Weak Averaging Principle and Weak Pullback Attractors for Mckean--Vlasov Stochastic Navier--Stokes Equations

We establish three weak averaging principles for distribution-dependent stochastic Navier–Stokes equations with rapidly oscillating coefficients on the torus ${\text{T}}^2$. First, solution laws are relatively compact on finite time intervals, and every limit point as $\text{var}\to 0$ is the path law of a variational martingale solution of the averaged equation. Under a dissipative condition, we can choose bounded complete variational solution laws of the original and averaged equations so that every sequence with $\text{var}\to 0$ has a subsequence converging weakly to a bounded complete variational solution law of the averaged equation. At the level of probability laws, the original nonautonomous equation admits a family of weak pullback attractors, while the averaged equation has a weak global attractor. The weak pullback attractors converge upper semicontinuously to the weak global attractor of the averaged equation, uniformly with respect to the coefficient hull.