This condition allows the degeneracy only where one of the two phases is exclusively present

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Volume 9, Numbers 11-12, Pages 1235–1278

ON A DEGENERATE PARABOLIC SYSTEM FOR COMPRESSIBLE, IMMISCIBLE, TWO-PHASE FLOWS IN

POROUS MEDIA

C´edric Galusinski and Mazen Saad

Mathématiques Appliquées de Bordeaux UMR CNRS 5466 Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France

(Submitted by: Roger Temam)

Abstract. The aim of this paper is to analyze a model of a degenerate nonlinear system arising from immiscible, compressible, two-phase, three-dimensional ﬂows occurring in porous media. A degenerate weighted formulation is introduced to take into account the compressibility and the degeneracy. Two existence results of such a degenerate weak solution are introduced. The ﬁrst result concerns the existence of solutions under a reasonable assumption on the capillary pressure.

This condition allows the degeneracy only where one of the two phases is exclusively present. The second result establishes, for suitable initial data, the existence of solutions when the degeneracy occurs where one or the other phase is exclusively present. Nevertheless, for suitable initial data, a classical weak solution is obtained when the degeneracy is not too strong and occurs only where the injected phase is exclusively present.

1. Introduction and model

We analyze a degenerate nonlinear system modeling a three-dimensional displacement of two immiscible, compressible ﬂuids in a porous medium.

The equations describing the immiscible displacement of two compressible ﬂuids are given by the mass conservation of each phase:

φ(x)∂t(ρisi)(t, x)+div(ρiVi)(t, x)+ρisig(t, x) =ρis_if(t, x) i= 1,2, (1.1) where φ is the porosity of the medium, and ρi and si are respectively the density and the saturation of the i^th ﬂuid. The velocity of each ﬂuid Vi is given by Darcy’s law:

Vi(t, x) =−K(x)ki(si(t, x))

µi ∇pi(t, x), i= 1,2, (1.2)

Accepted for publication: July 2004.

AMS Subject Classiﬁcations: 35K57, 35K55, 92D30.

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where K is the permeability tensor of the porous medium, ki the relative permeability of the i^th phase, µi the i-phase’s viscosity (considered to be constant) andpi thei-phase’s pressure. The eﬀects of gravity are neglected.

Here the functions f and g are respectively the injection and production terms. Note that the saturation of the injected ﬂuids is given (it appears in the term ρis_if in equation (1.1)), but the saturation of the produced ﬂuid is an unknown which appears in the termsρisig.

By deﬁnition of saturation, one has

s1(t, x) +s2(t, x) = 1. (1.3) We considers=s1 to be the saturation of the water phase or the gas phase, and s2 = 1−sthat of the oil. Thus, we deﬁne the capillary pressure as

p12(s(t, x)) =p1(t, x)−p2(t, x), (1.4) and the function s −→ p12(s) is nondecreasing (^dp_ds¹²(s) ≥ 0, for all s ∈ [0,1]). In this paper, the forced displacement of ﬂuids is modeled. It is used in many enhanced recovery processes: a ﬂuid such as water (or gas) is injected into some wells in a reservoir while the resident hydrocarbons are produced from other wells. Consequently, we consider the injected saturation s₁= 1 and s₂= 0.

By these formulations, we can see that the unknown functions are the saturation of one phase and one pressure. Let us denote as usual

Mi(s) =ki(s)/µi i-phase’s mobility, M(s) =M1(s) +M2(s) the total mobility,

ν(s) =M1(s)/M(s) the fractional ﬂow of the 1^st phase, 1−ν(s) =M2(s)/M(s) the fractional ﬂow of the 2^nd phase, V=V1+V2 the total velocity.

As in [5] and [16], we can express the total velocity in terms ofp2 andp12. We have

V(t, x) =−K(x)M(s) (∇p2(t, x) +ν(s)∇p12(s)),

deﬁning a function ˜p(s) such that ^d˜_ds^p(s) =ν(s)^dp_ds¹²(s), and settingp=p2+ ˜p (the so-called global pressure [5]), the total velocity becomes

V(t, x) =−K(x)M(s)∇p(t, x). (1.5) Thus, each phase velocity can be written as

V1 =ν(s)V−Kα(s)∇s (1.6)

V2 = (1−ν(s))V+Kα(s)∇s, (1.7) whereα(s) =M(s)ν(s)(1−ν(s))^dp_ds¹²(s)≥0.

