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

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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-1/4as 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.

Original languageEnglish
Pages (from-to)169-194
Number of pages26
JournalProceedings of the Royal Society of Edinburgh: Section A Mathematics
Volume106
Issue number1-2
DOIs
Publication statusPublished - 1987 Jan 1
Externally publishedYes

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Systems of Conservation Laws
Large Time Behavior
Parabolic Systems
Behavior of Solutions
Hyperbolic Systems
Linear Diffusion
Self-similar Solutions
Burgers Equation
Semilinear
Heat Equation
Unique Solution
Superposition
Initial Value Problem
Conservation
Linear equation
Viscosity
Heat
Converge
Fluid

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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