Theorems about the attractor for incompressible non-Newtonian flow driven by external forces that are rapidly oscillating in time but have a smooth average

This paper discusses the incompressible non-Newtonian fluid with rapidly oscillating external forces gɛ(x,t)=g(x,t,t/ɛ) possessing the average g0(x,t) as ɛ→0+, where 0<ɛ⩽ɛ0<1. Firstly, with assumptions (A1)-(A5) on the functions g(x,t,ξ) and g0(x,t), we prove that the Hausdorff distance between the uniform attractors Aɛ and A0 in space H, corresponding to the oscillating equations and the averaged equation, respectively, is less than O(ɛ) as ɛ→0+. Then we establish that the Hausdorff distance between the uniform attractors AɛV and A0V in space V is also less than O(ɛ) as ɛ→0+. Finally, we show Aɛ⊆AɛV for each ɛ∈[0,ɛ0].