# Existence of Weak Solutions to a Class of Degenerate Semiconductor Equations Modeling Avalanche Generation

Authors: Bin Wu
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

We consider the drift-diffusion model with avalanche generation for evolution
in time of electron and hole densitiesn, p coupled with the electrostatic potential ψ in a semiconductor device. We also assume that the diffusion term is degenerate. The existence of local weak solutions to this Dirichlet-Neumann mixed boundary value problem is obtained.


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## Body

## 1. Introduction

In this paper, we consider the following degenerate semiconductor equations modeling avalanche generation:(1.1)-∇·(∇ψ)=p-n+C(x),(1.2)nt-∇·Jn=R(n,p)+g,Jn=∇(nγ)-μ1n∇ψ,(1.3)pt+∇·Jp=R(n,p)+g,-Jp=∇(pγ)+μ2p∇ψ
with initial and boundary conditions
(1.4)(ψ,n,p)=(ψ̅,n̅,p̅),(x,t)∈ΣD≡ΓD×(0,T),(1.5)(∂ψ∂η,∂n∂η,∂p∂η)=(0,0,0),(x,t)∈ΣN≡ΓN×(0,T),(1.6)(n,p)=(n0,p0),x∈Ω,t=0.
Here the unknowns ψ,n, and p denote the electrostatic potential, the electron density, and the hole density, respectively. The boundary ∂Ω consists of two disjoint subsets ΓD and ΓN. The carrier densities and the potential are fixed at ΓD (Ohmic contacts), whereas ΓN models the union of insulating boundary segments. Jn represents the electron current, and Jp is the analogously defined physical quantity of the positively charged holes. Function C(x) denotes the doping profile (fixed charged background ions) characterizing the semiconductor under consideration, while the term g=α1(∇ψ)|Jn|+α2(∇ψ)|Jp| models the effect of impact ionization (avalanche generation of charged particles) (cf. [1, 2] for details). R(n,p)=r(n,p)(1-np) is the net recombination-generation rate, where r characterizes the mechanism of particle transition. The constant γ is the adiabatic or isothermal (if γ=1) exponent. The regime 0<γ<1 describes a fast diffusion process in the electron (hole) density, whereas 1<γ≤5 is related to slow diffusion.The standard drift-diffusion model corresponding toγ=1 has been mathematically and numerically investigated in many papers (see [3–6]). Existence and uniqueness of weak solutions have been shown. The standard model can be derived from Boltzmann's equation once assumed that the semiconductor device is in the low injection regime, that is, for small absolute values of the applied voltage. In [7] Jüngel showed that in the high-injection regime diffusion terms are no longer linear. A useful choice for γ is γ=5/3. In this case, the parabolic equations (1.2) and (1.3) become of degenerate type, and existence of solutions does not follow from standard theory. Recently, many authors [8–10] have studied the existence and uniqueness of weak solutions of this type of degenerate semiconductor equations without avalanche generation term. In [9], the degenerate semiconductor equations based on Fermi-Dirac statistics were introduced by Jüngel for the first time. The existence and uniqueness results are shown under the assumption that the solution ψ of Poisson equation with Dirichlet-Neumann mixed boundary conditions had the regularity ψ∈W2,r(Ω)(r>N), this amounts to a geometric condition on Ω, for example Ω∈C1,1 and ΓD∩ΓN=∅. ([11, Theorem  3.29]). Then Guan and Wu [8] obtain similar results without the assumption above.There are some papers concerning the semiconductor equations modeling avalanche generation. For instance, the existence of weak solutions of nondegenerate stationary problem has been investigated in [12, 13]. When γ=1, that is, the diffusion term is not degenerate, the authors [14] obtained the existence of local weak solutions of problem (1.1)–(1.6).Our main goal in this paper is to study the existence of weak solutions of problem (1.1)–(1.6). In contrast to the above works, degeneration of diffusion term we are going to study introduces significant new technical difficulties to estimate the avalanche term.We make the following assumptions:(H1)
Ω⊂RN(N=1,2,3) is bounded and ∂Ω∈C0,1, whose outward normal vector is η and ∂Ω¯=ΓD¯∪ΓN¯,ΓD∩ΓN=∅, measN-1(ΓD)>0;(H2)
C(x)∈L∞(Ω);(H3)
r(n,p) is a locally Lipschitz continuous function defined for (n,p) and 0≤r(n,p)≤r̅<∞;(H4)
αi∈C(RN), 0≤αi(ξ)≤αi0=  const.<+∞, forallξ∈RN(i=1,2);(H5)
n̅,p̅∈W1,∞(Ω)∩L∞(Ω), ψ¯∈H1(Ω)∩L∞(Ω), and n̅, p̅≥0 in Ω;(H6)
n0,p0∈L∞(Ω) and n0,p0≥0 a.e. in Ω.Let(1.7)Y={ω∈H1(Ω)|ω|ΓD=0}.Definition 1.1.
(ψ,n,p) is called the weak solution to the problem (1.1)–(1.6) if nγ∈n¯γ+L2(0,T;Y), pγ∈p¯γ+L2(0,T;Y), ψ∈ψ¯+L2(0,T;Y), nt,pt∈L2(0,T;Y*),n∣t=0=n0,p∣t=0=p0, and there hold
(1.8)∫Ω∇ψ·∇ζdx=∫Ω(p-n+C(x))ζdx,∀t∈(0,T),∀ζ∈Y∫0T〈nt,ζ〉Y*,Y+∫0T∫Ω(∇(nγ)-μ1n∇ψ)·∇ζdxdt=∫0T∫Ω[r(n,p)(1-np)+g]ζdxdt,∀ζ∈L2(0,T;Y),∫0T〈pt,ζ〉Y*,Y+∫0T∫Ω(∇(pγ)+μ2p∇ψ)·∇ζdxdt=∫0T∫Ω[r(n,p)(1-np)+g]ζdxdt,∀ζ∈L2(0,T;Y).
