# Nonlinear Stability of the Bipolar Navier-Stokes-Poisson System with Boundary

Authors: Tiantian Yu; Yong Li
Journal: Advances in Mathematical Physics
 (2023)

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## Abstract

The combined quasineutral and zero-viscosity limits of the bipolar Navier-Stokes-Poisson system with boundary are rigorously proved by establishing the nonlinear stability of the approximate solutions. Based on the conormal energy estimates, we showed that the solutions for the original system converge strongly inH3 space towards the solutions of the one-fluid compressible Euler system as long as the amplitude of the boundary layers is small enough.


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## Body

## 1. Introduction

In this paper, we are interested in the combined quasineutral and zero-viscosity limits for the following bipolar compressible Navier-Stokes-Poisson (NSP) system(1)nt+∇·nu=0,ut+u·∇u+Te∇lnn=−∇ϕ+μ′nΔu+μ′+ν′n∇∇·u,ht+∇·hv=0,vt+v·∇v+Ti∇lnh=∇ϕ+δ′hΔv+δ′+η′h∇∇·v,ε2Δϕ=h−n,for t>0, x=x1,x2,x3=y,x3∈ℝ2×0,1≕Ω, where n, h, u, and v are the densities of electrons and ions and velocities of electrons and ions, respectively, and the ϕ denotes the electric potential of fluid. Te and Ti are the average temperatures of electrons and ions, respectively, with Te≫Ti. ε=λD/L≪1 with the characteristic observation length for L and the Debye length for λD. The parameters μ′, ν′, δ′, and η′ are the constant visconsity coefficients satisfying μ′>0,δ′>0 and μ′+ν′>0,δ′+η′>0.In the case that the interaction of the two carriers is taken into consideration, system (1) consists of the compressible Navier-Stokes equation with the electrostatic potential force governed by the self-consistent Poisson equation, cf. [1]. We intend to discuss the quasineutral and zero-viscosity limits of this model. In fact, there are many signs of progress on the quasineutral limit or combined quasineutral and inviscid limits of the compressible NSP system. Among others, without boundary, we mention Donatelli and Marcati [3] and other works [4, 13, 14] on the isentropic NSP system; Ju and Li [10] and other works [12, 19, 21] on the nonisentropic NSP system; Yang [31] for the bipolar NSP system; Donatelli and Feireisl [2], Ju et al. [14], and Wang and Jiang [29] on the combined limits of the NSP system; and Kwon and Li [20] on the combined limits for the two-fluid NSP system.The emergence of the boundary layer will make the problem more complicated. However, we can peek some ideas from the study of the corresponding boundary layer problem of the Euler-Poisson system (see reference [5, 6, 11, 23, 25–28, 30]). For compressible NSP systems, recently, the combined quasineutral and vanishing viscosity limits in the half-space have been rigorously proved under a Navier-slip boundary condition for velocity and Dirichlet’s boundary condition for electric potential by Ju and Xu [18] and Ju et al. [15, 16], which was extended to the two-fluid compressible NSP system in [32].We consider the NSP system with the boundary conditions(2)u3=v3=0,ui−β1∂ui∂x3=0,vi−β2∂vi∂x3=0,i=1,2,ϕ=ϕb,on x3=0,1, where β1 and β2 are positive constants and the prescribed potential ϕb can be assumed that ϕb=ϕref+ϕy with ϕref being a constant and ϕy being a smooth function.Our purpose in this paper is to discuss the asymptotic behavior of smooth solutions to system (1) with boundary condition (2) by performing the combined quasineutral limit and inviscid limit as ε⟶0. Therefore, without loss of generality, we assume that
(3)μ′=με2,ν′=νε2,δ′=δε2,η′=ηε2.Formally, lettingε⟶0 in (1) yields that
(4)nt+∇·nu=0,ut+u·∇u+Te∇lnn=−∇ϕ,ht+∇·hv=0,vt+v·∇v+Ti∇lnh=∇ϕ,h=n.In particular, Ju and Li [11] justified rigorously that if n=h≕m and n,u0,x=h,v0,x, it holds ut,x=vt,x≕wt,x for all t∈0,T for some time interval T>0. Then, the above system gives us the following one-fluid compressible isothermal Euler equation for m,w under the same initial densities and initial velocities:
(5)mt+∇·mw=0,wt+w·∇w+Te+Ti2∇lnm=0,with
(6)ϕ=Ti−Te2lnm+ft.Naturally, we expect that system (1) converges to system (5) as ε⟶0, which physically appears as destruction of quasineutrality and disappearance of viscosity. Obviously, only one boundary condition u3=0 is needed for (5), which can not match the boundary condition (2). It is not difficult to find that there is a loss in the boundary conditions, which leads to the appearance of the boundary layer. Worthy of a special mention is the fact that the literature [16] has provided an analysis of the formation of strong density boundary layers and weak velocity field boundary layers.In this paper, we aim to obtain the nonlinear stability of error system for the approximate solution. An approximate solution involving two different kinds of boundary layers for the system and the linearized stability of error system have been established in [32]. However, the singular terms generated by the interaction between the strong boundary layer of density and the weak boundary layer of the velocity field are difficult to estimate, so the nonlinear stability was left open in [32]. The extremely singular terms, such as the normal derivative estimates, high-order normal derivatives, and mixed derivatives of the solutions, are the most challenging parts of the nonlinear analysis, which do not arise in the linear analysis in [32]. Furthermore, the lack of L2 estimation for the electric potential caused by the characteristics of the Poisson equation in the two-fluid model also increases the workload of estimating these mixed derivatives.The remainder of this paper is arranged as follows. In Section2, we describe the result of constructing approximate solution of (1) and (2) and state our main result. In Section 3, we give the proof of the main result.To end this introduction, we make some remarks and explanations as usual.Hs denotes the Sobolev space as usual, and the corresponding Sobolev norm is denoted by ·Hs. We will simply use ·=·L2Ω without any ambiguity. Especially, to define the Sobolev conormal space, we introduce Zi as a finite set of generators of vector fields which are tangent to ∂Ω and set
(7)Hcom=f∈L2ΩZαf∈L2Ω,forα≤m.The tangential vector fields involved in this article as(8)Z0=∂t,Z1=∂1,Z2=∂2,Z3=φx3∂3,where φx3 is a smooth bounded function with the property φ0=0, φ′0≠0, and φx3>0 for x3∈0,1, typically, φx3=x31−x3. For the sake of simplicity, we give the following definition:
(9)Zα=Z0α0Z1α1Z2α2Z3α3,forα=α0,α1,α2,α3.Based on these conormal derivatives, we introduce some seminorms(10)fHm=∑α≤m∂tα0∂1α1∂2α2∂3α3fL2Ω,fHcom=∑α≤mZ0α0Z1α1Z2α2Z3α3fL2Ω.In addition, we also define(11)fHm=∑α≤m∂1α1∂2α2∂3α3fL2Ω,Qm=n,h,u,v,ε∇ϕtHcomΩ.Throughout this paper,C denotes the general positive constants independent of time t and ε, and Ca depends only on the approximation solutions which may vary from line to line. We will use the notation u=uy,u3, uy=u1,u2. A,B=AB−BA represents the standard commutator of the operators A and B. Combined with the properties of the tangential vector fields, it gives [24]
(12)Zi,∂3=0,i=0,1,2,Z3,∂3=−ψ′x3∂3,(13)Z3m,∂3=∑β=0m−1ψβ,mx3Z3β∂3=∑β=0m−1ψβ,mx3∂3Z3β,(14)Z3m,∂33=∑β=0m−1ψ1,β,mx3Z3β∂3+∑β=0m−1ψ2,β,mx3Z3β∂33=∑β=0m−1ψ1,β,mx3∂3Z3β+∑β=0m−1ψ2,β,mx3∂33Z3β,where all the ψ−x3 or ψ−x3 are families of bounded smooth functions related to the derivatives of φx3. In particular, we consider φx3=x31−x3, and (12) will come in handy for us to calculate some commutators in the following text.


