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

Published online by Cambridge University Press:  14 November 2011

Shuichi Kawashima
Shuichi Kawashima
Affiliation:
Department of Mathematics, Nara Women's University, Nara 630, Japan
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Synopsis

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 − ¼ 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 − ½ +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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 106 , Issue 1-2 , 1987 , pp. 169 - 194
Copyright
Copyright © Royal Society of Edinburgh 1987

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