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

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.