## 2. Approximate Solution Form and the Main Result

In this section, we first give the approximate solutions involving the strong boundary layer of density and the weak boundary layer of velocity to the problem (1) and (2), which have been constructed in [32] as the following form:
(15)na,ha,ua,va,ϕa=∑i=0Kεini,hi,ui,vi,ϕit,x+∑i=0Kεiax3N+i,H+i,U+i,V+i,Φ+it,y,x3ε+bx3N−i,H−i,U−i,V−i,Φ−it,y,1−x3ε,where K is an arbitrarily large integer. ni,hi,ui,vi,ϕit,x describe the macroscopic behavior of solutions away from boundary. N+i,H+i,U+i,V+i,Φ+i and N−i,H−i,U−i,V−i,Φ−i denote the lower and the upper boundary layer functions, respectively. And we use z=x3/ε and ξ=1−x3/ε to denote the fast variables.Proposition 1 (see [32]).
LetK∈ℕ∗; suppose the initial profiles, similar to the expansion form in (15), are sufficiently smooth with n00x=h00x>0 and u00x=v00x, which satisfy the compatibility conditions with the boundary data. Then, there exists a T>0 independent of ε, and a smooth approximate solution of (1) of order K under the from (15) such that
(1)
We have smooth solutionsm0,w0satisfying the compressible isothermal Euler system (5) with initial datan00,u00andϕ0satisfying equation (6)(2)
For all1≤i≤K, we have smooth functions ni,hi,ui,vi,ϕi(3)
For all1≤i≤K, N+i,H+i,U+i,V+i,Φ+i and N−i,H−i,U−i,V−i,Φ−i are smooth functions and together with their derivatives, which belong to the set of uniformly exponentially decreasing functions with respect to the last variableWe consider a solutionnε,hε,uε,vε,ϕε of system (1) and (2) and define the error term n,h,u,v,ϕ as
(16)nε=na+n,hε=ha+h,uε=ua+u,vε=va+v,ϕε=ϕa+ϕ.Then, forn,h,u,v,ϕ, we get the following system:
(17)∂tn+ua+u·∇n+n∇·u+ua+∇·nau=εKRn,∂tu+ua+u·∇u+u·∇ua+Te∇nn+na−∇nanann+na=−∇ϕ+με2n+naΔu+μ+νε2n+na∇∇·u−με2nnan+naΔua−μ+νε2nnan+na∇∇·ua+εKRu,∂th+va+v·∇h+h∇·v+va+∇·hav=εKRh,∂tv+va+v·∇v+v·∇va+Ti∇hh+ha−∇hahahh+ha=∇ϕ+δε2h+haΔv+δ+ηε2h+ha∇∇·v−δε2hhah+haΔva−δ+ηε2hhah+ha∇∇·va+εKRv,ε2Δϕ=h−n+εK+1Rϕ,where the remainders Rn, Rh, Ru, Rv, and Rϕ are dependent on na,ha,ua,va,ϕa and satisfy
(18)supt∈0,T∇αRn,h,u,v,ϕL2Ω≤Cε−α3,∀α=α1,α2,α3∈ℕ3,α≤m.Without loss of generality, we set the slip-lengthβ1=β2=1.The boundary conditions and the initial data are given by(19)u3x3=0,1=0,v3x3=0,1=0,ϕx3=0,1=0,(20)ui−∂ui∂x3x3=0,1=0,i=1,2,vi−∂vi∂x3x3=0,1=0,i=1,2,(21)ut=0=εK+1u0,vt=0=εK+1v0,(22)nt=0=εK+1n0,ht=0=εK+1h0.We state the main result in this paper as follows.Theorem 2.
LetK∈N+ and K≥7. Assume that the initial data n00,u00x=h00,v00x, n0,h0,u0,v0∈H3Ω for (22) satisfying some compatibility conditions with the boundary conditions (20) and na,ha,ua,va,ϕa are smooth approximate solutions at order K given by Proposition 1 defined on 0,T. Then, there is a positive constant ε0 such that for every ε∈0,ε0, the solution of (17)–(22) is defined on 0,T and satisfies the estimates
(23)supt∈0,Tn,h,u,v,ε∇ϕH3Ω≤CaεK−6.


## 3. Proof of the Main Result

This section is aimed at proving a priori estimates, which is a crucial step to prove Theorem2. The local well-posedness of the system (17)–(22) for fixed ε can be obtained by using a similar method to the local well-posedness of the compressible Navier-Stokes equations with the Navier boundary condition, c.f. [7]. Therefore, we make a priori assumption as follows.Ifn,h,u,v,ϕ are assumed to be the solutions of the problem (17)–(22) on 0,T, it holds that
(24)supt∈0,Tn,h,u,v,ε∇ϕH32+ε2∫0T∇u,∇vH32≤Caε2.Then, we obtain the following estimates.Proposition 3.
