In this paper we study the large time behavior of solutions to the Navier-Stokes equations. We present a brief survey of results concerning energy decay, and discuss a related phenomenon of the large time energy concentration in the frequency space occurring in any turbulent solution. This leads us to the study of solutions in the Besov spaces and to proof that if we choose a suitable initial condition then in some Besov spaces the energy of the associated solution does not decrease asymptotically to zero.