Averaged Navier–Stokes model

An averaged Navier–Stokes model is

where, on divergence-free $H^{10}(\mathbb R^3)$ fields, the Euler bilinear operator $B$ is replaced through

Set $A_i=m_i(D)\operatorname{Rot}_{R_i}\operatorname{Dil}_{\lambda_i}$, with

The $R_i$ are random rotations, the $m_i$ are real order-zero multipliers, and $\mathbb E$ is expectation. Require $C^{-1}\leq\lambda_i\leq C$ almost surely and $\mathbb E\prod_iM_{k_i}(m_i)<\infty$ for every triple of orders. Pairings use the $L^2$ duality on the indicated Sobolev spaces.