AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/214546 214546 Research Article Global Existence of Solution to Initial Boundary Value Problem for Bipolar Navier-Stokes-Poisson System http://orcid.org/0000-0002-7030-0274 Liu Jian 1 Liu Haidong 2 Qin Xiaohong 1 College of Teacher Education Quzhou University Quzhou 324000 China 2 College of Mathematics, Physics and Information Engineering Jiaxing University Jiaxing 314001 China zjxu.edu.cn 2014 2882014 2014 02 06 2014 12 08 2014 28 8 2014 2014 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.

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: (1)ρt+div(ρU)=0,(ρU)t+div(ρUU)+P(ρ)=div(h(ρ)D(U))+(g(ρ)divU)+ρΦ,nt+div(nV)=0,(nV)t+div(nVV)+P(n)=div(h(n)D(V))+(g(n)divV)-nΦ,ΔΦ=ρ-n, where the unknown functions are the charges densities ρ(x,t), n(x,t), the velocities U(x,t), V(x,t), the pressure functions P(ρ)=ργ, P(n)=nγ(γ>1), and the electrostatic potential Φ(x,t). In (1), the strain tensors D(U) and D(V) are defined by D(U)=(1/2)(U+UT), D(V)=(1/2)(V+VT), and the Lamé viscosity coefficients satisfying h(ρ)0, h(ρ)+Ng(ρ)0, h(n)0, h(n)+Ng(n)0.

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 . 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  proved the global well-posedness of NSP in the Besov space. Li et al. in  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.  proved optimal L2 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  proved global existence of weak solution to free boundary value problem. Liu et al.  established global existence and asymptotic behavior of weak solution to initial boundary value problem in one-dimensional case. Lin et al.  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 h(ρ)=ρ, g(ρ)=0, h(n)=n, and g(n)=0, and the strain tensors are given by D(U)=U, D(V)=V. Then (1) is reduced to (2)ρt+div(ρU)=0,(ρU)t+div(ρUU)+ργ=ρΦ+·(ρU),nt+div(nV)=0,(nV)t+div(nVV)+nγ=-nΦ+·(nV),ΔΦ=ρ-n, for (x,t)Ω×[0,T] with T>0 and Ω being the unit ball in R3.

The boundary condition is taken as (3)m1(x,t)=ρ(x,t)U(x,t)=0,m2(x,t)=n(x,t)V(x,t)=0,hhhhhhhhhhhhhhΦ·ν=0,hhhhhhhhhhhhhhhlxΩ, where ν is outward pointing unit normal vector of Ω.

The initial data is (4)(ρ,U,n,V,Φ)(x,0)=(ρ0,U0,n0,V0,Φ0)(x),xΩ,m10(x,0)=ρ0(x)U0(x),m20(x,0)=n0(x)V0(x),xΩ.

Definition 1.

( ρ , U , n , V , Φ ) is said to be a weak solution to the initial boundary value problem (2)–(4) on Ω×[0,T], provided that (5)0ρ,nL(0,T;L1(Ω)Lγ(Ω)),ρU,nVL2(0,T;W1,-1(Ω)),ρ,nL(0,T;H1(Ω)),ρ  U,n  VL(0,T;L2(Ω)),ΦL(0,T;W1,1(Ω)), and the equations are satisfied in the sense of distributions. Namely, it holds for any t2>t10 and ϕC1(Ω¯×[0,T]) that (6)Ωρϕdx|t=t1t2=t1t2Ω(ρϕt+ρU·ϕ)dxdt,Ωnϕdx|t=t1t2=t1t2Ω(nϕt+nV·ϕ)dxdt, and for ψ=(ψ1,ψ2,ψ3)C1(Ω¯×[0,T]) satisfying ψ(x,t)=0 on Ω and ψ(x,T)=0 that (7)Ωm10·ψ(x,0)dx+0TΩ[ρ(ρ  U)·tψ+ρ  Uρ  U:ψ]dxdt+0TΩργdivψdxdt+0TΩρψ·Φdxdt-0TΩρU:ψdxdt=0,(8)Ωm20·ψ(x,0)dx+0TΩ[n(n  V)·tψ+n  Vn  V:ψ]dxdt+0TΩnγdivψdxdt-0TΩnψ·Φdxdt-0TΩnV:ψdxdt=0.

