On asymptotic dynamics of solutions of the homogeneous Navier-Stokes equations

As the main result of the paper we prove that if w is a strong global solution of the homogeneous Navier-Stokes equations in a smooth bounded domain Ω⊂R3 endowed with homogeneous Dirichlet boundary conditions then for every k,l,m∈N∪{0} there exist C=C(k,l,m), t0=t0(k,l,m)≥0 and δ0∈(0,1) such that ‖dkwdtk(t)‖m,2≤C‖dlwdtl(t+δ)‖,∀t≥t0 and δ∈[0,δ0].