Asymptotic behavior of solutions to the inflow problem for the bipolar compressible quantum Navier-Stokes-Poisson equations

This paper investigates the asymptotic behavior of solutions to the inflow problem for the one-dimensional bipolar compressible quantum Navier-Stokes-Poisson system, a model describing the motion of two-species charged particles, electrons and holes, in ultra-small sub-micron semiconductor devices where quantum effects are significant. First, with the aid of stable manifold theory and center manifold theory, we established the existence and spatial-decay properties of the boundary layer to the inflow problem for the transonic and subsonic cases. Next, under suitable assumptions on the boundary data and the space-asymptotic states, we proved the asymptotic stability of the boundary layer and the superposition of the boundary layer and the rarefaction wave in the case that the initial perturbation and the strength of the boundary layer are sufficiently small. The proof was completed by the $L^2$-energy method with the help of the spatial-decay properties of the boundary layer and the time-decay properties of the smooth approximate rarefaction wave.