Simple waves and pressure delta waves         for a Chaplygin gas in two-dimensions

Geng Lai; Wancheng Sheng; Yuxi Zheng

doi:10.3934/dcds.2011.31.489

Article Contents

2011, Volume 31, Issue 2: 489-523. Doi: 10.3934/dcds.2011.31.489

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Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions

1.
Department of Mathematics, Shanghai University, Shanghai 200444
2.
Department of Mathematical Sciences, Yeshiva University, New York, NY 10033

Received: January 31, 2010

Revised: February 28, 2011

Published: May 2011

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Abstract

We present two new types of self-similar solutions to the Chaplygin gas model in two space dimensions: Simple waves and pressure delta waves, which are absent in one space dimension, but appear in the solutions to the two-dimensional Riemann problems. A simple wave is a flow in a physical region whose image in the state space is a one-dimensional curve. The solutions to the interaction of two rarefaction simple waves are constructed. Comparisons with polytropic gases are made. Pressure delta waves are Dirac type concentration in the pressure variable, or impulses of the pressure on discontinuities. They appear in the study of Riemann problems of four rarefaction shocks. This type of discontinuities and concentrations are different from delta waves for the pressureless gas flow model, for which the delta waves are associated with convection and concentration of mass. By re-interpreting the terms in the Chaplygin gas system into new forms we are able to define distributional solutions that include the pressure delta waves. Generalized Rankine-Hugoniot conditions for pressure delta waves are derived.

Keywords:

Riemann problem,
self-similar,
characteristic,
characteristic direction,
characteristic angle,
sonic circle,
sonic edge,
sonic curve,
simple wave interaction,
generalized Rankine-Hugoniot conditions.

Mathematics Subject Classification: Primary: 35L65, 35J70, 35R35; Secondary: 35J65.

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