Conditions for asymptotic energy and enstrophy concentration in solutions to the Navier-Stokes equations

Let mu > 0, A(mu) be the power of the Stokes operator A and R (A(mu)) be the range of A(mu). We show as the main result of the paper that if w is a nonzero global weak solution to the Navier-Stokes equations satisfying the strong energy inequality and w(0) is an element of R(A(mu)), then the energy of the solution w concentrates asymptotically in frequencies smaller than or equal to the finite number C(1/2) = lim sup(t ->infinity) parallel to A(1/2)w(t)parallel to(2)/parallel to w(t)parallel to(2) in the sense that lim(t ->infinity) parallel to E(lambda)w(t)parallel to/parallel to w(t)parallel to = 1 for every lambda > C(1/2), where f {E(lambda); lambda >= 0} is the resolution of the identity of A. We also obtain an explicit convergence rate in the limit above and similar results for the enstrophy of w defined as parallel to A(1/2)w parallel to. (C) 2009 Elsevier Ltd. All rights reserved.