Global regularity for some classes of large solutions to the Navier-Stokes equations

In previous works by the first two 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 one of those studies 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 that setting to an ill prepared situation (the norm blows up as the small parameter goes to zero). As in those works, the proof uses the special structure of the nonlinear term of the equation.