Multi-dimensional shock fronts for second order wave equations.

*(English)*Zbl 0632.35047The authors study the short-time existence and structural stability of multidimensional shock fronts satisfying scalar second order quasi-linear wave equations. Their long rigorous proof given here for this special class of equations is simpler than those for general first order systems presented previously by the first author [Mem. Am. Math. Soc. 275, 95 p. (1983; Zbl 0506.76075) and 281, 92 p. (1982; Zbl 0517.76068)], although the basic ideas of these papers are here utilized.

The elegant strategy of their proof consists basically in the reduction of this shock front problem to an existence problem for a single scalar function defined on one side of the shock front and then in the introduction of a suitable partial hodograph transformation which leads to a nonlinear mixed problem for a single unknown function. This transformation (which has been used earlier by different authors in the study of elliptic free boundary value problems) is not only a remarkable simplification of the problem under consideration, but seems also to be appropriate tool for solving second order equations for which the previous techniques for the first order systems given in Majda’s papers quoted above are not successful. Then a fixed point iteration scheme is used. A special emphasis is given to the multidimensional shock fronts for the equations of isontropic irrotational compressible fluid which represents an important physical example for their theory.

The elegant strategy of their proof consists basically in the reduction of this shock front problem to an existence problem for a single scalar function defined on one side of the shock front and then in the introduction of a suitable partial hodograph transformation which leads to a nonlinear mixed problem for a single unknown function. This transformation (which has been used earlier by different authors in the study of elliptic free boundary value problems) is not only a remarkable simplification of the problem under consideration, but seems also to be appropriate tool for solving second order equations for which the previous techniques for the first order systems given in Majda’s papers quoted above are not successful. Then a fixed point iteration scheme is used. A special emphasis is given to the multidimensional shock fronts for the equations of isontropic irrotational compressible fluid which represents an important physical example for their theory.

Reviewer: P.Pucci

##### MSC:

35L70 | Second-order nonlinear hyperbolic equations |

35L67 | Shocks and singularities for hyperbolic equations |

76G25 | General aerodynamics and subsonic flows |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35B35 | Stability in context of PDEs |

##### Keywords:

conservation laws; Lax’s shock inequalities; multi-stability conditions; short-time existence; structural stability; shock fronts; quasi-linear wave equations; hodograph transformation; fixed point iteration; isontropic irrotational compressible fluid
PDF
BibTeX
XML
Cite

\textit{A. Majda} and \textit{E. Thomann}, Commun. Partial Differ. Equations 12, 777--828 (1987; Zbl 0632.35047)

Full Text:
DOI

##### References:

[1] | Courant R., Methods of Mathematical Physics (1962) · Zbl 0099.29504 |

[2] | Friedrichs K., Math. Ann 109 pp 560– (1934) |

[3] | Gårding L., Computes Rendus, Series A 285 pp 773– (1977) |

[4] | DOI: 10.2977/prims/1195194627 · Zbl 0207.10101 · doi:10.2977/prims/1195194627 |

[5] | Kinderlehrer D., Annals Scuola Norm. Sup. Pisa 5 (4) pp 373– (1977) |

[6] | Majda A., Proc. Intl (1224) |

[7] | Majda A., Memoir A.M.S. 275 (1983) |

[8] | Majda A., Memoirs A.M.S 281 (1983) |

[9] | DOI: 10.1007/978-1-4612-1116-7 · doi:10.1007/978-1-4612-1116-7 |

[10] | DOI: 10.1137/0143088 · Zbl 0544.76135 · doi:10.1137/0143088 |

[11] | DOI: 10.1002/sapm1984712117 · Zbl 0584.76075 · doi:10.1002/sapm1984712117 |

[12] | Miyatake S., Japanese J. Math 1 pp 111– (1975) |

[13] | Rauch J., Transactions A.M.S 189 pp 303– (1974) |

[14] | Schauder J., Fund. Math 24 pp 777– (1935) |

[15] | Thomann E., Ph.D. thesis (1985) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.