  This paper is concerned with the inflow problem for the one-dimensional
compressible Navier-Stokes equations. For such a problem, Matsumura and
Nishihara showed in [A. Matsumura and K. Nishihara, Large-time behaviors of
solutions to an inflow problem in the half space for a one-dimensional system
of compressible viscous gas. Comm. Math. Phys. 222 (2001), 449-474] that there
exists boundary layer solution to the inflow problem and both the boundary
layer solution, the rarefaction wave, and the superposition of boundary layer
solution and rarefaction wave are nonlinear stable under small initial
perturbation. The main purpose of this paper is to show that similar stability
results for the boundary layer solution and the supersonic rarefaction wave
still hold for a class of large initial perturbation which can allow the
initial density to have large oscillation. The proofs are given by an
elementary energy method and the key point is to deduce the desired lower and
upper bounds on the density function.
