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

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eρ g1 � x e , t � , x ∈ Ω R 2 , t ∈R,0 ρ 1. If the functions g 0(x, t) and g1 (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor A e is bounded in the space H, however, its normA e � H may be unbounded as e → 0+ since the magnitude of the external force is growing. Assuming that the function g1 (z, t) has a divergence represen- tation of the form g1 (z, t) = ∂z1 G1(z, t) + ∂z2 G2(z, t), z = (z1, z2) ∈ R 2 , where the functions G j (z, t) ∈ L b (R; Z ) (see Section 3), we prove that the global attractors A e of the N.-S. equations are uniformly bounded with respect to e :� A e � H C for all 0 <e 1. We also consider the "limiting" 2D N.-S. system with external force g0(x, t). We have found an estimate for the devia- tion of a solution u e (x, t) of the original N.-S. system from a solution u 0 (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 trans- lation compact in the corresponding spaces, and 0 ρ< 1, then we prove that the global attractors A e converges to the global attractor A 0 of the "limiting" system as e →0+ in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of A e from A 0 of the form:distH (A e ,A 0 ) C(ρ)e 1−ρ in the case, when the global attractor A 0 is exponential (the Grashof number of the "limiting" 2D N.-S. system is small).