.1016/j.jmaa.2012.07.015. [29] P. E. Kloeden and M. Rasmussem, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [30] P. E. Kloeden, Pullback attractors in nonautonomous difference equations, J. Difference Eqns. Appl., 6 (2000), 33-52. doi: 10.1080/10236190008808212. [31] P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dynamics Continuous Discrete and Impulsive Systems, 4 (1998), 211-226. [32] P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), 141-152. doi: 10.1023/A:1019156812251. [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, World Scientific Series in Applicable Analysis, 4 (1995), 397-439. doi: 10.1142/9789812796417_0026. [34] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418. [35] J. L. Lions, Quelques Méthodes de Résolutions des Probèmes aux Limites non Linéaires, Paris; Dunod, Gauthier-Villars, 1969. [36] J. Málek and D. Pražák, Large time behavior via the Method of l-trajectories, J. Diferential Equations, 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. [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), 2012-2030. doi: 10.1016/j.na.2010.11.008. [38] B. S. Razumikhin, Application of Liapunov's method to problems in the stability of systems with a delay, Automat. i Telemeh., 21 (1960), 740-748. [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, T. Redrich and N. J. Kosch), 1992, 185-192. [40] G. Sell, Non-autonomous differential equations and topological dynamics I, Trans. Amer. Math. Soc., 127 (1967), 241-262. [41] T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997. [42] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis, 2nd. ed., North Holland, Amsterdam, 1979. [43] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [44] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd Ed., SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050. [45] L. Wan and Q. Zhou, Asymptotic behaviors of stochastic two-dimensional Navier-Stokes equations with finite memory, Journal of Mathematical Physics, 52 (2011), 042703, 15pp. doi: 10.1063/1.3574630. [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), 141-155. doi: 10.1016/j.na.2012.10.001.

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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), 217-225. doi: 10.1007/978-1-4614-7333-6_15. [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Amsterdam, North Holland, 1992. [3] V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Z. angew. Math. Phys., 54 (2003), 449-461. doi: 10.1007/s00033-003-1087-y. [4] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. I, Birkhäuser, Boston-Basel-Berlin, 1992. [5] T. Caraballo and X. Han, Stability of stationary solutions to 2D-Navier-Stokes models with delays, Dyn. Partial Differ. Equ., 11 (2014), 345-359. doi: 10.4310/DPDE.2014.v11.n4.a3. [6] T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays, J. Math. Anal. Appl., 260 (2001), 421-438. doi: 10.1006/jmaa.2000.7464. [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), 1775-1802. doi: 10.1098/rspa.2000.0586. [8] T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41. doi: 10.1016/j.jde.2003.09.008. [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), 51-77. doi: 10.3934/dcds.2014.34.51. [10] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010. [11] 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. [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), 3181-3194. doi: 10.1098/rspa.2003.1166. [13] 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. [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), 1130-1145. doi: 10.1016/j.jmaa.2007.01.038. [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), 125-144. [16] H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays, Proc. Indian Acad. Sci. (Math. Sci.), 122 (2012), 283-295. doi: 10.1007/s12044-012-0071-x. [17] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. [18] P. Constantin and C. Foias, Navier Stokes Equations, The University of Chicago Press, Chicago, 1988. [19] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [20] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eq., 9 (1995), 307-341. doi: 10.1007/BF02219225. [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), 4085-4105. doi: 10.3934/dcds.2014.34.4085. [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), 181-201. doi: 10.3934/dcds.2014.34.181. [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), 203-227. doi: 10.3934/dcds.2014.34.203. [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), 331-357. [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), 225-238. doi: 10.3934/dcdsb.2011.16.225. [26] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, AMS, Providence, 1988. [27] X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467. doi: 10.3934/dcds.2011.31.445. [28] X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on $\mathbbZ^k$ in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254. doi: 10.1016/j.jmaa.2012.07.015. [29] P. E. Kloeden and M. Rasmussem, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176. [30] P. E. Kloeden, Pullback attractors in nonautonomous difference equations, J. Difference Eqns. Appl., 6 (2000), 33-52. doi: 10.1080/10236190008808212. [31] P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dynamics Continuous Discrete and Impulsive Systems, 4 (1998), 211-226. [32] P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), 141-152. doi: 10.1023/A:1019156812251. [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, World Scientific Series in Applicable Analysis, 4 (1995), 397-439. doi: 10.1142/9789812796417_0026. [34] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418. [35] J. L. Lions, Quelques Méthodes de Résolutions des Probèmes aux Limites non Linéaires, Paris; Dunod, Gauthier-Villars, 1969. [36] J. Málek and D. Pražák, Large time behavior via the Method of l-trajectories, J. Diferential Equations, 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. [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), 2012-2030. doi: 10.1016/j.na.2010.11.008. [38] B. S. Razumikhin, Application of Liapunov's method to problems in the stability of systems with a delay, Automat. i Telemeh., 21 (1960), 740-748. [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, T. Redrich and N. J. Kosch), 1992, 185-192. [40] G. Sell, Non-autonomous differential equations and topological dynamics I, Trans. Amer. Math. Soc., 127 (1967), 241-262. [41] T. Taniguchi, The exponential behavior of Navier-Stokes equations with time delay external force, Discrete Contin. Dyn. Syst., 12 (2005), 997-1018. doi: 10.3934/dcds.2005.12.997. [42] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis, 2nd. ed., North Holland, Amsterdam, 1979. [43] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [44] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd Ed., SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050. [45] L. Wan and Q. Zhou, Asymptotic behaviors of stochastic two-dimensional Navier-Stokes equations with finite memory, Journal of Mathematical Physics, 52 (2011), 042703, 15pp. doi: 10.1063/1.3574630. [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), 141-155. doi: 10.1016/j.na.2012.10.001.
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