Wellposedness and stability results for the Navier–Stokes equations in $R^3$

Abstract

In [J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D Navier–Stokes equations with large initial data, Annales Scientifiques de l'École Normale Supérieure de Paris, in press] a class of initial data to the three dimensional, periodic, incompressible Navier–Stokes equations was presented, 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 of [J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D Navier–Stokes equations with large initial data, Annales Scientifiques de l'École Normale Supérieure de Paris, in press] 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 [H. Koch, D. Tataru, Well-posedness for the Navier–Stokes equations, Advances in Mathematics 157 (2001) 22–35] 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 in [H. Bahouri, J.-Y. Chemin, I. Gallagher, Refined Hardy inequalities, Annali di Scuola Normale di Pisa, Classe di Scienze, Serie V 5 (2006) 375–391], thus generating a large number of different examples.