Under assumption (24), assume n,h,u,v,ϕ be the solutions of the problem (17)–(22) on 0,T; it holds that
(25)supt∈0,Tn,h,u,v,ε∇ϕH32+ε2∫0T∇u,∇vH32≤Caε2K−6.ConsideringK>7 and using a bootstrap argument, we could complete the proof of Theorem 2 with the help of Proposition 3. Thus, we just need to prove the above a priori estimates.We consider the following system which is converted from the system (17):
(26)∂tn+ua+u·∇n+na+n∇·u+u·∇na+n∇·ua=εKRn,na+n∂tu+ua+u·∇u+u·∇ua+∇n−n∇nana=−na+n∇ϕ+με2Δu+μ+νε2∇∇·u+Sn+εKRu,∂th+va+v·∇h+ha+h∇·v+v·∇ha+h∇·va=εKRh,ha+h∂tv+va+v·∇v+v·∇va+∇h−h∇haha=ha+h∇ϕ+δε2Δv+δ+ηε2∇∇·v+Sh+εKRv,ε2Δϕ=h−n+εK+1Rϕ,with
(27)Sn=−με2nnaΔua−μ+νε2nna∇∇·ua,(28)Sh=−δε2hhaΔva−δ+ηε2hha∇∇·va.Note that, for simplicity, we could setTi=Te=1, here.Lemma 4 (L2 estimate).
Under the assumptions of Theorem2 and assume n,h,u,v,ϕ be the solution of the problem (17)–(22) on 0,T satisfying a priori assumption (24), it holds that
(29)supt∈0,Tn,h,u,v,ε∇ϕ2+ε2∫0T∇u,∇v2≤Caε2K−1.Proof.
By taking (26)1×n/na+n+(26)2×u+(26)3×h/ha+h+(26)4×v, we get
(30)∂tn22na+n+ua+u2na+n·∇n2+nuna+n·∇na+n∇·u+n2na+n∇·ua+na+n∂tu22+ua+u2·∇u2+u·∇ua·u+u·∇n−nnau·∇na+∂th22ha+h+va+v2ha+h·∇h2+hvha+h·∇ha+h∇·v+h2ha+h∇·va+ha+h∂tv22+va+v2·∇v2+v·∇va·v+v·∇h−hhav·∇ha=ha+hv−na+nu·∇ϕ+με2uΔu+μ+νε2u·∇∇·u+δε2vΔv+δ+ηε2v·∇∇·v+Snu+Shv+nna+nεKRn+hha+hεKRh+εKuRu+vRv.
Integrating (30) over Ω with boundary conditions (20) and performing standard calculations, we have
(31)12ddt∫Ωn2na+n+h2ha+h+na+nu2+ha+hv2+ε2∫Ωμ∇u2+δ∇v2+ε2∫∂Ωμuy2+δvy2+ε2∫Ωμ+ν∇·u2+δ+η∇·v2−12∫Ωn2∂t+ua+u·∇1na+n+h2∂t+va+v·∇1ha+h−12∫Ωn2na+n∇·ua+u+h2ha+h∇·va+v−12∫Ωu2∂t+ua+u·∇na+n+v2∂t+va+v·∇ha+h−12∫Ωu2na+n∇·ua+u+v2ha+h∇·va+v+∫Ω1na+n−1nanu·∇na+1ha+h−1hahv·∇ha+∫Ωn2na+n∇·ua+h2ha+h∇·va+na+nu·u·∇ua+ha+hv·v·∇va=∫Ωha+hv−na+nu·∇ϕ+∫ΩSnu+Shv+∫Ωnna+nεKRn+hha+hεKRh+εKuRu+vRv.
We recall the properties of the approximate solution and the prior assumption (24),
(32)∂t+ua·∇naL∞+∂t+va·∇haL∞≤Ca,(33)∂t+ua+u·∇nL∞≤Ca∂tnL∞+∇nL∞≤Ca,(34)∂t+va+v·∇hL∞≤Ca∂thL∞+∇hL∞≤Ca.
Furthermore, the vanishing ofu3,v3 on the boundary gives us estimate u3≤x3∇uL∞,v3≤x3∇vL∞. Therefore, we have
(35)u·∇naL∞≤uy·∇ynaL∞+u3∂3naL∞≤Ca+∇uL∞z∂znaL∞≤Ca.
By combining (32) with (35), the following terms in (31) can be estimated as
(36)∂t+ua+u·∇na+nL∞≤∂t+ua·∇naL∞+u·∇naL∞+∂t+ua+u·∇nL∞≤Ca.
In a similar way, termsv·∇haL∞ and v2∂t+va+v·∇ha+hL∞ also can be controlled by Ca. Noting that the boundary layer of velocities U0=V0=0, thus, we get
(37)1na+n∇·ua+u+1ha+h∇·va+vL∞+na+n∇·ua+u+ha+h∇·va+vL∞≤Ca.
Combining the fast decay property of the boundary layer profiles inna,ha, ∇na,∇ha=Oε−1 and the priori assumptions (24), we have
(38)∫Ω1na−1na+nnu·∇na+1ha−1ha+hhv·∇ha≤nL22u∇naL∞+hL22v∇haL∞≤1εu,vL∞n,h2≤Can,h2.
Recall the expression ofSn and Sh, we have
(39)∫ΩSnu≤Ca∫Ωn2+u2,∫ΩShv≤Ca∫Ωh2+v2.
By plugging the above estimates (32)–(39) into (31) gives that
(40)12ddt∫Ωn2na+n+h2ha+h+na+nu2+ha+hv2+ε2∫Ωμ∇u2+δ∇v2+ε2∫∂Ωμuy2+δvy2+ε2∫Ωμ+ν∇·u2+δ+η∇·v2≤∫Ωha+hv−na+nu·∇ϕ+Can,h,u,v2+Caε2K.
Next, we turn to deal with the first term on the right-hand side of (40) with (26)1,3 and integrate by parts if necessary
(41)I=∫Ωha+hv−na+nu·∇ϕ=∫Ω∇·na+nu−ha+hvϕ=∫Ω∂th−∂tnϕ+∫Ω∇·hva−∇·nuaϕ+∫ΩεKRn−εKRhϕ=J1+J2+J3.
