Averaging of 2D Navier-Stokes equations with singularly oscillating forces

For ρ ∈ [0, 1) and e> 0, the nonautonomous 2D Navier–Stokes equations with singularly oscillating external force ∂t u − ν�u + (u ·∇ )u =− ∇p + g0(t) + e −ρ g1(t/e), ∇· u = 0 are considered, together with the averaged equations ∂t u − ν�u + (u ·∇ )u =− ∇p + g0(t), ∇· u = 0 formally corresponding to the limiting case e = 0. Under suitable assumptions on the external force, the uniform boundedness of the related uniform global attractors A e is established, as well as the convergence of the attractors A e of the first system to the attractor A 0 of the second one as e → 0 + . When the Grashof number of the averaged equations is small, the convergence rate of A e to A 0 is controlled by Ke 1−ρ .