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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


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/124


Received:2 November 2012
Accepted:26 April 2013
Published:14 May 2013

© 2013 Zhou and Li; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1 Introduction

In this paper, we are concerned with the following bipolar Navier-Stokes-Poisson equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M1">View MathML</a>

(1.1)

in a one-dimensional half space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M2">View MathML</a>. Here the unknown functions are the densities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M3">View MathML</a>, the velocities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5">View MathML</a>), and the electron field E. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5">View MathML</a>) is the pressure depending only on the density. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M8">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5">View MathML</a>) is viscosity coefficient. Throughout this paper, we assume that two fluids of electrons and ions have the same equation of state <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M10">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M13">View MathML</a>, and also they have the same viscosity coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M14">View MathML</a>. The bipolar Navier-Stokes-Poisson system is used to simulate the transport of charged particles (e.g., electrons and ions). It consists of the compressible Navier-Stokes equation of two-fluid under the influence of the electro-static potential force governed by the self-consisted Poisson equation. Note that when we only consider one particle in the fluids, we also have the unipolar Navier-Stokes-Poisson equations. For more details, we can refer to [1-4].

Recently, some important progress was made for the compressible unipolar Navier-Stokes-Poisson system. The local and/or global existence of a renormalized weak solution for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Poisson system were proved in [5-7]. Chan [8] also considered the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M15">View MathML</a>. Hao and Li [9] established the global strong solutions of the initial value problem for the multi-dimensional compressible Navier-Stokes-Poisson system in a Besov space. The global existence and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M16">View MathML</a>-decay rate of the smooth solution of the initial value problem for the compressible Navier-Stokes-Poisson system in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M17">View MathML</a> were achieved by Li and his collaborators in [10,11]. The pointwise estimates of the smooth solutions for the three-dimensional isentropic compressible Navier-Stokes-Poisson equation were obtained in [12]. The quasineutral limit of the compressible Navier-Stokes-Poisson system was studied in [13-15]. However, the results about the bipolar Navier-Stokes-Poisson equations are very few. Lastly, Li et al.[16] 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 [17] 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 Navier-Stokes-Poisson equations coincided with the one for the single Navier-Stokes system in the absence of the electric field. The consistency is also observed and proved between the bipolar Euler-Poisson system and the single damped Euler equation; for example, see [18-20] and the references therein.

In this paper, we are going to discuss the initial-boundary value problem for the one-dimensional bipolar Navier-Stokes-Poisson equations. Now we give the initial condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M18">View MathML</a>

(1.2)

and the boundary date

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M19">View MathML</a>

(1.3)

Here, we suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M20">View MathML</a> and further the compatibility condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M21">View MathML</a>. Moreover, for the unique existence, we also assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M22">View MathML</a>

(1.4)

In [17], the authors showed that the solution to (1.1)-(1.4) converges to the corresponding stationary solution of the single Navier-Stokes system in the absence of the electric field

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M23">View MathML</a>

(1.5)

as time tends to infinity. Then, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M24">View MathML</a> be the stationary solution to the system (1.5). We know that the stationary solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M25">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M26">View MathML</a>

(1.6)

and the boundary and spatial asymptotic conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M27">View MathML</a>

(1.7)

In this paper, we are mainly concerned with the decay rate of solutions to (1.1)-(1.4) toward the stationary solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M28">View MathML</a>. Now we state the main result in the following theorem.

Theorem 1.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M30">View MathML</a>hold. The initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M31">View MathML</a>is supposed to satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M32">View MathML</a>

(1.8)

and there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M34">View MathML</a>

(1.9)

(i) When<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35">View MathML</a>, in addition, the initial data also satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M40">View MathML</a>for a certain positive constantα, then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M41">View MathML</a>to (1.1)-(1.3) satisfies the decay estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M42">View MathML</a>

(1.10)

On the other hand, if the initial data satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M47">View MathML</a>for a certain positive constantζ, then there exists a positive constantαsuch that the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M41">View MathML</a>to (1.1)-(1.3) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M49">View MathML</a>

(1.11)

(ii) When<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50">View MathML</a>, and there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>such that if the initial data also satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M52">View MathML</a>for a certain constantαsatisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M53">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M54">View MathML</a>is a constant defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M55">View MathML</a>

then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M41">View MathML</a>to (1.1)-(1.3) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M57">View MathML</a>

(1.12)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M59">View MathML</a>andδare defined in Section 2, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M60">View MathML</a>.