Before stating the main result, we make the following assumptions on the initial data (4): (9)ρ0=ρ0(|x|),U0=u0(|x|)xr,n0=n0(|x|),V0=v0(|x|)xr,ρ00a.e.  in  Ω,ρ0W1,4(Ω),n00a.e.  in  Ω,n0W1,4(Ω),m10=0,a.e.on{xΩρ0(x)=0},    m104L1(Ω),m102+ηρ01+ηL1(Ω),m20=0,a.e.on{xΩn0(x)=0},    m204L1(Ω),  m202+ηn01+ηL1(Ω),ρ0L2(Ω),n0L2(Ω),Φ0L2(Ω), where η>0 is small enough. It follows from (9) that (10)ρ0L(Ω);ρ0U02+ηL1(Ω);ρ0U02L1(Ω),n0L(Ω);n0V02+ηL1(Ω);n0V02L1(Ω).

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

Theorem 2.

Let 1<γ<3. If the initial data satisfies (9), then the initial boundary value problem (2)–(4) has a global spherically symmetric weak solution (11)(ρ,U,n,V,Φ)(x,t)=(ρ(r,t),u(r,t)xr,n(r,t),v(r,t)xr,Φ(r,t)),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhr=|x|, which satisfies, for all T>0, (12)ρ(x,t)C([0,T];L3/2(Ω)),n(x,t)C([0,T];L3/2(Ω)),ρ  UL([0,T];L2(Ω)),n  VL([0,T];L2(Ω)),Ωρ(x,t) d x=Ωρ0(x) d x,Ωn(x,t) d x=Ωn0(x) d x,ΦL([0,T];W1,3/2(Ω)). Moreover, (13)supt[0,T]Ω(12ρ|U|2+12n  |V|2+1γ-1ργhhhhhhh+1γ-1nγ+12|Φ|2) d xC,supt[0,T]Ω(|ρ|2+|n|2) d xC, where C>0 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 L1 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): (14)ρt+div(ρU)=0,(15)(ρU)t+div(ρUU)-div((ρ+ɛρ3/4)U)+(ɛ4ρ3/4divU)+ργ=ρΦ,nt+div(nV)=0,(nV)t+div(nVV)-div((n+ɛn3/4)V)+(ɛ4n3/4divV)+nγ=-nΦ,ΔΦ=ρ-n, where 0<ɛ<1 is a constant.

Set ρ(x,t)=ρ(r,t), U(x,t)=u(r,t)(x/r), n(x,t)=n(r,t), V(x,t)=v(r,t)(x/r), and Φ(x,t)=Φ(r,t), and rewrite (15) in the form (16)ρt+(ρu)r+2ρur=0,(ρu)t+(ρu2+ργ)r+2ρu2r+2ur(ρ+ɛρ3/4)r=((ρ+3ɛ4ρ3/4)(ur+2ur))r+ρr2ɛr(ρ-n)s2ds,nt+(nv)r+2nvr=0,(nv)t+(nv2+nγ)r+2nv2r+2vr(n+ɛn3/4)r=((n+3ɛ4n3/4)(vr+2vr))r-nr2ɛr(ρ-n)s2ds, for r>0. We will first construct the smooth solution of (16) in the truncated region 0<ɛ<r<1 with the initial condition (17)(ρ,ρu)(r,0)=(ρ0+ɛ,m10),(n,nv)(r,0)=(n0+ɛ,m20), and the boundary condition (18)u(r,t)|r=ɛ=u(r,t)|r=1=0,v(r,t)|r=ɛ=v(r,t)|r=1=0.

