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On unbiased stochastic Navier–Stokes equations
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  • Published: 16 August 2011

On unbiased stochastic Navier–Stokes equations

  • R. Mikulevicius1 &
  • B. L. Rozovskii2 

Probability Theory and Related Fields volume 154, pages 787–834 (2012)Cite this article

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Abstract

A random perturbation of a deterministic Navier–Stokes equation is considered in the form of an SPDE with Wick type nonlinearity. The nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term \({u{\nabla}u}\) . This perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic Navier–Stokes equation. The perturbed equation is solved in the space of generalized stochastic processes using the Cameron–Martin version of the Wiener chaos expansion. It is shown that the generalized solution is a Markov process and scales effectively by Catalan numbers.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089-2532, USA

    R. Mikulevicius

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    B. L. Rozovskii

Authors
  1. R. Mikulevicius
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  2. B. L. Rozovskii
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Corresponding author

Correspondence to B. L. Rozovskii.

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Cite this article

Mikulevicius, R., Rozovskii, B.L. On unbiased stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 154, 787–834 (2012). https://doi.org/10.1007/s00440-011-0384-1

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  • Received: 12 October 2010

  • Revised: 06 July 2011

  • Published: 16 August 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0384-1

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Keywords

  • Stochastic Navier–Stokes
  • Unbiased perturbation
  • Second quantization
  • Skorokhod integral
  • Wick product
  • Kondratiev spaces
  • Catalan numbers

Mathematics Subject Classification (2000)

  • Primary 60H15
  • 35R60
  • 76N35
  • Secondary 35Q30
  • 15A18
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