# Existence of Weak Solution for a Free Boundary Lubrication Problem

Authors: Abdelhamid Laouar
Journal: ISRN Mathematical Analysis
 (2011)

---

## Abstract

This paper is interested in a free boundary problem modelling a phenomenon of cavitation in hydrodynamic lubrication. We reformulate the problem (see Boukrouche, (1993)) in a large context by introducing two positive parameters, namely,N0 and a. We build a weak formulation and establish the existence of the solution to the problem.


---

## Body

## 1. Introduction

The lubrication fields have many applications; one example is the study of the rotary mechanisms such as the bearing, joints. The study is concerned in looking for a moving free boundary problem related to the cavitation modelling in lubrication (see [1–4]). The experimental results make evidence of the occurrence of two distinct zones, one full of the fluid, namely, the saturated zone Ω(t); the other Ωc (Ωc=Ω∖Ω(t), where Ω is the global domain), is the cavitated zone (e.g., the mixture of fluid and air). Two approaches have been used to cope with phenomena. One of them [5] homogenizes the phenomena and considers it as a 2D phenomena, so introducing θ the saturation variable (lubricant concentration); the other one [6] takes full account of the three-dimensional character of this phenomena, with appearance of air bubbles and introduces in Ωc the relative height as supplementary unknown (for more details, see [1, 3]). We use here the first approach but both approaches lead to the same mathematical problem. In this paper, we take the problem studied in [1–3] and rewrite it, here, in a large context, by introducing two positive parameters, namely, N0 and a. This formulation of the problem gives as advantages a proportionality relation between N0 and the pressure p, and the parameter a which allows the control of a squeezing effects. The mathematical modelling is made according to the model of Jakobsson-Floberg (see [1, 3]), where the lubricant is not defined only by a pressure p but also by a saturation variable θ. This variable θ characterizes the cavitation phenomena, where θ≡1 in Ω(t) and 0≤θ<1 in Ωc. The interface between Ω(t) and Ωc constitutes the moving free boundary denoted by Γ(t). The problem is a convection-diffusion problem type, and the Reynolds equation is elliptic in the saturated zone and hyperbolic in the other one. We note that the study of existence and uniqueness point of views to the problem has been established in [3] in the particular case of the N0=0 and a=1. For this, the author proved the existence of solution for this kind of problem by way of an approximation by an elliptic problem. In our work, we followed, exactly the same way, and the aim of this study is to construct a weak formulation and establish the existence of solution to the problem.The plan of the paper is as follows. Section2 proposes a state of the problem and a weak formulation. Section 3 introduces an elliptic nonlinear problem and gives the existence and uniqueness of the solution to this problem. Section 4 proposes an approximation of the elliptic nonlinear problem by a family of linear problems and proves a priori estimate. Last section gives a theorem of existence of the solution to the problem.


## 2. Notations and State of the Problem

### 2.1. Description of the Phenomenon

We consider a global domainΩ with border ∂Ω. The fluid is injected at a given rate w over the fixed (internal) boundary ΓI (Γex=∂Ω∖ΓI). For each t∈]0;T[, the experimental results make evidence of the occurrence of two distinct zones: one full of the fluid is the saturated zone Ω(t), where the pressure p (p>0) and the saturation variable θ(θ≡1), and the other Ωc(Ωc=Ω∖Ω(t)) is the cavitated zone, where the pressure is constant (p=0) and 0≤θ<1 (e.g., the mixture of fluid and air). The free boundary of the region Ω(t) containing fluid is Γ(t) and the region Ω(0), with border ΓI, occupied by the fluid at t=0 being given (see Figure 1).Figure 1
A view of domainsΩ(t) and Ωc.