Notations Throughout this paper, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M61">View MathML</a> denotes the generic positive constant independent of time. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M62">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M63">View MathML</a>) denotes the space of measurable functions with the finite norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M64">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M65">View MathML</a> is the space of bounded measurable functions on ℝ with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M66">View MathML</a>. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M67">View MathML</a> to denote the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M16">View MathML</a>-norm. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M69">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M70">View MathML</a>) stands for the space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M71">View MathML</a>-functions f whose derivatives (in the sense of distribution) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M72">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M73">View MathML</a>) are also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M71">View MathML</a>-functions with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M75">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M76">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M77">View MathML</a>) denotes the space of the k-times continuously differentiable functions on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M78">View MathML</a> with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M79">View MathML</a>.

The rest of the paper is organized as follows. In Section 2, we review the results of the stationary solution and the non-stationary solutions, then we reformulate our problem. Finally, we give the a priori estimates for the cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50">View MathML</a> in Section 3 and 4, respectively.

2 Stationary solution and global existence of non-stationary solution

In this section we mainly review the property of a stationary solution, and the unique existence and asymptotic behavior of non-stationary solutions for (1.1)-(1.3). To begin with, we recall the stationary equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M82">View MathML</a>

(2.1)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M83">View MathML</a>

(2.2)

Integrating (2.1)1 over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M84">View MathML</a> yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M85">View MathML</a>, which implies by letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M87">View MathML</a>. Namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M88">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M90">View MathML</a>), which together with (2.1) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M91">View MathML</a>

(2.3)

Thus, the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M92">View MathML</a> has to be assumed whenever the outflow problem, i.e., the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M93">View MathML</a>, is consider. Moreover, let the strength of the boundary layer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M24">View MathML</a> be measured by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M95">View MathML</a>. Finally, we also define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M96">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M97">View MathML</a>

(2.4)

and denote the Mach number at infinity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M98">View MathML</a>. Then one has the following lemma.

Lemma 2.1 (see [21,22])

Assume that the condition (2.3) holds. The boundary problem (2.1)-(2.2) has a smooth solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M24">View MathML</a>, if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M30">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35">View MathML</a>, there exist two positive constantsλandCsuch that the stationary solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M25">View MathML</a>satisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M104">View MathML</a>

(2.5)

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50">View MathML</a>, the stationary solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M25">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M107">View MathML</a>

(2.6)

As to the stability of the stationary solution of (1.1)-(1.4), Duan and Yang showed the following results in [17].

Lemma 2.2 (see [17])

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M30">View MathML</a>hold. In addition, the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M110">View MathML</a>is supposed to satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M111">View MathML</a>

Then there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>such that if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M113">View MathML</a>

the initial boundary value problem (1.1)-(1.3) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M114">View MathML</a>for arbitrary<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M115">View MathML</a>. Moreover, the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M41">View MathML</a>converges to the stationary solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M28">View MathML</a>as time tends to infinity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M118">View MathML</a>

Here the solution space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M119">View MathML</a>is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M120">View MathML</a>

Finally, to enclose this section, we reformulate the original problem in terms of the perturbed variables. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M121">View MathML</a> from the stationary solution as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M122">View MathML</a>

Due to (1.1) and (1.6), we have the system of equations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M124">View MathML</a>

(2.7)

where the nonlinear terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M125">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M127">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5">View MathML</a>) are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M129">View MathML</a>

The initial and boundary condition to (2.7) are derived from (1.2), (1.3) and (1.4) as follows:

(2.8)

(2.9)

The uniform bound of the solutions in the weighted Sobolev space is derived later in Sections 3 and 4. For this purpose, we introduce the function spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M133">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M134">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M135">View MathML</a>

Here the two types of weight functions are considered: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M136">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M137">View MathML</a>. Also, we use the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M139">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M140">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M141">View MathML</a>

The following lemma, concerning the existence of the solution locally in time, is proved by the standard iteration method. Hence we omit the proof.

