A Stochastic Representation for Backward Incompressible Navier-Stokes Equations

Details A Stochastic Representation for Backward Incompressible Navier-Stokes Equations A Stochastic Representation for Backward Incompressible Navier-Stokes Equations

2008-10-26

arxiv.org/abs/0810.4664v2

Mathematics

Probability

17 Pages. Corrected some misprints and added a proof to the global existence for large viscosity by probability approach

Scientific paper

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 \cite{Co-Iy}. 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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