Asymptotic energy and enstrophy concentration in solutions to the Navier-Stokes equations in R3

Let A be the Stokes operator. We show as the main result of the paper that if w is a nonzero global weak solution to the Navier-Stokes equations in R3 satisfying the strong energy inequality, then the energy of the solution w concentrates asymptotically in frequencies smaller than or equal to the finite number (Formula presented.) in the sense that (Formula presented.) for every λ > C(1/2), where {Eλ; λ ≥ 0} is the resolution of identity of A. We also obtain an explicit convergence rate in the limit above and similar results for the enstrophy of w defined as {double pipe}A1/2w{double pipe}. © Università degli Studi di Ferrara 2009.