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H 2 Some results on global solutions to the Navier-Stokes equations

Auteurs : Gallagher, Isabelle (Auteur de la Conférence)
CIRM (Editeur )

Résumé : In this talk we shall present some results concerning global smooth solutions to the three-dimensional Navier-Stokes equations set in the whole space $(\mathbb{R}^3)$ :
$\partial_tu+u\cdot \nabla u-\Delta u = -\nabla p$, div $u=0$
We shall more particularly be interested in the geometry of the set $\mathcal{G}$ of initial data giving rise to a global smooth solution.
The question we shall address is the following: given an initial data $u_0$ in $\mathcal{G}$ and a sequence of divergence free vector fields converging towards $u_0$ in the sense of distributions, is the sequence itself in $\mathcal{G}$ ? The related question of strong stability was studied in [1] and [2] some years ago; the weak stability result is a recent work, joint with H. Bahouri and J.-Y. Chemin (see [3]-[4]). As we shall explain, it is necessary to restrict the study to sequences converging weakly up to rescaling (under the natural rescaling of the equation). Then weak stability can be proved, using profile decompositions in the spirit of P. Gerard's work [5], in an anisotropic context.

Codes MSC :
35Q30 - Stokes and Navier-Stokes equations
76E09 - Stability and instability of nonparallel flows

 Informations sur la Vidéo Langue : Anglais Date de publication : 25/08/14 Date de captation : 05/05/14 Collection : Research talks ; Geometry ; Mathematical Physics Format : quicktime ; audio/x-aac Durée : 00:51:37 Domaine : Mathematical Physics ; Geometry Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2014-05-05_Gallagher.mp4 Informations sur la rencontre Nom de la rencontre : Vorticity, rotation and symmetry (III) - approaching limiting cases of fluid flows / Vorticité, rotation et symétrie (III) - analyse des situations limites en théorie des fluidesOrganisateurs de la rencontre : Farwig, Reinhard ; Neustupa, Jiri ; Penel, PatrickDates : 05/05/14 - 09/05/14 Année de la rencontre : 2014 URL Congrès : https://www.cirm-math.fr/Archives/?EX=in...Citation Data DOI : 10.24350/CIRM.V.18593603 Cite this video as: Gallagher, Isabelle (2014). Some results on global solutions to the Navier-Stokes equations. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18593603 URI : http://dx.doi.org/10.24350/CIRM.V.18593603

Bibliographie

1. P. Auscher, S Dubois, and P. Tchamitchian, On the stability of global solutions to Navier-Stokes equations in the space, Journal de Mathématiques Pures et Appliquées, 83 (2004), pages 673- 697 - http://doi.org/10.1016/j.matpur.2004.01.003

2. I. Gallagher, D. Iftimie and F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, Annales de l'Institut Fourier,53 , 5 (2003), pages 1387-1424 - http://aif.cedram.org/item?id=AIF_2003__53_5_1387_0

3. H. Bahouri and I. Gallagher, On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations, Archive for Rational Mechanics and Analysis 209, 2 (2013), 569-629 - http://doi.org/10.1007/s00205-013-0623-y

4. H. Bahouri, J.-Y. Chemin and I. Gallagher, Stability by weak convergence for the Navier-Stokes equations, submitted, arXiv:1310.0256
- http://arxiv.org/abs/1310.0256

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