  In a previous work, we presented a class of initial data to the three
dimensional, periodic, incompressible Navier-Stokes equations, generating a
global smooth solution although the norm of the initial data may be chosen
arbitrarily large. The aim of this article is twofold. First, we adapt the
construction to the case of the whole space: we prove that if a certain
nonlinear function of the initial data is small enough, in a Koch-Tataru type
space, then there is a global solution to the Navier-Stokes equations. We
provide an example of initial data satisfying that nonlinear smallness
condition, but whose norm is arbitrarily large in $ C^{-1}$. Then we prove a
stability result on the nonlinear smallness assumption. More precisely we show
that the new smallness assumption also holds for linear superpositions of
translated and dilated iterates of the initial data, in the spirit of a
construction by the authors and H. Bahouri, thus generating a large number of
different examples.
