##
**Hyperbolic conservation laws with relaxation.**
*(English)*
Zbl 0633.35049

The effect of relaxation is important in many physical situations. It is present in the kinetic theory of gases, elasticity with memory, gas flow with thermo-non-equilibrium, water waves, etc. The governing equations often take the form of hyperbolic conservation laws with lower-order terms. In this article, we present and analyze a simple model of hyperbolic conservation laws with relaxation effects. Dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified. Stability criteria for diffusion waves, expansion waves and traveling waves are found and justified nonlinearly. Time-asymptotic expansion and the energy method are used in the analysis. For dissipative waves, the expansion is similar in spirit to the Chapman- Enskog expansion in the kinetic theory. For shock waves, however, a different approach is needed.

### MSC:

35L65 | Hyperbolic conservation laws |

35B40 | Asymptotic behavior of solutions to PDEs |

35B35 | Stability in context of PDEs |

### Keywords:

relaxation; hyperbolic conservation laws; lower-order terms; propagation of disturbances; Stability; diffusion waves; expansion waves; traveling waves; Time-asymptotic expansion; energy method; Chapman-Enskog expansion
Full Text:
DOI

### References:

[1] | Caflish, R., Liu, T.-P.: Nonlinear stability of shock waves for the Broadwell model (to appear) |

[2] | Cercignani, C.: Theory and application of the Boltzman equation. Scottish Academic Press 1975 · Zbl 0403.76065 |

[3] | Dafermos, C., Nohel, J.: A nonlinear hyperbolic Volterra equation in viscoelasticity, Contribution to Analysis and Geometry, pp. 87-116. Blatimore, MD.: John Hopkins University Press 1981 · Zbl 0588.35016 |

[4] | Greenberg, J., Hsiao, L.: The Riemann problem for systemu t+? x =0 and (??f(w) t)+(???f(u))=0. Arch. Ration. Mech. Anal.82, 87-108 (1983) · Zbl 0555.35083 |

[5] | Liu, T.-P.: The Riemann problem for general system of conservation laws. J. Differ. Equations18, 218-234 (1975) · Zbl 0297.76057 |

[6] | Liu, T.-P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc.328, No. 328 (1985) · Zbl 0576.35077 |

[7] | Whitham, J.: Linear and nonlinear waves. New York: Wiley 1974 · Zbl 0373.76001 |

[8] | Vicenti, W., Kruger, C.: Introduction to physical gas dynamics. Melbourne: Robert E. Krieger 1982 |

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.