Using the Poisson equation (26)5, we have ∂th−∂tn=ε2Δ∂tϕ−εK+1∂tRϕ; thus,
(42)J1=∫Ωε2Δ∂tϕ−εK+1∂tRϕϕ≤−ε22ddt∇ϕ2+Caε∇ϕ2+Caε2K,where the Poincaré inequality and integrate by parts are applied.
(43)J3=∫ΩεKRn−εKRhϕ≤Caε∇ϕ2+Caε2K−1.
By virtue of the fact thatua−va=Oε and the Poisson equation (26)5, J2 can be estimated as the following:
(44)J2=∫Ω∇ϕ·nua−hva=∫Ωnua−va·∇ϕ+∫Ωva·∇ϕn−h=∫Ωnua−va·∇ϕ+∫Ωε2∇ϕ·∇va·∇ϕ+ε2∫Ω∇∇ϕ22·va+∫ΩεK+1Rϕ∇ϕ·va≤Caε∇ϕ2+Can2+Caε2K.
Thus, we have that(45)I≤−ε22ddt∇ϕ2+Caε∇ϕ2+Can2+Caε2K−1.
Substituting (45) into (40) and integrating it over 0,T, we obtain
(46)n,h,u,v2+ε∇ϕ2+ε2∫0T∇u,∇v2≤Ca∫0Tn,h,u,v,ε∇ϕ2+Caε2K−1.
Using the Gronwall inequality, we deduceL2 estimate of the solution as that
(47)n,h,u,v,ε∇ϕ2+ε2∫0T∇u,∇v2≤Caε2K−1.Next, we expect to get the high order derivative estimates, and one of the tools used in the process is the conormal derivatives. In the following Lemma4, we will provide the uniform tangential estimates for solutions.Lemma 5 (tangential estimates).
Under the assumptions of Lemma4, it holds that
(48)supt∈0,TZjn,Zjh,Zju,Zjv,ε∇Zjϕ2+ε2∫0T∇Zju,∇Zjv2dτ≤Caε2K−j−1,0≤j≤3.Proof.
First of all, forj=0, the result in Lemma 5 has been proved in Lemma 4. We suppose that the above lemma is true for the conditions of j≤2; then, we can prove that Lemma 5 is true for the case of j=3. Now we set α=3 and apply Zα to equations (26) to acquire that
(49)∂tZαn+ua+u·∇Zαn+na+n∇·Zαu+Zαu·∇na+Zαn∇·ua=εKZαRn+Cnα,na+n∂tZαu+ua+u·∇Zαu+Zαu·∇ua+∇Zαn−Zαn∇nana=−na+n∇Zαϕ+με2ΔZαu+μ+νε2∇∇·Zαu+ZαSn+εKZαRu+Cuα,∂tZαh+va+v·∇Zαh+ha+h∇·Zαv+Zαv·∇ha+Zαh∇·va=εKZαRh+Chα,ha+h∂tZαv+va+v·∇Zαv+Zαv·∇va+∇Zαh−Zαh∇haha=ha+h∇Zαϕ+δε2ΔZαv+δ+ηε2∇∇·Zαv+ZαSh+εKZαRv+Cvα,ε2ΔZαϕ=Zαh−Zαn+εK+1ZαRϕ+Cϕα,where the commutators Cnα,Cuα,Chα,Cvα,Cϕα are given by
(50)Cnα=−Zα,ua+u·∇n−Zα,na+n∇·u−Zα,∇na·u−Zα,∇·uan,Cuα=−Zα,na+n∂tu−Zα,na+nua+u·∇u−Zα,na+n∇ua·u−Zα,∇n+Zα,∇nanan−Zα,na+n∇ϕ+με2Zα,Δu+μ+νε2Zα,∇∇·u,Chα=−Zα,va+v·∇h−Zα,ha+h∇·v−Zα,∇ha·v−Zα,∇·vah,Cvα=−Zα,ha+h∂tv−Zα,ha+hva+v·∇v−Zα,ha+h∇va·v−Zα,∇h+Zα,∇hahah+Zα,ha+h∇ϕ+δε2Zα,Δv+δ+ηε2Zα,∇∇·v,Cϕα=−ε2Zα,Δϕ.
Taking (49)1×Zαn/na+n+(49)2×Zαu+(49)3×Zαh/ha+h+(49)4×Zαv, we can obtain that
(51)12ddt∫ΩZαn2na+n+Zαh2ha+h+na+nZαu2+ha+hZαv2≤∫Ωha+hZαv−na+nZαu·∇Zαϕ+ε2∫Ωμ+νZαu·∇∇·Zαu+δ+ηZαv·∇∇·Zαv+ε2∫ΩμZαuΔZαu+δZαvΔZαv+∫ΩZαSnZαu+ZαShZαv+∫ΩZαnna+nCnα+Zαhha+hChα+∫ΩZαuCuα+ZαvCvα+CaZαn,Zαh,Zαu,Zαv2+Caε2K,which is similar as Lemma 4 in some sense.
Firstly, let us treat the first term on the right-hand side of (51). Using equations (49)1 and (49)3 to express ∇·na+nZαu and ∇·ha+hZαv, and integrating by parts, we get that
(52)I=∫Ω∇·na+nZαu−ha+hZαvZαϕ=∫Ω∂tZαh−∂tZαnZαϕ+∫Ω∇·va+vZαh−ua+uZαnZαϕ+∫ΩZαn∇·u−Zαh∇·v+Zαu·∇n−Zαv·∇hZαϕ+∫ΩεKZαRn−ZαRh+Cnα−ChαZαϕ=∑k=14Ik.
Taking time derivatives to the Poisson equation (49)5, we could estimate I1 as
(53)I1=∫Ωε2Δ∂tZαϕ−εK+1∂tZαRϕ−∂tCϕαZαϕ≤−ε22ddt∫Ω∇Zαϕ2+∫Ω∂tCϕαZαϕ+Ca∫Ωε∇Zαϕ2+Caε2K.
