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 by v_t - u_x = 0, u_t + p(v)_x = μ(u_x/v)_x on R¹ ₊ with boundary. The initial data of (v, u) have constant states (v₊, u₊) at +∞ and the boundary condition at x = 0 is given only on the velocity u, say u_-. By virtue of the boundary effect the solution is expected to behave as outgoing wave. Therefore, when u- < u₊, v_- is determined as (v₊, u₊) ∈ R₂(v_-, u_-), 2-rarefaction curve for the corresponding hyperbolic system, which admits the 2-rarefaction 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 system tends to the restriction of (v^r, u^r)(x/t) to R¹ ₊ as t → ∞ provided that both the initial perturbations and |(v₊ -v_-, u₊ - u_-)| are small. The result is given by an elementary L² energy method.