In this paper we prove the well known Clay Mathematical Institute conjecture of 3-dimensional global in time regularity of incompressible flows that satisfy the Navier-Stokes equations (viscous flows) with the standard hypotheses on initial conditions that include finite initial energy. It is proved something more difficult than the above celebrated Clay millennium conjecture as the global in time regularity is been proved for the more difficult case of Euler inviscid flows with the same hypotheses on the initial conditions. The proof is based on the well known necessary and sufficient condition for global in time existence (3D regularity), from local existence of the smooth solution, by the bounded accumulation of the vorticity on finite time intervals. And the bounded accumulation on finite intervals is proved by introducing an secondary (adjoint) different potential (irrotational) flow, which has the same evolution of deformations with the original flow, so that we can reduce the behavior of the original flow to that of the potential flow, which is easily provable to be globally in time smooth (3D regular). The existence of such an adjoin potential flow is a consequence of our main contribution in this paper, which is a new fundamental decomposition of the Euler and Navier-Stokes equations, based on the Helmholtz-Hodge decomposition.