For the approximate solution which will have lower bound of the density (see Lemma 8), the boundary condition of (18) is equivalent to ρu(r,t)|r=ɛ=ρu(r,t)|r=1=0, nv(r,t)|r=ɛ=nv(r,t)|r=1=0.

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

In the following, we will state the energy and entropy estimates for approximate solution (ρ,n,u,v,Φr). First, making use of similar arguments as in  with modifications, we can establish the following Lemma 3, of which we omit the details.

Lemma 3.

Let (ρ,n,u,v,Φr) be smooth solution of (16) defined on [ɛ,1]×[0,T] with boundary conditions (18) such that ρ>0, n>0. Then there exists a constant C, independent of ɛ, such that (19)ɛ1(ρ(r,t)+n(r,t))r2 d rC,(20)ɛ1(12ρu2+12nv2+1γ-1ργ+1γ-1nγ+12Φr2)r2 d r+0Tɛ1((ρ+ɛ4ρ3/4)(r2ur2+u2)hhhhhhhhh+(n+ɛ4n3/4)(r2vr2+v2)) d r d tC,(21)ɛ1(12ρ|u+(logρ)r+3ɛ4ρ-5/4ρr|2hhhhh+12n|v+(logn)r+3ɛ4n-5/4nr|2)r2 d r+0Tɛ1((γργ-2+3ɛ4γργ-9/4)ρr2hhhhhhhh+(γnγ-2+3ɛ4γnγ-9/4)nr2)r2 d r d t+0Tɛ1((ρ-n)2+ɛ(ρ3/4-n3/4)(ρ-n))r2 d r d tC.

Lemma 4.

Given ɛ>0, there is an absolute constant C, which is independent of ɛ, such that (22)0ρ(r,t)Cɛ2,0n(r,t)Cɛ2, for ɛr1 and t0.

Proof.

Define characteristic line: dr(t)/dt=u(r(t),t). Then, along the particle path, (16)1 can be solved to obtain(23)ρ(r(t),t)r2=ρ0(r(0))r(0)2e-0tru(r(s),s)ds, which implies that ρ0   provided that ρ00.

It follows from (20) and (21) that (24)ɛ1ρr2ρr2drC, for some absolute constant C independent of ɛ.

Then, it follows from (19) and (24) that (25)ρ(r,t)ɛ1ρ(r,t)dr+ɛ1|rρ(r,t)|dr1ɛ2ɛ1ρ(r,t)r2dr+1ɛ2(ɛ1ρr2dr+ɛ1ρr2ρr2dr)Cɛ2, for ɛr1 and t0.

Similarly, we also have (26)0n(r,t)Cɛ2. 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 ɛ>0 be fixed and define (27)x(r,t)=ɛrρr2dr,τ=t. Without loss of generality, we set ɛ1ρr2dr=1. Then, (28)xr=ρr2,xt=-ρr2u,τr=0,τt=1, and (16)1-(16)2 becomes (29)ρτ+ρ2(r2u)x=0,r-2uτ+(ργ)x=((ρ2+3ɛ4ρ7/4)(r2u)x)x-2ur(ρ+ɛρ3/4)x+ρΦx, for τ>0 and 0x1.

The corresponding initial data is (30)(ρ,ρu)(x,0)=(ρ0+ɛ,m10), and the boundary condition is (31)u(0,τ)=0,u(1,τ)=0.

For this system, the following a priori estimates hold.

Lemma 5.

For all τ[0,T], it holds that (32)01(12u2+1γ-1ργ-1) d x+0τ01(2u2r2+ρ2r4ux2) d x d shhhh+0τ01(ɛ2u2ρ1/4r2+ɛ4ρ7/4r4ux2) d x d s01(12u02+1γ-1ρ0γ-1) d x+C(ɛ,T),(33)0ρ(x,τ)C(ɛ,T),(34)ɛr(x,τ)1,(35)01u4 d x+0τ01(4u2r2+6ρ2r4u4ux2hhhhhhhhhhhhhhhh+2ɛu4ρ1/4r2+ɛρ7/4r4u2ux22ɛu4ρ1/4r2) d x d s01u04 d x+C(ɛ,T).

