Asymptotic stability of nonlinear wave for an inflow problem to the compressible Navier-Stokes-Korteweg system

In this paper, we are concerned with the inflow problem on the half line $(0,+\infty)$ for a one-dimensional compressible Navier-Stokes-Korteweg system, which is used to model compressible viscous fluids with internal capillarity, i.e., the liquid-vapor mixtures with phase interfaces. We first investigate that the asymptotic profile is a nonlinear wave: the superposition wave of a rarefaction wave and a boundary layer solution under the proper condition of the far fields and boundary values. The asymptotic stability on the nonlinear wave is shown under some conditions that the initial data are a small perturbation of the rarefaction wave and the strength of the stationary wave is small enough. The proofs are given by an elementary energy method.