The large-time energy concentration in solutions to the Navier-Stokes equations with nonzero external forces

It is known that the turbulent solutions to the unforced Navier-Stokes equations exhibit a large-time energy concentration in the frequency space. If we suppose the existence of a nonzero time dependent external force f∈L1((0,∞);Lσ2) in the equations, the situation is more complicated. It is possible to construct simple examples of solutions, in which the energy is asymptotically distributed over the entire spectrum of the Stokes operator. Further, we will show as the main result of this paper that there still exist wide classes of initial conditions and external forces yielding solutions with the large-time energy concentration analogical to the situation in the unforced Navier-Stokes equations. We will also show that the frequency spectrum of the solution does not generally relate to the frequency spectrum of the external force.