Résumé : In this talk we shall present some results concerning global smooth solutions to the three-dimensional Navier-Stokes equations set in the whole space $(\mathbb{R}^3)$ :
$\partial_tu+u\cdot \nabla u-\Delta u = -\nabla p$, div $u=0$
We shall more particularly be interested in the geometry of the set $\mathcal{G}$ of initial data giving rise to a global smooth solution.
The question we shall address is the following: given an initial data $u_0$ in $\mathcal{G}$ and a sequence of divergence free vector fields converging towards $u_0$ in the sense of distributions, is the sequence itself in $\mathcal{G}$ ? The related question of strong stability was studied in [1] and [2] some years ago; the weak stability result is a recent work, joint with H. Bahouri and J.-Y. Chemin (see [3]-[4]). As we shall explain, it is necessary to restrict the study to sequences converging weakly up to rescaling (under the natural rescaling of the equation). Then weak stability can be proved, using profile decompositions in the spirit of P. Gerard's work [5], in an anisotropic context.