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**Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping.**
*(English)*
Zbl 0763.35058

Summary: We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time- asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy’s law time-asymptotically. Our model may also be viewed as an elastic model with damping.

### MSC:

35L65 | Hyperbolic conservation laws |

35Q35 | PDEs in connection with fluid mechanics |

76S05 | Flows in porous media; filtration; seepage |

35B40 | Asymptotic behavior of solutions to PDEs |

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\textit{L. Hsiao} and \textit{T.-P. Liu}, Commun. Math. Phys. 143, No. 3, 599--605 (1992; Zbl 0763.35058)

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### References:

[1] | Duyn, C.T., Van Peletier, L.A.: Nonlinear analysis. T.M.A.1, 223–233 (1977) |

[2] | Liu, T.-P.: Nonlinear hyperbolic-parabolic partial differential equations. Nonlinear Analysis, Proceedings, 1989 Conference. Liu, F.C., Liu, T.P. (eds.), pp. 161–170. Academia Sinica, Taipei, R.O.C.: World Scientific · Zbl 0819.35101 |

[3] | Matzumura, A.: Nonlinear hyperbolic equations and related topics in fluid dynamics. Nishida, T. (ed.). Pub. Math. D’Orsay, 53–57 (1978) |

[4] | Nishida, T.: Nonlinear hyperbolic equations and related topics in fluid dynamics. Nishida, T. (ed.) Pub. Math. D’Orsay, 46–53 (1978) · Zbl 0392.76065 |

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