Next, we need to deal with the second term on the right-hand side of (53). Using (50)5, (12), the Poincaré inequality, and integrating by parts if necessary, we have that
(54)∫Ω∂tCϕαZαϕ=ε2∫Ω∂tZα,ΔϕZαϕ≤ε2∫Ω∑β=03ψ1,β,3∂3Zβϕ+ψ2,β,3∂33ZβϕZαϕ≤C∑β=03∫Ωε∂3Zβϕ2+∫ΩεZαϕ2+ε2∑β=03∫Ω∂3ψ2,β,3∂3ZβϕZαϕ+ψ2,β,3∂3Zβϕ∂3Zαϕ≤C∑β=03∫Ωε∂3Zβϕ2+∫ΩεZαϕ2+∫Ωε∂3Zαϕ2≤C∫Ωε∇Zαϕ2+Caε2K−3.
Thus,(55)I1≤−ε22ddt∫Ω∇Zαϕ2+Ca∫Ωε∇Zαϕ2+Caε2K−3.
With the help of the a priori assumption (24) and the fact that ua−va=Oε, I2 can be treated as
(56)I2=∫Ω∇Zαϕ·ua+uZαn−va+vZαh=∫Ω∇Zαϕ·ua+u−va+vZαn+∫Ω∇Zαϕ·va+vZαn−Zαh≤ε∇Zαϕ2+CaZαn2+1εu−vL∞ε∇Zαϕ2+Zαn2+∫Ω∇Zαϕ·va+vZαn−Zαh≤Caε∇Zαϕ2+CaZαn2+∫Ω∇Zαϕ·va+vZαn−Zαh.
Using the Poisson equation and integrating by parts if necessary, we estimate the last integral in (56) as
(57)∫Ω∇Zαϕ·va+vZαn−Zαh=−∫Ωε2ΔZαϕ−εK+1ZαRϕ−Cϕαva+v·∇Zαϕ≤∫Ωε2∇Zαϕ·∇va+v·∇Zαϕ+ε2∫Ω∇∇Zαϕ22·va+v+Ca+∇vL∞∫ΩεK+1ZαRϕ+Cϕα∇Zαϕ≤Ca∫Ωε∇Zαϕ2+Ca∫Ω∇ZαϕCϕα+Caε2K.
By virtue of the expression ofCϕα, similar to (54), we get that
(58)∫Ω∇ZαϕCϕα=∫Ω∇ZαϕZα,ε2Δϕ≤ε2∑β=02∫Ωψ1,β,3Zβ∂3ϕ+ψ2,β,3Zβ∂33ϕ∇Zαϕ≤∑β=02∫Ωε2ψ1,β,3Zβ,∂3ϕ+∂3Zβϕ+ψ2,β,3Zβ−ε2Δyϕ+h−n+εK+1Rϕ∇Zαϕ≤1ε2∫Ω∑β=02Zβ−ε2Δyϕ+h−n+εK+1Rϕ2+C∫Ωε∇Zαϕ2+ε2K−3≤C∫Ωε∇Zαϕ2+Caε2K−4,where we have used the commutators (14)2, the assumptions for j≤2 and the Poisson equation (26)5. Thus, it gives
(59)I2≤Ca∫Ωε∇Zαϕ2+Zαn2+Caε2K−4.
With the help of the Poincaré inequality, we can estimate the last two integrals in (52) directly
(60)I3+I4=∫ΩZαn∇·u−Zαh∇·v+Zαu·∇n−Zαv·∇hZαϕ+∫ΩεKZαRn−ZαRh+Cnα−ChαZαϕ≤∇·u,∇·vL∞Zαn,ZαhZαϕ+∇n,∇hL∞Zαu,ZαvZαϕ+εK−1ZαRn−ZαRhεZαϕ+Cnα−ChαZαϕ+Caε2K−1≤CaZαn,Zαh,Zαu,Zαv,ε∇Zαϕ2+CCnα,Chα2+C∇Zαϕ2+Caε2K−1.
Substituting the above estimates ofI1-I4 into (52), we have that
(61)I≤−ε22ddt∫Ω∇Zαϕ2+CaZαn,Zαh,Zαu,Zαv,ε∇Zαϕ2+CCnα,Chα2+C∇Zαϕ2+Caε2K−4.
In addition, it is easy to see that(62)∫ΩZαSnZαu+ZαShZαv≤CaZαu,Zαv2+ZαSn,ZαSh2.
Recalling the definitions ofSn and Sh in (27), we estimate them as follows:
(63)ZαSn,ZαSh2≤Cε2ZαΔuanan2+ε2Zα∇∇·uanan2+Cε2ZαΔvahah2+ε2Zα∇∇·vahah2.
Due to the assumptions ofj=1,2, using Lemma 5 and the fact that Δua,Δva=Oε−1, we estimate the first term on the right-hand side of (63) as
(64)ε2ZαΔuanan2≤Cε2Z3Δuanan2+ε2Z2ΔuanaZn2+Cε2Z1ΔuanaZ2n2+ε2ΔuanaZ3n2≤CεZ3ΔuanaL∞2εn2+εZ2ΔuanaL∞2εZn2+CεZΔuanaL∞2εZ2n2+εΔuanaL∞2εZ3n2≤CaZ3n2+Caε2K−3.
Similarly, the rest three terms of the right-hand side of (63) are estimated as
(65)ε2Zα∇∇·uanan2+ε2ZαΔvahah2+ε2Zα∇∇·vahah2≤CaZ3n,Z3h2+Caε2K−3.
Substituting the estimates (63)–(65) into (62),
(66)∫ΩZαSnZαu+ZαShZαv≤CaZαn,Zαh,Zαu,Zαv2+Caε2K−3.
With the help of the estimate (61) and (66), integrating (51) over 0,T×Ω, we obtain that
(67)Zαn2+Zαh2+Zαu2+Zαv2+ε∇Zαϕ2≤ε2∫0T∫ΩμZαuΔZαu+δZαvΔZαv+ε2∫0T∫Ωμ+νZαu∇∇·Zαu+δ+ηZαv∇∇·Zαv+1C∫0TCnα,Chα2+Ca∫0T∫ΩZαuCuα+ZαvCvα+Ca∫0TZαn,Zαh,Zαu,Zαv2+C∫0T∇Zαϕ2+Caε2K−4=∑k=14Jk+Ca∫0TZαn,Zαh,Zαu,Zαv2+C∫0T∇Zαϕ2+Caε2K−4.
It is particularly noted that some of the same processes are omitted in the estimation ofJk, since the structural form of Jk is consistent with that of Jk in reference [16].