Proof.

Multiplying (29)2 by r2u, using (29)1 and integration by parts, we can get (36)ddτ01(12u2+1γ-1ργ-1)dxhhhl+01(ρ2+3ɛ4ρ7/4)((r2u)x)2dxhhhh=01(ρ+ɛρ3/4)(2ru2)xdx+01ρr2uΦxdxhhhh=401(ρ+ɛρ3/4)uuxrdx+201(1+ɛρ-1/4)u2r2dxhhhl+01ρr2uΦxdx. Since (37)((r2u)x)2=(2urρ+r2ux)2=4u2ρ2r2+4ruuxρ+r4ux2, then from (36), we get (38)ddτ01(12u2+1γ-1ργ-1)dx+01(2u2r2+ρ2ux2r4)dxhhhh+01(ɛu2ρ1/4r2+3ɛ4ρ7/4ux2r4)dx=ɛ01ρ3/4uuxrdx+01ρr2uΦxdx1201ɛu2ρ1/4r2dx+1201ɛρ7/4ux2r4dx+(01ρr4Φx2dx)1/2(01ρu2dx)1/21201ɛu2ρ1/4r2dx+1201ɛρ7/4ux2r4dx+C(ɛ). Thus (32) holds.

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

Now, we prove (35). In fact, multiplying (29)2 by r2u3, using (29)1 and integration by parts, we get (39)14ddτ01u4dx+01(ρ2+3ɛ4ρ7/4)((r2u)x)2u2dxhhhh+201(ρ2+3ɛ4ρ7/4)u2ux2r4dx=-401(ρ+3ɛ4ρ3/4)u3uxrdx+01ργ(u3r2)xdxhhl+01(ρ+ɛρ3/4)(2u4r)xdx+01ρΦxr2u3dx. Thus (40)14ddτ01u4dx+01(2u4r2+3ρ2u2ux2r4)dxhhhh+01(ɛu4ρ1/4r2+9ɛ4ρ7/4u2ux2r4)dx=201ɛρ3/4u3uxrdx+01(2ργ-1u3r+3ργu2uxr2)dxhhl+01ρΦxr2u3dx. Using Hölder inequality, Young’s inequality, and Lemma 4, we estimate the right hand side of (40) as follows: (41)201ɛρ3/4u3uxrdx1201ɛu4ρ1/4r2dx+201ɛρ7/4u2ux2r4dx;  201ργ-1u3rdx2(01ρ4(γ-1)r2dx)1/4(01u4r2dx)3/41301u4r2dx+C;301ργu2uxr2dx3(01ρ2γ-2u2dx)1/2×(01ρ2u2ux2r4dx)1/23201ρ2γ-2u2dx+3201ρ2u2ux2r4dx3201ρ2u2ux2r4dx+1301u4r2dx+C,01ρΦxr2u3dx=ɛ1ρr2u3Φrdr=ɛ1ρu3(ɛr(ρ-n)s2ds)drC(ɛ,T)ɛ1ρu3dr=C(ɛ,T)01u3r2dx1301u4r2dx+C(ɛ,T)01u2r2dx. Putting the above estimates into (40) and using (32), one gets (42)01u4dxhhhh+0τ01(4u4r2+6ρ2u2ux2r4+2ɛu4ρ1/4r2+ɛρ7/4u2ux2r4)dxds01u04dx+C(ɛ,T). This proves (35).

Remark 6.

Consider (43)01u04dx=ɛ1m104(ρ0+ɛ)4ρr2drC(ɛ)m10L4(Ω)4.

Lemma 7.

There is a positive constant C=C(ɛ,T,ρ0W1,4(Ω),m10L4(Ω)) such that (44)01((ρ3/4)x)  4(x,τ) d xC,τ[0,T].

