Energy decay for a weak solution of the non-Newtonian fluid equations with slowly varying external forces

This chapter discusses the energy decay problem of the weak solution. Detailed studies on the initial-boundary value problem have been given earlier. In the chapter, those results are extended to more general external forces and establish nonuniform decay. For that purpose, a weak solution is constructed that satisfies a generalized energy inequality. The Fourier splitting method combined with an argument due to K. Masuda on the generalized energy inequality for the Navier-Stokes equations, yields nonuniform decay.