### Abstract

For simple models of hyperbolic systems of conservation laws, we study a new type of nonlinear hyperbolic wave, a delta-shock wave, which is a Dirac delta function supported on a shock. We prove that delta-shock waves are w*-limits in L^{1} of solutions to some reasonable viscous perturbations as the viscosity vanishes. Further, we solve the multiplication problem of a delta function with a discontinuous function to show that delta-shock waves satisfy the equations in the sense of distributions. Under suitable generalized Rankine-Hugoniot and entropy conditions, we establish the existence and uniqueness of solutions involving delta-shock waves for the Riemann problems. The existence of solutions to the Cauchy problem is also investigated.

Original language | English (US) |
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Pages (from-to) | 1-32 |

Number of pages | 32 |

Journal | Journal of Differential Equations |

Volume | 112 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1994 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

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## Cite this

*Journal of Differential Equations*,

*112*(1), 1-32. https://doi.org/10.1006/jdeq.1994.1093