Nonlinear compressible vortex sheets in two space dimensions

Jean-François Coulombel; Paolo Secchi

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 1, page 85-139
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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Coulombel, Jean-François, and Secchi, Paolo. "Nonlinear compressible vortex sheets in two space dimensions." Annales scientifiques de l'École Normale Supérieure 41.1 (2008): 85-139. <http://eudml.org/doc/272142>.

@article{Coulombel2008,
abstract = {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.},
author = {Coulombel, Jean-François, Secchi, Paolo},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {compressible Euler equations; vortex sheets; contact discontinuities; weak stability; loss of derivatives; Euler equations; hyperbolic; Lopatinski condition; Nash-Moser iteration scheme},
language = {eng},
number = {1},
pages = {85-139},
publisher = {Société mathématique de France},
title = {Nonlinear compressible vortex sheets in two space dimensions},
url = {http://eudml.org/doc/272142},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Coulombel, Jean-François
AU - Secchi, Paolo
TI - Nonlinear compressible vortex sheets in two space dimensions
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 1
SP - 85
EP - 139
AB - 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.
LA - eng
KW - compressible Euler equations; vortex sheets; contact discontinuities; weak stability; loss of derivatives; Euler equations; hyperbolic; Lopatinski condition; Nash-Moser iteration scheme
UR - http://eudml.org/doc/272142
ER -

References

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