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In [6] and [3] the authors consider an exponent state law to describe the displacement of one compressible fluid. We consider as in Aziz and Settari ([4], p. 13) that it is possible to assume that the fluid compressibility is also an exponent state law. The density of fluid depends on the pressure of the corresponding fluid, but taking advantage of the fact that this function varies slowly with capillary pressure (see Chavent et al. ([5], Chapter 4) for more details). Moreover, we assume thatρi=ρi(p) satisfies

dρi

dp(p) =γρi(p), γ >0. (1.8) Here we assumed that the compressibility factor γ is the same for the two ﬂuids. In [1], the authors consider a similar assumption for a compressible, miscible ﬂow.

In this case the system (1.1), with (1.6)–(1.7) and (1.8) and considering one ﬂuid being injected (i.e.,s₁= 1, s₂ = 0), can be reduced to

φ∂ts+γφs∂tp+ div(ν(s)V) +γν(s)V· ∇p−div(Kα(s)∇s) (1.9)

−γKα(s)∇s· ∇p+sg =f,

−φ∂ts+γφ(1−s)∂tp+ div((1−ν(s))V) +γ(1−ν(s))V· ∇p (1.10) + div(Kα(s)∇s) +γKα(s)∇s· ∇p+ (1−s)g= 0.

In order to obtain a nondegenerate pressure equation, we add the two equations and ﬁnd

γφ(x)∂tp+ divV+γV· ∇p=f−g. (1.11) Classically, in porous media the velocity is considered to be very small; thus, the quadratic terms in the velocity (γV· ∇p) are neglected as in [6]. The equations (1.11) and (1.9) are simpliﬁed as

γφ(x)∂tp+ divV=f−g (1.12) φ(x)∂ts + γφ(x)s∂tp+ div(ν(s)V)−div(K(x)α(s)∇s)

− γK(x)α(s)∇s· ∇p=f−sg (1.13)

V=−K(x)M(s)∇p. (1.14) The system is a direct generalization of the incompressible model, and it is consistent with the incompressible model in the sense that the system is the limiting form of the compressible system as the compressibility factor γ of the ﬂuids tends to zero.

Let T > 0 be ﬁxed, and let Ω be a bounded set of R³. We set QT = (0, T)×Ω and ΣT = (0, T)×∂Ω. To this system we add the initial conditions

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and we consider no ﬂux boundary conditions; in summary, we investigate the following nonlinear boundary-value problem of parabolic type in ΩT:

γφ(x)∂tp+ divV=f −g φ(x)∂ts−sdiv(V) + div(ν(s)V)

−div(K(x)α(s)∇s)−γK(x)α(s)∇s· ∇p= (1−s)g

V=−K(x)M(s)∇p (1.15)

V·n= 0, K(x)α(s)∇s·n= 0 on ΣT

p(0, x) =p0(x), s(0, x) =s0(x) in Ω, wheren represents the unit outward normal to∂Ω.

To our knowledge, such a complete model of compressible and immiscible displacement in porous media has never been studied from a mathematical point of view. On the other hand, several papers are devoted to the numerical [6] and mathematical ([1], [2], [3], [8], [11], [12]) studies of compressible, miscible fluids. Our model is close to that of [2]. Nevertheless, the model arising from miscible fluids leads to a nondegenerate parabolic system. Furthermore, the authors consider a constant fluid viscosity, allowing a more regular pressure field. As a matter of fact, their model on the pressure reduces to

φ(x)a(u)∂tp−div(k(x)∇p) =qⁱ−q^s,

which allows one to exhibit a regular pressure field. In our model of immiscible fluid, even if the fluid viscosity is assumed to be constant, we have

γφ(x)∂tp−div(K(x)M(s)∇p) =f −g.

It would not be relevant to assume a constant total mobility. We are then limited by an (L²(QT))^N regularity on the velocity ﬁeldV =−KM(s)∇p.

This makes the nonlinear terms such assdivVandγK(x)α(s)∇s· ∇p more diﬃcult to control in the equation governing the saturations. Furthermore, the dissipation term degenerates in this equation, obliging one, in most cases, to add a degenerate weight to deﬁne solutions to our problem. These solutions are called degenerate weak solutions.

