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Attractors for Navier–Stokes equation with fractional operator in the memory term

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Abstract

In this paper, we study the long-time dynamics of Navier–Stokes equations with a fractional operator in the memory term and external forces

$$\begin{aligned} \partial _tu-\nu \Delta u +\displaystyle \int _{0}^{\infty }g(s)(-\Delta )^\alpha u(t-s)\,\mathrm{d}s +(u\cdot \nabla )u+\nabla p=\epsilon f(x) \end{aligned}$$

on a bounded domain \(\Omega \) in \(\mathbb {R}^2\) with smooth boundary. We establish the existence of global attractor for the associated dynamical system. We prove the continuity of global attractors with respect to forcing parameter \(\epsilon \) in a residual dense set. Moreover, the upper-semicontinuity with respect to parameter \(\epsilon \) of global attractors is shown.

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Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors would like to thank the referee for his critical review and valuable comments which allowed to improve the paper.

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Authors and Affiliations

  1. University of Pará, Raimundo Santana Street s/n, Salinópolis, PA, 68721-000, Brazil

    M. M. Freitas

  2. Institute of Exact and Natural Sciences, Doctoral Program in Mathematics, Federal University of Pará, Augusto corrêa Street, Number 01, Belém, PA, 66075-110, Brazil

    G. M. Araújo & J. S. Fonseca

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  1. M. M. Freitas
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  2. G. M. Araújo
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  3. J. S. Fonseca
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Freitas, M.M., Araújo, G.M. & Fonseca, J.S. Attractors for Navier–Stokes equation with fractional operator in the memory term. Ann Univ Ferrara 67, 269–284 (2021). https://doi.org/10.1007/s11565-021-00373-7

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