Espaces critiques pour le système des equations de Navier-Stokes incompressibles

Pascal Auscher (1) , Philippe Tchamitchian (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

In this work, we exhibit abstract conditions on a functional space E who insure the existence of a global mild solution for small data in E or the existence of a local mild solution in absence of size constraints for a class of semi-linear parabolic equations, which contains the incompressible Navier-Stokes system as a fundamental example. We also give an abstract criterion toward regularity of the obtained solutions. These conditions, given in terms of Littlewood-Paley estimates for products of spectrally localized elements of $E$, are simple to check in all known cases: Lebesgue, Lorents, Besov, Morrey... spaces. These conditions also apply to non-invariant spaces E and we give full details in the case of some 2-microlocal spaces. The following comments did not show on the first version: This article was written around 1998-99 and never published, because at that time, Koch and Tataru announced their result on well-posedness of Navier-stokes equations with initial data in $BMO^{-1}$. We believe though that some results and counterexamples here are of independent interest and we make them available electronically.

Mots clés: Navier-Stokes systems; mild solutions; Littlewood-Paley decomposition; maximal spaces

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Pascal Auscher, Philippe Tchamitchian. Espaces critiques pour le système des equations de Navier-Stokes incompressibles. 1999. ⟨hal-00344728v2⟩