A Stochastic Representation for Backward Incompressible Navier-Stokes Equations

TL;DR

This paper introduces a stochastic representation for backward incompressible Navier-Stokes equations using stochastic Lagrangian paths, providing new proofs for local and global existence of solutions and analyzing particle trajectory deviations as viscosity diminishes.

Contribution

It develops a novel stochastic backward formulation for Navier-Stokes equations and offers new proofs for solution existence, including large viscosity and two-dimensional cases, plus a large deviation estimate.

Findings

Provided a stochastic backward representation for Navier-Stokes equations.

Proved local existence of solutions in Sobolev spaces.

Established a large deviation estimate for particle trajectories as viscosity approaches zero.

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.