Unique ergodicity of projective cocycles over the 2D stochastic Navier-Stokes equations

We consider the linear cocycles generated by the linearized vorticity equation and by passive scalar advection diffusion, both driven by the two dimensional stochastic Navier-Stokes flow on the torus with non degenerate additive forcing, and we prove uniqueness of the stationary measures for the projective process associated to such linear dynamics. The proof relies on a localized asymptotic strong coupling construction, and on the non-degeneracy of the Malliavin matrix of the projective process. Establishing this non-degeneracy for passive scalar advection poses additional challenges, and requires a proof (based on Cameron-Martin analyticity arguments) that generically the passive scalar is nowhere one-dimensional.