Asymptotic stability of the rarefaction wave for the compressible quantum Navier-Stokes-Poisson equations. (English) Zbl 1367.35124
Summary: 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 \(t \rightarrow + \infty\). 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.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76Y05 | Quantum hydrodynamics and relativistic hydrodynamics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
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