### Abstract

Given a smooth curve as a sonic line in the plane, we construct a local smooth supersonic solution on one side of the curve for the steady compressible Euler system of equations in two space dimensions. Our construction hinges on a new set of coordinates introduced here to handle the inherent degeneracy of the system at the sonic curve. We analyze the streamlines of the solutions to illustrate that the shock-free portion of the solutions may be combined with known results of existence of sonic-subsonic solutions of Xie and Xin [33] on the other side of the curve to form shock-free transonic flows in a channel. The existence result is also a partial generalization of the exact solution of Ringleb [28, ZAMM (1940)] toward a flexible existence.

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

Pages (from-to) | 1785-1817 |

Number of pages | 33 |

Journal | Indiana University Mathematics Journal |

Volume | 63 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*63*(6), 1785-1817. https://doi.org/10.1512/iumj.2014.63.5434

}

*Indiana University Mathematics Journal*, vol. 63, no. 6, pp. 1785-1817. https://doi.org/10.1512/iumj.2014.63.5434

**Sonic-supersonic solutions for the steady Euler equations.** / Zhang, Tianyou; Zheng, Yuxi.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Sonic-supersonic solutions for the steady Euler equations

AU - Zhang, Tianyou

AU - Zheng, Yuxi

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Given a smooth curve as a sonic line in the plane, we construct a local smooth supersonic solution on one side of the curve for the steady compressible Euler system of equations in two space dimensions. Our construction hinges on a new set of coordinates introduced here to handle the inherent degeneracy of the system at the sonic curve. We analyze the streamlines of the solutions to illustrate that the shock-free portion of the solutions may be combined with known results of existence of sonic-subsonic solutions of Xie and Xin [33] on the other side of the curve to form shock-free transonic flows in a channel. The existence result is also a partial generalization of the exact solution of Ringleb [28, ZAMM (1940)] toward a flexible existence.

AB - Given a smooth curve as a sonic line in the plane, we construct a local smooth supersonic solution on one side of the curve for the steady compressible Euler system of equations in two space dimensions. Our construction hinges on a new set of coordinates introduced here to handle the inherent degeneracy of the system at the sonic curve. We analyze the streamlines of the solutions to illustrate that the shock-free portion of the solutions may be combined with known results of existence of sonic-subsonic solutions of Xie and Xin [33] on the other side of the curve to form shock-free transonic flows in a channel. The existence result is also a partial generalization of the exact solution of Ringleb [28, ZAMM (1940)] toward a flexible existence.

UR - http://www.scopus.com/inward/record.url?scp=84918524139&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84918524139&partnerID=8YFLogxK

U2 - 10.1512/iumj.2014.63.5434

DO - 10.1512/iumj.2014.63.5434

M3 - Article

VL - 63

SP - 1785

EP - 1817

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 6

ER -