Almost-sure enhanced dissipation and uniform-in-diffusivity exponential mixing for advection-diffusion by stochastic Navier-Stokes

TL;DR

This paper proves that certain stochastic velocity fields induce almost-sure exponential mixing and enhanced dissipation in advection-diffusion equations, uniformly in diffusivity, with optimal time scales, advancing understanding of scalar mixing in turbulence.

Contribution

It establishes the first rigorous example of stochastic velocity fields causing uniform-in-diffusivity exponential mixing and enhanced dissipation, with optimal time scales.

Findings

Almost-sure exponential mixing with a deterministic rate.

Uniform-in-$ppa$ exponential decay in $L^2$ after $t \u2265 |\,log ppa|$.

Optimal time scale $O(|\,log ppa|)$ for Lipschitz velocity fields.

Abstract

We study the mixing and dissipation properties of the advection-diffusion equation with diffusivity $0 < \kappa \ll 1$ and advection by a class of random velocity fields on $\mathbb T^d$, $d=\{2,3\}$, including solutions of the 2D Navier-Stokes equations forced by sufficiently regular-in-space, non-degenerate white-in-time noise. We prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity $\kappa$. Namely, that there is a deterministic, exponential rate (independent of $\kappa$) such that all mean-zero $H^1$ initial data decays exponentially fast in $H^{-1}$ at this rate with probability one. This implies almost-sure enhanced dissipation in $L^2$. Specifically that there is a deterministic, uniform-in-$\kappa$, exponential decay in $L^2$ after time $t \gtrsim |\log \kappa|$. Both the $O(|\log \kappa|)$ time-scale and the uniform-in-$\kappa$ exponential mixing are optimal for Lipschitz velocity fields and, to our knowledge, are the first rigorous examples of velocity fields satisfying these properties (deterministic or stochastic). This work is also a major step in our program on scalar mixing and Lagrangian chaos necessary for a rigorous proof of the Batchelor power spectrum of passive scalar turbulence.