Convergence rate of solutions toward stationary solutions to the bipolar Navier-Stokes-Poisson equations in a half line: [Doc 376]

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.[PUBLICATION ABSTRACT]

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R E S E A R C H Open Access

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

Fang Zhou1 and Yeping Li2*

*Correspondence: mailto:[email protected]

Web End [email protected]

2Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. ChinaFull list of author information is available at the end of the article

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 ow at spatial

innity, 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 innity. For the transonic ow at

spatial innity, 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:

(.)

in a one-dimensional half space R+ := (, ). Here the unknown functions are the den

sities i (i = , ) > , the velocities ui (i = , ), and the electron eld E. Pi(i) (i = , ) is the pressure depending only on the density. i (i = , ) is viscosity coecient. Throughout this paper, we assume that two uids of electrons and ions have the same equation of state P() = P() = P() with P() = K for K > and , and also they have the

same viscosity coecients = = . 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-uid under the inuence of the electro-static potential force governed by the self-consisted Poisson equation. Note that when we only

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t + x(u) = ,

t(u) + x(u + P()) = uxx + E, t + x(u) = ,

t(u) + x(u + P()) = uxx E,

Ex = ,

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consider one particle in the uids, we also have the unipolar Navier-Stokes-Poisson equations. For more details, we can refer to [].

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 []. Chan [] also considered the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in RN. Hao and Li [] established the global strong solutions of the initial value problem for the multi-dimensional compressible Navier-Stokes-Poisson system in a Besov space. The global existence and L-decay rate of the smooth solution of the initial value problem for the compressible Navier-Stokes-Poisson system in R were achieved by Li and his collaborators in [, ]. The point-wise estimates of the smooth solutions for the three-dimensional isentropic compressible Navier-Stokes-Poisson equation were obtained in []. The quasineutral limit of the compressible Navier-Stokes-Poisson system was studied in []. However, the results about the bipolar Navier-Stokes-Poisson equations are very few. Lastly, Li et al. [] showed the global existence and asymptotic behavior of smooth solutions for the initial value problem of the bipolar Navier-Stokes-Poisson equations. Duan and Yang [] studied the unique existence and asymptotic stability of a stationary solution for the initial boundary value problem, and they showed that the large-time behavior of solutions for the bipolar Navier-Stokes-Poisson equations coincided with the one for the single Navier-Stokes system in the absence of the electric eld. The consistency is also observed and proved between the bipolar Euler-Poisson system and the single damped Euler equation; for example, see [] 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

(, u, , u)(x, ) = (, u, , u)(x) (+, u+, +, u+) as x , (.)

and the boundary date

u(, t) = u(, t) = ub < . (.)

Here, we suppose infxR+ i (i = , ) > and further the compatibility condition ub =

u() = u(). Moreover, for the unique existence, we also assume

E(+, t) = . (.)

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

t + x(u) = ,

t(u) + x(u + P()) = uxx,

(.)

as time tends to innity. Then, let (

,)(x) be the stationary solution to the system (.).

We know that the stationary solution (

,) satises

(

)x = , (

+ P(

))x =

xx,

(.)

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and the boundary and spatial asymptotic conditions

() = ub, lim

x(

,) = (+, u+), inf

x

R+

(x) > . (.)

In this paper, we are mainly concerned with the decay rate of solutions to (.)-(.) toward the stationary solution (

,,

,, ). Now we state the main result in the following

theorem.

Theorem . Suppose that M+ and ub < u hold. The initial data (, u, , u,

E)(x) is supposed to satisfy

(i, ui) (i = , ) H(

R+), E(x) L(

R+), inf

x

R+(, ) > , (.)

and there exists a positive constant such that

(

, u ,

, u )

+ E + < . (.)

(i) When M+ > , in addition, the initial data also satises ( + x) (

), ( + x) (u

), ( + x)

(

), ( + x) (u ), ( + x)

E L(

R+) for a certain positive constant

, then the solution (, u, , u, E) to (.)-(.) satises the decay estimate

(

, u ,

, u , E)

L C( + x)

. (.)

