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Probability Theory and Related Fields

, Volume 148, Issue 1–2, pp 305–332 | Cite as

A stochastic representation for backward incompressible Navier-Stokes equations

  • Xicheng ZhangEmail author
Article

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.

Keywords

Backward Navier-Stokes equation Stochastic representation Global existence Large deviation 

Mathematics Subject Classification (2000)

60H30 35Q30 76D05 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia

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