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

This paper is concerned with asymptotic behavior of solutions of a one-dimensional barotropic flow governed byvt −ux = 0,ut +p(v)x = μ(ux/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+) ∈R2(v−,u−), 2-rarefaction curve for the corresponding hyperbolic system, which admits the 2-rarefaction wave (vr,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 (vr,ur)(x/t) toR+1 as t → ∞ provided that both the initial perturbations and ¦(v+ −v−,u+-ut-) are small. The result is given by an elementaryL2 energy method.