On the stability of global solutions to the three-dimensional Navier-Stokes equations

Bahouri, Hajer; Chemin, Jean-Yves; Gallagher, Isabelle

doi:10.5802/jep.84

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]

Bahouri, Hajer ; Chemin, Jean-Yves ; Gallagher, Isabelle

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 843-911.

Résumé
Abstract

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 ${(u_{0, n})}_{n \in ℕ}$ , 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 $u_{0, n} = n ϕ_{0} (n \cdot)$ ou $u_{0, n} = ϕ_{0} (\cdot - x_{n})$ avec $| x_{n} | \to \infty$ . 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 ${(u_{0, n})}_{n \in ℕ}$ 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 $u_{0, n} = n ϕ_{0} (n \cdot)$ or $u_{0, n} = ϕ_{0} (\cdot - x_{n})$ with $| x_{n} | \to \infty$ . 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 : 2018-02-14
Accepté le : 2018-10-14
Publié le : 2018-11-06

MR 3877168 | Zbl 06988594

DOI : https://doi.org/10.5802/jep.84
Classification : 35Q30, 42B37
Mots clés : Équations de Navier-Stokes, anisotropie, espaces de Besov, décompositions en profils

BibTeX
RIS

@article{JEP_2018__5__843_0,
     author = {Bahouri, Hajer and Chemin, Jean-Yves and Gallagher, Isabelle},
     title = {On the stability of global solutions to the three-dimensional {Navier-Stokes} equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {843--911},
     publisher = {Ecole polytechnique},
     volume = {5},
     year = {2018},
     doi = {10.5802/jep.84},
     zbl = {06988594},
     mrnumber = {3877168},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jep.84/}
}

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AU  - Bahouri, Hajer
AU  - Chemin, Jean-Yves
AU  - Gallagher, Isabelle
TI  - On the stability of global solutions to the three-dimensional Navier-Stokes equations
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2018
DA  - 2018///
SP  - 843
EP  - 911
VL  - 5
PB  - Ecole polytechnique
UR  - http://archive.numdam.org/articles/10.5802/jep.84/
UR  - https://zbmath.org/?q=an%3A06988594
UR  - https://www.ams.org/mathscinet-getitem?mr=3877168
UR  - https://doi.org/10.5802/jep.84
DO  - 10.5802/jep.84
LA  - en
ID  - JEP_2018__5__843_0
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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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