Nonlinear compressible vortex sheets in two space dimensions
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 1, p. 85-139

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.

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.

DOI : https://doi.org/10.24033/asens.2064
Classification:  76N10,  35Q35,  35L50,  76E17
Keywords: compressible Euler equations, vortex sheets, contact discontinuities, weak stability, loss of derivatives
@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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {1},
     year = {2008},
     pages = {85-139},
     doi = {10.24033/asens.2064},
     zbl = {1160.35061},
     mrnumber = {2423311},
     language = {en},
     url = {http://www.numdam.org/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, Serie 4, Volume 41 (2008) no. 1, pp. 85-139. doi : 10.24033/asens.2064. http://www.numdam.org/item/ASENS_2008_4_41_1_85_0/

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