Proof.

We rewrite (29)1 in the form (45)(ρ+ɛρ3/4)xτ=-[(ρ2+3ɛ4ρ7/4)(r2u)x]x. Then substituting (45) into (29)2, one gets (46)r-2uτ+(ργ)x=-(ρ+ɛρ3/4)xτ-2ur(ρ+ɛρ3/4)x+ρΦx; that is, (47)r2(ρ+ɛρ3/4)xτ+2ur(ρ+ɛρ3/4)x=-uτ-r2(ργ)x+ρr2Φx. Since r/τ=u, the above equation can be rewritten as (48)(r2(ρ+ɛρ3/4)x)τ=-uτ-r2(ργ)x+ρr2Φx. Integrating it over [0,τ], one gets (49)u(x,τ)-u0(x)+0τr2(ργ)x(x,s)ds=r02(43ρ01/4+ɛ)x(ρ03/4)-(43ρ1/4+ɛ)r2x(ρ3/4)+0τρr2Φxds. Multiplying (49) by (r2x(ρ3/4))3 and integrating over [0,1] with respect to x, one gets (50)01(43ρ1/4+ɛ)(r2x(ρ3/4))4dx=01r02(43ρ01/4+ɛ)x(ρ03/4)(r2x(ρ3/4))3dx+01(r2x(ρ3/4))30τρr2Φxdsdx-01{u-u0+0τr2(ργ)x(x,s)ds}(r2x(ρ3/4))3dxC(01(r2x(ρ3/4))4dx)3/4×{(0τxργL44ds)1/4u-u0L4+x(ρ03/4)L4hhh+(0τρr2ΦxL44ds)1/4+(0τxργL44ds)1/4}, in which (51)0τρr2ΦxL44ds=0τ01(ρr2Φx)4dxds=0τɛ1ρr2Φr4drds=0τɛ1ρr2(1r2ɛr(ρ-n)s2ds)4drdsC(ɛ,T). Using Lemma 5 and Young’s inequality, we deduce from (50) that there is a positive constant C, depending on ρ0W1,4[0,1], u0L4[0,1], ɛ, and T, such that (52)ɛ01(r2x(ρ3/4))4dxɛ201(r2x(ρ3/4))4dx+C0τ01(xργ)4dxds+C; that is, (53)01(x(ρ3/4))4dxC+C0τmax[0,1](ρ4γ-3)01(x(ρ3/4))4dxds. Applying Gronwall’s inequality to (53), we have (54)01(x(ρ3/4))4dxC. This proves (44).

Now we can obtain the lower bound of the density ρ.

Lemma 8.

There is a positive constant C=C(ɛ,T,ρ0W1,4(Ω),m10L4(Ω)) such that (55)ρC,x[0,1],τ[0,T].

Proof.

Set v(x,τ)=1/ρ(x,τ) and V(τ)=max[0,1]×[0,τ]v(x,s). Equation (29)1 can be written as vτ=(r2u)x, which implies that 01v(x,τ)dx=01v(x,0)dxC0. Then it follows from Sobolev’s embedding W1,1([0,1])L([0,1]) that, for any 0<β<1, (56)vβ(x,τ)01vβ(x,τ)dx+01|xvβ|dx(01vdx)β+β01vβ+3/4|ρxρ1/4|dxC+Cβ(01(vβ+3/4)4/3dx)3/4×(01((ρ3/4)x)4dx)1/4C+CβVβ(01vdx)3/4(01((ρ3/4)x)4dx)1/4C+CβVβ. Choosing β>0 small enough, which may depend on ɛ and T, we obtain (57)V(τ)C, where C=C(ɛ,T,ρ0W1,4(Ω),m10L4(Ω)). The proof of the lemma is completed.

4. Proof of Theorem <xref ref-type="statement" rid="thm2.1">2</xref> Proof.

With the estimates obtained in Section 3, we can apply the method in  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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