We study a weak stability problem for the three-dimensional Navier–Stokes system: if a sequence of initial data, bounded in some scaling invariant space, converges weakly to an initial data which generates a global regular solution, does generate a global regular solution? Because of the invariances of the Navier–Stokes equations, a positive answer in general to this question would imply global regularity for any data, so we 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.
On étudie la stabilité faible pour le système de Navier–Stokes : si une suite de données de Cauchy , bornée dans un espace invariant par échelle, converge faiblement vers une donnée engendrant une solution globale régulière, est-ce que engendre une solution globale régulière ? À cause des invariances de l'équation de Navier–Stokes, une réponse positive en toute généralité à cette question impliquerait la régularité globale pour toutes les données. Dans ce travail, nous fournissons une réponse positive dans le cadre d'un nouveau concept de convergence faible. La preuve est basée sur des décompositions en profils dans des espaces anisotropes et leur propagation par les équations de Navier–Stokes.
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Hajer Bahouri 1; Jean-Yves Chemin 2; Isabelle Gallagher 3
@article{CRMATH_2014__352_4_305_0, author = {Hajer Bahouri and Jean-Yves Chemin and Isabelle Gallagher}, title = {Stability by rescaled weak convergence for the {Navier{\textendash}Stokes} equations}, journal = {Comptes Rendus. Math\'ematique}, pages = {305--310}, publisher = {Elsevier}, volume = {352}, number = {4}, year = {2014}, doi = {10.1016/j.crma.2014.02.007}, language = {en}, }
TY - JOUR AU - Hajer Bahouri AU - Jean-Yves Chemin AU - Isabelle Gallagher TI - Stability by rescaled weak convergence for the Navier–Stokes equations JO - Comptes Rendus. Mathématique PY - 2014 SP - 305 EP - 310 VL - 352 IS - 4 PB - Elsevier DO - 10.1016/j.crma.2014.02.007 LA - en ID - CRMATH_2014__352_4_305_0 ER -
Hajer Bahouri; Jean-Yves Chemin; Isabelle Gallagher. Stability by rescaled weak convergence for the Navier–Stokes equations. Comptes Rendus. Mathématique, Volume 352 (2014) no. 4, pp. 305-310. doi : 10.1016/j.crma.2014.02.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.02.007/
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