Global solutions for the Navier-Stokes equations with some large initial data

We consider the three dimensional Navier-Stokes equations with a large initial data and

we prove the existence of a global smooth solution. The main feature of the initial data

is that it varies slowly in the vertical direction and has a norm which blows up as the

small parameter goes to zero. In the language of geometrical optics, this type of

initial data can be seen as the ``ill prepared" case. Using analytical-type estimates

and the special structure of the nonlinear term of the equation we obtain the existence

of a global smooth solution generated by this large initial data. This talk is based on a

work in collaboration with J.-Y. Chemin and I. Gallagher and on a joint work with Z.

Zhang.