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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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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DOI: https://doi.org/10.1007/s00440-009-0234-6
Keywords
- Backward Navier-Stokes equation
- Stochastic representation
- Global existence
- Large deviation
Mathematics Subject Classification (2000)
- 60H30
- 35Q30
- 76D05