Optimal decay rate of solutions to compressible quantum Navier-Stokes-Poisson system

Abstract In this paper, we investigate the optimal decay-in-time rate of solutions to the Cauchy problem of the compressible quantum Navier-Stokes-Poisson (abbr. QNSP) equations, which had been used to model the motion of electrons in the semiconductor devices on the nanometer scale. Since the quantum effect in the QNSP equations is described by the third-order spatial derivatives of density, this fact results in some essential difficulties in the investigation of the optimal decay-in-time rate of solutions by the spectral analysis method. To avoid the essential difficulties, we use the method of pure energy estimates to substitute the spectral analysis method as in [1], and thus establish the decay-in-time rates of solutions. We believe that the decay-in-time rates obtained in this paper are optimal, since they are consistent with the ones of the linear equations.