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**On a weak solution for a transonic flow problem.**
*(English)*
Zbl 0615.76070

The objective of this paper is to establish the existence of weak solutions for steady state two-dimensional inviscid irrotational compressible flow in an infinitely long channel or in the exterior of an airfoil given the speed upstream at \(\infty\). If the speed upstream is sufficiently small there exists a strong smooth flow because the governing equations are elliptic. This means that the speed of sound always exceeds the fluid speed. But beyond a certain speed at infinity the flow ceases to be purely elliptic and shocks may and do arise. This paper shows how to establish a weak existence theorem for such a flow by the addition of artificial viscosity v. The existence of a certain viscous flow with free boundary is presumed. The speed of flow is then assumed to be bounded away from cavitation speed and stagnation and the angle of flow is assumed bounded independent of the viscosity. Then weak existence of the flow solution in the limit \(v\to 0\) is proved.

### MSC:

76H05 | Transonic flows |

35L67 | Shocks and singularities for hyperbolic equations |

76N20 | Boundary-layer theory for compressible fluids and gas dynamics |

76L05 | Shock waves and blast waves in fluid mechanics |

35B40 | Asymptotic behavior of solutions to PDEs |

### Keywords:

singularly perturbed solution; Bernoulli’s law; boundary value problem; limiting process; Lax entropy pairs; weak solutions; steady state two- dimensional inviscid irrotational compressible flow; infinitely long channel; exterior of an airfoil; speed upstream; speed of sound; shocks; weak existence theorem; free boundary; cavitation speed; stagnation; angle of flow
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\textit{C. S. Morawetz}, Commun. Pure Appl. Math. 38, 797--817 (1985; Zbl 0615.76070)

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### References:

[1] | DiPerna, Arch. Rat. Mech. Anal. 82 pp 27– (1983) |

[2] | Compensated Compactness and General Systems of Conservation Laws, Preprint. · Zbl 0606.35052 |

[3] | Murat, Ann. Scuola Norm. Sup. Pisa 5 pp 489– (1978) |

[4] | Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, IV, 1979, pp. 136–192. Research Notes in Mathematics, Pitman. |

[5] | Osher, NASA Tech. Mem. 85751 (1984) |

[6] | Synge, Quart. App. Math. 13 pp 271– (1955) |

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