### 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 ow 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 ow 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.

Original language | English (US) |
---|---|

Pages (from-to) | 489-523 |

Number of pages | 35 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 31 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1 2011 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems*,

*31*(2), 489-523. https://doi.org/10.3934/dcds.2011.31.489

}

*Discrete and Continuous Dynamical Systems*, vol. 31, no. 2, pp. 489-523. https://doi.org/10.3934/dcds.2011.31.489

**Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions.** / Lai, Geng; Sheng, Wancheng; Zheng, Yuxi.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions

AU - Lai, Geng

AU - Sheng, Wancheng

AU - Zheng, Yuxi

PY - 2011/10/1

Y1 - 2011/10/1

N2 - 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 ow 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 ow 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.

AB - 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 ow 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 ow 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.

UR - http://www.scopus.com/inward/record.url?scp=84859266875&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859266875&partnerID=8YFLogxK

U2 - 10.3934/dcds.2011.31.489

DO - 10.3934/dcds.2011.31.489

M3 - Article

VL - 31

SP - 489

EP - 523

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 2

ER -