Large-time behavior of the solutions for an inflow problem of the full compressible Navier-Stokes-Korteweg equations

In this article, we are concerned with the large-time behavior of solutions to an inflow problem for the one-dimensional full compressible Navier-Stokes-Korteweg equations, which can be used to model the compressible viscous fluids with internal capillarity and heat conductivity. Under some suitable assumptions of the far fields and boundary values of the density, the velocity and the absolute temperature, the large-time behavior of the solutions for the initial boundary value problem is govern by the stationary solution and basic waves of the corresponding hyperbolic system. Then the asymptotic stability of the boundary layer, and the superposition of the boundary layer and the three-rarefaction wave are shown provided that the initial perturbation and the strength of the rarefaction wave and stationary wave are small. The proof is mainly based on L-2-energy method, some time-decay estimates in L-p-norm for the smoothed rarefaction wave and the space variable decay estimates of the boundary layer. This can be viewed as the first result about the nonlinear stability of the combination of two different wave patterns for the inflow problem of the full compressible Navier-Stokes-Korteweg equations.