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Global attractors and determining modes for the 3D Navier-Stokes-Voight equations

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Abstract

The authors investigate the long-term dynamics of the three-dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid. Specifically, upper bounds for the number of determining modes are derived for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view the authors consider the Navier-Stokes-Voight model as a non-viscous (inviscid) regularization of the three-dimensional Navier-Stokes equations. Furthermore, it is also shown that the weak solutions of the Navier-Stokes-Voight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient ν → 0.

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Authors and Affiliations

  1. Department of Mathematics, Koç University, Rumelifeneri Yolu, Sariyer, 34450, Istanbul, Turkey

    Varga K. Kalantarov

  2. Department of Mathematics and Department of Mechanical and Aerospace Engineering, University of California, Irvine, California, 92697, USA

    Edriss S. Titi

  3. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Edriss S. Titi

Authors
  1. Varga K. Kalantarov
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  2. Edriss S. Titi
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Correspondence to Varga K. Kalantarov.

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Dedicated to Professor Andrew Majda on the Occasion of his 60th Birthday with Friendship and Admiration

Project supported by the Scientific and Research Council of Turkey (No. 106T337), the ISF Grant (No. 120/6), the BSF Grant (No. 2004271), and the National Science Foundation (Nos. DMS-0504619, DMS-0708832).

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Kalantarov, V.K., Titi, E.S. Global attractors and determining modes for the 3D Navier-Stokes-Voight equations. Chin. Ann. Math. Ser. B 30, 697–714 (2009). https://doi.org/10.1007/s11401-009-0205-3

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