Asymptotic behavior of a one-dimensional compressible viscous gas with free boundary

We consider the initial-boundary value problem for a one-dimensional compressible viscous gas with free boundary, which is modeled in the Eulerian coordinate as (IBVP) { ρ + (ρu)_x = 0, x > x(t), t > 0, (ρu)_t + (ρu² + p)_x = μu_xx, x > x(t), t > 0, (p - μu_x)|_x=x(t) = p0, dx(t)/dt = u(x(t),t), t ≥ 0, (ρ, u)|_t=0 = (ρ0, u0)(x), x ≥ x(0). Here, ρ(> 0) is the density, u is the velocity, p = p(ρ) = ρ^γ (γ ≥ 1: the adiabatic constant) is the pressure, and μ(> 0) is the viscosity constant. At the boundary the flow is attached to the atmosphere with pressure p₀ (> 0) and the boundary condition is derived by the balance law. The initial data have constant states (ρ₊, u₊) at x = ∞. The flow has no vacuum state so that ρ₀(x) > 0 and ρ₊ > 0 are assumed. Our main purpose is to investigate the asymptotic behaviors of solutions for (IBVP), which are closely related to those for the corresponding Cauchy problem and hence the corresponding Riemann problem. Depending on p₀ and the endstates (ρ₊, u₊), the solutions are shown to tend to the outgoing rarefaction wave or the outgoing viscous shock wave as t tends to infinity. The proof is given under the weakness assumption of the waves. The analysis will be done by changing (IBVP) into the problem in the Lagrangian coordinate.