Lemma 2.3If the initial data satisfies (1.8) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M142">View MathML</a>, there exists a positive constantTsuch that the initial boundary value problem (2.7)-(2.9) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M143">View MathML</a>. Moreover, if the initial data satisfies (1.8), (1.9) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M144">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M145">View MathML</a>, there exists a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M133">View MathML</a>.

3 A priori estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35">View MathML</a>

In this section, we derive the a priori estimates of the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123">View MathML</a> for the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35">View MathML</a> holds in some Sobolev space. To summarize the a priori estimate, we use the following notation (see [23]) for a weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M151">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M152">View MathML</a>

Proposition 3.1Suppose that the same assumptions as in Theorem 1.1 hold.

(i) (Algebraic decay) Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M153">View MathML</a>is a solution to (2.7)-(2.9) for certain positive constantsαandT. Then there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>andCsuch that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M155">View MathML</a>, then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123">View MathML</a>satisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M157">View MathML</a>

(3.1)

for arbitrary<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M158">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M159">View MathML</a>.

(ii) (Exponential decay) Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M160">View MathML</a>is a solution to (2.7)-(2.9) for certain positive constantsζandT. Then there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>, C, β (<ζ) andαsatisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M162">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M155">View MathML</a>, then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123">View MathML</a>satisfies the estimate

(3.2)

For the sake of clarity, we divide the proof of Proposition 4.1 into the following lemmas. We first derive the basic energy estimate.

Lemma 3.2Suppose that the same assumptions as in Theorem 1.1 hold. Then there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M167">View MathML</a>, it holds that

(3.3)

Proof From (2.7), a direct computation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M169">View MathML</a>

(3.4)

here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M170">View MathML</a>

Owing to Lemmas 2.1 and 2.2, we see that the energy form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M171">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M172">View MathML</a>. That is, there exist positive constants c and C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M173">View MathML</a>

(3.5)

We also have positive bounds of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M174">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5">View MathML</a>) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M176">View MathML</a>

(3.6)

Further, multiplying (3.4) by a weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M177">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M178">View MathML</a>

(3.7)

Due to the boundary conditions (1.3) and (2.9), the integration of the second term on the left-hand side of (3.7) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M179">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M180">View MathML</a>

(3.8)

where we have used the estimates (3.5) and (3.6). Next, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M181">View MathML</a> can be computed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M182">View MathML</a>

(3.9)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M183">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M184">View MathML</a>

The conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M92">View MathML</a> yield that the quadratic form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M187">View MathML</a> is positive definite since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M188">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M189">View MathML</a>

(3.10)

Using (2.5), (3.5) and the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M191">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M192">View MathML</a>, we have the estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M193">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M194">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M195">View MathML</a>

(3.11)

Moreover, the positive bound of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M174">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5">View MathML</a>), (3.6) and the Schwarz inequality yield the estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M198">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M199">View MathML</a>

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M200">View MathML</a>

(3.12)

Therefore, integrating (3.7) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201">View MathML</a>, substituting the above inequalities (3.8)-(3.12) into the resultant equality and then taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M202">View MathML</a> suitably small, we obtain the desired estimate (3.3). □

Next, we obtain the estimate for the first-order derivatives of the solution for (2.7)-(2.9). As the existence of the higher-order derivatives of the solution is not supposed, we need to use the difference quotient for the rigorous derivation of the higher-order estimates. Since the argument using the difference quotient is similar to that in the paper [21,22], we omit the details and proceed with the proof as if it verifies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M203">View MathML</a>

Lemma 3.3There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M167">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M206">View MathML</a>

