An alternative approach to regularity for the Navier–Stokes equations in critical spaces

Kenig, Carlos E. ; Koch, Gabriel S.

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011), p. 159-187 / Harvested from Numdam

Résumé

Dans cet exposé, nous présentons un point de vue différent sur les études récentes concernant la régularité des solutions des équations de Navier–Stokes dans les espaces critiques. En particulier, nous démontrons que les solutions faibles qui restent bornées dans lʼespace $ {\dot{H}}^{\frac{1}{2}}$ ne deviennent pas singulières en temps fini. Ce résultat a été démontré dans un cas plus général par L. Escauriaza, G. Seregin et V. Šverák en utilisant une approche différente. Nous utilisons la méthode de « concentration-compacité » + « théorème de rigidité » utilisant des « éléments critiques » qui a été récemment développée par C. Kenig et F. Merle pour traiter les équations dispersives critiques. À la connaissance des auteurs, cʼest la première fois que cette méthode est appliquée à une équation parabolique.

Abstract

In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier–Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space $ {\dot{H}}^{\frac{1}{2}}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Šverák using a different approach. We use the method of “concentration-compactness” + “rigidity theorem” using “critical elements” which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authorsʼ knowledge, this is the first instance in which this method has been applied to a parabolic equation.

Détail

Publié le : 2011-01-01

MR 2784068 | Zbl 1220.35119

DOI : https://doi.org/10.1016/j.anihpc.2010.10.004

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