On the other hand, if the initial data satises e

x(

), e

x(u ), e

x(

),

e

x(u ), e

xE L(

R+) for a certain positive constant , then there exists a positive constant such that the solution (, u, , u, E) to (.)-(.) satises

(

, u ,

, u , E)

L Cet. (.)

(ii) When M+ = , and there exists a positive constant such that if the initial data also satises (+x)

(

, u ,

, u ) + (+x)

E < for a certain constant

satisfying [, ), where is a constant dened by

=

+ and

> ,

then the solution (, u, , u, E) to (.)-(.) satises

(

, u ,

, u , E)

L C( + t)

, (.)

where M+, u and are dened in Section , and E(x) =

x ( )(y) dy.

Notations Throughout this paper, C > denotes the generic positive constant independent of time. Lp(R) ( p < ) denotes the space of measurable functions with the nite

norm L

p = (

R| |p dx)p , and L is the space of bounded measurable functions on R with the norm L = ess supx | |. We use to denote the L-norm. Hk(R) (k )

stands for the space of L(R)-functions f whose derivatives (in the sense of distribution)

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Dlxf (l k) are also L(

kl= Dl ) . Moreover, Ck([, T]; Hl(R)) (k, l ) denotes the space of the k-times continuously dierentiable

functions on the interval [, T] with values in Hl(R).

The rest of the paper is organized as follows. In Section , 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 M+ > and M+ = in Section and , 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 (.)-(.). To begin with, we recall the stationary equation

(

R)-functions with the norm k = (

)x = , (

+ P(

))x =

xx

(.)

with

() = ub < , lim

x(

,) = (+, u+), inf

x

R+

(x) > . (.)

Integrating (.) over (x, ) yields

(x) = +u+((x)), which implies by letting x +,

b =

+ ), which together with (.) implies

u+ = v+

v()ub < . (.)

Thus, the condition u+ < has to be assumed whenever the outow problem, i.e., the case ub < , is consider. Moreover, let the strength of the boundary layer (

,)(x) be measured

() = +u+(ub). Namely, = u+v+ v (v = , v+ =

by := |u+ ub|. Finally, we also dene (v, u) as follows:

u+ = u+

v+ v

v+ v P

, u = u+v+ v, (.)

and denote the Mach number at innity M+ =: |u+|

P (+) . Then one has the following lemma.

Lemma . (see [, ]) Assume that the condition (.) holds. The boundary problem (.)-(.) has a smooth solution (

v

P

v+

,)(x), if and only if M+ and ub < u. Moreover, if

M+ > , there exist two positive constants and C such that the stationary solution (

,)

satises the estimate

k

x (x) +,(x) u+

Cex for k = , , , . . . . (.)

If M+ = , the stationary solution (

,) satises

k

x (x) +, u+

C

k+

( + x)k+ for k = , , , . . . . (.)

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

Lemma . (see []) Suppose that M+ and ub < u hold. In addition, the initial data

(, u, , u, E) is supposed to satisfy

(

, u ,

, u ) H(

R+), E L(

R+), inf

x

R+(, ) > .

Then there exists a positive constant such that if

(

, u ,

, u )

+ E + < ,

the initial boundary value problem (.)-(.) has a unique solution (, u, , u, E) X(, T) for arbitrary T > . Moreover, the solution (, u, , u, E) converges to the stationary solution (

,,

,, ) as time tends to innity:

lim

t

, u , E| = .

Here the solution space X(, T) is dened by

X(, T) = (, u, , u, E) : , u , , u C

, T; H

sup

x

R+ |

, u ,

,

(

)x, (

)x L

, T; L

, (u )x, (u )x L

, T; H

,

E C

, T; L

, (u )(t, ) = (u )(t, ) = ( t T)

.

