A stochastic representation for backward incompressible Navier-Stokes equations

Xicheng Zhang*

*Corresponding author for this work

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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.

Original languageEnglish
Pages (from-to)305-332
Number of pages28
JournalProbability Theory and Related Fields
Volume148
Issue number1
DOIs
Publication statusPublished - 2010
Externally publishedYes

Keywords

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

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Zhang, X. (2010). A stochastic representation for backward incompressible Navier-Stokes equations. Probability Theory and Related Fields, 148(1), 305-332. https://doi.org/10.1007/s00440-009-0234-6