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Random Kick-Forced 3D Navier–Stokes Equations in a Thin Domain

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Abstract

We consider the Navier–Stokes equations in the thin 3D domain \({\mathbb{T}}_2 \times (0, \epsilon)\) , where \({\mathbb{T}}_2\) is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly, when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as ε → 0) to a unique stationary measure for the Navier–Stokes equation on \({\mathbb{T}}_2\) . Thus, the 2D Navier–Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains.

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody Sq., Kharkov, 61077, Ukraine

    Igor Chueshov

  2. Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, UK

    Sergei Kuksin

Authors
  1. Igor Chueshov
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  2. Sergei Kuksin
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Correspondence to Igor Chueshov.

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Communicated by V. Sverak

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Chueshov, I., Kuksin, S. Random Kick-Forced 3D Navier–Stokes Equations in a Thin Domain. Arch Rational Mech Anal 188, 117–153 (2008). https://doi.org/10.1007/s00205-007-0068-2

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  • DOI: https://doi.org/10.1007/s00205-007-0068-2

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