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

This paper focuses on the 3D shear thickening non-Newtonian fluid equation with the nonlinear constitutive relations τijv=2(μ0+μ1|e(u)|r−2)eij(u)−2μ2Δeij(u) for r≥3. We consider the difference between a weak solution u of the aforementioned equation with the initial data u0 and the weak solution u˜ of the same equation with perturbed initial data u0+w0. 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 thatC1(1+t)−5−ϵ4≤‖u˜(t)−u(t)‖L2(R3)≤C2(1+t)−5−ϵ4,for t>1large, where ϵ>0 is taken to be sufficiently small. The initial perturbation w0 is not required to be small.