The large-time energy concentration in solutions to the Navier–Stokes equations in the frequency space

In the paper we study the large-time behavior of solutions to the Navier–Stokes equations in the frequency space. We describe in detail the large-time energy concentration which occurs in every (turbulent) solution. If the energy of the solution decreases exponentially then it concentrates in frequencies localized in an annulus in the frequency space. The annulus can be taken arbitrarily narrow and its diameter determines the rate of the exponential decay. All the other solutions are characterized by the concentration of the energy in the frequencies localized in a ball with an arbitrarily small diameter centered at the origin of the coordinates. It will follow from the presented results that the frequencies outside the annulus or the ball and especially the higher frequencies die out very quickly. We will further observe the concentration occurring in any time derivative of the solution or in the vorticity and its time derivatives with the same annulus or the ball for the particular solution.