Stability of rarefaction waves of the Navier–Stokes–Poisson system

Abstract

In the paper we are concerned with the large time behavior of solutions to the one-dimensional NavierStokes-Poisson system in the case when the potential function of the self-consistent electric field may take distinct constant states at x = ±∞. Precisely, it is shown that if initial data are close to a constant state with asymptotic values at far fields chosen such that the Riemann problem on the corresponding quasineutral Euler system admits a rarefaction wave whose strength is not necessarily small, then the solution exists for all time and tends to the rarefaction wave as t → +∞. The construction of the nontrivial large-time profile of the potential basing on the quasineutral assumption plays a key role in the stability analysis. The proof is based on the energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.

Large-time behavior for fluid and kinetic plasmas with collisions

“…The main idea is based on the study of the same problem on the Navier-StokesPoisson system; see [6]. Here we remark that the similar results should also hold true for the two-fluid models.…”

“…We may start from the simple case when only the potential force is present, and mainly focus on the interesting situation where the potential function is non-trivial, cf. Duan-Liu [6,7].…”

“…Here we remark that the similar results should also hold true for the two-fluid models. Indeed, we also considered in [6] the two-fluid Navier-Stokes-Poisson system:…”

Stability of the superposition of rarefaction wave and contact discontinuity for the non‐isentropic Navier–Stokes–Poisson system

“…Duan and Yang firstly proved the stability of rarefaction wave and boundary layer for outflow problem on the two‐fluid NSP system. Duan‐Liu then improved the results of in the sense that all physical constants appearing in the model can be taken in a general way, and the large time profile of the electric potential is nontrivial on the basis of the quasineutral assumption. Recently, Duan et al extended the results in to the non‐isentropic case under an extra assumption which implies that the ratio of the thermal speeds of ions and electrons at both far fields is not less than one half if one introduces the thermal speed , where k B is the Boltzmann constant.…”

“…Duan‐Liu then improved the results of in the sense that all physical constants appearing in the model can be taken in a general way, and the large time profile of the electric potential is nontrivial on the basis of the quasineutral assumption. Recently, Duan et al extended the results in to the non‐isentropic case under an extra assumption which implies that the ratio of the thermal speeds of ions and electrons at both far fields is not less than one half if one introduces the thermal speed , where k B is the Boltzmann constant. Liu et al overcame the difficult term coming from the self‐consistent force to prove the stability of the contact discontinuity of the NSP system with free boundary.…”