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

Abstract

Let μ > 0 , A μ be the power of the Stokes operator A and R ( A μ ) be the range of A μ . 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 ) ∈ R ( A μ ) , 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 → ∞ ‖ A 1 / 2 w ( t ) ‖ 2 / ‖ w ( t ) ‖ 2 in the sense that lim t → ∞ ‖ E λ w ( t ) ‖ / ‖ w ( t ) ‖ = 1 for every λ > C ( 1 / 2 ) , where { E λ ; λ ≥ 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 ‖ A 1 / 2 w ‖ .