(3.13)

Proof By differentiating the first and third equations of (2.7) in x, and then multiplying them by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M207">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M208">View MathML</a>, respectively, one has for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M210">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M211">View MathML</a>

(3.14)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M212">View MathML</a>. On the other hand, multiplying the second and fourth equations of (2.7) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M214">View MathML</a>, respectively, gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M215">View MathML</a>

Further, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M216">View MathML</a>

(3.15)

here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M217">View MathML</a>

Combining (3.14) and (3.15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M218">View MathML</a>

(3.16)

The second term on the right-hand side of (3.16) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M219">View MathML</a>

Under the assumption (3.6) on the densities, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M220">View MathML</a>

(3.17)

Moreover, owing to the Schwarz inequality with the aid of (2.5), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M221">View MathML</a> is estimated as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M222">View MathML</a>

(3.18)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M223">View MathML</a>

(3.19)

Multiplying (3.16) by a weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M224">View MathML</a>, we get

(3.20)

The boundary condition (2.9) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M226">View MathML</a>

(3.21)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M227">View MathML</a>

(3.22)

Integrating (3.20) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M228">View MathML</a>, substituting (3.17), (3.18), (3.19), (3.21), (3.22) and the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M229">View MathML</a>

(3.23)

which is proved by the similar computation as in [21-24], in the resultant equality, and take ε and δ suitably small. These computations together with (3.3) give the desired estimate (3.13). □

Lemma 3.4There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M167">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M232">View MathML</a>

(3.24)

Proof Multiplying (2.7)2 by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M233">View MathML</a>, and (2.7)4 by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M234">View MathML</a>, respectively, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M5">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M236">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M237">View MathML</a>

(3.25)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M238">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M239">View MathML</a>, and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M240">View MathML</a> is estimated by using (2.5) and Schwarz inequality as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M241">View MathML</a>

(3.26)

where ε is an arbitrary positive constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M242">View MathML</a> is a positive constant depending on ε. Then, multiplying (3.25) by a weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M224">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M244">View MathML</a>

(3.27)

Integrate (3.27) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M228">View MathML</a>, substitute (3.26) as well as the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M246">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M247">View MathML</a>

in the resultant equality, and take ε suitably small. These computations together with (3.3), (3.13) and (3.23) give the desired estimate (3.24). □

Proof of Proposition 3.1 Summing up the estimates (3.3), (3.13) and (3.24) and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M202">View MathML</a> suitably small, we have

(3.28)

First, we prove the estimate (3.1). Noting the Poincaré-type inequality (3.23), and substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M250">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M251">View MathML</a> in (3.28) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M253">View MathML</a> gives

(3.29)

Therefore, applying an induction to (3.29) gives the desired estimate (3.1). Since this computation is similar those in [23,25], we omit the details.

Next, we prove the estimate (3.2). Substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M255">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M256">View MathML</a> in (3.28) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M257">View MathML</a> to obtain

(3.30)

Here, we have used the Poincaré-type inequality (3.23) again. Thus, taking δ, β and α suitably small, we obtain the desired a priori estimate (3.2). □

4 A priori estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50">View MathML</a>

In the section we proceed to consider the transonic case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M50">View MathML</a>. To state the a priori estimate of the solution precisely, here we use the notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M261">View MathML</a>

Proposition 4.1Suppose that the same assumption as in Theorem 1.1 holds. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M262">View MathML</a>is a solution to (2.7)-(2.9) for certain positive constantsαandT. Then there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>andCsuch that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M264">View MathML</a>, then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123">View MathML</a>satisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M266">View MathML</a>

(4.1)

In order to prove Proposition 4.1, we need to get a lower estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M267">View MathML</a> and the Mach number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M268">View MathML</a> on the stationary solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M25">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M270">View MathML</a>.

Lemma 4.2 (see [23])

The stationary solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M271">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M272">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M273">View MathML</a>. Moreover, there exists a positive constantCsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M274">View MathML</a>

Based on Lemma 4.2, we obtain the weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M16">View MathML</a> estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M123">View MathML</a>.