### 2.2. State of the Problem

The strong formulation of the problem described the phenomenon is written as follows.For eacht≥0, find a pair (p,θ)∈L2(0,T;H1(Ω))×L∞(Q)∩H1(0,T;H-1(Ω)) and Γ(t) such that(2.1)∇(K(h,t,N0)∇p)=∇(Vhθ)+a∂(hθ)∂t,(θ≡1)inΩ(t),(2.2)∇(Vhθ)+a∂(hθ)∂t=0inΩc,(2.3)K(h,t,N0)∂p∂n=h(1-θ)(V-aυ)nonΓ(t),(2.4)p=0onΓ(t),(2.5)K(h,t,N0)∂p∂n̲-hθVn̲=wonΓI,(2.6)p=0onΓex,(2.7)p(1-θ)=0inΩ,(2.8)θ|t=0=θ0,
where(2.9)K(h,t,N0)=h31-N02Φ(h,t,N0)with0≤N0<1,Φ(h,t,N0)=112+14h2(1-N02)-14hN021-N02coth⁡(N0h1-N02).
h(t,x) is the thickness of the thin film supposed a regular and given function of the problem. V is the speed of the axis supposed being given. υ is the moving free boundary (with υ=0 on ΓI). n (resp., n̲) is the normal vector along Γ(t) (resp., ΓI) exterior to Ω(t) (resp., Ω(0)). The saturation variable θ can be represented by a graph (see, Figure 2).Figure 2
Graph of saturation variableθ.In (2.1)–(2.3), there are the diffusion term ∇(K(h,t,N0)∇p), the shearing term ∇(hV), and the squeezing term ∂h/∂t.


### 2.3. Weak Formulation

Before starting the construction of a weak formulation of the problem (2.1)–(2.8), we denote by(2.10)Q=Ω×]0,T[,ΣI=ΓI×]0,T[,Σt=Γ(t)×]0,T[,Σex=Γex×]0,T[,E={φ∈H1(Q):φ=0onΣexandt=0}.
Indeed, multiplying (2.1) by φ∈E and integrating over [0,T]×Ω(t), we obtain(*)-∫0T∫Ω(t)K∇P∇φdxdt+∫0T∫Γ(t)K∇Pφn+∫0T∫ΓIK∇Pφn̲=-∫0T∫Ω(t)Vh∇φdxdt+∫0T∫Γ(t)Vhφn+∫0T∫ΓIVhφn̲+∫0T∫ΓexVhφ⋅nex+a{∫0Tddt∫Ω(t)hφdxdt-∫0T∫Γ(t)VhΦ⋅ndt-∫0T∫ΓIVhφn̲dt-∫0T∫Ω(t)hdφdtdxdt},
where nex is the normal vector along ∂Ω exterior to Ω.In the same way, we apply to (2.2)(**)-∫0T∫ΩcVh∇φdxdt+∫0T∫Γ(t)Vhφ⋅n+∫0T∫ΓIVhφ⋅n̲+∫0T∫ΓexVhφ⋅nex+a{∫0Tddt∫Ωchφdxdt-∫0T∫Γ(t)Vhφ⋅ndt-∫0T∫ΓIVhφn̲dt-∫0T∫Ωchdφdtdxdt}=0.
By adding (*) and (**) in all Ω=Ω(t)∪Ωc, and using (2.3)–(2.6) then the weak formulation can be written as follows.Find a pair(p,θ)∈L2(0,T;H1(Ω))xL∞(Q)∩H1(0,T;H-1(Ω))(2.11)0≤θ<1,p=0onΣex,p(1-θ)=0,a.e.inQ,(2.12)∫Q(K∇p-Vhθ)∇φ+a{∫Ω(T)hθφ-∫Ω(0)hθφ-∫Qhθdφdt}=∫ΣIwφ,∀φϵE,θ|t=0=θ0∈H-1(Ω).
As φ|t=0=0 implies that ∫Ω(0)hθφ=0.


## 2.1. Description of the Phenomenon

We consider a global domainΩ with border ∂Ω. The fluid is injected at a given rate w over the fixed (internal) boundary ΓI (Γex=∂Ω∖ΓI). For each t∈]0;T[, the experimental results make evidence of the occurrence of two distinct zones: one full of the fluid is the saturated zone Ω(t), where the pressure p (p>0) and the saturation variable θ(θ≡1), and the other Ωc(Ωc=Ω∖Ω(t)) is the cavitated zone, where the pressure is constant (p=0) and 0≤θ<1 (e.g., the mixture of fluid and air). The free boundary of the region Ω(t) containing fluid is Γ(t) and the region Ω(0), with border ΓI, occupied by the fluid at t=0 being given (see Figure 1).Figure 1
A view of domainsΩ(t) and Ωc.


