The decay and formation of one-dimensional nonconservative shocks.

*(English)*Zbl 0644.73033The method of singular surfaces is used to obtain explicit conditions under which a one-dimensional acceleration wave develops into a shock when some dissipation mechanism is present. The conditions which secure the initial growth of the strong shock wave propagating into an undeformed nonlinear and dissipative medium are also derived. The analysis is presented for a single-balance law, one-dimensional elasticity, and the nonlinear Maxwellian continuum.

##### MSC:

74J99 | Waves in solid mechanics |

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

35L67 | Shocks and singularities for hyperbolic equations |

74M20 | Impact in solid mechanics |

74B20 | Nonlinear elasticity |

##### Keywords:

nonconservative hyperbolic systems; existence of global solutions; shock formation; nonlinear partial differential equations; one-dimensional acceleration wave; dissipation; single-balance law; one-dimensional elasticity; nonlinear Maxwellian continuum
PDF
BibTeX
XML
Cite

\textit{M. Elżanowski} and \textit{M. Epstein}, Appl. Math. Modelling 12, No. 3, 280--284 (1988; Zbl 0644.73033)

Full Text:
DOI

##### References:

[1] | Kosinski, W, Gradient catastrophe in the solution of nonconservative hyperbolic systems, J. math. anal. appl., 61, 672-688, (1977) · Zbl 0369.35043 |

[2] | Nishida, T. Global smooth solutions for the second order quasilinear wave equations with the first order dissipation (unpublished) |

[3] | Slemrod, M, Instability of steady shearing flows in a nonlinear viscoelastic fluid, Arch. rational mech. anal., 68, 211-225, (1978) · Zbl 0393.76004 |

[4] | Dafermos, C.M, Development of singularities in the motion of materials with fading memory, Arch. rational mech. anal., 91, 192-205, (1985) · Zbl 0595.73026 |

[5] | Dafermos, C.M, Can dissipation prevent the breaking of waves?, () · Zbl 0506.73023 |

[6] | Dafermos, C.M; Hrusa, W.J, Energy methods for quasihyperbolic initial-boundary value problems, (), 267-292 · Zbl 0586.35065 |

[7] | Smoller, J, Shock waves and reaction-diffusion equations, (1983), Springer-Verlag Berlin · Zbl 0508.35002 |

[8] | Chen, P.J, Growth and decay of waves in solids, () |

[9] | Elżanowski, M; Epstein, M, Decay of strong shocks in nonlinear elasticity, J. sound and vibration, 103, 371-378, (1985) |

[10] | Elżanowski, M; Epstein, M, A note on breaking of waves of hyperbolic balance laws, (), No. 342 |

[11] | Lax, P.D, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, (), Siam · Zbl 0108.28203 |

[12] | Elżanowski, M; Epstein, M, Attenuation of shocks by viscoelastic support, ASCE. J. engg. mech., 112, 587-592, (1986) |

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.