Uniqueness of the invariant measure

Determine whether the stochastic three-dimensional Navier–Stokes equation with superlinear multiplicative noise, additive noise, and time-independent deterministic forcing, defined by equation (SNS) under assumptions [A1]–[A3], has a unique invariant measure.

Background

The paper proves existence of at least one invariant measure for the Markov semigroup associated with the stochastic three-dimensional Navier–Stokes equation and establishes moment bounds for every invariant measure. It then proves existence of at least one ergodic invariant measure by showing that the convex set of invariant measures is tight and applying the characterization of ergodic measures as extreme points.

Uniqueness is not established. Consequently, determining whether the invariant measure is unique remains unresolved for the equation with the specified superlinear multiplicative noise, additive noise, and deterministic forcing.

References

So far we have proved the existence but not the uniqueness of the invariant measures of the stochastic Navier-Stokes equation SNS.

SNS:

du(t)+[Au(t)+B(u(t))] dt=f dt+g(u(t)) dw0(t)+G dw(t),{\rm d}u(t)+[Au(t)+B(u(t))]\,{\rm d}t= f\,{\rm d}t+g(u(t))\,{\rm d}w_0(t) +G\,{\rm d}w(t),

— Existence of ergodic invariant measures for 3D Navier-Stokes eqs with superlinear noise  (2610.13266 - Brzezniak et al., 9 Oct 2026) in Section 5, “Existence of an ergodic invariant measure,” immediately before Theorem 5.2 (theorem label \ref{ergodic})