Finally, to enclose this section, we reformulate the original problem in terms of the perturbed variables. Set (, , , ) from the stationary solution as

i = i

, i = ui , i = , .

Due to (.) and (.), we have the system of equations for (, , , , E) as

t + ux + x = f,

t + P ()x + ux = xx g + E,

t + ux + x = f,

t + P ()x + ux = xx g E,

E =

(.)

x ( )(y, t) dy,

where the nonlinear terms fi (i = , ) and gi (i = , ) are given by

fi =xi +

xi, gi =x(i + ii) +

x

P (i) P ( )

.

The initial and boundary condition to (.) are derived from (.), (.) and (.) as follows:

(i, i)(x, ) = (i, i)(x) := (i

, ui ), i = , , (.) (, , , )(t, ) = . (.)

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The uniform bound of the solutions in the weighted Sobolev space is derived later in Sections and . For this purpose, we introduce the function spaces X(, T) and X(, T) dened by

X(, T) = (, , , , E) : (, , , , E) C

, T; L(R+)

and

X(, T) =

(, , , , E) : (, , , ) C

, T; H(R+)

;

, T; L(R+)

.

Here the two types of weight functions are considered: (x) := ( + x), or (x) = ex. Also, we use the norms | |,, | |a,, and | |e, dened by

|f |, :=

E C

(x)f (x) dx

, |f |a, := |f |,(+x), |f |e, := |f |,ex .

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 . If the initial data satises (.) and , , , , E L(R+), there exists a positive constant T such that the initial boundary value problem (.)-(.) has a unique solution (, , , , E) X(, T). Moreover, if the initial data

satises (.), (.) and , , , H(

R+) and E L(

R+),

there exists a unique solution (, , , , E) in X(, T).

3 A priori estimates for M+ > 1

In this section, we derive the a priori estimates of the solution (, , , , E) for the

case that M+ > holds in some Sobolev space. To summarize the a priori estimate, we use the following notation (see []) for a weight function W(x, t) = (t)(x):

N(t) = sup

t (

, , , )()

,

t

M(t) =

()

(

x, x, Ex)()

+

(

x, x)()

d

t

+

()

(, ) + (, ) + E(, )

d,

t

L(t) =

t()

(

, , , , E)()

, +

(

x, x, x, x)()

+ ()

(,

)()

,xx +

(

, , , , E)()

,|x|

d.

(, T) is a solution to (.)-(.) for certain positive constants and T. Then there exist positive constants and C

Proposition . Suppose that the same assumptions as in Theorem . hold.(i) (Algebraic decay) Suppose that (, , , , E) X(+x)

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such that if N(T) + , then the solution (, , , , E) satises the estimate

( + t)+

(

, , , )(t)

+

E(t)

t

+ ( +

)+

(

x, x, Ex)()

+

(

x, x)()

d +

t

( +

)+

(

, , E, x, x)(, )

d

C

(

, , , )

+

(

, , , , E)

a,

( + t) (.)

for arbitrary t [, T] and > .

(ii) (Exponential decay) Suppose that (, , , , E) Xe

x (, T) is a solution to (.)-(.) for certain positive constants and T. Then there exist positive constants , C,

(< ) and satisfying such that if N(T) + , then the solution (, , , , E)

satises the estimate

et

(

, , , )(t)

+

(

, , , , E)(t)

e,

+

e (

x, x, Ex)()

t

+

(

x, x)()

d +

t

e (

, , E, x, x)(, )

d

t

+

e (

, , , , E, x, x)()

e, d

C

(

, , , )

+

(

, , , , E)

e,

. (.)

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

Lemma . Suppose that the same assumptions as in Theorem . hold. Then there exists a positive constant such that N(T) + < , it holds that

(t)

(

, , , , E)

,w +

t

()

(

, , , , E)()

,wx +

(

x, x)()

,w

d

t

+

()

i=i(, ) + E(, )

d

C

(

, , , , E)

,w + CL(t). (.)