Our main result in this paper is as follows.Theorem 1.2.
Under hypotheses (H1)–(H6), there exists at least one local weak solution to the problem (1.1)–(1.6).


## 2. Approximate Problem

For simplicity, we assume thatμi=1,i=1,2. As [14], we first construct the following bound approximate sequence gτ of avalanche generation term g:
(2.1)gτ(n,p,∇n,∇p,∇ψ)=(α1(∇ψ)|Jn|+α2(∇ψ)|Jp|)·[1+τ(α1(∇ψ)|Jn|+α2(∇ψ)|Jp|)]-1,
here 0<τ<1. Obviously, 0<gτ<1/τ .Now we introduce the following approximate problem with the initial and boundary conditions (1.4)–(1.6):(2.2)-∇·(∇ψ)=p-n+C(x),(2.3)nt-∇·(∇nγ-n∇ψ)=r(n,p)(1-np)+gτ(n,p,∇n,∇p,∇ψ),(2.4)pt-∇·(∇pγ+p∇ψ)=r(n,p)(1-np)+gτ(n,p,∇n,∇p,∇ψ).This section is devoted to the proof of global existence of weak solutions to the above approximate problem (2.2)–(2.4), (1.4)–(1.6). We will prove the following existence theorem.Theorem 2.1.
Under hypotheses (H1)–(H6), there exists at least one global weak solution to the problem (2.2)–(2.4), (1.4)–(1.6).The proof is based on Schauder's fixed pointed theorem. The main difficulty in the proof is that problem (2.2)–(2.4), (1.4)–(1.6) is degenerated at points where n,p=0. This difficulty leads us to consider the following auxiliary regularized problem with the initial and boundary conditions (1.4)–(1.6):
(2.5)-∇·(∇ψ)=pk-nk+C(x),(2.6)nt-∇·((γnkγ-1+ε)∇n-nk∇ψ)=r(n,p)(1-npk)+h(n,p,∇n,∇p,∇ψ),(2.7)pt-∇·((γpkγ-1+ε)∇p+pk∇ψ)=r(n,p)(1-nkp)+h(n,p,∇n,∇p,∇ψ),
where sk=min{max{0,s},k} and
(2.8)h(ξ,η,ξ′,η′,ζ)=(α1(ζ)|γξkγ-1ξ′-ξkζ|+α2(ζ)|γηkγ-1η′+ηkζ|)×[1+τ(α1(ζ)|γξkγ-1ξ′-ξkζ|+α2(ζ)|γηkγ-1η′+ηkζ|)]-1
for any ξ,η∈R and ξ′,η′,ζ∈RN.Let(2.9)𝒦={x∈L2(QT)∣∥x∥L2(QT)≤R},
and ñ,p̃∈𝒦. It is obvious that 𝒦 is a closed convex set and weakly compact in L2(QT).The theory of linear elliptic boundary value problems [15] gives a unique ψ such that
(2.10)-∇·(∇ψ)=p̃k-ñk+C(x),(2.11)ψ∣ΣD=ψ̅,∂ψ∂η|ΣN=0,(2.12)∥∇ψ∥L2(QT)+∥ψ∥L∞(QT)≤C,
where C is dependent on k,QT and the L∞ norms for C(x) and ψ̅, but not on R.Next, for the unique weak solutionψ to problem (2.10)–(2.11), we consider the following problem:
(2.13)nt-∇·((γñkγ-1+ε)∇n-ñk∇ψ)=r(ñ,p̃)(1-np̃k)+h(ñ,p̃,∇n,∇p,∇ψ),(2.14)pt-∇·((γp̃kγ-1+ε)∇p+p̃k∇ψ)=r(ñ,p̃)(1-ñkp)+h(ñ,p̃,∇n,∇p,∇ψ),(2.15)(n,p)∣ΣD=(n̅,p̅),(∂n∂η,∂p∂η)|ΣN=(0,0),(2.16)(n,p)=(n0,p0),x∈Ω,t=0.Lemma 2.2.
Under hypotheses (H1)–(H6), there exists one global and unique weak solution to the problem (2.13)–(2.16).
Further there are bounds on∥n∥L2(0,T;H1(Ω)) and ∥nt∥L2(0,T;Y*) which depend on ε,k,τ,QT, and the known data, but not on R. Similar estimates also hold for p.Proof.
We begin by choosing a constantρ such that
(2.17)ρ≥max{α10,α20}2γ2k2γ-22min{α11,α21}
and observe that (n,p) satisfies (2.13)–(2.16) if and only if (U,V)=(e-ρtn,e-ρtp) satisfies
(2.18)Ut-∇·(α11∇U-e-ρtñk∇ψ)+α12U=e-ρtH(eρt∇U,eρt∇V)+e-ρtr(ñ,p̃),(2.19)Vt-∇·(α21∇V+e-ρtp̃k∇ψ)+α22V=e-ρtH(eρt∇U,eρt∇V)+e-ρtr(ñ,p̃),(2.20)(U,V)∣ΣD=(e-ρtn̅,e-ρtp̅),(∂U∂η,∂V∂η)|ΣN=(0,0),(2.21)(U,V)=(n0,p0),x∈Ω,t=0,
or if and only if (u,v)=(U-e-ρtn̅,V-e-ρtp̅) satisfies
(2.22)ut-∇·(α11∇u)+α12u=e-ρtH(eρt∇u+∇n̅,eρt∇v+∇p̅)+F1,(2.23)vt-∇·(α21∇v)+α22v=e-ρtH(eρt∇u+∇n̅,eρt∇v+∇p̅)+F2,(2.24)(u,v)∣ΣD=(0,0),(∂u∂η,∂v∂η)|ΣN=(-e-ρt∂n̅∂η,-e-ρt∂p̅∂η),(2.25)(u,v)=(n0-n̅,p0-p̅),x∈Ω,t=0.
where
(2.26)(αij)2×2=(γñkγ-1+ερ+r(ñ,p̃)p̃kγp̃kγ-1+ερ+r(ñ,p̃)ñk),H(∇u,∇v)=h(ñ,p̃,∇u,∇v,∇ψ),(F1F2)=(e-ρt(∇·(α11∇n̅-ñk∇ψ)+(ρ-α12)n̅+r(ñ,p̃))e-ρt(∇·(α21∇p̅+p̃k∇ψ)+(ρ-α22)p̅+r(ñ,p̃))).
