Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise

TL;DR

This paper proves that the 3D stochastic Navier-Stokes equations with mildly degenerate noise have a unique ergodic behavior, ensuring long-term statistical stability under certain conditions.

Contribution

It establishes the unique ergodicity of the 3D stochastic Navier-Stokes equations driven by finitely forced Fourier modes, using strong Feller regularity and irreducibility.

Findings

Proves unique ergodicity for the equations

Demonstrates strong Feller regularity

Shows irreducibility of the Markov solutions

Abstract

We prove that the any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e.all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.