Asymptotics toward rarefaction wave for an inflow problem of the compressible Navier–Stokes–Korteweg equation

. In this article, we are concerned with the large-time behavior of solutions to an inflow problem in one-dimensional case for the Navier-Stokes- Korteweg equation, which models compressible fluids with internal capillarity. We first investigate that the asymptotic state is the rarefaction wave under the proper condition of the far fields and boundary values. The asymptotic stability of the rarefaction wave under some smallness conditions is shown. The proof is completed by the energy method with the help of time-decay estimate for the rarefaction wave.