Clearly, (F1,F2)∈(L2(0,T;Y*))2.
First, we prove the uniqueness of weak solution to the problem (2.13)–(2.16) which is equivalent to (2.18)–(2.21). Let (Ui,Vi),i=1,2 be two weak solutions to the problem (2.18)–(2.21), then (N,P)=(U1-U2,V1-V2) satisfies
(2.27)Nt-∇·(α11∇N)+α12N=e-ρtG(∇U1,∇U2,∇V1,∇V2),(2.28)Pt-∇·(α21∇P)+α22P=e-ρtG(∇U1,∇U2,∇V1,∇V2),(2.29)(N,P)∣ΣD=(0,0),(∂N∂η,∂P∂η)|ΣN=(0,0),(2.30)(U,V)=(0,0),x∈Ω,t=0,
here
(2.31)G(∇U1,∇U2,∇V1,∇V2)=H(eρt∇U1,eρt∇V1)-H(eρt∇U2,eρt∇V2)
such that
(2.32)G(∇U1,∇U2,∇V1,∇V2)≤α1(∇ψ)||γñkγ-1eρt∇U1-ñk∇ψ|-|γñkγ-1eρt∇U2-ñk∇ψ||+α2(∇ψ)||γp̃kγ-1eρt∇V1+p̃k∇ψ|-|γp̃kγ-1eρt∇V2+p̃k∇ψ||≤max{α10,α20}γkγ-1eρt(|∇N|+|∇P|).
Take N,P as test functions in (2.27), (2.28), respectively. By (2.32) and Hölder inequality,
(2.33)12ddt∫Ω(|N(t)|2+|P(t)|2)+min{α11,α21}∫Ω(|∇N|2+|∇P|2)+ρ∫Ω(N2+P2)≤max{α10,α20}γkγ-1∫Ω(|∇N|+|∇P|)(|N|+|P|)≤12min{α11,α21}∫Ω(|∇N|2+|∇P|2)+max{α10,α20}2γ2k2γ-22min{α11,α21}∫Ω(N2+P2).
Thus the uniqueness is established by Gronwall's inequality.
We are now in a position to prove the existence result. DefineW=L2(0,T;Y), 𝒱=W×W. Clearly, 𝒱 is a Hilbert space with respect to the scalar product
(2.34)((u,v),(ξ,η))=∫0T∫Ω(∇u·∇ξ+∇v·∇η).
Set, for (u,v),(ξ,η)∈𝒱,
(2.35)〈𝒜(u,v),(ξ,η)〉=∫0T∫Ω(α11∇u·∇ξ+α21∇v·∇η)+∫0T∫Ω(α12uξ+α22vη)+∫0T∫Ωe-ρtH(eρt∇u+∇n̅,eρt∇v+∇p̅)(ξ+η).
The operator 𝒜:𝒱→𝒱* is well defined and bounded (because 0≤H≤1/τ). To prove the existence result by using [16, Theorem  30.A], it suffices to verify that the operator 𝒜 is hemicontinuous, monotone, and coercive.
Note that
(2.36)e-ρt|H(eρt∇(u+λφ)+∇n̅,eρt∇(v+λϕ)+∇p̅)-H(eρt∇u+∇n̅,eρt∇v+∇p̅)|≤max{α10,α20}λγkγ-1(|∇φ|+|∇ϕ|)
for any (u,v),(φ,ϕ)∈𝒱. The hemicontinuity of 𝒜 is easily obtained by the standard method.
For the monotone, we first notice that
(2.37)∫0T‍∫Ω[H(eρt∇u1+∇n̅,eρt∇v1+∇p̅)-H(eρt∇u2+∇n̅,eρt∇v2+∇p̅)][(u1-u2)+(v1-v2)]≥-max{α10,α20}γkγ-1∫0T∫Ω(|∇(u1-u2)|+|∇(v1-v2)|)(|u1-u2|+|v1-v2|)≥-min{α11,α21}2∫0T∫Ω(|∇(u1-u2)|2+|∇(v1-v2)|2)-max{α10,α20}2γ2k2γ-22min{α11,α21}∫0T∫Ω[(u1-u2)2+(v1-v2)2].
Hence
(2.38)〈𝒜(u1,v1)-𝒜(u2,v2),(u1-u2,v1-v2)〉≥min{α11,α21}2∫0T‍∫Ω(|∇(u1-u2)|2+|∇(v1-v2)|2)+(ρ-max{α10,α20}2γ2k2γ-22min{α11,α21})∫0T∫Ω[(u1-u2)2+(v1-v2)2].
By the choice of ρ, we can easily obtain the monotone of 𝒜. Moreover, from (2.38) we also know that the operator 𝒜 is coercive.
Therefore, there exists a unique(u,v)∈𝒱 with (ut,vt)∈𝒱* such that (u0,v0)=(n0-n̅,p0-p̅) in Ω and
(2.39)〈(ut,vt)+𝒜(u,v),(ξ,η)〉=〈(F1,F2),(ξ,η)〉,∀(ξ,η)∈𝒱.
Especially,
(2.40)〈(ut,vt)+𝒜(u,v),(ξ,0)〉=〈(F1,F2),(ξ,0)〉,〈(ut,vt)+𝒜(u,v),(0,η)〉=〈(F1,F2),(0,η)〉.
That is, (u,v) is a weak solution to the problem (2.22)–(2.25).
