Asymptotic behavior of solutions for the 1-D isentropic Navier-Stokes-Korteweg equations with free boundary

In this paper, we consider the problem with a gas-gas free boundary for the one dimensional isentropic compressible Navier-Stokes-Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Also, we show the asymptotic stability toward rarefaction wave without the smallness on the strength if the initial data around the rarefaction wave are sufficiently small. (C) 2020 Elsevier Ltd. All rights reserved.