From the degenerate quantum compressible Navier-Stokes-Poisson system to incompressible Euler equations

In this paper, we study the convergence from the degenerate quantum compressible Navier-Stokes-Poisson system on a unbounded domain R2×T with general initial data to the incompressible Euler equation with the damping term. We prove rigorously that the weak solutions of the degenerate quantum compressible Navier-Stokes-Poisson system converge to the strong solution of the incompressible Euler equations with a linear damping term, and the result is proven by applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of velocity. Furthermore, the convergence rates are obtained. To handle the oscillations of velocity, we use the dispersive estimates of acoustic systems in the work of D. Donatelli, E. Feireisl, and A. Novotny, Math. Models Methods Appl. Sci. 25(2), 371–394 (2015).