Lemma 4.3There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M278">View MathML</a>, then

(4.2)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M252">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M253">View MathML</a>.

Proof First, from (2.7), similar as (3.4), we also have

(4.3)

Notice that from the second and fourth equations of (1.1), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M283">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M284">View MathML</a>

Plugging the above equality into (4.3), we arrive at

(4.4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M286">View MathML</a>

Further, multiplying (4.4) by a weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M287">View MathML</a>, we have

(4.5)

Where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M290">View MathML</a> are defined

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M291">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M292">View MathML</a>

By the same computation as in deriving (3.9), we rewrite the terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M290">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M295">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M296">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M297">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M298">View MathML</a>

By utilizing Lemma 4.2 with the aid of the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M299">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M92">View MathML</a>, we obtain the lower estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M301">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M302">View MathML</a>

(4.6)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M303">View MathML</a>. On the other hand, the estimates (2.6), (3.5) and (3.6) yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M304">View MathML</a>

(4.7)

For the first term on the right-hand side of (4.5), we estimate it as

(4.8)

Similarly, we get

(4.9)

In the same way, we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M307">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M308">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M309">View MathML</a>

(4.10)

and

(4.11)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M311">View MathML</a>, we have

(4.12)

Moreover, it is easy to compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M313">View MathML</a>, which implies

(4.13)

Finally, integrate (4.5) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201">View MathML</a>, substitute (4.6)-(4.13) in the resultant equality, and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M316">View MathML</a> and δ suitably small. This procedure yields the desired estimate (4.2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M303">View MathML</a>.

Next, we prove (4.2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M318">View MathML</a>. Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M319">View MathML</a> in (4.5) and integrating the resultant equality over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M321">View MathML</a>

Here, we have used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M322">View MathML</a> holds. Therefore, we obtain the estimate (4.2) for the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M318">View MathML</a>. □

In order to complete the proof of Proposition 4.1, we need to obtain the weighted estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M324">View MathML</a>.

Lemma 4.4There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M33">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M278">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M252">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M253">View MathML</a>,

(4.14)

Proof Since the derivation of the estimate (4.14) is similar to that of (3.13) and (3.24), we only give the outline of the proof. Multiplying (3.16) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M330">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M331">View MathML</a>

(4.15)

Integrating (4.15) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201">View MathML</a> and substituting (4.2) gives the estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M334">View MathML</a> as

(4.16)

Here, we have used the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M336">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M337">View MathML</a>

where ε is an arbitrary positive constant. We note that the third term on the right-hand side of the above inequality is estimated by applying the Poincaré-type inequality (3.23) for the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M318">View MathML</a>.

Next, we prove the estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M339">View MathML</a>. Multiply (3.25) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M340">View MathML</a> to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M341">View MathML</a>

(4.17)

Integrate (4.17) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M201">View MathML</a> and substitute (4.2) and (4.16) in the resultant equality with the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M343">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M344">View MathML</a>

This procedure yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M345">View MathML</a>

(4.18)

Finally, adding (4.16) to (4.18) and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M346">View MathML</a> suitably small give the desired estimate (4.14). □

By the same inductive argument as in deriving (4.1), we can prove Proposition 4.1, which immediately yields the decay estimate (1.12).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

The research of Li is partially supported by the National Science Foundation of China (Grant No. 11171223) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 13ZZ109).

