Stability of stationary solutions in the inflow problem for full quantum hydrodynamic equations

Abstract

In this paper, we are concerned with the inflow problem in the half line $(0,\infty )$ to the full hydrodynamic equations with quantum effects. We first give some necessary and sufficient conditions for the existence of the stationary solutions with the aid of center manifold theory. We also show the stability of the stationary solutions under smallness assumptions on the initial perturbation in the Sobolev space. The analysis is based on the elementary $L^2-$energy method, but various techniques are introduced to establish the uniform energy estimates.