Effective diffusivity conjecture for critical fractional stochastic Navier–Stokes equations

Let $d\geq 3$, let $\theta=1$, and let $u^N$ denote the rescaled stationary solution of the truncated fractional stochastic Navier–Stokes equation. The mollifier $\rho^1$ has Fourier transform equal to the indicator function of the unit ball, and

$$\omega_d=\frac{d\pi^{\frac d2}}{\Gamma(1+\frac d2)}.$$

Effective diffusivity conjecture. The sequence $u^N$ converges weakly to the stationary solution $u$ of

$$\partial_t u=\nu_{\operatorname{eff}}\Delta u+\sqrt{2\nu_{\operatorname{eff}}}(-\Delta)^{\frac12}\pmb{\Pi}\xi,$$

where

$$\nu_{\operatorname{eff}}=\sqrt{1+\frac{\hat\lambda^2\omega_d}{4\pi^2(d-2)}}.$$

At the critical exponent, the nonlinear term has a nontrivial large-scale effect. The conjecture predicts that this effect is captured by a linear limiting equation with explicitly computable effective diffusivity strictly greater than $1$; the proposed higher-dimensional proof is expected to require methods beyond those used in dimension two.