  In these notes we consider solutions u to the evolution three dimensional
Navier-Stokes equations in the whole space. Set w= curl u, the vorticity of u.
Our study concerns mainly the relation between Holdercontinuity assumptions on
the direction of the vorticity and induced space-time integrability regularity
results for w. We discuss some related open problems, consistent with our
calculations. We hope that our approach should help readers interesting in
carrying on the investigation.
