, Volume 16, Issue 3, pp 431-441

Asymptotic stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas with boundary

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Abstract

This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed byv tu 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.

To complete this work the first and second authors were supported in part by the Guangxi Education Committee Foundation GJK1998-69 and the KD Science Foundation of the State Council Office of Overseas Chinese Affairs. Research of the third author was supported in part by Waseda University Grant for Special Research Project 97A-242.