Finally, noting that0≤h≤1/τ we can easily establish the bounds on ∥∇u∥L2(0,T;H1(Ω)) and ∥ut∥L2(0,T;Y*) by the standard energy estimate.Lemma 2.3.
Under hypotheses (H1)–(H6), there exists at least one global weak solution to the problem (2.5)–(2.7), (1.4)–(1.6).Proof.
We define the mappingS as
(2.41)S:𝒦2→(L2(QT))2,(ñ,p̃)↦(n,p)
with (n,p) solution of (2.13)–(2.16). From Lemma 2.2, we know that S is well defined and compact. Indeed, (n,p) lies in a bounded sunsets of (L2(0,T;H1(Ω)))2, and (nt,pt) lies in a bounded subset of (L2(0,T;Y*))2. Since the injection H1(Ω)↪L2(Ω) is compact, we conclude from Aubin's lemma that S is relatively compact in (L2(QT))2. And for given τ and k, S(𝒦2)↪𝒦2 holds if we choose R large enough.
To apply Schauder's fixed point theorem, we still need to prove that the mappingS is continuous. Consider any sequence (ñj,p̃j)⊂𝒦2→(ñ,p̃) strongly in (L2(QT))2 and let S(ñj,p̃j)=(nj,pj). Since S(𝒦2) is relatively compact in (L2(QT))2 and bounded in (L2(0,T;H1(Ω)))2, we can extract subsequences such that
(2.42)((nj)t,(pj)t)→(nt,pt)weaklyinL2(0,T;Y*),nj→n,pj→pstronglyinL2(QT),nj→n,pj→pweaklyinL2(0,T;H1(Ω)),nj→n,pj→pa.e.inQT.
We only have to show S(ñ,p̃)=(n,p). To do this, we only need to prove
(2.43)∫0T‍∫Ωh(ñj,p̃j,∇nj,∇pj,∇ψj)·ζ→∫0T∫Ωh(ñ,p̃,∇n,∇p,∇ψ)·ζ,
where ζ∈L2(0,T;Y) is a test function, and ψj,ψ are solutions of (2.10)–(2.11) corresponding to (ñj,p̃j),(ñ,p̃), respectively. The reminder of convergence proof is standard (details see [8] or [9]). Use nj-n∈L2(0,T;Y) as test function in a modification of (2.13) in which the functions ñ,p̃ have been replaced by ñj,p̃j, respectively. Then we have
(2.44)ε∫0T∫Ω|∇(nj-n)|2≤-∫0T〈nt,nj-n〉Y*,Y-ε∫0T∫Ω∇n·∇(nj-n)+∫0T∫Ω(ñj)k∇ψj·∇(nj-n)+∫0T∫Ωr(ñj,p̃j)(1-nj(p̃j)k)(nj-n)+max{α01,α02}max{γkγ-1,2k}∫0T∫Ω(|∇nj|+|∇pj|+|∇ψj|)(nj-n)≤-∫0T〈nt,nj-n〉Y*,Y-ε∫0T‍∫Ω∇n·∇(nj-n)+ε4(∥∇(nj-n)∥L2(QT)2+∥∇(pj-p)∥L2(QT)2)+C∫0T∫Ω(1+|nj|)|nj-n|+C(ε)(∥∇(ψj-ψ)∥L2(QT)2+|∫0T∫Ω∇ψ·∇(nj-n)|+∥nj-n∥L2(QT)2)+C(ε)∫0T∫Ω(|∇n|+|∇p|+|∇ψ|)|nj-n|.
A similar estimate holds for ε∫0T∫Ω|∇(pj-p)|2. Then adding the two inequalities and using ∇ψj→∇ψ strongly in L2(QT) (details see [9]) and (2.42), we conclude that ∥∇(nj-n)∥L2(QT)+∥∇(pj-p)∥L2(QT)→0. This implies that (∇nj,∇pj)→(∇n,∇p) a.e. in QT. Then we can easily prove (2.43) by using Vitali's theorem.
Now existence of a fixed point ofS follows which is a solution of (2.5)–(2.7), (1.4)–(1.6).To obtain the existence result of problem (2.2)–(2.4), (1.4)–(1.6), the following L∞ estimates on n,p,ψ uniformly in ε are necessary.Lemma 2.4.
The solutions of problem (2.5)–(2.7), (1.4)–(1.6) satisfy the estimates
(2.45)0≤n(x,t),p(x,t)≤C,a.e.(x,t)∈QT,
where C is dependent on τ,QT and the known data, but not on ε.Proof.
By takingn-=min{n,0}∈L2(0,T;Y) as test function in (2.6), we have
(2.46)12∫Ωn-(t)2+ε∫0T‍∫Ω|∇n-|2≤∫0T∫Ωnk∇ψ·∇n--∫0T∫Ωr(n,p)pkn-2+∫0T∫Ω(r(n,p)+h)n-.
By taking into account nk=0 in {n≤0} and the nonnegativity of r and h we obtain
(2.47)12∫Ωn-(t)2≤0,
and thus n(x,t)≥0 a.e. in QT. Similarly, we have p(x,t)≥0 a.e. in QT.
To obtain the upper bound set
(2.48)M=max{∥n̅∥L∞(Ω),∥n0∥L∞(Ω),∥p̅∥L∞(Ω),∥p0∥L∞(Ω)},k≥M
and use (nk-M)+q as test function in (2.6), then
(2.49)1q+1∫Ω(nk-M)+(q+1)+ε∫0T∫Ωq(nk-M)+(q-1)|∇(nk-M)+|2≤∫0T∫Ω∇ψ·∇(qq+1(nk-M)+(q+1)+M(nk-M)+q)+∫0T∫Ω(r(n,p)(1-npk)+h)(nk-M)+q≤∫0T∫Ω(pk-nk+C(x))(qq+1(nk-M)+(q+1)+M(nk-M)+q)+∫0T∫Ω(r̅+1τ)(nk-M)+q.
