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

We study the mixing and dissipation properties of the advection–diffusion equation with diffusivity 0 < κ ≪ 1 and advection by a class of random velocity fields on 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 κ . Namely, that there is a deterministic, exponential rate (independent of κ ) 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- κ , exponential decay in L 2 after time t ≳ log κ . Both the O ( log κ ) time-scale and the uniform-in- κ exponential mixing are optimal for Lipschitz velocity fields. 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.