  In three previous papers by the two first authors, classes of initial data to
the three dimensional, incompressible Navier-Stokes equations were presented,
generating a global smooth solution although the norm of the initial data may
be chosen arbitrarily large. The main feature of the initial data considered in
the last paper is that it varies slowly in one direction, though in some sense
it is ``well prepared'' (its norm is large but does not depend on the slow
parameter). The aim of this article is to generalize the setting of that last
paper to an ``ill prepared'' situation (the norm blows up as the small
parameter goes to zero).The proof uses the special structure of the nonlinear
term of the equation.

Analysis of PDEs