Now, let us estimate them one by one. Integrating by parts if necessary, we estimateJ1 as
(68)J1=−∫0T∫Ωμε2∇Zαu2−∫0T∫ℝ2με2∂3ZαuyZαuy−∫0T∫Ωδε2∇Zαv2−∫0T∫ℝ2δε2∂3ZαvyZαvy.
It is worth noting that(69)∂3Zαuy=Zα∂3uy+∂3,Zαuy=Zαuy+∂3,Zαuy,∂3Zαvy=Zα∂3vy+∂3,Zαvy=Zαvy+∂3,Zαvy,which is contributed by the boundary condition (20).
By the property of commutators, the boundary condition (20), and trace theorem, we get that
(70)−∫0T∫ℝ2με2∂3ZαuyZαuy≤−με2∫0TZαuyL2ℝ22−με2∫0T∂3,ZαuyL2ℝ2ZαuyL2ℝ2≤−με2∫0TZαuyL2ℝ22+∫0TCε2∂3,ZαuyL2ℝ22+με24∇Zαuy2+CZαuy2≤−με2∫0TZαuyL2ℝ22+C∫0Tε2∇Z2uy2+uHco22+∫0Tμε24∇Zαuy2+CZαuy2,where the term ∂3,ZαuyL2ℝ22=∑β=02ψβ,3Zβ∂3uyL2ℝ22, and the term Zβ∂3uy=Zβuy that the trace theorem is used again. Similarly, we can obtain the estimation for ∫0T∫ℝ2δε2∂3ZαvyZαvy.
Then, with the help of Lemma5 for j=2, we can rewrite (68) as
(71)J1+∫0T∫Ω3με24∇Zαu2+∫0T∫ℝ2με2Zαuy2+∫0T∫Ω3δε24∇Zαv2+∫0T∫ℝ2δε2Zαvy2≤C∫0TZαu,Zαv2+Cε2K−3.
Integrating by parts and using the boundary condition (20), we get that
(72)J2=−ε2∫0T∫Ωμ+ν∇·Zαu2−ε2∫0T∫Ωδ+η∇·Zαv2.
Next, we turn to the term ofJ3. By the expression of Cnα, we treat the first integral of J3 more carefully.
(73)∫0TCnα2=∫0TZα,ua+u·∇n2+∫0TZα,na+n∇·u2+∫0TZα,∇na·u2+∫0TZα,∇·uan2=∑k=14J3k.
For the termJ31,
(74)J31=∫0TZα,ua+u·∇n2=∫0TZαua+u·∇n−ua+u·∇Zαn2=∫0TZ3ua+u·∇n+Z2ua+u·Z∇n+Zua+u·Z2∇n+ua+u·Zα,∇n2≤∫0TZ3ua·∇n+Z2ua·Z,∇+∇Zn+Zua·Z2,∇+∇Z2n2+Z3u·∇n2+∫0TZ2uL42Z∇nL42+Zu·Z2,∇+∇Z2n2+ua+u∑β=02ψβ,3∂3Zβn2≤∫0T∇Z3uaL∞2Zn2+∇Z2uaL∞2−ψ′x3Z+Z2n2+∫0T∇ZuaL∞2∑β=02ψβ,2Zβ+Z3n2+∫0TZ3u2nH32+Z2uL2Z2uH1nH32+∫0T∇ZuL∞2∑β=02ψβ,2Zβ+Z3n2+∫0T∇ua+uL∞∑β=03ψβ,3Zβn2≤Caε2K−3+∫0TnH32Z3u2+∇Z2u2+Z2u2+∫0TuH32∑β=03ψβ,3Zβn2+supt∈0,T∑β=02ψβ,2Zβ+Z3n2∫0TZ∇−Z,∇uL∞2≤Ca∫0TZ3n,Z3u2+supt∈0,TZ3n2+Caε2K−3,where we deal some estimates with the help of the Poincaré inequality, Gagliardo-Nirenberg interpolation inequality, and the property of normal operator, for example,
(75)Z3ua·∇n2≤Z3uay·∇yn2+Z3ua3∂3n2≤CaZn2+∇Z3ua3L∞2ϕx3∂3n2≤CaZn2.
Similarly, we can estimate the termJ32, which can be controlled by Ca∫0TZ3n,Z3u2+Caε2K−4. Noticing that ∇na=Oε−1, J33 and J34 can be controlled by Ca∫0TZ3n,Z3u2+Caε2K−4 and Ca∫0TZ3n,Z3u2+Caε2K−3, respectively.
Therefore, substitutingJ31-J34 into (73), we have
(76)∫0TCnα2≤Ca∫0TZ3n,Z3u2+supt∈0,TZ3n2+Caε2K−4.
Since the structural form ofChα is the same as that of Cnα, we get
(77)∫0TChα2≤Ca∫0TZ3h,Z3v2+supt∈0,TZ3h2+Caε2K−4.
Collecting the above estimates, we obtain(78)J3≤Ca∫0TZ3n,Z3h,Z3u,Z3v2+1Csupt∈0,TZ3n,Z3h2+Caε2K−4.
For the termJ4, by virtue of the expression of the Cuα, we get that
(79)∫0T∫ΩZαuCuα≤∫0T∫ΩZα,na+n∂tuZαu+∫0T∫ΩZα,na+nua+u·∇uZαu+∫0T∫ΩZα,na+n∇ua·uZαu+∫0T∫ΩZα,∇nananZαu+∫0T∫ΩZα,na+n∇ϕZαu+∫0T∫ΩZα,∇nZαu+ε2∫0T∫ΩμZα,ΔuZαu+ε2∫0T∫Ωμ+νZα,∇∇·uZαu=∑k=18J4k.
Similar toJ31, for the term J42, one has
(80)J42=∫0TZα,na+nua+u·∇uZαu≤∫0TZ3na+nua+u·∇u+Z2na+nua+u·Z∇uZαu+∫0TZna+nua+u·Z2∇u+na+nua+u·Zα,∇uZαu≤Ca∫0TZ3naua∇u+Z3nau+uan+nu∇uL∞Zαu+∫0TZ2nauaZ,∇+∇Zu+Z2nau+uan+nuL4Z∇uL4Zαu+∫0TZna+nu+uaL∞Z2,∇+∇Z2u+na+nua+u∑β=02ψβ,3∂3ZβuZαu≤Ca∫0TZ3u2+Caε2K−3+∫0Tn,uH3Z3u,Z3n2+∇Z2nau+uan+nuL22uH32+∫0TCa1+n,uH32∑β=02ψβ,2Zβ+Z3u2+∫0TCa1+n,uH32∑β=03ψβ,3Zβu2≤∫0TZ3n,Z3u2+Caε2K−4.
