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

Let μ>0μ>0, AμAμ be the power of the Stokes operator AA and R(Aμ)R(Aμ) be the range of AμAμ. We show as the main result of the paper that if ww is a nonzero global weak solution to the Navier–Stokes equations satisfying the strong energy inequality and w(0)∈R(Aμ)w(0)∈R(Aμ), then the energy of the solution ww concentrates asymptotically in frequencies smaller than or equal to the finite number C(1/2)=lim supt→∞‖A1/2w(t)‖2/‖w(t)‖2C(1/2)=lim supt→∞‖A1/2w(t)‖2/‖w(t)‖2 in the sense that View the MathML sourcelimt→∞‖Eλw(t)‖/‖w(t)‖=1 Turn MathJax on for every λ>C(1/2)λ>C(1/2), where {Eλ;λ≥0}{Eλ;λ≥0} is the resolution of the identity of AA. We also obtain an explicit convergence rate in the limit above and similar results for the enstrophy of ww defined as ‖A1/2w‖‖A1/2w‖.