Asymptotic behavior of generic Riemann solutions to one-dimensional full compressible Navier-Stokes-Poisson equations

Abstract

This paper investigates the time-asymptotic stability of the one-dimensional full compressible Navier-Stokes-Poisson equations, assuming that the corresponding quasi-neutral Euler equations admit a generic Riemann solution consisting of a shock, a contact discontinuity, and a rarefaction wave. It is proved that there exists a unique global-in-time solution, which converges in $L^\infty$ norm, as time goes to infinity, to the ansatz composed of a viscous shock with a time-dependent shift, a viscous contact wave, and an inviscid rarefaction wave. The main difficulty arises from the non-conservative lower-order source terms induced by the electric potential in the momentum and energy equations, which significantly complicates the proof of the existence of individual waves (especially the viscous shock) and the derivation of the relative entropy estimates. To overcome this, we make full use of the Poisson equation satisfied by the electric potential, and combine the weighted relative entropy method with the $a$-contraction method with shifts to close the estimates within the $H^2$-perturbation framework.