Exponential mixing and limit theorems of quasi-periodically forced 2D stochastic Navier-Stokes Equations in the hypoelliptic setting

TL;DR

This paper proves that 2D stochastic Navier-Stokes equations with quasi-periodic forcing have a unique quasi-periodic invariant measure attracting all solutions exponentially, and establishes quantitative limit theorems with explicit convergence rates.

Contribution

It introduces a novel analysis of the asymptotic behavior of quasi-periodically forced stochastic Navier-Stokes equations, including exponential mixing and explicit limit theorems.

Findings

Existence of a quasi-periodic invariant measure for all viscosities.

Exponential attraction of solutions to the invariant measure.

Quantitative strong law of large numbers and central limit theorem with explicit rates.

Abstract

We consider the incompressible 2D Navier-Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a quasi-periodic invariant measure that exponentially attracts the law of all solutions. The result is true for any value of the viscosity $\nu>0$ and does not depend on the strength of the external forces. By utilizing this quasi-periodic invariant measure, we establish a quantitative version of the strong law of large numbers and central limit theorem for the continuous time inhomogeneous solution processes with explicit convergence rates. It turns out that the convergence rate in the central limit theorem depends on the time inhomogeneity through the Diophantine approximation property on the quasi-periodic frequency of the quasi-periodic force.