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-dime nsional barotropic flow governed by v(t) - u(x) = 0, u(t) + p(v)(x) = mu(u( x)/v)(x) on R-+(1) with boundary. The initial data of (v, u) have constant states (v(+), u(+)) at +infinity and the boundary condition at x = 0 is giv en only on the velocity u, say u_. By virtue of the boundary effect the sol ution is expected to behave as outgoing wave. Therefore, when u_ < u(+), v( -) is determined as (v(+), u(+)) is an element of R-2(v(-), u(-)), 2-rarefa ction curve for the corresponding hyperbolic system, which admits the a-rar efaction wave (v(r), u(r))(x/t) connecting two constant states (v(-), u(-)) and (v(+), u(+)). Our assertion is that the solution of the original syste m tends to the restriction of (v(r), u(r))(x/t) to R-+(1) as t --> infinity provided that both the initial perturbations and \(v(+) - v(-), u(+) - u(- ))\ are small. The result is given by an elementary L-2 energy method.