Uniqueness of 2-D compressible vortex sheets

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July 2009, 8(4): 1439-1450. doi: 10.3934/cpaa.2009.8.1439

Uniqueness of 2-D compressible vortex sheets

Jean-françois Coulombel ^1, and Paolo Secchi ^2,

1.	CNRS, Université Lille 1 and Team Project SIMPAF of INRIA Lille Nord Europe, Laboratoire Paul Painlevé, Bâtiment M2, Cité Scientifique, 59655 VILLENEUVE D'ASCQ CEDEX, France
2.	Dipartimento di Matematica, Facoltà di Ingegneria, Università di Brescia, Via Valotti, 9, 25133 Brescia, Italy

Received June 2008 Revised November 2008 Published March 2009

Abstract
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We consider compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. Under a supersonic condition that precludes violent instabilities, in previous papers [3, 4] we have studied the linearized stability and proved the local existence of piecewise smooth solutions to the nonlinear problem. This 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. In the present paper we prove that sufficiently smooth solutions are unique.

Keywords: vortex sheets, Compressible Euler equations, loss of derivatives., weak Lopatinskii condition, contact discontinuities.

Mathematics Subject Classification: Primary: 76N10; Secondary: 35Q35, 35L50, 76E1.

Citation: Jean-françois Coulombel, Paolo Secchi. Uniqueness of 2-D compressible vortex sheets. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1439-1450. doi: 10.3934/cpaa.2009.8.1439

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