Asymptotic stability of the stationary solution to an out-flow problem for the Navier–Stokes–Korteweg equations of compressible fluids

We shall investigate the large-time behavior of solutions to an out-flow problem in one-dimensional half space for the Navier–Stokes–Korteweg equations which models compressible fluids with internal capillarity. Applying the center manifold theory, we prove the existence of the stationary solution under a smallness condition on the boundary data and a proper relation between the capillary coefficient κ and Mach number M+ at far field. Then making use of the energy method and the inequality of Poincaré type, we show that the stationary solution is asymptotically stable under smallness assumptions on the boundary data and the initial perturbation in Sobolev space.