Proof From (.), a direct computation yields

E + E +E t Gx + x + x = (x + x)x + R, (.)

here

Ei := E(i, ui) (i = , ),

E(, u) := (,

) +

|u |,

(,

) :=

P(s) P(

)s ds,

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G = uE uE

P() P( )

P() P( )

E,

P() P( ) P ( ) + (u )

+ P() P(

) P (

) + (u

)

R = x

E

P(

)x

P(

)x.

Owing to Lemmas . and ., we see that the energy form E(, u) is equivalent to |( , u )|. That is, there exist positive constants c and C such that

c

i + i

Ei C

, i = , . (.)

We also have positive bounds of i (i = , ) as

< c i (i = , ) C, (t, x) [, T]

i + i

R+. (.)

Further, multiplying (.) by a weight function W(t, x) = (t)(x), we have

WE + WE +WE t (WG)x + WxG + Wx + Wx

= Wt

E + E + E +

i=

Wiix Wx i

x

+ + WR. (.)

Due to the boundary conditions (.) and (.), the integration of the second term on the left-hand side of (.) over R+

R+

W

+

Wxx

uE + uE +

P() P( )

+

P() P( )

+ E x dx

= (t)(t, )ubE(t, ) (t)(t, )ubE(t, )

ub

(t)E(t, )

, (.)

where we have used the estimates (.) and (.). Next, G can be computed as

G = G + G (.)

with

G = K

+|u+|

C(t)

(t, ) + (t, ) + E(t, )

+

+

+|u+|

+

K+( + ) u+ E

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and

G =

K+u+ i

+i

i + K+u+

i

i

+ K

+

ii + K

i

i

i + (iui +u+)Ei

( u+)E.

The conditions M+ > and u+ < yield that the quadratic form G is positive denite since

G =

i=

P (+)

+

i +

++

i

(M+ )

+

P (+) +

P (

+)i +i

u+E

C

+ + + + E

,

which yields

t

t

,x d. (.)

Using (.), (.) and the inequalities | (s) r(s )| C|s |, |s (s )|

C|s | for |s | , we have the estimate for G as

|G| C

N(t) + + + + + E

R+ WxG dx d C

()

(

, , , , E)

,

which implies

t

R+ WxG dx d C

N(t) + t()

(

, , , , E)

,x d. (.)

Moreover, the positive bound of i (i = , ), (.) and the Schwarz inequality yield the estimate for R as

|R| C|x|

+ + + + E

,

then we have

t

t

,|x| d. (.)

Therefore, integrating (.) over R+ (, t), substituting the above inequalities (.)-(.)

into the resultant equality and then taking N(T) + suitably small, we obtain the desired estimate (.).

R+ WR dx d C

(t)

(

, , , , E)

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

(, , , ) C

[, T]; H(R+)

, x, x L

[, T]; H(R+)

,

Ex L

[, T]; L(R+)

, x, x L

[, T]; H(R+)

.

Lemma . There exists a positive constant such that if N(T) + < , then

(t) (

x, x)

+

t

()

(

x, x, Ex)

+ x(, ) + x(, )

d

+ CL(t) + CM(t). (.)

Proof By dierentiating the rst and third equations of (.) in x, and then multiplying them by x

and x , respectively, one has for i = , ,

ix i

C

(

, )x

+

(

, , , , E)

,

t + uiix i

x xix i +

x

i

ixix +

i

ixixx = fix

ix

i ,

which yields

x

+

x

t + ux + ux

x +xxx + xxx = R (.)

with R =x

x xx fx x fx x . On the other hand, multiplying the second and fourth equations of (.) by x

and x , respectively, gives

iiix t iiit + xi i

x ix + P (i)iix ixiix + xxii

+

x

i

x

x xx +x

x

iix +x(

xi

ix)i i

= gi

ix

i + ()i

E

i

ix +

ixixx

i , i = , .

Further, we have

x +x

t t + x +t + x

x x x

+ P (

)

x + P (

)

x

= E

x E

x + R +

xxx

+

xxx

, (.)

