Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor

We study the global attractor Aε of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form g0(x, t)+ 1 ερ g1 ( x ε , t ) , x ∈Ω R2, t ∈R, 0 ρ 1. If the functions g0(x, t) and g1 (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor Aε is bounded in the space H , however, its norm ‖Aε‖H may be unbounded as ε → 0+ since the magnitude of the external force is growing. Assuming that the function g1 (z, t) has a divergence representation of the form g1 (z, t)= ∂z1G1(z, t)+ ∂z2G2(z, t), z = (z1, z2) ∈R2, where the functions G j (z, t) ∈ L2(R; Z) (see Section 3), we prove that the global attractors Aε of the N.–S. equations are uniformly bounded with respect to ε : ‖Aε‖H C for all 0 < ε 1. We also consider the “limiting” 2D N.–S. system with external force g0(x, t). We have found an estimate for the deviation of a solution uε(x, t) of the original N.–S. system from a solution u0(x, t) of the “limiting” N.–S. system with the same initial data. If the function g1 (z, t) admits the divergence representation, the functions g0(x, t) and g1 (z, t) are translation compact in the corresponding spaces, and 0 ρ < 1, then we prove that the global attractors Aε converges to the global attractor A0 of the “limiting” system as ε→0+ in the norm of H . In the last section, we present an estimate for the Hausdorff deviation of Aε from A0 of the form:distH (Aε,A0) C(ρ)ε1−ρ in the case, when the global attractor A0 is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).