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.