Adding the equality (2.49) for n and a similar inequality for p, we get
(2.50)1q+1∫Ω((nk-M)+(q+1)+(pk-M)+(q+1))≤qq+1∫0T∫Ω(pk-nk)((nk-M)+(q+1)-(pk-M)+(q+1))+M∫0T∫Ω(pk-nk)((nk-M)+q-(pk-M)+q)+qq+1∫0T∫ΩC(x)((nk-M)+(q+1)-(pk-M)+(q+1))+∫0T∫Ω(r̅+1τ+MC(x))((nk-M)+q-(pk-M)+q).
Noticing that
(2.51)(pk-nk)((nk-M)+θ-(pk-M)+θ)≤0,∀θ>1
and applying Hölder inequality, we further have
(2.52)∥(nk-M)+∥Lq+1(Ω)q+1+∥(pk-M)+∥Lq+1(Ω)q+1≤C(q+1)∫0T(∥(nk-M)+∥Lq+1(Ω)q+1+∥(pk-M)+∥Lq+1(Ω)q+1)+C,
where C is independent of ε and q. Gronwall's inequality then implies that
(2.53)∥(nk-M)+∥Lq+1(Ω)+∥(pk-M)+∥Lq+1(Ω)≤CeCt
for all q≥1 and k≥M. Since the right-hand side of this equality does not depend on k, we can let k→∞ and then q→∞ to obtain the desired upper bound.Thus, takingk large enough, we see that (ψε,nε,pε) solves
(2.54)-∇·(∇ψε)=pε-nε+C(x),(2.55)(nε)t-∇·((γnεγ-1+ε)∇nε-nε∇ψε)=r(nε,pε)(1-nεpε)+gτ(nε,pε,∇nε,∇pε,∇ψε),(2.56)(pε)t-∇·((γpεγ-1+ε)∇pε+pε∇ψ)=r(nε,pε)(1-nεpε)+gτ(nε,pε,∇nε,∇pε,∇ψε)
subject to the initial and boundary conditions (1.4)–(1.6).Proof of Theorem2.1.
Noticing that the functiongτ is bound, we can obtain the following convergence properties by using the same method as the proof in [8, Theorem 1.1] and Lemma 3.2:
(2.57)(nε(γ-1)/2,pε(γ-1)/2)→(n(γ-1)/2,p(γ-1)/2)stronglyinL2(QT),(nε,pε)→(n,p),a.e.inQT,(∇nεγ,∇pεγ)→(∇nγ,∇pγ)weaklyinL2(0,T;H1(Ω)),((nε)t,(pε)t,)→(nt,pt)weaklyinL2(0,T;Y0*),ψε→ψweaklyinL∞(0,T;H1(Ω)),(nε∇ψε,pε∇ψε)→(n∇ψ,p∇ψ)weaklyinL2(QT).
In addition, a standard elliptic estimate gives
(2.58)∥∇(ψε-ψ)∥L2(QT)≤∥(pε-p)-(nε-n)∥L2(QT),
from which we conclude
(2.59)∇ψε→∇ψstronglyinL2(QT),
and furthermore
(2.60)∇ψε→∇ψa.e.inQT.
Next, using (nεγ-nγ) as test function in (2.55), we get
(2.61)∫0T∫Ω|∇(nεγ-nγ)|2≤-∫0T〈nt,nεγ-nγ〉Y*,Y+ε∫0T∫Ω∇nε·∇(nεγ-nγ)+∫0T∫Ω∇nγ·∇(nεγ-nγ)+∫0T∫Ωnε∇ψε·∇(nεγ-nγ)+max{α10,α20}∫0T∫Ω(|∇nεγ|+|∇pεγ|+(nε+pε)|∇ψε|)|nεγ-nγ|→0
as ε→0, where we have used
(2.62)ε∫0T∫Ω∇nε·∇(nεγ-nγ)≤ε1/2(∫0T∫Ωε|∇nε|2)1/2(∫0T∫Ω|∇(nεγ-nγ)|2)1/2≤ε1/2C→0,∫0T∫Ωnε∇ψε·∇(nεγ-nγ)=1γ+1∫0T∫Ω(pε-nε+C(x))(nεγ+1-nγ+1)-∫0T∫Ωnε∇ψε·∇nγ+1γ+1∫0T∫Ω∇ψε·∇nγ+1→0,
and (2.57)–(2.60). The same argument shows that ∥∇pεγ-∇pγ∥L2(QT)→0 as ε→0.
Thus, there exists a subsequence (not relabeled) such that∇nεγ→∇nγ,∇pεγ→∇pγ almost everywhere in QT as ε→0. Then it follows from Vitali's theorem that
(2.63)gτ(nε,pε,∇nε,∇pε,∇ψε)→gτ(n,p,∇n,∇p,∇ψ)stronglyinL2(QT).
Now we can conclude that(ψ,n,p) is the solution of the problem (2.2)–(2.4), (1.4)–(1.6) from the above convergence by standard method and then complete the proof of Theorem 2.1.


## 3. Proof of the Main Result

In the last section, we prove that there is at least one global weak solution(ψτ,nτ,pτ) to the problem (2.2)–(2.4), (1.4)–(1.6) for every given τ. In the following what we need to do is to prove that the limit of (nτ,pτ,ψτ) is a solution of (1.1)–(1.6). To this end, we first give some uniform estimates for the problem (2.2)–(2.4), (1.4)–(1.6). For simplicity, we drop the subscript τ of (nτ,pτ,ψτ) and set αi=1,i=1,2.Lemma 3.1.
Forallu∈H1(Ω), there holds
(3.1)∥u∥Ls(Ω)≤C∥u∥L2(Ω)α∥u∥H1(Ω)1-α,
where α,s satisfy
(3.2)0<1-α=N(12-1s)<1,1<s<2NN-2.This is the well-known Gagliardo-Nirenberg Inequality [15].Lemma 3.2.
