Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise

TL;DR

This paper proves unique ergodicity and exponential convergence to the invariant measure for Markov solutions of the 3D Navier-Stokes equations with additive noise, and provides criteria for well-posedness based on invariant measures.

Contribution

It establishes unique ergodicity and exponential convergence for Markov solutions of the 3D Navier-Stokes equations with additive noise, and introduces a well-posedness criterion involving invariant measures.

Findings

Unique ergodicity of Markov solutions

Exponential convergence to invariant measure

Energy balance analysis

Abstract

We prove that every Markov solution to the three dimensional Navier-Stokes equation with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a well-posedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance.