### Abstract

We study the large-time behaviour of solutions to the initial value problem for hyperbolic-parabolic systems of conservation equations in one space dimension. It is proved that under suitable assumptions a unique solution exists for all time t ≥ 0, and converges to a given constant state at the rate t_{-1/4}as t→∞ Moreover, it is proved that the solution approaches the superposition of the non-linear and linear diffusion waves constructed in terms of the self-similar solutions to the Burgers equation and the linear heat equation at the rate t_{-1/2+α}, α <0, as t→∞ The proof is essentially based on the fact that for t→∞, the solution to the hyperbolic-parabolic system is well approximated by the solution to a semilinear uniformly parabolic system whose viscosity matrix is uniquely determined from the original system. The results obtained are applicable straightforwardly to the equations of viscous (or inviscid) heat-conductive fluids.

Original language | English |
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Pages (from-to) | 169-194 |

Number of pages | 26 |

Journal | Proceedings of the Royal Society of Edinburgh: Section A Mathematics |

Volume | 106 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1987 Jan 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Large-time behaviour of solutions to hyperbolic–parabolic systems of conservation laws and applications.** / Kawashima, Shuichi.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Large-time behaviour of solutions to hyperbolic–parabolic systems of conservation laws and applications

AU - Kawashima, Shuichi

PY - 1987/1/1

Y1 - 1987/1/1

N2 - We study the large-time behaviour of solutions to the initial value problem for hyperbolic-parabolic systems of conservation equations in one space dimension. It is proved that under suitable assumptions a unique solution exists for all time t ≥ 0, and converges to a given constant state at the rate t-1/4as t→∞ Moreover, it is proved that the solution approaches the superposition of the non-linear and linear diffusion waves constructed in terms of the self-similar solutions to the Burgers equation and the linear heat equation at the rate t-1/2+α, α <0, as t→∞ The proof is essentially based on the fact that for t→∞, the solution to the hyperbolic-parabolic system is well approximated by the solution to a semilinear uniformly parabolic system whose viscosity matrix is uniquely determined from the original system. The results obtained are applicable straightforwardly to the equations of viscous (or inviscid) heat-conductive fluids.

AB - We study the large-time behaviour of solutions to the initial value problem for hyperbolic-parabolic systems of conservation equations in one space dimension. It is proved that under suitable assumptions a unique solution exists for all time t ≥ 0, and converges to a given constant state at the rate t-1/4as t→∞ Moreover, it is proved that the solution approaches the superposition of the non-linear and linear diffusion waves constructed in terms of the self-similar solutions to the Burgers equation and the linear heat equation at the rate t-1/2+α, α <0, as t→∞ The proof is essentially based on the fact that for t→∞, the solution to the hyperbolic-parabolic system is well approximated by the solution to a semilinear uniformly parabolic system whose viscosity matrix is uniquely determined from the original system. The results obtained are applicable straightforwardly to the equations of viscous (or inviscid) heat-conductive fluids.

UR - http://www.scopus.com/inward/record.url?scp=84976041027&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976041027&partnerID=8YFLogxK

U2 - 10.1017/S0308210500018308

DO - 10.1017/S0308210500018308

M3 - Article

VL - 106

SP - 169

EP - 194

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 1-2

ER -