Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect

The initial-boundary value problem on the half-line R₊ = (0, ∞) for a system of barotropic viscous flow v_t-u_x = 0, u_t+p(v)_x = μ(^ux/v)_x is investigated, where the pressure p(v) = v^-γ (γ≥1) for the specific volume v>0. Note that the boundary value at x = 0 is given only for the velocity u, say u_-, and that the initial data (v₀, u₀)(x) have the constant states (v₊, u₊) at x = +∞ with v₀(x)>0, v₊>0. If u_-<u₊, then there is a unique v_- such that (v₊,u₊)∈ R₂(v_-,u_-) (the 2-rarefaction curve) and hence there exists the 2-rarefaction wave (v₂ ^R, u₂ ^R)(x/t) connecting (v_-,u_-) with (v₊,u₊). Our assertion is that, if u_-<u₊, then there exists a global solution (v, u)(t,x) in C⁰([0, ∞); H¹(R₊)), which tends to the 2-rarefaction wave (v₂ ^R, u₂ ^R)(x/t)|_x≥0 as t→∞ in the maximum norm, with no smallness condition on |u₊-u_-| and ∥(v₀-v₊, u₀-u₊)∥_H(1), nor restriction on γ(≥1). A similar result to the corresponding Cauchy problem is also obtained. The proofs are given by an elementary L²-energy method.