## 2.2. State of the Problem

The strong formulation of the problem described the phenomenon is written as follows.For eacht≥0, find a pair (p,θ)∈L2(0,T;H1(Ω))×L∞(Q)∩H1(0,T;H-1(Ω)) and Γ(t) such that(2.1)∇(K(h,t,N0)∇p)=∇(Vhθ)+a∂(hθ)∂t,(θ≡1)inΩ(t),(2.2)∇(Vhθ)+a∂(hθ)∂t=0inΩc,(2.3)K(h,t,N0)∂p∂n=h(1-θ)(V-aυ)nonΓ(t),(2.4)p=0onΓ(t),(2.5)K(h,t,N0)∂p∂n̲-hθVn̲=wonΓI,(2.6)p=0onΓex,(2.7)p(1-θ)=0inΩ,(2.8)θ|t=0=θ0,
where(2.9)K(h,t,N0)=h31-N02Φ(h,t,N0)with0≤N0<1,Φ(h,t,N0)=112+14h2(1-N02)-14hN021-N02coth⁡(N0h1-N02).
h(t,x) is the thickness of the thin film supposed a regular and given function of the problem. V is the speed of the axis supposed being given. υ is the moving free boundary (with υ=0 on ΓI). n (resp., n̲) is the normal vector along Γ(t) (resp., ΓI) exterior to Ω(t) (resp., Ω(0)). The saturation variable θ can be represented by a graph (see, Figure 2).Figure 2
Graph of saturation variableθ.In (2.1)–(2.3), there are the diffusion term ∇(K(h,t,N0)∇p), the shearing term ∇(hV), and the squeezing term ∂h/∂t.


## 2.3. Weak Formulation

Before starting the construction of a weak formulation of the problem (2.1)–(2.8), we denote by(2.10)Q=Ω×]0,T[,ΣI=ΓI×]0,T[,Σt=Γ(t)×]0,T[,Σex=Γex×]0,T[,E={φ∈H1(Q):φ=0onΣexandt=0}.
Indeed, multiplying (2.1) by φ∈E and integrating over [0,T]×Ω(t), we obtain(*)-∫0T∫Ω(t)K∇P∇φdxdt+∫0T∫Γ(t)K∇Pφn+∫0T∫ΓIK∇Pφn̲=-∫0T∫Ω(t)Vh∇φdxdt+∫0T∫Γ(t)Vhφn+∫0T∫ΓIVhφn̲+∫0T∫ΓexVhφ⋅nex+a{∫0Tddt∫Ω(t)hφdxdt-∫0T∫Γ(t)VhΦ⋅ndt-∫0T∫ΓIVhφn̲dt-∫0T∫Ω(t)hdφdtdxdt},
where nex is the normal vector along ∂Ω exterior to Ω.In the same way, we apply to (2.2)(**)-∫0T∫ΩcVh∇φdxdt+∫0T∫Γ(t)Vhφ⋅n+∫0T∫ΓIVhφ⋅n̲+∫0T∫ΓexVhφ⋅nex+a{∫0Tddt∫Ωchφdxdt-∫0T∫Γ(t)Vhφ⋅ndt-∫0T∫ΓIVhφn̲dt-∫0T∫Ωchdφdtdxdt}=0.
By adding (*) and (**) in all Ω=Ω(t)∪Ωc, and using (2.3)–(2.6) then the weak formulation can be written as follows.Find a pair(p,θ)∈L2(0,T;H1(Ω))xL∞(Q)∩H1(0,T;H-1(Ω))(2.11)0≤θ<1,p=0onΣex,p(1-θ)=0,a.e.inQ,(2.12)∫Q(K∇p-Vhθ)∇φ+a{∫Ω(T)hθφ-∫Ω(0)hθφ-∫Qhθdφdt}=∫ΣIwφ,∀φϵE,θ|t=0=θ0∈H-1(Ω).
As φ|t=0=0 implies that ∫Ω(0)hθφ=0.


