Asymptotic stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas with boundary
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This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed byv t −u x = 0,u t +p(v) x = μ(u x/v) x onR + 1 with boundary. The initial data of (v,u) have constant states (v +,u+) at +∞ and the boundary condition atx = 0 is given only on the velocityu, say u−. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, whenu − <u +,v − is determined as (u +,u +) ∈R 2(v −,u −), 2-rarefaction curve for the corresponding hyperbolic system, which admits the 2-rarefaction wave (v r,ur)(x/t) connecting two constant states (v −,u −) and (v +,u +). Our assertion is that the solution of the original system tends to the restriction of (v r,ur)(x/t) toR + 1 as t → ∞ provided that both the initial perturbations and ¦(v + −v −,u +-u t-) are small. The result is given by an elementaryL 2 energy method.
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- Asymptotic stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas with boundary
Japan Journal of Industrial and Applied Mathematics
Volume 16, Issue 3 , pp 431-441
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- asymptotic behavior
- rarefaction wave
- compressible viscous gas
- Author Affiliations
- 1. Department of Mathematics, Guangxi University, 530004, Nanning, P.R. China
- 2. Department of Mathematics, Jinan University, 510632, Guangzhou, P.R. China
- 3. School of Political Science and Economics, Waseda University, 169-50, Tokyo, Japan