Large-Time Behavior of Solutions to an Inflow Problem for the Compressible Navier-Stokes-Korteweg Equations in the Half Space

This article is concerned with the large-time behavior of solutions to an inflow problem of the one-dimensional compressible Navier-Stokes-Korteweg equations, which models the compressible fluids with internal capillarity. Applying the center manifold theory, we firstly prove the existence of the boundary layer solution. Then making use of the energy method, the inequality of Poincare type and the spatial decay estimates for the boundary layer solution, we give the rigorous proofs of the stability results on the boundary layer solution under some smallness conditions. This is the first result on the stability of nonlinear wave patterns for the inflow problem of the compressible Navier-Stokes-Korteweg equations.