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Ergodicity of 2D singular stochastic Navier-Stokes equations

Martin Hairer, Wenhao Zhao

TL;DR

This work proves ergodicity for the 2D stochastic Navier–Stokes equations driven by a rough noise with near-white-noise regularity. By introducing a novel two-frequency ansatz and a refined Da Prato–Debussche-type decomposition, the authors obtain a uniform-in-time Lyapunov bound and build invariant measures with stretched exponential tails. They then establish exponential mixing under a non-degeneracy condition on the noise, using strong Feller properties and a support theorem to obtain irreducibility. The results advance the understanding of singular SPDEs without strong damping, offering a framework potentially applicable to stochastic Yang–Mills-type models and other critical SPDEs.

Abstract

We consider the 2D stochastic Navier-Stokes equations driven by noise that has the regularity of space-time white noise but doesn't exactly coincide with it. We show that, provided that the intensity of the noise is sufficiently weak at high frequencies, this systems admits uniform bounds in time, so that it has an invariant measure, for which we obtain stretched exponential tail bounds.

Ergodicity of 2D singular stochastic Navier-Stokes equations

TL;DR

This work proves ergodicity for the 2D stochastic Navier–Stokes equations driven by a rough noise with near-white-noise regularity. By introducing a novel two-frequency ansatz and a refined Da Prato–Debussche-type decomposition, the authors obtain a uniform-in-time Lyapunov bound and build invariant measures with stretched exponential tails. They then establish exponential mixing under a non-degeneracy condition on the noise, using strong Feller properties and a support theorem to obtain irreducibility. The results advance the understanding of singular SPDEs without strong damping, offering a framework potentially applicable to stochastic Yang–Mills-type models and other critical SPDEs.

Abstract

We consider the 2D stochastic Navier-Stokes equations driven by noise that has the regularity of space-time white noise but doesn't exactly coincide with it. We show that, provided that the intensity of the noise is sufficiently weak at high frequencies, this systems admits uniform bounds in time, so that it has an invariant measure, for which we obtain stretched exponential tail bounds.

Paper Structure

This paper contains 20 sections, 31 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notations
  4. Preliminaries and strategies
  5. Preliminaries
  6. Itô's formula
  7. Ansatz
  8. Uniform in time bound
  9. Stochastic objects
  10. Energy estimate
  11. Lower bound for stopping time
  12. Strong Markov property and iteration
  13. Moment bounds
  14. Energy estimate
  15. Concentration effect
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $P_t$ be the Markov semigroup of e:SNS. Let $V_{\alpha}: \mathcal{C}^{-\kappa} \rightarrow \mathbf{R}$ defined above be the Lyapunov function for the Markov process. There exist some positive constants $0 < \alpha_0\leq 1$ and $C > 0$ such that, if the noise $\xi$ satisfies Assumption as:noise a As a result, the solution to e:SNS admits at least one invariant measure $\mu_\star$. Moreover, for

Theorems & Definitions (74)

  • Theorem 1.2
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 64 more