The martingale problem for Markov solutions to the Navier-Stokes equations

TL;DR

This paper establishes that under certain conditions, Markov solutions to stochastic Navier-Stokes equations are uniquely characterized by their generators through the martingale problem framework, providing a rigorous foundation for their analysis.

Contribution

It proves the uniqueness of Markov solutions to stochastic Navier-Stokes equations via the associated martingale problem under regularity and non-degeneracy assumptions.

Findings

Markov solutions have associated diffusion generators.

Uniqueness of solutions is established through the martingale problem.

Elementary examples illustrate the theoretical results.

Abstract

Under suitable assumptions of regularity and non-degeneracy on the covariance of the driving additive noise, any Markov solution to the stochastic Navier-Stokes equations has an associated generator of the diffusion and is the unique solution to the corresponding martingale problem. Some elementary examples are discussed to interpret these results.