Stability of Steady States of the Navier--Stokes--Poisson Equations with Non-Flat Doping Profile

Abstract

We consider the stability of the steady state of the compressible Navier-Stokes-Poisson equations with the non-flat doping profile. We prove the global existence of classical solutions near the steady state for the large doping profile. For the small doping profile, we prove the time decay rates of the solution provided that the initial perturbation belongs to L p with 1 p < 3/2.

Well-posedness on a new hydrodynamic model of the fluid with the dilute charged particles

“…In particular, some fine interpolation estimates are used. Unlike the pure Navier-Stokes-Poisson equations [55], we also need to deal carefully with the terms involved with the electrostatic potential ϕ. This is because that here the electrostatic potential ϕ is related to the microscopic charge density V, W by the Poisson equation…”

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

“…In this section, we are devoted to establishing the existence and uniqueness of stationary solutions to (1.2) of Theorem 1.1 by using the iteration method. First, we state several elementary lemmas, which will be needed later, compared with [12].…”

“…Wang [9] observed the special construction of the (NSP) equations and posed some stronger conditions on the initial value and then proved the global existence and asymptotic decay of solutions in three-space dimensions under smallness condition on the initial data. Recently, for the non-flat doping profile, Tan et al [12] study the stability of the steady state of the compressible (NSP) equations, where they prove the global existence near the steady state for the large doping profile. From those work, a common feature shows that the momentum of the (NSP) system decays at the slower rate than that of the compressible Navier-Stokes system in the absence of the electric field, which thus implies that the electric field could affect the large time behavior of the solution and produce some additional difficulties of analysis.…”

Large time behavior of solutions to wave equations arising from the linearized compressible Navier-Stokes-Poisson system

“…Their results were improved by Shi and Xu in [14] later. If the doping profile are functions respect to space variable, Tan, Wang and Wang [12] proved the time decay rates for the solution provided that the initial perturbation in L p with 1 ≤ p < 3 2 . Feng and Liu [4] recently proved the stability of steady-state solutions based on the antisymmetric matrix techniques and induction argument.…”

“…As the CNSP system has strong physical background, which attract mathematicians and physicians to study it and many results have been established, here we mainly pay attention to the results related to the asymptotic behavior. On one hand, some researchers focused on establishing the optimal time decay estimates of strong solutions to the CNSP system, see [2,4,10,11,12,13,14,16,19]. On the other hand, some researchers investigated the quasineutral limit and inviscid limit of CNSP system, see [3,7,8,9,15].…”