A Stochastic Representation for Backward Incompressible Navier-Stokes Equations
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 \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.
Keywords
Cite
@article{arxiv.0810.4664,
title = {A Stochastic Representation for Backward Incompressible Navier-Stokes Equations},
author = {Xicheng Zhang},
journal= {arXiv preprint arXiv:0810.4664},
year = {2008}
}
Comments
17 Pages. Corrected some misprints and added a proof to the global existence for large viscosity by probability approach