Global in time existence of strong solution to 3D periodic Navier-Stokes equations

The purpose of this paper is to bring to light a method through which the global in time existence for arbitrary large in $H^1$ initial data of a strong solution to 3D periodic Navier-Stokes equations follows. The method consists of subdividing the time interval of existence into smaller sub-intervals carefully chosen. These sub-intervals are chosen based on the hypothesis that for any wavenumber $m$, one can find an interval of time on which the energy quantized in low-frequency components (up to $m$) of the solution $u$ is lesser than the energy quantized in high-frequency components (down to $m$) or otherwise the opposite. We associate then a suitable number $m$ to each one of the intervals and we prove that the norm $\|u(t)\|_{H^1}$ is bounded in both mentioned cases. The process can be continued until reaching the maximal time of existence $T_{max}$ which yields the global in time existence of strong solution.