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here

R =

x

x

xx

x

x x(

x

x) g

x

+

x

x

xx

x(

x

x) g

x

.

Combining (.) and (.), we have

x

+

x

+

x

+

x

t + ux t

x

+

ux

t

x

x +P ()x + P ()x

= x + x + E

x x

+ R + R. (.)

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

E

x x =

(ln ln )x

E ( )E x

=

x (ln ln )Ex +ExE x .

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

(ln i ln e)Ex CEx,

ExE

(ln ln )E

x

. (.)

Moreover, owing to the Schwarz inequality with the aid of (.), R is estimated as

R |x|

x + x + x + x

C| x|

E + Ex

+ | x|

x + x + x + x

+ | xx|

+ x + + x

+ |xx|

+ x + + x

C

+ + + + x + x + x + x

. (.)

Similarly, we have

R

x + x

+ C

x + + x +

+ C

+ + + + x + x + x + x

. (.)

Multiplying (.) by a weight function (t), we get

(t)

i=

ix i +

ixi i

t

+ (t)

uiixi itii

xi

i

x +

i= (t)P (i)ix i

= t(t)

i=

ix i +

ixi i

+ (t)

i=ix + E

x x

+ (t)R + (t)R. (.)

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The boundary condition (.) gives

(t)

i=

uiixi itii

i i

x

dx C(t)

x(t, ) + x(t, )

(.)

and

(ln ln )E

x dx C

(t, ) + (t, ) + E(t, )

. (.)

Integrating (.) over [, t]

R+, substituting (.), (.), (.), (.), (.) and the

estimate

(t, x)

+

(t, x)

+

(t, x)

+

(t, x)

+

E(t,

x)

(t, )

+

(t, )

+

E(t, )

+ x

(

x, x, x, x, Ex)(t)

,

(.)

which is proved by the similar computation as in [], in the resultant equality, and take

and suitably small. These computations together with (.) give the desired estimate (.).

Lemma . There exists a positive constant such that if N(T) + < , then

(t) (

x, x)

+

t

()

(

xx, xx)

d

C

(

, )x

+

(

, , , , E)

,

M(t). (.)

Proof Multiplying (.) by xx , and (.) by xx , respectively, we have for i = , ,

ix

t itix + ui ix

x + ixx i

+ CL(t) + C

N(t) +

=

ix

xix

+

P (i) i

ixixx + gi

ixx

i + ()iE

ixx,

which yields

x +x

t

i=

itix + ui ix

x +xx +xx = E(xx xx) + R (.)

with

R =

x

xx

+

P ()

xxx

x

xx

+

P ()

xxx + g

xx

+

gxx

.

Note that Exx +Exx = (E( )x)x +Ex( )x, and the function R is estimated by using (.) and Schwarz inequality as

R

xx + xx

+ C

x + x + x + x + x + x

+ C|x|

+ + + + x + x

+ C| x|

+

, (.)

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where is an arbitrary positive constant and C is a positive constant depending on . Then, multiplying (.) by a weight function (t), we get

ix

t

itix + ui ix

x

+

ixx i

= t

i=ix

E(xx xx) R

. (.)

Integrate (.) over [, t]

R+, substitute (.) as well as the estimate

x + x dx C

x x + x x

CN(t)

(

x, xx, x, xx)

and

t

E(

)x(, )

d

t

d

t

E(,

)

d +

(

)x(, )

t

d +

t

E(,

)

d + C

(

, )x

t

(

, )xx

d

in the resultant equality, and take suitably small. These computations together with (.), (.) and (.) give the desired estimate (.).

Proof of Proposition . Summing up the estimates (.), (.) and (.) and taking N(T) + suitably small, we have

(t)

i=

(i,

i)

,w +

i=

(

ix, ix)

+ |E|,w

t

()

i=

(

i, i)

,wx +

ix()

,w + i(, ) + ix(, ) +

(

ix, ixx)

+

E(

+ ) ,wx + Ex + E(, ) d

C

(

, , , , E)

,w +

(

, , , )x

+ CL(t). (.)

