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. This work was done when the first author was at Université de Picardie.

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-00344728v1⟩