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Open Access Research

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

Fang Zhou1 and Yeping Li2*

Author Affiliations

1 Department of Mathematics, Hubei University of Science and Technology, Xianning, 437100, P.R. China

2 Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. China

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Boundary Value Problems 2013, 2013:124  doi:10.1186/1687-2770-2013-124

Published: 14 May 2013

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.

Keywords:
convergence rate; Navier-Stokes-Poisson equation; stationary wave; weighted energy method