Convergence rate of solutions toward stationary solutions to the bipolar Navier-Stokes-Poisson equations in a half line

Fang Zhou,Yeping Li

doi:10.1186/1687-2770-2013-124

Abstract

In this paper, we show the convergence rate of a solution toward the stationary solution to the initial boundary value problem for the one-dimensional bipolar compressible Navier-Stokes-Poisson equations. For the supersonic flow at spatial infinity, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. For the transonic flow at spatial infinity, the solution converges to the stationary solution in time with the lower rate than that of the initial perturbation in the spatial. These results are proved by the weighted energy method. MSC: 35M31; 35Q35

Highlights

We are concerned with the following bipolar Navier-Stokes-Poisson equations:
Duan and Yang [ ] studied the unique existence and asymptotic stability of a stationary solution for the initial boundary value problem, and they showed that the large-time behavior of solutions for the bipolar NavierStokes-Poisson equations coincided with the one for the single Navier-Stokes system in the absence of the electric field
In Section, we review the results of the stationary solution and the non-stationary solutions, we reformulate our problem

Summary

Introduction

We are concerned with the following bipolar Navier-Stokes-Poisson equations:. Hao and Li [ ] established the global strong solutions of the initial value problem for the multi-dimensional compressible Navier-Stokes-Poisson system in a Besov space. Li et al [ ] showed the global existence and asymptotic behavior of smooth solutions for the initial value problem of the bipolar Navier-Stokes-Poisson equations. Duan and Yang [ ] studied the unique existence and asymptotic stability of a stationary solution for the initial boundary value problem, and they showed that the large-time behavior of solutions for the bipolar NavierStokes-Poisson equations coincided with the one for the single Navier-Stokes system in the absence of the electric field. We are going to discuss the initial-boundary value problem for the onedimensional bipolar Navier-Stokes-Poisson equations.

Mach number at infinity

Multiplying by ψ xx ρ