If(ψ,n,p) is the solution of the problem (2.2)–(2.4), (1.4)–(1.6), the following estimate holds:
(3.3)∫0T‍∫Ω(nγ+pγ)|∇ψ|2≤δ∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+C(δ)[∫0T(∫Ω(nγ+1+pγ+1)dx)2+1],∀γ≥1,
where δ is a sufficiently small constant.Proof.
First of all, the followingL∞ estimate of ψ(3.4)∥ψ(t)∥L∞(Ω)≤C{1+(∫Ω(n2(t)+p2(t))dx)1/2},∀t≥0
can follow from the standard techniques in [17]. Then by taking ψ-ψ¯ as test function in (2.2), we have
(3.5)∫Ω|∇(ψ-ψ¯)|2+∫Ω∇ψ¯·∇(ψ-ψ¯)=∫Ω(p-n+C(x))(ψ-ψ¯).
The assumptions we have made and Poincaré inequality yield
(3.6)∥∇(ψ-ψ¯)∥L2(Ω)≤C(1+∥p-n∥L2(Ω))≤C(1+∥n∥L2(Ω)+∥p∥L2(Ω)).
Using (nγ-n¯γ)ψ as test function for (2.2), and noting that 2<γ+1 we get
(3.7)∫0T∫Ωnγ|∇ψ|2=∫0T∫Ωn¯γ|∇ψ|2+∫0T∫Ω(p-n+C(x))[(nγ-n¯γ)ψ]-∫0T∫Ωψ∇(nγ-n¯γ)·∇ψ≤C∥∇ψ∥L2(QT)2+∫0T∥ψ∥L∞(Ω)∫Ω(nγ+1+pγ+1)+∫0T∥ψ∥L∞(Ω)∥∇ψ∥L2(Ω)∥∇(nγ-n¯γ)∥L2(Ω)≤C∫0T∫Ω(n2+p2)+∫0T[(∫Ω(n2+p2))1/2∫Ω(nγ+1+nγ+1)]+δ∫0T∫Ω|∇(nγ)|2+C(δ)[∫0T(∫Ω(n2+p2))2+1]≤δ∫0T∫Ω|∇(nγ)|2+C(δ)[∫0T(∫Ω(nγ+1+pγ+1))2+1].
A similar estimate for ∫0T∫Ωpγ|∇ψ|2 follows from a same procedure. Then the proof is completed.Lemma 3.3.
If(ψ,n,p) is the solution of the problem (2.2)–(2.4), (1.4)–(1.6), there holds that
(3.8)∥n∥L∞(0,T0;Lγ+1(Ω))+∥∇(nγ)∥L2(QT0)≤C,(3.9)∥p∥L∞(0,T0;Lγ+1(Ω))+∥∇(pγ)∥L2(QT0)≤C,(3.10)∥ψ∥L2(0,T0;H1(Ω))+∥ψ∥L∞(QT0)≤C,
for some sufficiently small T0; here positive constant C is independent of τ.Proof.
Without loss of generality, we assumeN=3 and the N=1 or 2 case is easier.Case 1.
2≤γ≤5. In this case, from Hölder inequality and (3.3), we have
(3.11)∫0T∫Ω(n2+p2)|∇ψ|2≤C(∫0T∫Ω(nγ+pγ)|∇ψ|2+∫0T∫Ω|∇ψ|2)≤δ∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+C(δ)[∫0T(∫Ω(nγ+1+pγ+1)dx)2+1].
Noting that 2≤γ≤5, we can choose a constant q such that 6γ/5≤2q≤γ+1. Then by taking n2q-γ-n¯2q-γ and p2q-γ-p¯2q-γ as test function in (2.3) and (2.4), respectively, and adding them together, we get
(3.12)12q-γ+1∫Ω(n2q-γ+1+p2q-γ+1)+γ(2q-γ)q2∫0T∫Ω[|∇(nq)|2+|∇(pq)|2]=12q-γ+1∫Ω(n02q-γ+1+p02q-γ+1)+∫Ω[n¯2q-γ(n-n0)+p¯2q-γ(p-p0)]+∫0T∫Ω[∇ψ·(n∇(n2q-γ-n¯2q-γ)-p∇(p2q-γ-p¯2q-γ))]+∫0T∫Ω[∇(nγ)·∇(n¯2q-γ)+∇(pγ)·∇(p¯2q-γ)]+∫0T∫Ω(R(n,p)+gε)[(n2q-γ-n¯2q-γ)+(p2q-γ-p¯2q-γ)]=I1+⋯+I4.
We estimate the right-hand side term by term. Due to the equation of ψ and (3.6), we obtain
(3.13)I2=2q-γ2q-γ+1∫0T∫Ω∇ψ·[∇(n2q-γ+1-n¯2q-γ+1)-∇(p2q-γ+1-p¯2q-γ+1)]+∫0T∫Ω∇ψ·[(n¯-n)∇(n¯2q-γ)-(p¯-p)∇(p¯2q-γ)]≤2q-γ2q-γ+1∫0T∫Ω(p-n+C(x))[(n2q-γ+1-p2q-γ+1)-(n¯2q-γ+1-p¯2q-γ+1)]+C∫0T∥∇ψ∥L2(Ω)(∥n¯-n∥L2(Ω)+∥p¯-p∥L2(Ω))≤C[∫0T∫Ω(nγ+1+pγ+1)+1],
where we use that (p-n)(n2q-γ+1-p2q-γ+1)≤0 and 2q-γ+1≤γ+1 when γ≥2. I3 and I4 can be bounded as
(3.14)I3≤λ14∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+C(λ1),(3.15)I4≤λ14∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+C∫0T∫Ω(n2+p2)+C(λ1)[∫0T∫Ω(n4q-2γ+p4q-2γ)+∫0T∫Ω(n2+p2)|∇ψ|2],
where we have used the nonnegativity of n and p. Inserting (3.13)–(3.15) into (3.12) we conclude that
(3.16)∫Ω(n2q-γ+1+p2q-γ+1)+∫0T∫Ω[|∇(nq)|2+|∇(pq)|2]≤λ1[∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+∫Ω(nγ+1+pγ+1)]+C(λ1)[∫0T∫Ω(nγ+1+pγ+1)+∫0T∫Ω(n2+p2)|∇ψ|2+1].