In the same way to estimate the termJ45, it can be controlled by Ca∫0TZ3u,Z3n2+Caε2K−4. Noticing that ∇na=Oε−1, for the simpler structures, J41, J43, and J44 can be controlled by Ca∫0TZ3u,Z3n2+Caε2K−3 and Ca∫0TZ3u2+Caε2K−4 with the direct commutator calculation.
ForJ46,
(81)∫0T∫ΩZα,∇nZαu=∫0T∫Ω∑β=02ψβ,3x3∂3ZβnZαu=∫0T∫Ω∑β=02ψβ,3x3ψx3∂3Zβn∂3Zα−1u≤Ca∫0TZ3n2+∂3Z2u2+Caε2K−3≤Ca∫0TZ3n2+Caε2K−4.
Next, with the help of the property of commutators, the boundary condition (20), and the trace theorem, we estimate J47 as
(82)∫0T∫Ωμε2Zα,ΔuZαu=με2∫0T∫Ω∑β=02ψ1,β,3∂3ZβuZαu+ψ2,β,3∂33ZβuZαu≤C∫0TZαu2+Cε2K−3+με2∑β=02∫0T∫∂Ωψ2,β,3∂3ZβuyZαuy+με2∑β=02∫0T∫Ω∂3ψ2,β,3∂3ZβuZαu+ψ2,β,3∂3Zβu∂3Zαu≤με2∑β=02∫0T∫∂Ωψ2,β,3Zβuy−Zβ,∂3uyZαuy+με216∫0T∇Zαu2+C∫0TZαu2+Cε2K−3≤C∫0TZαu2+με28∫0T∇Zαu2+Cε2K−3,where the term
(83)∫∂ΩZβ,∂3uyZαuy=∫∂Ω∑β′=0β−1ψβ′,2Zβ′∂3uyZαuy=∫∂Ω∑β′=0β−1ψβ′,2Zβ′uyZαuy≤∇∑β′=0β−1ψβ′,2Zβ′uy2+∑β′=0β−1ψβ′,2Zβ′uy2+CZαuy2+με216∇Zαuy2.
In a similar way to estimate theJ48, we can also get
(84)J48=∫0T∫Ωμ+δε2Zα,∇∇·uZαu≤C∫0TZαu2+με28∫0T∇Zαu2+Cε2K−3.
Therefore, substitutingJ41-J48 into (79), we have
(85)∫0T∫ΩZαuCuα≤Ca∫0TZαn2+Zαu2+με24∫0T∇Zαu2+Caε2K−4.
Subsequently, we can use a similar argument as used before to obtain that(86)∫0T∫ΩZαvCvα≤Ca∫0TZαh2+Zαv2+δε24∫0T∇Zαv2+Caε2K−4.
Collecting the above two estimates, we obtain(87)J4≤Ca∫0TZαn,Zαh,Zαu,Zαv2+ε2∫0Tμ4∇Zαu2+δ4∇Zαv2+Caε2K−4.
SubstitutingJ1-J4 into (67) can be written as
(88)Zαn2+Zαh2+Zαu2+Zαv2+ε∇Zαϕ2+ε2∫0Tμ2∇Zαu2+δ2∇Zαv2+ε2∫0T∫ℝ2μZαuy2+δZαvy2+ε2∫0Tμ+ν∇·Zαu2+δ+η∇·Zαv2≤Ca∫0TZαn,h,u,v,ε∇ϕ2+C∫0T∫Ω∇Zαϕ2+Caε2K−4,where the term 1/Csupt∈0,TZ3n,Z3h2 in (78) can be absorbed by the left-hand side.
To handle the second term on the right-hand side of (88), similar approaches as in Lemma 4.3 of [11, 32] are adopted in the following process.
We consider the indexα into two cases.
Forα1≠0 or α2≠0 or α3≠0, multiplying the Poisson equation (49)5 by Zαϕ, integrating it over Ω, and using integration by parts, we get that
(89)∇Zαϕ2=−∫ΩΔZαϕZαϕ=ε−2∫ΩZαn−hZαϕ−ε−2∫ΩεK+1Rϕ+CϕZαϕ≤ε−2Zα−1h−n∇Zαϕ+εK−1ZαRϕZαϕ+∑β=02∫Ωψ1,β,3Zβ∂3ϕ+ψ2,β,3Zβ∂33ϕZαϕ≤ε−4h−nHcom−1Ω2+12∇Zαϕ2+Caε2K−2+Cε2ε∇ϕHcom−1Ω2+112∇Zαϕ2−∑β=02∫Ω∂ψ2,β,3Zβ−1∂3ϕZαϕ+ψ2,β,3Zβ−1∂33ϕ∂Zαϕ≤ε−4h−nHcom−1Ω2+Cε2ε∇ϕHcom−1Ω2+34∇Zαϕ2+Caε2K−2+Cε4∑β=02Zβ−1−ε2Δyϕ+h−n+εK+1Rϕ2≤ε−4h−nHcom−1Ω2+Cε2ε∇ϕHcom−1Ω2+34∇Zαϕ2+Caε2K−2,where the application of some fundamental inequality and the commutator characteristics are essential. And in a natural manner, we get
(90)∇Zαϕ2≤Cε−4n−hHcom−1Ω2+ε−2ε∇ϕHcom−1Ω2+Caε2K−2≤Caε−4Qm−12t+ε2K−2.
Forα1=α2=α3=0, the characteristics of commutators give us α=α0 and Cϕ=0. Acting operator Z0α0−1 on equation (26)1, one has
(91)Z0α0n+Z0α0−1∇·ua+u·n+Z0α0−1∇·nau=εKZ0α0−1Rn.
