Asymptotic stability of the rarefaction wave for the compressible quantum Navier–Stokes–Poisson equations

In this study, we consider the large time behavior of the solution to the one-dimensional isentropic compressible quantum Navier–Stokes–Poisson equations. The system describes a compressible particle fluid under quantum effects with the potential function of the self-consistent electric field. We show that if the initial data are close to a constant state with asymptotic values at far fields selected such that the Riemann problem on the corresponding Euler system admits a rarefaction wave with a strength that is not necessarily small, then the solution exists for all time and it tends to the rarefaction wave as t→+∞. The proof is based on the energy method by considering the effect of the self-consistent electric field and quantum potential in the viscous compressible fluid. In addition, we compare the quantum compressible Navier–Stokes–Poisson equations and the corresponding compressible Navier–Stokes–Poisson equations based on the large-time behavior of these two classes of models.