Similarly, taking nγ-n¯γ and pγ-p¯γ as test function in (2.3) and (2.4), we conclude that
(3.17)1γ+1∫Ω(nγ+1+pγ+1)+∫0T∫Ω[|∇(nγ)|2+|∇(pγ)|2]=1γ+1∫Ω(n0γ+1+p0γ+1)+∫Ω[n¯γ(n-n0)+p¯γ(p-p0)]+∫0T∫Ω[∇(nγ)·∇(n¯γ)+∇(pγ)·∇(p¯γ)]+∫0T∫Ω∇ψ·[(n¯-n)∇(n¯γ)-(p¯-p)∇(p¯γ)]+γγ+1∫0T∫Ω∇ψ·[∇(nγ+1-n¯γ+1)-∇(pγ+1-p¯γ+1)]+∫0T∫Ω(R(n,p)+gε)[(nγ-n¯γ)+(pγ-p¯γ)]≤λ1[∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+∫Ω(nγ+1+pγ+1)]+C(λ1)[∫0T∫Ω(nγ+1+pγ+1)+∫0T∫Ω(n2γ+p2γ)+∫0T∫Ω(n2+p2)|∇ψ|2+1].
Choosing λ1 sufficiently small, and then summing (3.16) and (3.17), we have
(3.18)∫Ω(nγ+1+pγ+1)+∫0T∫Ω[|∇(nγ)|2+|∇(pγ)|2]+∫Ω(n2q-γ+1+p2q-γ+1)+∫0T∫Ω[|∇(nq)|2+|∇(pq)|2]≤C(λ1)[∫0T∫Ω(nγ+1+pγ+1)+∫0T∫Ω(n2γ+p2γ)+∫0T∫Ω(n2+p2)|∇ψ|2+1].
Now we estimate the term ∫0T∫Ωn2γ (or ∫0T∫Ωp2γ ). Due to Lemma 2.2 we get
(3.19)∫0T‍∫Ωn2γ=∫0T∫Ω(nq)s≤∫0T∫Ω∥nq∥H1(Ω)(1-α)s∥nq∥L2(Ω)αs≤λ2∫0T∥∇(nq)∥L2(Ω)2+C(λ2)[∫0T(∫Ωn2qdx)β+1],
where s,α,β satisfy
(3.20)s=2γq,0<1-α=3(12-q2γ)<1,0<(1-α)s<2,β=αs2-(1-α)s
Consequently we obtain, taking into account (3.11) and the choice of q,
(3.21)∫Ω(nγ+1+pγ+1)+∫0T∫Ω[|∇(nγ)|2+|∇(pγ)|2]≤C(λ1,λ2,δ)[1+∫0T(∫Ω(nγ+1+pγ+1)dx)max{2,β}]
for sufficiently small λ2 and δ such that (λ2+δ)C(λ1)<1, where β only depends on γ.
This proves that
(3.22)maxt∈[0,T0]∫Ω[nγ+1(t)+pγ+1(t)]≤C
for some sufficiently small T0 by Lemma 2.2 of the appendix in [14] , and thus
(3.23)∫0T0∫Ω[|∇(nγ)|2+|∇(pγ)|2]≤C.
Using the above estimates and (3.4)–(3.6), we can obtain (3.10).Case 2.
1<γ<2. In this case, the estimates on I1,I2,I3 are the same as Case 1, and
(3.24)I4≤λ14∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+C∫0T∫Ω(n2+p2)+C(λ1)[∫0T∫Ω(n4q-3γ+2+p4q-3γ+2)+∫0T∫Ω(nγ+pγ)|∇ψ|2].
Then we have
(3.25)∫Ω(n2q-γ+1+p2q-γ+1)+∫0T∫Ω[|∇(nq)|2+|∇(pq)|2]≤λ1[∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+∫Ω(nγ+1+pγ+1)]+C(λ1)[∫0T∫Ω(nγ+1+pγ+1)+∫0T∫Ω(n4q-3γ+2+p4q-3γ+2)+∫0T∫Ω(nγ+pγ)|∇ψ|2+1].
Next, by a small change in (3.17), we obtain
(3.26)1γ+1∫Ω(nγ+1+pγ+1)+∫0T∫Ω[|∇(nγ)|2+|∇(pγ)|2]≤λ1[∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+∫Ω(nγ+1+pγ+1)]+C(λ1)[∫0T∫Ω(nγ+1+pγ+1)+∫0T∫Ω(nγ+2+pγ+2)+∫0T∫Ω(n2γ+p2γ)+∫0T∫Ω(nγ+pγ)|∇ψ|2+1]≤λ1[∫0T∫Ω(|∇(nγ)|2+|∇(pγ)|2)+∫Ω(nγ+1+pγ+1)]+C(λ1)[∫0T∫Ω(nγ+2+pγ+2)+∫0T∫Ω(nγ+pγ)|∇ψ|2+1].
Choose q such that
(3.27)310(γ+2)<q<γ+12,
and reset
(3.28)s=γ+2q,0<1-α=3(12-qγ+2)<1,0<(1-α)s<2,β=αs2-(1-α)s.
By a discuss similar to (3.19), we obtain
(3.29)∫0T∫Ωnγ+2≤λ2∫0T∥∇(nq)∥L2(Ω)2+C(λ2)[∫0T(∫Ωn2qdx)β+1].
Finally, noting 4q-3γ+2<γ+2, from (3.25)–(3.29) and estimate (3.3) we deduce that (3.21) also holds for 1<γ<2. Therefore, we can also obtain estimates (3.8)–(3.10) when 1<γ<2.The following lemma is indispensable to prove theL2(0,T0;Y*) estimates of nt and pt uniformly in τ.Lemma 3.4.
