Stability of stationary solutions and viscous shock wave in the inflow problem for isentropic Navier-Stokes-Korteweg system

Abstract

In this paper, we are concerned with the inflow problem in the half line (0,∞) to the isentropic compressible Navier-Stokes-Korteweg system. We first show existence and asymptotic stability of the stationary solutions to the inflow problem. Also, for shock wave, asymptotic profile of the inflow problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and we prove that if the initial data around the shifted viscous shock profile and the strength of shifted viscous shock profile are sufficiently small, then the inflow problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Moreover, we show the stability of superposition viscous shock profiles and stationary solutions. The analysis is based on the elementary L2-energy method, but various techniques are introduced to establish the uniform boundary and energy estimates.