Abstract and Applied Analysis

Volume 2014 (2014), Article ID 214546, 8 pages

http://dx.doi.org/10.1155/2014/214546

Research Article

Global Existence of Solution to Initial Boundary Value Problem for Bipolar Navier-Stokes-Poisson System

Jian Liu¹ and Haidong Liu²

¹College of Teacher Education, Quzhou University, Quzhou 324000, China

²College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China

Received 2 June 2014; Accepted 12 August 2014; Published 28 August 2014

Academic Editor: Xiaohong Qin

Copyright © 2014 Jian Liu and Haidong Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper concerns initial boundary value problem for 3-dimensional compressible bipolar Navier-Stokes-Poisson equations with density-dependent viscosities. When the initial data is large, discontinuous, and spherically symmetric, we prove the global existence of the weak solution.

1. Introduction

Bipolar Navier-Stokes-Poisson (BNSP) has been used to simulate the transport of charged particles under the influence of electrostatic force governed by the self-consistent Poisson equation. In this paper, we consider the initial boundary value problem (IBVP) for 3-dimensional isentropic compressible BNSP with density-dependent viscosities:

where the unknown functions are the charges densities , , the velocities , , the pressure functions , , and the electrostatic potential . In (1), the strain tensors and are defined by , , and the Lamé viscosity coefficients satisfying , , , .

There have been extensive studies on the global existence and asymptotic behavior of weak solution to the unipolar Navier-Stokes-Poisson system (NSP). The global existence of weak solution to NSP with general initial data was proved in [1, 2]. The quasineutral and some related asymptotic limits were studied in [3–5]. In the case when the Poisson equation describes the self-gravitational force for stellar gases, the global existence of weak solution and asymptotic behavior were also investigated together with the stability analysis; refer to [6, 7] and the references therein. In addition, Hao and Li [8] proved the global well-posedness of NSP in the Besov space. Li et al. in [9] proved the global existence and the optimal time convergence rates of the classical solution.

For bipolar Navier-Stokes-Poisson system, there are also abundant results on the existence and asymptotic behavior of the global solution. Li et al. [10] proved optimal time convergence rate for the global classical solution for a small initial perturbation of the constant equilibrium state. The optimal time decay rate of global strong solution was established in [11, 12]. Liu and Lian in [13] proved global existence of weak solution to free boundary value problem. Liu et al. [14] established global existence and asymptotic behavior of weak solution to initial boundary value problem in one-dimensional case. Lin et al. [15] studied the global existence and uniqueness of the strong solution in hybrid Besov spaces with the initial data close to an equilibrium state. As a continuation of the study in this direction, in this paper, we will deal with the initial boundary value problem for BNSP.

The rest of this paper is as follows. In Section 2, we state the main results of this paper. In Section 3, we give the entropy estimates and the pointwise bounds of the density of the smooth approximate solution. In Section 4, we prove the global existence of weak solution.

2. Main Results

For the sake of simplicity, the viscosity terms are assumed to satisfy , , , and , and the strain tensors are given by , . Then (1) is reduced to

for with and being the unit ball in .

The boundary condition is taken as

where is outward pointing unit normal vector of .

The initial data is

Definition 1. is said to be a weak solution to the initial boundary value problem (2)–(4) on , provided that

and the equations are satisfied in the sense of distributions. Namely, it holds for any and that

and for satisfying on and that

Before stating the main result, we make the following assumptions on the initial data (4):

where is small enough. It follows from (9) that

Then, we have the following results for global weak solution.

Theorem 2. Let . If the initial data satisfies (9), then the initial boundary value problem (2)–(4) has a global spherically symmetric weak solution

which satisfies, for all ,

Moreover,

where is a constant.

3. Approximate Solutions and Their Estimates

The key point of the proof of Theorem 2 is to construct smooth approximate solution satisfying the a priori estimates required in the stability analysis. The crucial issue is to obtain lower and upper bounds of the density. To this end, we study the following approximate system of (2):

where is a constant.

Set , , , , and , and rewrite (15) in the form

for . We will first construct the smooth solution of (16) in the truncated region with the initial condition

and the boundary condition

For the approximate solution which will have lower bound of the density (see Lemma 8), the boundary condition of (18) is equivalent to , .

We assume that the initial data is smooth and satisfies (9) with constants independent of .

In the following, we will state the energy and entropy estimates for approximate solution . First, making use of similar arguments as in [16] with modifications, we can establish the following Lemma 3, of which we omit the details.

Lemma 3. Let be smooth solution of (16) defined on with boundary conditions (18) such that , . Then there exists a constant , independent of , such that

Lemma 4. Given , there is an absolute constant , which is independent of , such that

for and .

Proof. Define characteristic line: . Then, along the particle path, (16)₁ can be solved to obtain

which implies that provided that .

It follows from (20) and (21) that

for some absolute constant independent of .

Then, it follows from (19) and (24) that

for and .

Similarly, we also have

The proof of the lemma is finished.

To derive a priori estimates about the velocity of the approximate solution, the crucial step is to obtain lower bounds of the density. For this purpose and for simplicity, we solve the IBVP (16) in Lagrangian coordinates. Since the process is the same, we just deal with (16)₁-(16)₂.

Let be fixed and define

Without loss of generality, we set . Then,

and (16)₁-(16)₂ becomes

for and .

The corresponding initial data is

and the boundary condition is

For this system, the following a priori estimates hold.

Lemma 5. For all , it holds that

Proof. Multiplying (29)₂ by , using (29)₁ and integration by parts, we can get

Since

then from (36), we get

Thus (32) holds.

Next, (33) follows from Lemma 4 and (34) holds trivially.

Now, we prove (35). In fact, multiplying (29)₂ by , using (29)₁ and integration by parts, we get

Thus

Using Hölder inequality, Young’s inequality, and Lemma 4, we estimate the right hand side of (40) as follows:

Putting the above estimates into (40) and using (32), one gets

This proves (35).

Remark 6. Consider

Lemma 7. There is a positive constant such that

Proof. We rewrite (29)₁ in the form

Then substituting (45) into (29)₂, one gets

that is,

Since , the above equation can be rewritten as

Integrating it over , one gets

Multiplying (49) by and integrating over with respect to , one gets

in which

Using Lemma 5 and Young’s inequality, we deduce from (50) that there is a positive constant , depending on , , , and , such that

that is,

Applying Gronwall’s inequality to (53), we have

This proves (44).

Now we can obtain the lower bound of the density .

Lemma 8. There is a positive constant such that

Proof. Set and . Equation (29)₁ can be written as , which implies that . Then it follows from Sobolev’s embedding that, for any ,

Choosing small enough, which may depend on and , we obtain

where . The proof of the lemma is completed.

4. Proof of Theorem 2

Proof. With the estimates obtained in Section 3, we can apply the method in [16] and references therein with modifications to prove the existence of weak solution to the IBVP (2). The details are omitted.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed to each part of this work equally.

Acknowledgments

The authors are grateful to Professor Hai-Liang Li for his helpful discussions and suggestions about the problem. The research of Jian Liu is supported by NNSFC no. 11326140 and the Doctoral Starting up Foundation of Quzhou University no. BSYJ201314 and no. XNZQN201313.

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