Large-time behavior of solutions to outflow problem of full compressible Navier–Stokes–Korteweg equations

In this article, we investigate the large-time behavior of the solution to outflow problem for full compressible Navier–Stokes–Korteweg equations in the one-dimensional half space. The full compressible Navier–Stokes–Korteweg equations model compressible fluids with viscosity, heat-conductivity and internal capillarity, and include the Korteweg stress effects into the dissipative structure of the hyperbolic–parabolic system and turn out to be more complicated than that in the simpler full compressible Navier–Stokes equations. Under some suitable assumptions of the far fields and the boundary values of the density, the velocity and the absolute temperature, the asymptotic stability of the boundary layer, the 3-rarefaction wave, and the superposition of the boundary layer and the 3-rarefaction wave are shown provided that the initial perturbation and the strength of the nonlinear wave are small. The proof is mainly based on $L^2$-energy method, some time-decay estimates of the smoothed rarefaction wave and the space-decay estimates of the boundary layer. This can be viewed as the first result about the stability of basic wave patterns for the outflow problem of the full compressible Navier–Stokes–Korteweg equations.