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**Generalized characteristics in hyperbolic systems of conservation laws.**
*(English)*
Zbl 0714.35046

In the paper the Cauchy problem for a strictly hyperbolic nonlinear system of conservation laws is considered. Because the solutions of this problem are generally discontinuous in a finite time, they may exist in the large as weak solutions. The author’s goal is to make a first step in directly studying those. It is known that behaviour of the solutions in question can be described in terms of their generalized characteristics and entropy estimates. In particular, any generalized characteristic is proved to propagate either with appropriate classical characteristic speed or with appropriate shock speed. In addition, the estimates of the second derivatives of entropy with respect to the Riemann invariants are obtained for the genuinely nonlinear system of two conservation laws. References, 17 in number, fully cover the stated problem.

Reviewer: V.Chernyatin

### MSC:

35L65 | Hyperbolic conservation laws |

35B40 | Asymptotic behavior of solutions to PDEs |

35L67 | Shocks and singularities for hyperbolic equations |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

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\textit{C. M. Dafermos}, Arch. Ration. Mech. Anal. 107, No. 2, 127--155 (1989; Zbl 0714.35046)

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### References:

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[12] | Lax, P. D., Shock waves and entropy. Contributions to Functional Analysis, ed. E. A. Zarantonello, New York: Academic Press 1971, pp. 603-634. |

[13] | Liu, T.-P., Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws. Comm. Pure Appl. Math. 30 (1977), 585-610. · Zbl 0357.35059 |

[14] | Liu, T.-P., Pointwise convergence to N-waves for solutions of hyperbolic conservation laws. (To appear.) |

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