Asymptotic stability of the stationary wave for the quantum Navier–Stokes–Poisson system

Abstract

We shall investigate the asymptotic behavior of solutions to the Cauchy problem for the three-dimensional quantum Navier–Stokes–Poisson system. We first establish the stationary wave, then by means of the energy method, we show that the smooth solutions to the Cauchy problem exist uniquely and globally, and time-asymptotically converge to the stationary wave when the initial perturbation around the stationary wave is small enough. Finally, based on the detailed analysis for the corresponding linear problem and the energy estimates for the nonlinear system, the L2-decay rate of the solution toward the stationary wave is also obtained.