Résultats récents sur la limite incompressible

Gallagher, Isabelle

Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 926, pp. 29-57.

La compréhension du passage des équations de la mécanique des fluides compressibles aux équations incompressibles a fait de grands progrès ces vingt dernières années. L'objectif de cet exposé est de présenter l'évolution des méthodes mathématiques mises en œuvre pour étudier ce passage à la limite, depuis les travaux de S. Klainerman et A. Majda dans les années quatre-vingts, jusqu'à ceux récents de G. Métivier et S. Schochet (pour les équations non isentropiques). Suivant les conditions initiales et les conditions aux bords, les méthodes utilisées sont variées, et nous décrirons des résultats de type optique géométrique aussi bien que d'autres liés à la théorie des mesures de défaut, à des estimations de Strichartz ou encore à des calculs de petits diviseurs.

In the last two decades, a great amount of progress has been made in the understanding of the passage from the equations governing compressible fluids, to the incompressible equations. The aim of this talk is to present the evolution of the mathematical methods used to study that limit, from the works of S. Klainerman and A. Majda in the eighties, to the recent studies of G. Métivier and S. Schochet (for the non isentropic equations). The methods followed are different according to the initial conditions as well as the boundary conditions; we will describe methods of geometrical optics as well as others linked to the theory of defect measures, Strichartz estimates, and also small divisor type computations.

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