## 3. An Elliptic Nonlinear Problem

To solve the problem (2.11)-(2.12), we will approximate it by an elliptic problem in the same way as Boukrouche [3] and Gilardi [7].Letβ be a real function (see Figure 3) satisfying the following assumptions:(3.1)β∈C∞(R)∀ξ∈R;0≤β(ξ)≤1,withβ′(ξ)≥0,β(ξ)=0forξ≤0.
Put(3.2)∂0Q=Σex∪Ω0.
consider now the problem, given ε>0, find p∈H1(Q) such that(3.3)∫Qε∂p∂t∂φ∂t+∫Q(K∇p-Vhβ(p))∇φ+a{∫Ω(T)hβ(p)φ-∫Qhβ(p)dφdt}=∫∑Iwφ,p=0  on ∂0Q, for all φ∈H1(Q) vanishing on ∂0Q.Figure 3
Graph of the nonlinear functionβ.Introducing the operatorτ is as follows:(3.4)τ:H1(Q)⟶H1(Q),p⟼q=τ(p).
If p∈H1(Q),τ(p) is a unique solution q to the linear problem(3.5)∀q∈H1(Q),q=0,on∂0Q∫Qε∂q∂t∂φ∂t+∫QK∇q∇φ-∫Qhβ(p)(V∇φ+adφdt)+a∫ΩThβ(p)φ=∫∑Iwφ
for every φ∈H1(Q) vanishing on ∂0Q.Lemma 3.1.
The operatorτ is continuous from H1(Q) with the weak topology into H1(Q) with the strong topology. Moreover τ(H1(Q)) is bounded in H1(Q).Proof.
Letpi∈H1(Q) with qi=τ(pi) for i=1,2.
Takingφ=(q1-q2)in (3.5), we have
(3.6)∫Qε(∂∂t(q1-q2))2+∫QK|∇(q1-q2)|2=∫Qh(β(p1)-β(p2))(V∇(q1-q2)+ad(q1-q2)dt)-a{∫Ω(T)h(β(p1)-β(p2))(q1-q2)}.
Using Cauchy-Schwarz's inequality, we obtain
(3.7)α‖q1-q2‖H1(Q)2≤C1{‖q1-q2‖H1(Q)‖β(p1)-β(p2)‖L2(Q)}+C2{‖q1-q2‖L2(Ω(T))‖β(p1)-β(p2)‖L2(Ω(T))},
where α is constant depending on h, N0, and ε. C1 and C2 are two constants depending on h, V, and a.
As(3.8)H1/2(Ω(T))⟶L2(Ω(T)),H1(Q)⟶H1/2(Ω(T)),H1/2(ΣI)⟶L2(ΣI),H1(Q)⟶H1/2(ΣI)
and β is Lipschitz continuous function, there exists a constant C depending onh, V, N0, a, and ɛ such that
(3.9)‖q1-q2‖H1(Q)≤C{‖p1-p2‖L2(Q)+‖p1-p2‖L2(Ω(T))}.
If pε→pconverge weakly in H1(Q), then pε→p,L2(Q) and pε(T)→p(T),L2(Ω(T)).  Thus τ(pɛ)→τ(p), then the continuity of τ is shown.
Takingφ=q in (3.5) and using Cauchy-Schwarz's inequality, we obtain ∥q∥H1(Q)≤k1, where k1 is a constant depending on h, V, N0, β, T, a, and ε.Theorem 3.2.
If the functionβ satisfies hypothesis (3.1), then, for every ε>0, there exists a solution to the problem (3.3).Proof.
Use Lemma3.1 and Schauder fixed-point theorem.Theorem 3.3 (cf. [2, 3]).
If the functionβ satisfies hypothesis (3.1), then for every ε>0, the solution of the problem (3.3) is unique.


