A solution of the 4th clay millennium problem about the Navier-Stokes equations.

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 and by applying the regularity of the Poisson equation which is a well- studied linear PDE, involving the also well studies harmonic functions. The Poisson equation either in scalar or vector form, relates many magnitudes of the flow, like pressures and velocities, velocity and vorticity and velocities and viscosity forces. It is also proved 5 new necessary and sufficient conditions of regularity based on the pressures, viscosity forces, trajectories lengths, pressure forces etc. The final key result to derive the regularity is that the pressures are bounded in finite time intervals, as proved after projecting the work of the pressures forces on specially chosen bundles of paths