Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 2249–2274 | Cite as

Analysis of a Stochastic 2D-Navier–Stokes Model with Infinite Delay

  • Linfang LiuEmail author
  • Tomás Caraballo


Some results concerning a stochastic 2D Navier–Stokes system when the external forces contain hereditary characteristics are established. The existence and uniqueness of solutions in the case of unbounded (infinite) delay are first proved by using the classical technique of Galerkin approximations. The local stability analysis of constant solutions (equilibria) is also carried out by exploiting two approaches. Namely, the Lyapunov function method and by constructing appropriate Lyapunov functionals. The asymptotic stability and hence, the uniqueness of equilibrium solution are obtained by constructing Lyapunov functionals. Moreover, some sufficient conditions ensuring the polynomial stability of the equilibrium solution in a particular case of unbounded variable delay will be provided. Exponential stability for other special cases of infinite delay remains as an open problem.


Stochastic Navier–Stokes equation Equilibrium solution Polynomial stability Unbounded variable delay 



We would like to thank the referee for the helpful and valuable comments, remarks and suggestions which allowed us to greatly improve the presentation of our paper.


  1. 1.
    Appleby, J.A.D., Buckwar, E.: Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation. Electron. J. Qual. Theory Differ. Equ. 2, 32 (2016)zbMATHGoogle Scholar
  2. 2.
    Babin, A., Mahalov, A., Nicolaenko, B.: Global regularity of 3D rotating Navier–Stokes equations for resonant domains. Appl. Math. Lett. 13(4), 51–57 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Babin, A.V.: Attractors of Navier-Stokes equations. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. 2, pp. 169–222. North-Holland, Amsterdam (2003)CrossRefGoogle Scholar
  4. 4.
    Babin, A., Mahalov, A., Nicolaenko, B.: Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48(3), 1133–1176 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Caraballo, T., Márquez-Durán, A.M., Real, J.: Asymptotic behaviour of the three-dimensional \(\alpha \)-Navier–Stokes model with delays. J. Math. Anal. Appl. 340(1), 410–423 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Caraballo, T., Márquez-Durán, A.M., Real, J.: Asymptotic behaviour of the three-dimensional \(\alpha \)-Navier–Stokes model with locally Lipschitz delay forcing terms. Nonlinear Anal. 71(12), e271–e282 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Caraballo, T., Real, J.: Attractors for 2D-Navier–Stokes models with delays. J. Differ. Equ. 205(2), 271–297 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Caraballo, T., Real, J.: Asymptotic behaviour and attractors of 2D-Navier–Stokes models with delays. In: EQUADIFF 2003, pp. 827–832. World Scientific Publications, Hackensack, NJ (2005)Google Scholar
  9. 9.
    Caraballo, T., Real, J., Shaikhet, L.: Method of Lyapunov functionals construction in stability of delay evolution equations. J. Math. Anal. Appl. 334(2), 1130–1145 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Caraballo, T., Han, X.: Stability of stationary solutions to 2D-Navier–Stokes models with delays. Dyn. Partial Differ. Equ. 11(4), 345–359 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Caraballo, T., Han, X.: A survey on Navier–Stokes models with delays: existence, uniqueness and asymptotic behavior of solutions. Discrete Contin. Dyn. Syst. Ser. S 8(6), 1079–1101 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Caraballo, T., Kloeden, P., Schmalfuß, B.: Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl. Math. Optim. 50(3), 4183–207 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Caraballo, T., Langa, J.A., Taniguchi, T.: The exponential behaviour and stabilizability of stochastic 2D-Navier–Stokes equations. J. Differ. Equ. 179(2), 714–737 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Caraballo, T., Real, J.: Navier-Stokes equations with delays. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457(2014), 2441–2453 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Caraballo, T., Real, J.: Asymptotic behaviour of two-dimensional Navier–Stokes equations with delays. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459(2040), 3181–3194 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Caraballo, T., Shaikhet, L.: Stability of delay evolution equations with stochastic perturbations. Commun. Pure Appl. Anal. 13(5), 2095–2113 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Chen, H.: Asymptotic behavior of stochastic two-dimensional Navier–Stokes equations with delays. Proc. Indian Acad. Sci. Math. Sci. 122(2), 283–295 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Constantin, P., Foias, C.: Navier–Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988)Google Scholar