## 4. Approximating Problems

In order to solve the problem (2.11)-(2.12), we consider a new family of problems of type (3.3) in which the function β is an approximation of the Heaviside function (see Figure 4). Therefore we consider a family of functions(4.1)Hε:[0,+∞[⟶R∖Hε∈C∞(R),(4.2)0≤Hε≤1,Hε′≥0,(4.3)Hε(0)=0,lim⁡ε-→0inf⁡⁡{ζ>0:Hε(ζ)=1}=0,(4.4)lim⁡ε-→0Lεε=0,whereLε=sup⁡⁡{Hε′(ζ):ζ>0}.
Consider now the following approximating problem.Figure 4
Graph of  Heaviside functionHε.For fixedε and a (where ε, a∈]0;1[), find pε such that(4.5)pε∈H1(Q),pε=0on∂0Q,∫Qε∂pε∂t∂φ∂t+∫QK∇pε∇φ-∫QhHε(pε)(V∇φ+adφdt)+a∫Ω(T)hHε(pε)φ=∫∑Iwφ
for every φ∈H1(Q) vanishing on ∂0Q.From Theorems3.2 and 3.3, we deduce the following theorem.Theorem 4.1 (cf., [3]).
For everyε and a∈]0,1[, there exists at least one solution to the problem (4.5). Moreover if h and V are sufficiently regular, every solution belongs to H1(Q)∩L∞(Q).Lemma 4.2.
If the functionHε verifies  (4.1)-(4.2) and  dh/dt≥0, then one has
(4.6)∫Ω(T)hHε(pε)pε-∫QhHε(pε)dpεdt≥0.Proof.
PuttingH̃ε(ζ)=∫0ζHε(τ)dτ, we have then H̃ε(ζ)≤ζHε(ζ) and ∫Ω(T)hHε(pε)pε-∫QhHε(pε)(dpɛ/dt)=∫Ω(T)hHε(pε)pε-∫Qh(d/dt)(H̃ε(pε)).
Using the integration by parts, we obtain(4.7)∫Ω(T)hHε(pε)pε-∫Qhddt(H̃ε(pε))=∫Ω(T)h[Hε(pε)pε-H̃ε(pε)]+∫QdhdtH̃ε(ζ)
as Hε(pε)pε-H̃ε(pε)≥0, then Lemma 4.2 is shown.The following proposition gives some a priori estimates for pressurepε.Proposition 4.3.
There exists a constantC independent of  ε,  a,N0, and  pε(4.8)∫Qε|∂pε∂t|2+∫QK|∇pε|2+a{∫Ω(T)hHεpε-∫QhHεdpεdt}≤C.Proof.
Takingφ=pε in (4.5), we obtain
(4.9)∫Qε|∂Pε∂t|2+∫QK|∇pε|2+a{∫Ω(T)ThHεpε-∫QhHεdpεdt}=∫QhVHε(pε)∇pε+∫ΣIwpε.
By using Lemma 4.2, we have
(4.10)∫Qε|∂pε∂t|2+∫QK|∇pε|2≤mes(Q)β‖∇pε‖L2(Q)‖V‖(L∞)2+‖pε‖L2(ΣI)‖w‖L2(ΣI).
Applying Poincare's inequality, we have
(4.11)‖pε‖L2(Q)2=∫0T∫Ω|pε|2≤α1‖∇pε‖L2(Q)2,
where α1 is constant depending on Ω.
As∥pε∥L2(ΣI)≤α2∥∇pε∥L2(ΣI)2, we finally deduce the result.Proposition 4.4.
For everyφnonnegative and φ∈D(Ω), there exists a constant  C(φ) such that
(4.12)∫Ωφ|ε∂pε(0,x)∂t|2≤C(φ)(1+Lε)ε.Proof.
From (4.5), we have
(4.13)ε∂2Pε∂t2+∇⋅(K∇pε)=∇⋅(VhHε(pε))+a∂∂t(hHε(pε))inΩ,K(h,t,N0)∇pεn-hHεVn=wonΣI,pε=0on∂0Q,∂pε∂t(T)=0onΩ.
Multiplying (4.13) by φ(∂pε/∂t) and integrating over ]0,t[×Ω, we have
(4.14)ε2∫Ωφ|∂pε∂t(0,x)|2+12∫ΩKφ|∇pε|2+∫0t∫ΩφhHε′(pε)|∂pε∂t(t)|2=-∫0t∫Ω{K∇pε∇φ+φ∇(hHε(pε)V)+aφHε(pε)∂h∂t}∂pε∂t(t)+ε2∫Ωφ|∂pε∂t(t)|2+12∫0t∫Ωφ∂∂t(K)|∇pε|2.
Taking
(4.15)A=ε2∫Ωφ|∂pε∂t(t)|2+12∫0t∫Ωφ∂∂t(K)|∇pε|2,B=-∫0t∫Ω{K∇pε∇φ+φ∇(hHε(pε)V)+aφHε(pε)∂h∂t}∂pε∂t(t),
then we have
(4.16)ε2∫Ωφ|∂pε∂t(0,x)|2≤A+B,ε22T∫Ωφ|∂pε∂t(0,x)|2≤∫0T(εA+εB)dt,ε∫0TAdt=ε22∫0T∫Ωφ|∂pε∂t(t)|2+ε2∫0T∫Ωφ∂∂t(K)|∇pε|2≤‖φ‖L∞ε2∫Qε|∂pε∂t(t)|2+ε‖∂∂t(K)‖L∞‖φ‖L∞∫0T∫Ω|∇pε|2.
Using Proposition 4.3., we obtain
(4.17)ε∫0TAdt≤c1(φ)ε,
where c1(φ)=Cmax{(1/2)∥φ∥L∞,∥(∂/∂t)(K)∥L∞∥φ∥L∞} and C is constant of Proposition 4.3:
(4.18)ε∫0TBdt=-ε{∫0T∫ΩK∇φ∇pεε∂pε∂t+φhVHε′(pε)|∂pε∂t(t)|2ε+∫0T∫ΩφHε(pε)∇(hV)ε∂pε∂t+aφHε(pε)∂h∂t∂pε∂tε}.
We obtain finally
(4.19)T2∫Ωφ|ε∂pε∂t(0,x)|2≤∫0T(εA+εB)dt≤C(φ)(1+Lε)ε.


