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

Auscher, Pascal ; Tchamitchian, Philippe

HAL, hal-00344728 / Harvested from HAL

Résumé

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.

Détail

Publié le : 1999-05-07
Classification:  Navier-Stokes systems,  mild solutions,  Littlewood-Paley decomposition,  maximal spaces,  AMS: 35K55, 35Q30, 35R05, 35S50, 42B25,  [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP],  [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA]