Fluctuation dynamics in randomly advected Navier-Stokes equations below critical scaling

Abstract

We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations. We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale ${\epsilon }^2$ and has spatial correlation length $\hat{I}´$; the critical regime $\epsilon =\hat{I}´$ corresponds to the natural parabolic scaling of the Navier-Stokes equation. In the full subcritical regime $\epsilon =o(\hat{I}´)$, we prove a law of large numbers in dimensions $d=2,3$: the solutions converge to a deterministic Navier--Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula. In two space dimensions, under the slightly stronger assumption $\epsilon =o(\hat{I}{´}^{1+{\hat{I}}^1})$ for some ${\hat{I}}^1>0$, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.