Upper and lower convergence rates for weak solutions of the 3D non-Newtonian flows

Abstract This paper focuses on the 3D shear thickening non-Newtonian fluid equation with the nonlinear constitutive relations τ i j v = 2 ( μ 0 + μ 1 | e ( u ) | r − 2 ) e i j ( u ) − 2 μ 2 Δ e i j ( u ) for r ≥ 3 . We consider the difference between a weak solution u of the aforementioned equation with the initial data u 0 and the weak solution u ˜ of the same equation with perturbed initial data u 0 + w 0 . The goal is to find the exact large-time behavior of the difference u ˜ ( t ) − u ( t ) . By invoking some new observations on the nonlinear parts of the aforementioned non-Newtonian fluid equation and using an iterative argument together with a generalized Fourier splitting method, we are able to show that C 1 ( 1 + t ) − 5 − ϵ 4 ≤ ‖ u ˜ ( t ) − u ( t ) ‖ L 2 ( R 3 ) ≤ C 2 ( 1 + t ) − 5 − ϵ 4 , for  t > 1 large , where ϵ > 0 is taken to be sufficiently small. The initial perturbation w 0 is not required to be small.