HAL : hal-00344728, version 2

arXiv : 0812.1158

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v1 (05-12-2008)

v2 (30-12-2008)

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

Pascal Auscher 1, Philippe Tchamitchian 2

(07/05/1999)

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.

1 :	Laboratoire de Mathématiques d'Orsay (LM-Orsay)
	CNRS : UMR8628 – Université Paris XI - Paris Sud
2 :	Laboratoire d'Analyse, Topologie, Probabilités (LATP)
	CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III

Domaine

:

Mathématiques/Equations aux dérivées partielles Mathématiques/Analyse fonctionnelle

Mots Clés : Navier-Stokes systems – mild solutions – Littlewood-Paley decomposition – maximal spaces

hal-00344728, version 2

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Contributeur : Pascal Auscher <>

Soumis le : Lundi 29 Décembre 2008, 23:49:46

Dernière modification le : Mardi 30 Décembre 2008, 08:03:20