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Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise

Huaxiang Lü, Xiangchan Zhu

Abstract

We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on $\mathbb{T}^2$ \begin{align*} \dif u+÷(u\otimes u)\dif t+\nabla p\dif t&=Δu\dif t + (-Δ)^{\fa/2}\dif B_t,\ \ \ \ ÷u=0,\notag \end{align*} driven by derivative of space-time white noise, where $\fa\in[0,\frac13)$. In this setting, the solutions are not function valued and probabilistic renormalization is required to give a meaning to the equations. Finally, we show that the stationary distributions are not Gaussian distribution $N(0,\frac12(-Δ)^{\fa-1})$. The proof relies on a time-dependent decomposition and a stochastic version of the convex integration method which provides uniform moment bounds in some function spaces.

Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise

Abstract

We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on \begin{align*} \dif u+÷(u\otimes u)\dif t+\nabla p\dif t&=Δu\dif t + (-Δ)^{\fa/2}\dif B_t,\ \ \ \ ÷u=0,\notag \end{align*} driven by derivative of space-time white noise, where . In this setting, the solutions are not function valued and probabilistic renormalization is required to give a meaning to the equations. Finally, we show that the stationary distributions are not Gaussian distribution . The proof relies on a time-dependent decomposition and a stochastic version of the convex integration method which provides uniform moment bounds in some function spaces.
Paper Structure (29 sections, 16 theorems, 256 equations)

This paper contains 29 sections, 16 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries
  4. Function spaces
  5. Paraproducts
  6. Anti-divergence operators
  7. Probabilistic elements
  8. Analytic decomposition and estimate for stochastic objects
  9. Analytic construction
  10. Estimate of $v_q^1$
  11. Proof of (\ref{['3.13.1']}) and (\ref{['3.13.2']})
  12. Proof of (\ref{['3.12']})
  13. Proof of (\ref{['4.4.1']})
  14. Detailed iteration – proof of Proposition \ref{['prop:3.1']}
  15. Choice of parameters
  16. ...and 14 more sections

Key Result

Theorem 1.5

There exist $(1)$ infinitely many stationary solutions; $(2)$ infinitely many ergodic stationary solutions; to the stochastic Navier–Stokes equation (1.1). Moreover, these stationary distributions are not Gaussian distribution $N(0,\frac{1}{2}(-\Delta)^{ {\mathfrak a}-1})$.

Theorems & Definitions (24)

  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • ...and 14 more