## 5. An Existence Theorem of the Problem (2.11)-(2.12)

Theorem 5.1.
There exists at least one solution to the problem (2.11)-(2.12).Proof.
Ifa∈]0,1] and from Proposition 4.3, we can extract a subsequence of pɛ, still denoted by(pɛ), such that
(5.1)pεε→0⟶pinL2(0,T;H1(Ω))weakly,(5.2)ε∂pε∂tε→0⟶0inL2(Q)weakly.
Moreover Hε(pε)∈[0,1], for all ε>0, then there exists θϵL∞(Q) such that
(5.3)Hε(pε)⟶θinL2(Q)weakly.
We can now proof that the (p,θ) given in (5.1) and (5.3) is solution of the problem (2.11)-(2.12).
We denote by(5.4)k={φϵL2(0,T;H1(Ω)):φ=0onΣexandφ≥0inQ},k′={φϵL2(Q):0≤φ≤1},k×k′isconvexspaceofL2(0,T;H1(Ω))×L2(Q),thenk×k′isweaklyclosed.
Therefore pϵL2(0,T;H1(Ω)), p≥0 a.e. in Q and θϵL2(Q)∩L∞(Q) with θϵ[0,1] a.e. in Q.
Thus ifε→0 in (4.5), we find the problem (2.11)-(2.12). In order to prove 0≤θ<1 and  p(1-θ)=0, a.e in Q, we give a brief demonstration (for more details, see [7, pages 1113-1114]).
First, we need to prove∫Qp(1-θ)=0. We notice that lim ∫Qpε(1-Hε(pε))=0(5.5)since0≤lim⁡∫Qpε(1-Hε(pε))≤mesQ⋅sup⁡{ζ≥0:Hε(pε)<1}.
Now we have to prove that
(5.6)∫Qp(1-θ)=lim⁡∫Qpε(1-Hε(pε)).
We define wε=ε(∂pε/∂t)pε-Hε(pε) and use (4.13).
Then the couple (p,θ) is solution of the problem (2.11)-(2.12).Theorem 5.2.
Ifw≥0 on ΓI×[0,T] and θ0≥0, then p≥0 a.e. in Ω×[0,T].Proof.
Following [7], we construct a sequence solution of the problem
(5.7)ε∂pε∂t+pε=p-on[0,T],pε(T)=0,
where p-=sup(-p,0), p+=sup(p,0) and p=p+-p-.
From the classical Cauchy-Lipschitz-Picard theorem [2], there exists a unique solution pεϵC1[0,T]. Multiplying (5.7) by pε (resp., by ε(∂pε/∂t)) and integrating over Q, we obtain
(5.8)‖pε‖L2(Q)≤‖p-‖L2(Q),
respectively,
(5.9)‖ε∂pε∂t‖L2(Q)≤‖p-‖L2(Q).
Deriving (5.7) with respect to x, multiplying by ∇pε, and integrating over Q, we obtain
(5.10)‖∇pε‖L2(Q)≤‖∇p-‖L2(Q).
We deduce that there exists p̃ϵL2(0,T;H1(Ω)) and ζϵL2(Q) such that
(5.11)pε⟶p-weaklyinL2(0,T;H1(Ω)),(5.12)ε∂pε∂t⟶ζweaklyinL2(0,T;H1(Ω)).
From (5.11) we have
(5.13)εpε⟶0inL2(0,T;H1(Ω)),thereforeζ=0.
Passing to the limit in (5.7), we deduce that p̃=p- a.e. in Q.
AsE=H01(0,T;V′) is dense in L2(0,T;H1(Ω)), (2.12) can be rewritten in the following form:
(5.14)a∫0T<V′∂(θh)∂t,φ>V+∫Q(K∇p-Vhθ)∇φ=∫ΣIwφ,∀φϵV,a.e.in[0,T].
Taking now pε as test function in (5.14) and passing to the limit over ε, we deduce
(5.15)∫QK|∇p-|2≥∫Σwp-
as w≥0, therefore p-=0 a.e. in Q, that is, p≥0   a.e. in Q.Next work will consist in finding some existence of relationship between the pressurep and parameter N0 and in completing numerical analysis study to the problem (2.1)–(2.8).



---

*Source: 460602-2010-12-22.xml*