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Long time behavior of the stochastic 2D Navier-Stokes equations

Benedetta Ferrario, Margherita Zanella

TL;DR

The paper analyzes the long-time behavior of the 2D stochastic Navier–Stokes equations with both additive and multiplicative noise, focusing on the existence, uniqueness, and asymptotic stability of invariant measures. It surveys and unifies results across elliptic, effectively elliptic, and hypoelliptic regimes, employing Krylov–Bogoliubov arguments, generalized asymptotic coupling, Foias–Prodi estimates, and Girsanov transformations to establish ergodicity and statistical equilibrium. Key contributions include demonstrating uniqueness under linear-growth multiplicative noise via finite-dimensional nudging and coupling, and proving asymptotic convergence to the invariant measure under various noise-growth conditions. These findings illuminate how randomness shapes the turbulent-like long-time statistics of 2D flows and provide a rigorous foundation for statistical steady states in stochastic fluid dynamics.

Abstract

We review some basic results on existence and uniqueness of the invariant measure for the two-dimensional stochastic Navier-Stokes equations. A large part of the literature concerns the additive noise case; after revising these models, we consider our recent result, arXiv:2307.03483, with a multiplicative noise.

Long time behavior of the stochastic 2D Navier-Stokes equations

TL;DR

The paper analyzes the long-time behavior of the 2D stochastic Navier–Stokes equations with both additive and multiplicative noise, focusing on the existence, uniqueness, and asymptotic stability of invariant measures. It surveys and unifies results across elliptic, effectively elliptic, and hypoelliptic regimes, employing Krylov–Bogoliubov arguments, generalized asymptotic coupling, Foias–Prodi estimates, and Girsanov transformations to establish ergodicity and statistical equilibrium. Key contributions include demonstrating uniqueness under linear-growth multiplicative noise via finite-dimensional nudging and coupling, and proving asymptotic convergence to the invariant measure under various noise-growth conditions. These findings illuminate how randomness shapes the turbulent-like long-time statistics of 2D flows and provide a rigorous foundation for statistical steady states in stochastic fluid dynamics.

Abstract

We review some basic results on existence and uniqueness of the invariant measure for the two-dimensional stochastic Navier-Stokes equations. A large part of the literature concerns the additive noise case; after revising these models, we consider our recent result, arXiv:2307.03483, with a multiplicative noise.
Paper Structure (10 sections, 18 theorems, 67 equations)

This paper contains 10 sections, 18 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Mathematical setting
  3. Solutions
  4. Well posedness
  5. Existence of invariant measures
  6. Uniqueness of the invariant measure
  7. Effectively elliptic setting
  8. Elliptic setting
  9. Hypoelliptic setting
  10. Asymptotic stability of the invariant measure

Key Result

Lemma 2.1

Let $0 \le \delta < 1$. Then there exists some constant $C=C(\rho, \theta, \delta)$ such that where $\theta >0,\: \rho >0,\: \delta + \theta + \rho \ge 1,\: \delta + \rho > {\frac{1}{2}}$.

Theorems & Definitions (34)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Remark 3.4
  • Proposition 4.1
  • Theorem 4.2
  • proof
  • ...and 24 more