Nous construisons des nappes de tourbillon supersoniques pour les équations d'Euler compressibles isentropiques en deux dimensions d'espace. Il s'agit d'un problème non-linéaire hyperbolique à frontière libre présentant deux difficultés principales : la frontière libre est caractéristique et la condition dite de Lopatinskii n'est satisfaite que dans un sens faible, ce qui induit des estimations à perte. Néanmoins nous montrons l'existence de telles solutions régulières par morceaux des équations d'Euler en utilisant un schéma itératif de type Nash-Moser palliant les pertes de régularité. Notre analyse s'étend au cas de discontinuités non-caractéristiques et faiblement stables comme certaines ondes de choc pour les équations d'Euler ou les transitions de phase liquide- vapeur.
We consider supersonic compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. The problem is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. Nevertheless, we prove the local existence of such piecewise smooth solutions to the Euler equations. Since the a priori estimates for the linearized equations exhibit a loss of regularity, our existence result is proved by using a suitable modification of the Nash-Moser iteration scheme. We also show how a similar analysis yields the existence of weakly stable shock waves in isentropic gas dynamics, and the existence of weakly stable liquid/vapor phase transitions.
@article{ASENS_2008_4_41_1_85_0,
author = {Coulombel, Jean-Fran\c cois and Secchi, Paolo},
title = {Nonlinear compressible vortex sheets in two space dimensions},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
volume = {41},
year = {2008},
pages = {85-139},
doi = {10.24033/asens.2064},
mrnumber = {2423311},
zbl = {1160.35061},
language = {en},
url = {http://dml.mathdoc.fr/item/ASENS_2008_4_41_1_85_0}
}
Coulombel, Jean-François; Secchi, Paolo. Nonlinear compressible vortex sheets in two space dimensions. Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008) pp. 85-139. doi : 10.24033/asens.2064. http://gdmltest.u-ga.fr/item/ASENS_2008_4_41_1_85_0/
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