Similarly, applying the operatorZ0α0−1 to equation (26)3 gives us the expression about Z0α0h. Combining the expressions of Z0α0n and Z0α0h and applying operator Z0α0 to Poisson equation (26)5, with the help of integration by parts, then one has
(92)∫ΩZ0α0n−hZ0α0ϕ=∫ΩZ0α0−1∇·va+v·h−ua+u·nZ0α0ϕ+∫ΩZ0α0−1∇·hav−nauZ0α0ϕ+∫ΩεKZ0α0−1Rn−RhZ0α0ϕ≤∫ΩZ0α0−1ua+u·n−va+v·h+nau−hav∇Z0α0ϕ+∫ΩεKZ0α0−1Rn−RhZ0α0ϕ≤Caε−2Qm−12t+ε2K−1+ε24∇Zαϕ2,∇Z0α0ϕ2=−∫ΩΔZ0α0ϕZ0α0ϕ=ε−2∫ΩZ0α0n−hZ0α0ϕ−∫ΩεK−1Z0α0RϕZ0α0ϕ≤Caε−4Qm−12t+ε2K−2+12∇Z0α0ϕ2+Caε2K−2.
Hence, the estimates in both cases give us that(93)∇Zαϕ2≤Caε−4Qm−12t+ε2K−2.
Injecting (93) into (88) and using Gronwall’s inequality, we obtain that
(94)n,h,u,v,ε∇ϕHcomΩ2+ε2∫0T∇Zαu,∇Zαv2≤Ca∫0Tε−4Qm−12t+Caε2K−4.
Next, we define the following weighted norm:(95)Q~mt=∑α≤mε2αZ0α0Z1α1Z2α2Z3α3n,h,u,v,ε∇ϕL2Ω.
Obviously,Q~m−12t≤Q~m2t, (94) can be written as
(96)Q~m2t+ε2+4α∫0T∇Zαu,∇Zαv2≤Ca∫0TQ~m2t+Caε2K−4+4α.
Subsequently, using Gronwall’s inequality, we get(97)Q~m2t≤Caε2K−4+4α1+CateCat,∀t∈0,T.
Then, it implies(98)Zαn,h,u,v,ε∇ϕ2+ε2∫0T∇Zαu,∇Zαv2≤Caε2K−4.
That ends the proof of Lemma5.After the tangential estimates of the solution for the system (17)–(22) completed, next, we will give the estimates of the normal derivative for the solution.Lemma 6.
Under the assumptions of Lemma4, it holds that
(99)ε2supt∈0,TZαR,ZαW2+∫0TZαR,ZαW2≤Caε2K−α−2,α=0,1,2,(100)ε2supt∈0,T∂3αR,∂3αW2+∫0T∂3αR,∂3αW2≤Caε2K−α−2,α=1,2,(101)ε2supt∈0,TZ∂3R,Z∂3W2+∫0TZ∂3R,Z∂3W2≤Caε2K−4,(102)supt∈0,TZα∂3u,Zα∂3v2+ε2∫0TZα∂33u,Zα∂33v2≤Caε2K−α−3,α=0,1,2,(103)supt∈0,T∂32u,∂32v2+ε2∫0T∂33u,∂33v2≤Caε2K−4,(104)supt∈0,T∂33u,∂33v2+ε2∫0T∂34u,∂34v2≤Caε2K−6,(105)supt∈0,TZ∂32u,Z∂32v2+ε2∫0TZ∂33u,Z∂33v2≤Caε2K−5,where R=∂3n,W=∂3h.Proof.
Since the continuity and momentum equations of the electron have the similar structure as the corresponding equations of the ions, the above results can be available by the similar way in the proof of Lemma 3.5, Lemma 3.6, and Lemma 3.7 in [16].Next, based on the existing proof results, we will give the procedure of the high-order normal derivatives and mixed derivative estimate of theϕ.Lemma 7.
Under the assumptions of Lemma4, it holds that
(106)∂32ϕ2+ε2Z∂32ϕ2+ε4Z2∂32ϕ2≤Caε2K−3.Proof.
From (26)5, the Poisson equation can be rewritten as the following form:
(107)ε2∂32ϕ=−ε2Δyϕ+h−n+εK+1Rϕ.
In theL2 estimate to (107) and the tangential estimates in Lemma 5, we have
(108)ε2∂32ϕ2≤ε2Δyϕ2+h−n2+εK+1Rϕ2≤ε2ε∇Zϕ2+Caε2K−1≤Caε2K−1.
Applying the operatorZ to equation (107), we obtain that
(109)ε2Z∂32ϕ=−ε2ZΔyϕ+Zh−Zn+εK+1ZRϕ.
Let us take theL2 estimate to (109), combining the tangential estimates in Lemma 5, then we have
(110)ε2Z∂32ϕ2≤ε2ZΔyϕ2+Zh−Zn2+εK+1ZRϕ2≤ε2ε∇Z2ϕ2+Caε2K−2≤Caε2K−2.
Applying the operatorZ2 to equation (107) and taking the L2 estimate give us that
(111)ε2Z2∂32ϕ2≤−ε2Z2Δyϕ+Z2h−Z2n+εK+1Z2Rϕ2≤ε2ε∇Z3ϕ2+Z2h−Z2n2+Caε2K+1≤Caε2K−3.
Thus, the proof of Lemma7 ends here.Lemma 8.
Under the assumptions of Lemma4, it holds that
(112)∂33ϕ2+ε2Z∂33ϕ2≤Caε2K−5,∂34ϕ2≤Caε2K−6.Proof.
Applying the operator∂3 to equation (107), it is easily to get
(113)ε2∂33ϕ=−ε2∂3Δyϕ+∂3h−n+εK+1∂3Rϕ.
And taking theL2 estimate of (113) and using (99), one has
(114)ε2∂33ϕ2≤−ε2∂3Δyϕ+∂3h−n+εK+1∂3Rϕ2≤ε2ε∇Z2ϕ2+∂3h−n2+Caε2K+1≤Caε2K−3.
In order to obtain the estimation of the mixed derivative ofϕ, applying the operator Z to (113), taking its L2 norm, and using (101), we get
(115)ε2Z∂33ϕ2≤−ε2Z∂3Δyϕ+Z∂3h−n+εK+1Z∂3Rϕ2≤ε2εZ,∂3+∂3ZΔyϕ2+Z∂3h−n2+Caε2K+1≤ε2ε∇Z3ϕ2+Caε2K−4≤Caε2K−4.
Similarly, applying the operator∂3 to equation (113) and using (100), we can obtain the following estimate:
(116)ε2∂34ϕ2≤−ε2∂32Δyϕ+∂32h−n+εK+1∂32Rϕ2≤ε2ε∇Z3ϕ2+∂32h−n2+Caε2K+1≤Caε2K−4.
This completes the proof of Lemma8.Proposition3 can be proved by combining Lemmas 4–7 with Lemma 8.



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