If(ψ,n,p) is the solution of the problem (2.2)–(2.4), (1.4)–(1.6), there holds that
(3.30)∥n∥Lm(QT0)+∥p∥Lm(QT0)≤C,103<m<10.Proof.
Denote byV2(QT) a Banach space in which function v satisfies
(3.31)∥v∥V2(QT)=sup0≤t≤T∥v(t)∥L2(Ω)+(∫0T∫Ω|∇v|2)1/2<∞.
The following proof is base on Moser's iteration technique [15]. We insert (n-s)+ into (2.3), (p-s)+ into (2.4), integrate over the interval (0,T0), and add the equations to obtain
(3.32)12∫Ω[(n(t)-s)+2+(p(t)-s)+2]+γ∫0T0∫Ωnγ-1|∇(n-s)+|2+γ∫0T0∫Ωpγ-1|∇(p-s)+|2=∫0T0∫Ωn∇ψ·∇(n-s)+-∫0T0∫Ωp∇ψ·∇(p-s)++∫0T0∫Ω(R(n,p)+gε)[(n-s)++(p-s)+]≤12∫0T0∫Ω[p-n+C(x)][(n2-s2)+-(p2-s2)+]+∫0T0∫Ω(gε+r¯)[(n-s)++(p-s)+].
Since (p-n)[(n2-s2)+-(p2-s2)+] is negative, and choosing s≥c0=max{supΩ{n0,p0},supΣD{n¯,p¯}}+1, we have
(3.33)12∫Ω[(n(t)-s)+2+(p(t)-s)+2]+∫0T0∫Ω[|∇(n-s)+|2+|∇(p-s)+|2]≤∫0T0∫ΩC(x)[n(n-s)++p(p-s)+]+∫0T0∫Ω(gε+r¯)[(n-s)++(p-s)+]=I1+I2
for all s≥c0.
Applying Hölder inequality, we obtain
(3.34)I1≤∥C(x)∥L∞(Ω)∥n∥L2(QT0)∥(n-s)++(p-s)+∥L2(N+2)/N(QT0)|QT0∩[n>s,p>s]|1/(N+2)≤λ3∥(n-s)++(p-s)+∥L2(N+2)/N(QT0)2+C(λ3)|QT0∩[n>s,p>s]|2/(N+2),(3.35)I2≤(∥∇(nγ+pγ)∥L2(QT0)+∥(n+p)∇ψ∥L2(QT0))×∥(n-s)++(p-s)+∥L2(N+2)/N(QT0)|QT0∩[n>s,p>s]|1/(N+2)+r¯∥(n-s)++(p-s)+∥L2(N+2)/N(QT0)|QT0∩[n>s,p>s]|(N+4)/2(N+2)≤λ3∥(n-s)++(p-s)+∥L2(N+2)/N(QT0)2+C(λ3)|QT0∩[n>s,p>s]|2/(N+2),
where |·| denotes measure, and [n>s,p>s]={(x,t)∣n(x,t)>s,p(x,t)>s}.
Chooseλ3 sufficiently small, together with (3.33)–(3.35) and V2(QT0)↪L2(N+2)/N(QT0), then
(3.36)∥(nε-s)++(pε-s)+∥L2(N+2)/N(QT0)2≤C(λ3)|QT0∩[nε>s,pε>s]|2/(N+2).
On the other hand, we have
(3.37)∫0T0∫Ω[(n-s)++(p-s)+]≤∥(nε-s)++(pε-s)+∥L2(N+2)/N(QT0)|QT0∩[nε>s,pε>s]|(N+4)/2(N+2).
Let χ(s)=|QT0∩[nε>s,pε>s]|. From (3.36) and (3.37), we obtain
(3.38)∫0T0∫Ω[(n-s)++(p-s)+]≤C(λ3)χ(s)(N+6)/2(N+2),∀s≥c0,
which proves (3.30) by [15, Lemma 5.2 in Chapter 2].Lemma 3.5.
If(ψ,n,p) is the solution of the problem (2.2)–(2.4), (1.4)–(1.6), there holds that
(3.39)∥nt∥L2(0,T0;Y*)≤C,(3.40)∥pt∥L2(0,T0;Y*)≤C.Proof.
We only prove (3.39). The proof of (3.40) is completely the same as that for (3.39). Take test function ϕ∈L2(0,T0;Y) for (2.3). Then by Lemma 3.2–3.4 we conclude from
(3.41)|∫0T0〈nt,ϕ〉Y*,Y|≤(∥∇(nγ)∥L2(QT0)+∥n∇ψ∥L2(QT0))∥∇ϕ∥L2(QT0)+[∥∇(nγ+pγ)∥L2(QT0)+∥(n+p)|∇ψ|∥L2(QT0)+∥n∥L4(QT0)∥p∥L4(QT0)]∥ϕ∥L2(QT0)
that ∥nt∥L2(0,T1;Y0*)≤c. The proof is complete.Proof of Theorem1.2.
By passing to a subsequence if necessary, from Lemma3.2–3.5, together with compact lemma of Lions [18], we infer that
(3.42)((nτ)γ,(pτ)γ)→(nγ,pγ)stronglyinL2(QT0),(nτ,pτ)→(n,p),a.e.inQT0,((nτ)γ,(pτ)γ)→(nγ,pγ)weaklyinL2(0,T0;H1(Ω)),((nτ)t,(pτ)t)→(nt,pt)weaklyinL2(0,T0;Y*),(nτ∇ψτ,pτ∇ψτ)→(n∇ψ,p∇ψ)weaklyinL2(QT0),ψτ→ψweakly*inL∞(0,T0;H1(Ω)).
Then by the same argument as the proof of Theorem2.1, we can conclude that (ψ,n,p) is the solution of the problem (1.1)–(1.6) from the convergence of (3.42) and then complete the proof of Theorem 1.2.



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*Source: 873683-2009-07-30.xml*