  In this work we investigate the question of preventing the three-dimensional,
incompressible Navier-Stokes equations from developing singularities, by
controlling one component of the velocity field only, in space-time scale
invariant norms. In particular we prove that it is not possible for one
component of the velocity field to tend to~$0$ too fast near blow up.
  We also introduce a space "almost" invariant under the action of the scaling
such that if one component of the velocity field measured in this space remains
small enough, then there is no blow up.
