In the first part of the paper the author studies the large-time behavior of the higher-order space derivatives of solutions to the Navier-Stokes equations in ℝ3. Specifically, they show that if w is a nonzero global weak solution to the Navier-Stokes equations satisfying the strong energy inequality and 0 < α < β < ∞, then there exist C = C(α,β) > 1, δ0 = δ0(α,β) ∈ (0,1) and t0 = t0(α,β) > 0 such that
for every t > t0 and every δ ∈ [0,δ0]. A denotes the Stokes operator and Aβ its powers. In the second part of the paper the author derives several consequences of the above inequality concerning the large-time energy concentration in w.