Abstract: 
In this paper I prove that the global in time regularity of the flows in the 4th
Clay millennium problem, under its official formulation hypotheses, is equivalent to
the corresponding regularity (bounded accumulation in finite intervals) of the
deformations of the flow. It is well known that it is also equivalent to the
corresponding regularity of the vorticity of the flow. We also prove, using some
relatively recent ideas suggested by T. Tao , that the Schwartz initial conditions of the
its official formulation of the problem in the direction of regularity are equivalent to
the simpler compact support initial conditions. Finally I prove using the HelmholtzHodge orthogonal decomposition of vector fields, a powerful fundamental
decomposition of the Euler and NavierStokes equations which is significant for the
internal symmetries of the equations.
