Abstract: In the study of long time asymptotic behaviors of the solutions to a class system of the incompressible non-Newtonian fluid flows in R3, it is proved that the weak solutions decay in L2 norm at (1 + t)-34 and the error of difference between non-Newtonian fluid and linear equation is also investigated. The findings are mainly based on the classic Fourier splitting methods.

Key words: decay rates, Fourier splitting methods, non-Newtonian flows