  19. 19.
    Foias, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Encyclopedia of Mathematics and its Applications, vol. 83. Cambridge University Press, Cambridge (2001)zbMATHCrossRefGoogle Scholar
  20. 20.
    García-Luengo, J., Marín-Rubio, P., Real, J.: Pullback attractors for 2D Navier–Stokes equations with delays and their regularity. Adv. Nonlinear Stud. 13(2), 331–357 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    García-Luengo, J., Marín-Rubio, P., Real, J.: Regularity of pullback attractors and attraction in \(H^1\) in arbitrarily large finite intervals for 2D Navier–Stokes equations with infinite delay. Discrete Contin. Dyn. Syst. 34(1), 181–201 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Hale, J.K.: Retarded equations with infinite delays. In: Functional differential equations and approximation of fixed points (Proceedings of Summer School and Conference, University Bonn, Bonn, 1978), volume 730 of Lecture Notes in Mathematics, pp. 157–193. Springer, Berlin (1979)CrossRefGoogle Scholar
  23. 23.
    Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21(1), 11–41 (1978)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hino, Y., Murakami, S., Naito, T.: Functional-Differential Equations with Infinite Delay. Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991)zbMATHCrossRefGoogle Scholar
  25. 25.
    Kato, T.: Asymptotic behavior of solutions of the functional differential equation \(y^{^{\prime } }(x)=ay(\lambda x)+by(x)\), pp. 197–217 (1972)Google Scholar
  26. 26.
    Kato, T., McLeod, J.B.: The functional-differential equation \(y^{\prime } \,(x)=ay(\lambda x)+by(x)\). Bull. Am. Math. Soc. 77, 891–937 (1971). (Lyapunov’s methods in stability and control)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Kolmanovskii, V., Shaikhet, L.: Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results. Math. Comput. Modell. 36(6), 691–716 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Liu, L., Caraballo, T., Marín Rubio, P.: Asymptotic behavior of 2d-Navier–Stokes equations with infinite delay. J. Differ. Equ. (Accepted)Google Scholar
  29. 29.
    Marín-Rubio, P., Márquez-Durán, A.M., Real, J.: Three dimensional system of globally modified Navier–Stokes equations with infinite delays. Discrete Contin. Dyn. Syst. Ser. B 14(2), 655–673 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Marín-Rubio, P., Márquez-Durán, A.M., Real, J.: Pullback attractors for globally modified Navier–Stokes equations with infinite delays. Discrete Contin. Dyn. Syst. 31(3), 779–796 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Marín-Rubio, P., Real, J., Valero, J.: Pullback attractors for a two-dimensional Navier–Stokes model in an infinite delay case. Nonlinear Anal. 74(5), 2012–2030 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Read, C.C.: Markov Inequality. Encyclopedia of Statistical Sciences, vol. 7. Wiley, New York (2006)Google Scholar
  33. 33.
    Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Cham (2013)zbMATHCrossRefGoogle Scholar
  34. 34.
    Sritharan, S.S., Sundar, P.: Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise. Stoch. Process. Appl. 116(11), 1636–1659 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Taniguchi, T.: The existence and asymptotic behaviour of energy solutions to stochastic 2D functional Navier–Stokes equations driven by Levy processes. J. Math. Anal. Appl. 385(2), 634–654 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Temam, R.: Navier-Stokes Equations. AMS Chelsea Publishing, Providence (2001). (Theory and numerical analysis, Reprint of the 1984 edition)zbMATHCrossRefGoogle Scholar
  37. 37.
    Wei, M.J., Zhang, T.: Exponential stability for stochastic 2D-Navier–Stokes equations with time delay. Appl. Math. J. Chin. Univ. Ser. A 24(4), 493–500 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anPeople’s Republic of China
  2. 2.Depto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de MatemáticasUniversidad de SevillaSevilleSpain

Personalised recommendations