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Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity
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  • Published: 19 July 2008

Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity

  • Michael Röckner1,2 &
  • Xicheng Zhang1,3,4 

Probability Theory and Related Fields volume 145, pages 211–267 (2009)Cite this article

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Abstract

In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly (Ann. Math. 164:993–1032, 2006), we prove the uniqueness of invariant measures for the corresponding transition semigroup.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Michael Röckner & Xicheng Zhang

  2. Departments of Mathematics and Statistics, Purdue University, W. Laffayette, IN, 47907, USA

    Michael Röckner

  3. School of Mathematics and Statistics, The University of New South Wales, Sydney, 2052, Australia

    Xicheng Zhang

  4. Department of Mathematics, Huazhong University of Science and Technology, 430074, Wuhan, People’s Republic of China

    Xicheng Zhang

Authors
  1. Michael Röckner
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  2. Xicheng Zhang
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Correspondence to Xicheng Zhang.

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Röckner, M., Zhang, X. Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity. Probab. Theory Relat. Fields 145, 211–267 (2009). https://doi.org/10.1007/s00440-008-0167-5

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  • Received: 28 February 2008

  • Revised: 28 May 2008

  • Published: 19 July 2008

  • Issue Date: September 2009

  • DOI: https://doi.org/10.1007/s00440-008-0167-5

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Keywords

  • Navier–Stokes equation
  • Invariant measure
  • Ergodicity
  • Asymptotic strong Feller property

Mathematics Subject Classification (2000)

  • 60H15
  • 37A25
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