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
In this paper, we are concerned with the following bipolar Navier-Stokes-Poisson equations:
in a one-dimensional half space . Here the unknown functions are the densities , the velocities ( ), and the electron field E. ( ) is the pressure depending only on the density. ( ) is viscosity coefficient. Throughout this paper, we assume that two fluids of electrons and ions have the same equation of state with for and , and also they have the same viscosity coefficients . 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  also considered the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in . 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 -decay rate of the smooth solution of the initial value problem for the compressible Navier-Stokes-Poisson system in 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 . 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. 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 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
and the boundary date
Here, we suppose and further the compatibility condition . Moreover, for the unique existence, we also assume
In , 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
as time tends to infinity. Then, let be the stationary solution to the system (1.5). We know that the stationary solution satisfies
and the boundary and spatial asymptotic conditions
In this paper, we are mainly concerned with the decay rate of solutions to (1.1)-(1.4) toward the stationary solution . Now we state the main result in the following theorem.
Theorem 1.1Suppose that and hold. The initial data is supposed to satisfy
and there exists a positive constant such that
(i) When , in addition, the initial data also satisfies , , , , for a certain positive constantα, then the solution to (1.1)-(1.3) satisfies the decay estimate
On the other hand, if the initial data satisfies , , , , for a certain positive constantζ, then there exists a positive constantαsuch that the solution to (1.1)-(1.3) satisfies
(ii) When , and there exists a positive constant such that if the initial data also satisfies for a certain constantαsatisfying , where is a constant defined by
then the solution to (1.1)-(1.3) satisfies
where , andδare defined in Section 2, and .
Notations Throughout this paper, denotes the generic positive constant independent of time. ( ) denotes the space of measurable functions with the finite norm , and is the space of bounded measurable functions on ℝ with the norm . We use to denote the -norm. ( ) stands for the space of -functions f whose derivatives (in the sense of distribution) ( ) are also -functions with the norm . Moreover, ( ) denotes the space of the k-times continuously differentiable functions on the interval with values in .
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 and 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
Integrating (2.1)1 over yields , which implies by letting , . Namely, ( , ), which together with (2.1) implies
Thus, the condition has to be assumed whenever the outflow problem, i.e., the case , is consider. Moreover, let the strength of the boundary layer be measured by . Finally, we also define as follows:
and denote the Mach number at infinity . Then one has the following lemma.
Assume that the condition (2.3) holds. The boundary problem (2.1)-(2.2) has a smooth solution , if and only if and . Moreover, if , there exist two positive constantsλandCsuch that the stationary solution satisfies the estimate
If , the stationary solution satisfies
As to the stability of the stationary solution of (1.1)-(1.4), Duan and Yang showed the following results in .
Lemma 2.2 (see )
Suppose that and hold. In addition, the initial data is supposed to satisfy
Then there exists a positive constant such that if
the initial boundary value problem (1.1)-(1.3) has a unique solution for arbitrary . Moreover, the solution converges to the stationary solution as time tends to infinity:
Here the solution space is defined by
Finally, to enclose this section, we reformulate the original problem in terms of the perturbed variables. Set from the stationary solution as
Due to (1.1) and (1.6), we have the system of equations for as
where the nonlinear terms ( ) and ( ) are given by
The initial and boundary condition to (2.7) are derived from (1.2), (1.3) and (1.4) as follows:
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 and defined by
Here the two types of weight functions are considered: , or . Also, we use the norms , , and defined by
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 , there exists a positive constantTsuch that the initial boundary value problem (2.7)-(2.9) has a unique solution . Moreover, if the initial data satisfies (1.8), (1.9) and and , there exists a unique solution in .
3 A priori estimates for
In this section, we derive the a priori estimates of the solution for the case that holds in some Sobolev space. To summarize the a priori estimate, we use the following notation (see ) for a weight function :
Proposition 3.1Suppose that the same assumptions as in Theorem 1.1 hold.
(i) (Algebraic decay) Suppose that is a solution to (2.7)-(2.9) for certain positive constantsαandT. Then there exist positive constants andCsuch that if , then the solution satisfies the estimate
for arbitrary and .