First, we prove the estimate (.). Noting the Poincar-type inequality (.), and substituting (x) = ( + x) and (t) = ( + t) in (.) for [, ] and gives

( + t)

(

, , , , E)(t)

t

+ ( +

a, +

(

, , , )(t)

)

(, ) + (, ) + x(, ) + x(, ) + E(, ) d

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+

t

( +

)

(

, , , , E)()

a, d

t

+ ( +

)

(

x, x)()

a, +

(

x, x, xx, xx)()

d

C

(

, , , Ex)

+

(

, , , , E)

a,

+ C( )

t

( +

)

(

, )

a, d

+ C

t

( +

)

(

, , , , E)

a, +

(

x, x, x, x)

d. (.)

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

Next, we prove the estimate (.). Substitute (x) = ex and (t) = et in (.) for < to obtain

et

(

, , , , E)(t)

e, +

(

x, x, x, x)(t)

t

+

e

(, ) + (, ) + x(, ) + x(, ) + E(, )

d

t

+ e (

, , , , E)()

e,

+

(

x, x, Ex, xx, xx)()

+

(

x, x)()

e,

d

C

(

, , , )x

+

(

, , , , E)

e,

+ C

+

t

e (

, , , , E)

e, d

t

+ C

e (

x, x, x, x)

d

t

+ C

e

(, ) + (, ) + E(, )

+

(

x, x, Ex, x, x)()

d. (.)

Here, we have used the Poincar-type inequality (.) again. Thus, taking , and suitably small, we obtain the desired a priori estimate (.).

4 A priori estimate for M+ = 1

In the section we proceed to consider the transonic case M+ = . To state the a priori estimate of the solution precisely, here we use the notations:

N(t) = sup

t

(

+ x)/(, , , )()

,

t

M(t) =

( + t) (

x, x, Ex, x, x, xx, xx)()

a, d.

Proposition . Suppose that the same assumption as in Theorem . holds. Suppose that (, , , , E) X(+x)(, T) is a solution to (.)-(.) for certain positive constants

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and T. Then there exist positive constants and C such that if N(T) + , then the

solution (, , , , E) satises the estimate

( + t)/+

(

, , , )(t)

t

+ ( +

+

E(t)

)/+

(

, , x, x, E)(, )

d

t

+ ( +

)/+

(

x, x, Ex)()

+

(

x, x)()

d

C

(

, , , )x

+

(

, , , , E)

a,

( + t). (.)

In order to prove Proposition ., we need to get a lower estimate forx and the Mach

number M on the stationary solution (

,) dened by M(x) :=

|(x)|

P (

(x)) .

Lemma . (see []) The stationary solution(x) satises

x(x) A

u+ ub

+ ( + Bx) , A :=( + )+ , B := A

for x (, ). Moreover, there exists a positive constant C such that

+ |u+|

+ Bx C

( + Bx) M(x) C

+ Bx.

Based on Lemma ., we obtain the weighted L estimate of (, , , , E).

Lemma . There exists a positive constant such that if N(T) + < , then

( + t) (

, , , , E)

a, +

t

( +

)

(, ) + (, ) + E(, ) d

t

+ ( +

)

(

, , , )()

a,

+

E(

) a, +

(

x, x)()

a,

d

C

(

, , , , E)

a, + C

t

( +

)

(

, , , , E)

a, d

+ C

t

( +

)

(

x, x, Ex, xx, xx)

a, d (.)

for [, ] and .

Proof First, from (.), similar as (.), we also have

E + E + E t+

uE +

P() P( )

x + uE

+ P() P( )

x + E x+x

P() P( ) P ( ) +

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+ P() P(

) P (

) +

E

+ x + x

= xx

[ + ]. (.)

