In this paper it is solved the 4th Clay Millennium problem about the Navier-Stokes equations, in the direction of regularity. It is done so by utilizing the hypothesis of finite initial energy The final key result to derive the regularity is that the pressures are bounded in finite time intervals, as proved after projecting the bounded by the conservation of energy, virtual work of the pressures forces on specially chosen bundles of instantaneous paths. It is proved that not only there is no Blow-up in finite time but not even at the time T=+∞.