Comptes Rendus
Partial differential equations
Stability by rescaled weak convergence for the Navier–Stokes equations
Comptes Rendus. Mathématique, Volume 352 (2014) no. 4, pp. 305-310.

We study a weak stability problem for the three-dimensional Navier–Stokes system: if a sequence (u0,n)nN of initial data, bounded in some scaling invariant space, converges weakly to an initial data u0 which generates a global regular solution, does u0,n 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 (u0,n)nN, bornée dans un espace invariant par échelle, converge faiblement vers une donnée u0 engendrant une solution globale régulière, est-ce que u0,n 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.02.007

Hajer Bahouri 1; Jean-Yves Chemin 2; Isabelle Gallagher 3

1 Université Paris-Est Créteil, UMR 8050, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France
2 Université Pierre-et-Marie-Curie, Paris 6, UMR 7598, 4, place Jussieu, 75252 Paris cedex 05, France
3 Université Paris-Diderot, UMR 7586, Bâtiment Sophie-Germain, 75205 Paris cedex 13, France
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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/

[1] P. Auscher; S. Dubois; P. Tchamitchian On the stability of global solutions to Navier–Stokes equations in the space, J. Math. Pures Appl., Volume 83 (2004), pp. 673-697

[2] H. Bahouri; J.-Y. Chemin; I. Gallagher Stability by rescaled weak convergence for the Navier–Stokes equations | arXiv

[3] H. Bahouri; J.-Y. Chemin; R. Danchin Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, vol. 343, Springer, Berlin, Heidelberg, 2011

[4] H. Bahouri; A. Cohen; G. Koch A general wavelet-based profile decomposition in the critical embedding of function spaces, Confluentes Math., Volume 3 (2011), pp. 1-25

[5] H. Bahouri; I. Gallagher On the stability in weak topology of the set of global solutions to the Navier–Stokes equations, Arch. Ration. Mech. Anal., Volume 209 (2013), pp. 569-629

[6] H. Bahouri; P. Gérard High-frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., Volume 121 (1999), pp. 131-175

[7] J.-Y. Chemin; I. Gallagher Large, global solutions to the Navier–Stokes equations, slowly varying in one direction, Trans. Amer. Math. Soc., Volume 362 (2010) no. 6, pp. 2859-2873

[8] J.-Y. Chemin; I. Gallagher; P. Zhang Sums of large global solutions to the incompressible Navier–Stokes equations, J. Reine Angew. Math., Volume 681 (2013), pp. 65-82

[9] J.-Y. Chemin; I. Gallagher; C. Mullaert The role of spectral anisotropy in the resolution of the three-dimensional Navier–Stokes equations | arXiv

[10] I. Gallagher; D. Iftimie; F. Planchon Asymptotics and stability for global solutions to the Navier–Stokes equations, Ann. Inst. Fourier, Volume 53 (2003), pp. 1387-1424

[11] P. Gérard Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., Volume 3 (1998), pp. 213-233

[12] D. Iftimie Resolution of the Navier–Stokes equations in anisotropic spaces, Rev. Mat. Iberoam., Volume 15 (1999) no. 1, pp. 1-36

[13] S. Jaffard Analysis of the lack of compactness in the critical Sobolev embeddings, J. Funct. Anal., Volume 161 (1999), pp. 384-396

[14] P.-G. Lemarié-Rieusset Recent Developments in the Navier–Stokes Problem, Research Notes in Mathematics, vol. 43, Chapman and Hall/CRC Press, 2002

[15] J. Leray Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., Volume 63 (1933), pp. 193-248

[16] J. Leray Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl., Volume 12 (1933), pp. 1-82

[17] M. Paicu Équation anisotrope de Navier–Stokes dans des espaces critiques, Rev. Mat. Iberoam., Volume 21 (2005) no. 1, pp. 179-235

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