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

E = t(u u) +

x

u u +

P ()x P ()x

uxx uxx

,

which implies

x

E =

x

(u u)E t

x

(u u)Et +

x

E

x

u u

+ A

x

( )

xE +x E uxx uxx

x

= (u u)E t

+

x

E(

)

x + ( + )( )

xx

E(

) +x

Ex

+ Ax ( )

xE

+x

E

x + uxx uxx

.

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

E + E x (u u)E + E t+

uE + uE

+ P() P( )

x

+ P() P( )

x +x ( )E + E x+ x + x

+x

P() P( ) P ( ) + ( ) + P()

P(

) P (

) +

(

)

= xx

[ + ] + R, (.)

where

R = xx

E(

) +x

Ex

+x E(x x)

+ A

x

( )

xE +x

uxx uxx

E

=: R + R + R + R + R.

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Further, multiplying (.) by a weight function W(t, x) := ( + Bx)( + t) , we have

WE + WE x W(u u)E + WE t

+

WG Wx Wx + Wx + Wx

x

+ WxG + Wx + Wx + G

= Wt

E + E x (u u)E +E Wxx

[ + ] + WR. (.)

Where G and G are dened

G = uE uE

P() P( )

P() P( )

x

(

)E

E

and

P() P( ) P ( ) + ( ) + P() P( )

P (

) +

G := Wx

(

)

Wxx

Wxx

.

By the same computation as in deriving (.), we rewrite the terms G and G to G = G + G, G = G + G with

G =

i=

P (+)/

+

i +

+

P (+)i

( M ) +P ( )

P (

)i i

x

(

)E

u+E,

G =

P ( )

i + P( )

i

i

+ (iui

)Ei +

P(i) P( ) P ( )i

i

P (

)/

P (+)/

+

i

+

P ( ) +

P (+)

i

( M ) ( u+)E,

G = Wx

+ + P (+) Wxx

+ Wx

+ + P (+) Wxx

and

G = Wx

(i+)i +

P ( )P (+)

i +P(i)P( )P ( )i P ( )i

.

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By utilizing Lemma . with the aid of the fact that < and u+ < , we obtain the lower estimate of WxG + G as

WxG + G

K+A

( + ) +

u+ ub

+( )

( + t)( + Bx)i

+

+A

u+ ub

+ ( + )( )

( + t)( + Bx)i

C( + t)( + Bx)

+ + +

+ + +

+ C( + t)( + Bx)E (.)

for (, ]. On the other hand, the estimates (.), (.) and (.) yield

|WxG + G| C

N(t) +

C( C)( + t)( + Bx)

+ + +

+ C( + t)( + Bx)E. (.)

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

( + t)( + Bx)

R+ W xx

[ + ] dx d

t

t

R+ W xx

+ + +

dx d

C

t

( +

)

(, ) + (, ) d

+ C

t

( +

)

R+ ( + Bx) x + x + x + x

dx d. (.)

Similarly, we get

t

R+ WR dx d

R+ Wxx E+ +

t

dx d

C

t

( +

)E(, ) d

+ C

t

( +

)

R+ ( + Bx) x + x + Ex

dx d. (.)

In the same way, we estimate R and R as follows:

t

R+ WR dx d

C

t

( +

)

R+ ( + Bx) x + x + Ex

dx d (.)

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and

t

R+ WR dx d

C

t

( +

)E(, ) d + C

t

( +

)

R+ ( + Bx)Ex dx d

+ CN(t)

t

( +

)

R+ ( + Bx) x + x

dx d. (.)

Since ( )x = ( )[( )

x + x x], we have

t

R+ WR dx d

C

t

( +

)E(, ) d

+ C

t

( +

)

R+ ( + Bx) x + x + Ex

dx d. (.)

Moreover, it is easy to compute R =x(Exx Exx ) +xxx E( ), which implies

t

R+ WR dx d

C

t

( +

)

E(, ) + (, ) + (, ) d

+ C

t

( +

)

R+ ( + Bx) x + x + Ex + xx + xx

dx d. (.)

