Nonlinear compressible vortex sheets in two space dimensions

[Nappes de tourbillon compressibles]

Coulombel, Jean-François ; Secchi, Paolo

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 1, pp. 85-139.

Abstract (VO)

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.

Résumé

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.

Détail

DOI : 10.24033/asens.2064

Classification : 76N10, 35Q35, 35L50, 76E17
Keywords: compressible Euler equations, vortex sheets, contact discontinuities, weak stability, loss of derivatives

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