Uniqueness of invariant measures for stochastic damped anisotropic Navier--Stokes equations

TL;DR

This work addresses the long-time behavior of stochastic damped anisotropic Navier–Stokes equations on $\mathbb{R}^2$ by proving uniqueness of invariant measures when the linear damping dominates the noise intensity. The authors develop an asymptotic coupling framework together with anisotropic energy estimates and exponential $H^1$-energy bounds, compensating for the lack of a Poincaré inequality on the unbounded plane. A Maslowski–Seidler type criterion is used to establish existence, while a carefully crafted energy-driven coupling argument yields uniqueness under a quantitative damping-noise balance condition. The results hold for general additive noise, including the deterministic case, highlighting the crucial role of damping for ensuring unique long-time statistical behavior in unbounded domains.

Abstract

We study a two-dimensional Navier--Stokes system with anisotropic viscosity, linear damping term, and an additive noise on the whole space $\mathbb{R}^2$. For this model we prove uniqueness of invariant measures when the damping coefficient is sufficiently large compared to the noise intensity. The argument is based on an asymptotic coupling method and relies on anisotropic energy estimates together with exponential-type estimates for the $H^1$-energy. Since no Poincaré inequality is available on $\mathbb{R}^2$, the damping term is essential even for the existence of invariant measures. Our result applies to general additive noise without any non-degeneracy condition and remains valid even in the deterministic case $σ\equiv0$.

Theorems & Definitions (25)

• Lemma 3.1
• proof
• Theorem 3.2: Well-posedness and $H^1$-energy estimates
• Theorem 4.1
• Proposition 4.2
• Lemma 4.3
• proof
• Proposition 4.4
• Remark 4.5
• Theorem 4.6