Finally, integrate (.) over R+ (, t), substitute (.)-(.) in the resultant equality, and

take N(t) and suitably small. This procedure yields the desired estimate (.) for

(, ].

Next, we prove (.) for = . Substituting W = ( + t) in (.) and integrating the resultant equality over R+ (, t), we get

( + t)

(

, , , , E)(t)

t

+ ( +

)

i=i(, ) + E(, ) +

(

x, x)

d

C

(

, , , , E)

+ C

t

( +

)

(

, , , , E)

d

+ C

t

( +

)

(

x, x, Ex, xx, xx)

dx d.

Here, we have used the fact that G holds. Therefore, we obtain the estimate (.) for

the case of = .

In order to complete the proof of Proposition ., we need to obtain the weighted estimate of (x, x, x, x).

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Lemma . There exists a positive constant such that if N(T)+ < , then for [, ]

and ,

( + t) (

x, x, x, x)

a, +

t

( +

)

x(, ) + x(, ) d

t

+ ( +

)

(

x, xx, x, xx, Ex)()

a, d

C

(

, , , , E)

a, +

(

x, x, x, x)

a,

+ C

t

( +

)

a, d. (.)

Proof Since the derivation of the estimate (.) is similar to that of (.) and (.), we only give the outline of the proof. Multiplying (.) by W = ( + Bx)( + t) , we have

i=

(

, , x, x, , , x, x)()

W

ix i +

ixi i

t

+

W

uiixi itii

xi

i

x

+ W P (

i)ix

i

=

Wt ixi +ixii + W

ix + ()i+Eixi + R + R

Wx

uiixi itii

xi

i

. (.)

Integrating (.) over R+ (, t) and substituting (.) gives the estimate for x and x

as

( + t)

(

x, x)

a, +

t

( +

)

x(, ) + x(, ) +

(

x, x, Ex)

a,

d

C

(

, , , , E, x, x)

a,

+ C

t

( +

)

(

, , x, , , x)

a, d + C

N(t) +

M(t). (.)

Here, we have used the inequalities

( + Bx)

x

ux

t

x

+

ux

t

x

C( + Bx)

x + x + C( + Bx)

x + x

+ C( + Bx)

+ + +

and

( + Bx)|R + R| ( + C)( + Bx)

x + x + C( + Bx)

x + x

+ C( + Bx)

+ + +

,

where 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 Poincar-type inequality (.) for the case of = .

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Next, we prove the estimate for (x, x). Multiply (.) by W = ( + t)( + Bx) to get

Wx + W x

t

i=

Witix + W ui ix

x + W

i=ixx i

=

Wt

i=ix Wx

itix + ui ix

EWxx + EWxx + WR. (.)

Integrate (.) in R+ (, t) and substitute (.) and (.) in the resultant equality with

the inequalities

( + Bx) x

tx

+ u

x +

tx + u

x

+ ( + Bx)|R|

( + Bx)

i=ixx + C( + Bx)

ix + ix + ix

+ C( + Bx)

i + i

and

( + Bx)

x + x

dx CN(t)

(

x, xx, x, xx)

a, .

This procedure yields

( + t)

(

x, x)

a, +

t

( +

)

(

xx, xx)

a, d

C

(

, , , , E)

a, +

(

x, x, x, x)

a,

+ C

t

( +

)

i=

(

M(t). (.)

Finally, adding (.) to (.) and taking N(t) + suitably small give the desired estimate (.).

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

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors typed, read and approved the nal manuscript.

Author details

1Department of Mathematics, Hubei University of Science and Technology, Xianning, 437100, P.R. China. 2Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R. China.

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).

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

i, i, ix, ix)

a, d + C

N(t) +

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doi:10.1186/1687-2770-2013-124Cite this article as: Zhou and Li: Convergence rate of solutions toward stationary solutions to the bipolar Navier-Stokes-Poisson equations in a half line. Boundary Value Problems 2013 2013:124.