Abstract
We consider a Navier–Stokes–Voigt fluid model where the instantaneous kinematic viscosity has been completely replaced by a memory term incorporating hereditary effects, in presence of Ekman damping. Unlike the classical Navier–Stokes–Voigt system, the energy balance involves the spatial gradient of the past history of the velocity rather than providing an instantaneous control on the high modes. In spite of this difficulty, we show that our system is dissipative in the dynamical systems sense and even possesses regular global and exponential attractors of finite fractal dimension. Such features of asymptotic well-posedness in absence of instantaneous high modes dissipation appear to be unique within the realm of dynamical systems arising from fluid models.
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Acknowledgements
F. Di Plinio was partially supported by the National Science Foundation under the Grant Nos. NSF-DMS-1500449 and NSF-DMS-1650810. R. Temam was supported by NSF DMS Grant 1510249 and by the Research Fund of Indiana University.
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Communicated by Edriss S. Titi.
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Di Plinio, F., Giorgini, A., Pata, V. et al. Navier–Stokes–Voigt Equations with Memory in 3D Lacking Instantaneous Kinematic Viscosity. J Nonlinear Sci 28, 653–686 (2018). https://doi.org/10.1007/s00332-017-9422-1
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DOI: https://doi.org/10.1007/s00332-017-9422-1