Almost-sure exponential mixing of passive scalars by the stochastic Navier–Stokes equations

Abstract

We deduce almost-sure exponentially fast mixing of passive scalars advected by solutions of the stochastically-forced 2D Navier–Stokes equations and 3D hyper-viscous Navier–Stokes equations in ${\mathbb{T}}^{\mathit{d}}$ subjected to nondenegenerate ${\mathit{H}}^{\mathit{\sigma }}$-regular noise for any σ sufficiently large. That is, for all $\mathit{s}>0$ there is a deterministic exponential decay rate such that all mean-zero ${\mathit{H}}^{\mathit{s}}$ passive scalars decay in ${\mathit{H}}^{-\mathit{s}}$ at this same rate with probability one. This is equivalent to what is known as quenched correlation decay for the Lagrangian flow in the dynamical systems literature. This is a follow-up to our previous work, which establishes a positive Lyapunov exponent for the Lagrangian flow—in general, almost-sure exponential mixing is much stronger than this. Our methods also apply to velocity fields evolving according to finite-dimensional models, for example, Galerkin truncations of Navier–Stokes or the Stokes equations with very degenerate forcing. For all $0\le \mathit{k}<\infty$, this exhibits many examples of ${\mathit{C}}_{\mathit{t}}^{\mathit{k}}{\mathit{C}}_{\mathit{x}}^{\infty }$ random velocity fields that are almost-sure exponentially fast mixers.