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Polynomial mixing for the white-forced Navier-Stokes system in the whole space

Vahagn Nersesyan, Meng Zhao

Abstract

We study the mixing properties of the white-forced Navier-Stokes system in the whole space $\mathbb{R}^2$. Assuming that the noise is sufficiently non-degenerate, we prove the uniqueness of stationary measure and polynomial mixing in the dual-Lipschitz metric. The proof combines the coupling method with a Foiaş-Prodi type estimate, weighted growth estimates for trajectories, and an estimate for the Leray projector involving Muckenhoupt $A_2$-class weights.

Polynomial mixing for the white-forced Navier-Stokes system in the whole space

Abstract

We study the mixing properties of the white-forced Navier-Stokes system in the whole space . Assuming that the noise is sufficiently non-degenerate, we prove the uniqueness of stationary measure and polynomial mixing in the dual-Lipschitz metric. The proof combines the coupling method with a Foiaş-Prodi type estimate, weighted growth estimates for trajectories, and an estimate for the Leray projector involving Muckenhoupt -class weights.

Paper Structure

This paper contains 17 sections, 26 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Construction of a mixing extension
  3. Growth estimates for solutions
  4. L2-estimate for the velocity field
  5. L2-estimate for the vorticity
  6. Weighted estimate for the velocity field
  7. Weighted estimate for the vorticity
  8. Estimate for the weighted parabolic energy
  9. Stability of solutions
  10. Foiaş--Prodi type estimate
  11. Growth estimate for an auxiliary process
  12. Proof of Theorem \ref{['theorem2']}
  13. Recurrence
  14. Polynomial squeezing
  15. Moment estimates
  16. ...and 2 more sections

Key Result

Theorem 2.1

Under the assumptions of the Main Theorem, the family $(u(t),\mathbb{P}_u)$ has a unique stationary measure $\mu\in\mathscr{P}(H)$. Moreover, for any $q>1$, there is a constant $C_q>0$ such that for any $\lambda\in\mathscr{P}(H)$.

Theorems & Definitions (43)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 33 more