(ii) (Exponential decay) Suppose that is a solution to (2.7)-(2.9) for certain positive constantsζandT. Then there exist positive constants , C, β (<ζ) andαsatisfying such that if , then the solution satisfies the estimate
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 such that , it holds that
Proof From (2.7), a direct computation yields
Owing to Lemmas 2.1 and 2.2, we see that the energy form is equivalent to . That is, there exist positive constants c and C such that
We also have positive bounds of ( ) as
Further, multiplying (3.4) by a weight function , we have
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
where we have used the estimates (3.5) and (3.6). Next, can be computed as
The conditions and yield that the quadratic form is positive definite since
Using (2.5), (3.5) and the inequalities , for , we have the estimate for as
Moreover, the positive bound of ( ), (3.6) and the Schwarz inequality yield the estimate for as
then we have
Therefore, integrating (3.7) over , substituting the above inequalities (3.8)-(3.12) into the resultant equality and then taking 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
Lemma 3.3There exists a positive constant such that if , then
Proof By differentiating the first and third equations of (2.7) in x, and then multiplying them by and , respectively, one has for ,
with . On the other hand, multiplying the second and fourth equations of (2.7) by and , respectively, gives
Further, we have
Combining (3.14) and (3.15), we have
The second term on the right-hand side of (3.16) can be rewritten as
Under the assumption (3.6) on the densities, it holds that
Moreover, owing to the Schwarz inequality with the aid of (2.5), is estimated as
Similarly, we have
Multiplying (3.16) by a weight function , we get
The boundary condition (2.9) gives
Integrating (3.20) over , substituting (3.17), (3.18), (3.19), (3.21), (3.22) and the estimate
Lemma 3.4There exists a positive constant such that if , then
Proof Multiplying (2.7)2 by , and (2.7)4 by , respectively, we have for ,
Note that , and the function is estimated by using (2.5) and Schwarz inequality as
where ε is an arbitrary positive constant and is a positive constant depending on ε. Then, multiplying (3.25) by a weight function , we get
Integrate (3.27) over , substitute (3.26) as well as the estimate
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 suitably small, we have
First, we prove the estimate (3.1). Noting the Poincaré-type inequality (3.23), and substituting and in (3.28) for and gives
Next, we prove the estimate (3.2). Substitute and in (3.28) for to obtain
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
In the section we proceed to consider the transonic case . To state the a priori estimate of the solution precisely, here we use the notations:
Proposition 4.1Suppose that the same assumption as in Theorem 1.1 holds. Suppose that is a solution to (2.7)-(2.9) for certain positive constantsαandT. Then there exist positive constants andCsuch that if , then the solution satisfies the estimate
In order to prove Proposition 4.1, we need to get a lower estimate for and the Mach number on the stationary solution defined by .
Lemma 4.2 (see )
The stationary solution satisfies
for . Moreover, there exists a positive constantCsuch that
Based on Lemma 4.2, we obtain the weighted estimate of .
Lemma 4.3There exists a positive constant such that if , then
for and .
Proof First, from (2.7), similar as (3.4), we also have
Notice that from the second and fourth equations of (1.1), one has
Plugging the above equality into (4.3), we arrive at
Further, multiplying (4.4) by a weight function , we have
Where and are defined
By the same computation as in deriving (3.9), we rewrite the terms and to , with
By utilizing Lemma 4.2 with the aid of the fact that and , we obtain the lower estimate of as
for . On the other hand, the estimates (2.6), (3.5) and (3.6) yield
For the first term on the right-hand side of (4.5), we estimate it as
Similarly, we get
In the same way, we estimate and as follows:
Since , we have
Moreover, it is easy to compute , which implies
Finally, integrate (4.5) over , substitute (4.6)-(4.13) in the resultant equality, and take and δ suitably small. This procedure yields the desired estimate (4.2) for .
Next, we prove (4.2) for . Substituting in (4.5) and integrating the resultant equality over , we get
Here, we have used the fact that holds. Therefore, we obtain the estimate (4.2) for the case of . □
In order to complete the proof of Proposition 4.1, we need to obtain the weighted estimate of .
Lemma 4.4There exists a positive constant such that if , then for and ,
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 , we have