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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
Article

Abstract

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.

Keywords

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

Notes

Acknowledgements

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.

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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

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