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

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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.

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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. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Linfang Liu

  2. Depto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012, Seville, Spain

    Linfang Liu & Tomás Caraballo

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  1. Linfang Liu
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  2. Tomás Caraballo
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Correspondence to Linfang Liu.

Additional information

Partially supported by the Projects MTM2015-63723-P (MINECO/ FEDER, EU) and P12-FQM-1492 (Junta de Andalucía), and by NSF of China (Nos. 11671142, 11371087, 11571125 and 11871022), Science and Technology Commission of Shanghai Municipality (No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407), respectively.

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Liu, L., Caraballo, T. Analysis of a Stochastic 2D-Navier–Stokes Model with Infinite Delay. J Dyn Diff Equat 31, 2249–2274 (2019). https://doi.org/10.1007/s10884-018-9703-x

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  • DOI: https://doi.org/10.1007/s10884-018-9703-x

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