L2-Decay Rates for Weak Solutions of a Perturbed Navier-Stokes System in R3

We study global weak solutions u (with generalized energy inequality) of the Cauchy problem for the Navier-Stokes-type system ut−Δu+v(0)•∇u+u•∇v(0)+u•∇u+∇π=0, div u=0. Here v(0) is a bounded smooth solenoidal vector field in R3 with v(0)→v∞ as |x|→∞, v∞∈R3 is a constant vector. We assume v(0)−v∞∈Lp(R3) for some p∈[2, 3) and that the energetic Reynolds number of v(0) is small. In this case we derive explicit decay rates for the kinetic energy of u.