On the global wellposedness of the 3-D Navier-Stokes equations with large initial data

Jean-Yves Chemin (1) , Isabelle Gallagher (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 IMJ - Institut de Mathématiques de Jussieu

We give a condition for the periodic, three dimensional, incompressible Navier-Stokes equations to be globally wellposed. This condition is not a smallness condition on the initial data, as the data is allowed to be arbitrarily large in the scale invariant space~$ B^{-1}_{\infty,\infty}$, which contains all the known spaces in which there is a global solution for small data. The smallness condition is rather a nonlinear type condition on the initial data; an explicit example of such initial data is constructed, which is arbitrarily large and yet gives rise to a global, smooth solution.

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Yves Chemin, Isabelle Gallagher. On the global wellposedness of the 3-D Navier-Stokes equations with large initial data. Annales Scientifiques de l'École Normale Supérieure, 2006. ⟨hal-00008043⟩