On the stability of global solutions to the three-dimensional Navier-Stokes equations
[Sur la stabilité de solutions globales aux équations de Navier-Stokes tridimensionnelles]
Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 843-911.

On démontre un résultat de stabilité faible pour les équations de Navier-Stokes tridimensionnelles, incompressibles et homogènes. Plus précisément on étudie le problème suivant : si une suite de données initiales (u 0,n ) n , bornée dans un espace invariant d’échelle, converge faiblement vers une donnée u 0 qui engendre une solution globale régulière, est-ce que u 0,n engendre une solution globale régulière ? Une réponse affirmative à cette question en général aurait pour conséquence la régularité globale pour toute donnée initiale, via les exemples u 0,n =nϕ 0 (n·) ou u 0,n =ϕ 0 (·-x n ) avec |x n |. On introduit donc un nouveau concept de convergence faible (convergence faible remise à l’échelle) sous lequel on peut donner une réponse affirmative. La démonstration repose sur des décompositions en profils dans des espaces de régularité anisotrope, et leur propagation par les équations de Navier-Stokes.

We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem: if a sequence (u 0,n ) n of initial data, bounded in some scaling invariant space, converges weakly to an initial data u 0 which generates a global smooth solution, does u 0,n generate a global smooth solution? A positive answer in general to this question would imply global regularity for any data, through the following examples u 0,n =nϕ 0 (n·) or u 0,n =ϕ 0 (·-x n ) with |x n |. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.84
Classification : 35Q30, 42B37
Keywords: Navier-Stokes equations, anisotropy, Besov spaces, profile decomposition
Mot clés : Équations de Navier-Stokes, anisotropie, espaces de Besov, décompositions en profils
Bahouri, Hajer 1 ; Chemin, Jean-Yves 2 ; Gallagher, Isabelle 3

1 Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, Université Paris-Est Créteil 61, avenue du Général de Gaulle, 94010 Créteil Cedex, France
2 Laboratoire Jacques Louis Lions - UMR 7598, Sorbonne Université Boîte courrier 187, 4 place Jussieu, 75252 Paris Cedex 05, France
3 DMA, École normale supérieure, CNRS, PSL Research University 75005 Paris and UFR de mathématiques, Université Paris-Diderot, Sorbonne Paris-Cité 75013 Paris, France
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Bahouri, Hajer; Chemin, Jean-Yves; Gallagher, Isabelle. On the stability of global solutions to the three-dimensional Navier-Stokes equations. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 843-911. doi : 10.5802/jep.84. http://archive.numdam.org/articles/10.5802/jep.84/

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