References

  1. Besse, C, Claudel, J, Degond, P: A model hierarchy for ionospheric plasma modeling. Math. Models Methods Appl. Sci.. 14, 393–415 (2004). Publisher Full Text OpenURL

  2. Degond, P: Mathematical modelling of microelectronics semiconductor devices. Some Current Topics on Nonlinear Conservation Laws, pp. 77–110. Am. Math. Soc., Providence (2000)

  3. Jüngel, A: Quasi-hydrodynamic Semiconductor Equations, Birkhäuser, Basel (2001)

  4. Sitnko, A, Malnev, V: Plasma Physics Theory, Chapman & Hall, London (1995)

  5. Ducomet, B: A remark about global existence for the Navier-Stokes-Poisson system. Appl. Math. Lett.. 12, 31–37 (1999)

  6. Ducomet, B: Local and global existence for the coupled Navier-Stokes-Poisson problem. Q. Appl. Math.. 61, 345–361 (2003)

  7. Zhang, Y-H, Tan, Z: On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow. Math. Methods Appl. Sci.. 30, 305–329 (2007). Publisher Full Text OpenURL

  8. Chan, D: On the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M15">View MathML</a>. Commun. Partial Differ. Equ.. 35, 535–557 (2010). Publisher Full Text OpenURL

  9. Hao, C-C, Li, H-L: Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions. J. Differ. Equ.. 246, 4791–4812 (2009). Publisher Full Text OpenURL

  10. Li, H-L, Matsumura, A, Zhang, G-J: Optimal decay rate of the compressible Navier-Stokes-Poisson system in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M17">View MathML</a>. Arch. Ration. Mech. Anal.. 196, 681–713 (2010). Publisher Full Text OpenURL

  11. Zhang, G-J, Li, H-L, Zhu, C-J: Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/124/mathml/M17">View MathML</a>. J. Differ. Equ.. 250, 866–891 (2011). Publisher Full Text OpenURL

  12. Wang, W-K, Wu, Z-G: Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions. J. Differ. Equ.. 248, 1617–1636 (2010). PubMed Abstract | Publisher Full Text OpenURL

  13. Donatelli, D, Marcati, P: A quasineutral type limit for the Navier-Stokes-Poisson system with large data. Nonlinearity. 21, 135–148 (2008). Publisher Full Text OpenURL

  14. Ju, Q-C, Li, F-C, Li, Y, Wang, S: Rate of convergence from the Navier-Stokes-Poisson system to the incompressible Euler equations. J. Math. Phys.. 50, Article ID 013533 (2009)

  15. Wang, S, Jiang, S: The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equ.. 31, 571–591 (2006). Publisher Full Text OpenURL

  16. Li, H-L, Yang, T, Zou, C: Time asymptotic behavior of the bipolar Navier-Stokes-Poisson system. Acta Math. Sci., Ser. B. 29, 1721–1736 (2009). Publisher Full Text OpenURL

  17. Duan, R-J, Yang, X-F: Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Commun. Pure Appl. Anal.. 12, 985–1014 (2013)

  18. Gasser, I, Hsiao, L, Li, H-L: Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors. J. Differ. Equ.. 192, 326–359 (2003). Publisher Full Text OpenURL

  19. Huang, F-M, Li, Y-P: Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum. Discrete Contin. Dyn. Syst.. 24, 455–470 (2009)

  20. Huang, F-M, Mei, M, Wang, Y: Large time behavior of solution to n-dimensional bipolar hydrodynamic model for semiconductors. SIAM J. Math. Anal.. 43, 1595–1630 (2011). Publisher Full Text OpenURL

  21. Kawashima, S, Nishibata, S, Zhu, P: Asymptotic stability of the stationary solution to the compressible Navier-Stokes-Poisson equations in half space. Commun. Math. Phys.. 240, 483–500 (2003)

  22. Kawashima, S, Zhu, P: Asymptotic stability of nonlinear wave for the compressible Navier-Stokes-Poisson equations in half space. J. Differ. Equ.. 224, 3151–3179 (2008)

  23. Nakamura, T, Nishibata, S, Yuge, T: Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line. J. Differ. Equ.. 241, 94–111 (2007). Publisher Full Text OpenURL

  24. Nikkuni, Y, Kawashima, S: Stability of stationary solutions to the half-space problem for the discrete Boltzmann equation with multiple collisions. Kyushu J. Math.. 54, 233–255 (2000). Publisher Full Text OpenURL

  25. Matsumura, A, Nishihara, K: Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity. Commun. Math. Phys.. 165, 83–96 (1994). Publisher Full Text OpenURL