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 g ɛ ( x , t ) = g ( x , t , t / ɛ ) possessing the average g 0 ( x , t ) as ɛ → 0 + , where 0 < ɛ ⩽ ɛ 0 < 1 . Firstly, with assumptions ( A 1 ) – ( A 5 ) on the functions g ( x , t , ξ ) and g 0 ( x , t ) , we prove that the Hausdorff distance between the uniform attractors A ɛ and A 0 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 A ɛ V and A 0 V in space V is also less than O ( ɛ ) as ɛ → 0 + . Finally, we show A ɛ ⊆ A ɛ V for each ɛ ∈ [ 0 , ɛ 0 ] .