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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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References

1Friedrichs, K. O. and Lax, P. D.. Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 1686–1688.CrossRefGoogle ScholarPubMed
2Hopf, E.. The partial differential equation ut + uux = μuxx. Comm. Pure Appl. Math. 3 (1950), 201–230.CrossRefGoogle Scholar
3Kato, T.. Perturbation Theory for Linear Operators, 2nd edn. (New York: Springer, 1976).Google Scholar
4Kawashima, S.. The asymptotic equivalence of the Broadwell model equation and its Navier-Stokes model equation. Japan J. Math. (N.S.) 7 (1981), 1–43.CrossRefGoogle Scholar
5Kawashima, S.. Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Doctoral thesis, Kyoto University, 1983.Google Scholar
6Kawashima, S. and Okada, N.. Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Japan Acad. Ser. A. 58 (1982), 384–387.CrossRefGoogle Scholar
7Lax, P. D.. Hyperbolic systems of conservation laws, II. Comm. Pure Appl. Math. 10 (1957), 537–566.CrossRefGoogle Scholar
8Liu, T.-P.. Linear and non-linear large-time behaviour of solutions of general systems of hyperbolic conservation laws. Comm. Pure Appl. Math. 30 (1977), 767–796.CrossRefGoogle Scholar
9Liu, T.-P.. Non-linear stability of shock waves for viscous conservation laws. Mem. Math. Soc. 328 56 (1985).Google Scholar
10Nishida, T.. Equations of motion of compressible viscous fluids. Patterns and Waves, Qualitative Analysis of Nonlinear Differential Equations, Studies in Mathematics and Its Applications 18 (North-Holland, Kinokuniya, 1986).Google Scholar
11Shizuta, Y. and Kawashima, S.. Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14 (1985), 249–275.CrossRefGoogle Scholar
12Umeda, T., Kawashima, S. and Shitzuta, Y.. On the decay of solutions to the linearized equations of electro-magnetofluid dynamics. Japan J. Appl. Math. 1 (1984), 435–457.CrossRefGoogle Scholar