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A stochastic representation for backward incompressible Navier-Stokes equations
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  • Published: 10 June 2009

A stochastic representation for backward incompressible Navier-Stokes equations

  • Xicheng Zhang1,2 

Probability Theory and Related Fields volume 148, pages 305–332 (2010)Cite this article

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  • 17 Citations

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Abstract

By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer’s forward formulations in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.

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Author information

Authors and Affiliations

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

    Xicheng Zhang

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

    Xicheng Zhang

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  1. Xicheng Zhang
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Correspondence to Xicheng Zhang.

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Zhang, X. A stochastic representation for backward incompressible Navier-Stokes equations. Probab. Theory Relat. Fields 148, 305–332 (2010). https://doi.org/10.1007/s00440-009-0234-6

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  • Received: 02 November 2008

  • Revised: 18 May 2009

  • Published: 10 June 2009

  • Issue Date: September 2010

  • DOI: https://doi.org/10.1007/s00440-009-0234-6

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Keywords

  • Backward Navier-Stokes equation
  • Stochastic representation
  • Global existence
  • Large deviation

Mathematics Subject Classification (2000)

  • 60H30
  • 35Q30
  • 76D05
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