Models dealing with immiscible fluids (so with degenerate dissipative terms) have already been studied from a mathematical point of view, but only in the case of an incompressible fluid; see for example [5], [9], [7], [10], and [13]. Thus, these models do not contain the additional nonlinear terms we met in our model since divV = f −g and γ = 0. Classical weak solutions of parabolic problems can be defined for these models of incompressible fluids.

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For our problem, we ﬁnd classical weak solutions only whenαdegenerates at most like a quadratic function around s= 1 (see assumption (H7a)).

Next we are going to introduce some physically relevant assumptions on the coeﬃcients of the system. We consider in the assumptions (H7b) and (H7c) two cases of a degenerate problem, the ﬁrst with a degenerate dissipation only at s = 0 and the second a degenerate dissipation ats= 0 and s= 1.

(H1) ∃φ0 >0, φ1 >0 such thatφ0 ≤φ(x)≤φ1.

(H2) ∃k0 >0, k_∞ > 0 such that (K(x)ξ, ξ) ≥k0|ξ|², for all ξ ∈ R^N, and almost every x∈Ω, K(L^∞(Ω))^N×N ≤k_∞.

(H3) M ∈ C⁰([0,1]),∃m0>0 such thatM(s)≥m0 for all s∈[0,1].

(H4) ν ∈ C⁰([0,1]),ν(0) = 0, ν(1) = 1 and∃C >0; ν(s)≤Cs.

(H5) (f, g)∈(L²(QT))²,g(t, x)≥0 for almost every (t, x)∈QT. (H6) (p0, s0)∈(L²(Ω))², 0≤s0(t, x)≤1 for almost every x∈Ω.

The assumptions (H1)–(H6) are classical for porous media. A major diﬃ- culty of system (1.15) is the degeneracy of the diﬀusion terms. Let us state the assumption on the function α:

(H7a) α∈ C¹([0,1]),α(s)>0 for 0< s <1,α(0)>0,α(1) = 0,

there existα0>0, 0< r2 ≤2,s1 <1,m1 and M1>0 such that α(s)≥α0 for all s∈[0, s1],

m1(1−s)^r² ≤α(s)≤M1(1−s)^r², for all s∈[s1,1].

Furthermore, lims→1 1−ν(s) 1−s = 0.

(H7b) α∈ C¹([0,1]),α(s)>0 for 0< s <1,α(0) = 0, α(1)>0.

Furthermore, there existr1 >0 andm1, M1 >0 such that m1r1s^r¹⁻¹ ≤α(s)≤M1r1s^r¹⁻¹ for all 0≤s≤1.

(H7c) α∈ C¹([0,1]),α(s)>0 for 0< s <1,α(0) = 0, α(1) = 0, there existr1>0,r2 >0,s1 <1,m1 and M1>0 such that m1r1s^r¹⁻¹ ≤α(s)≤M1r1s^r¹⁻¹, for all s∈[0, s1].

−r2M1(1−s)^r²⁻¹ ≤α(s)≤ −r2m1(1−s)^r²⁻¹, for all s∈[s1,1], and lims→11−ν(s)

1−s = 0.

The assumption (H7a) indicates essentially that the function α does not degenerate near s = 0 and behaves like the function (1−s)^r² near s = 1.

The assumption (H7b) indicates essentially that the functionα behaves like the function s^r¹ near s= 0 and is not vanishing at s= 1. These situations correspond to some models of capillary pressure which ensure these conditions. In [9], the authors assume a similar condition on the incompressible compositional model; see also [4]. The assumption (H7c) corresponds essentially to models on relative permeabilities. For example, if we consider

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Corey’s model with relative permeabilitiesr₁ >1 and r₂ >1, we have 1−ν(s) = (1−s)^r²/(µ2

µ1

s^r¹ + (1−s)^r²), α(s) =s^r¹(1−s)^r²dp12

ds (s)/(µ2s^r¹+µ1(1−s)^r²).

Assumption (H7c) allows a bounded capillary-pressure derivative, whereas assumption (H7a) (respectively (H7b)) obliges one to consider models where the capillary pressure grows quickly to inﬁnity near s = 0 (respectively s= 1).

In the next section we introduce ﬁrst the existence of classical weak solutions under assumptions (H1)–(H6) and (H7a) for a particular choice of initial data. Next, we introduce the existence of a solution in a weaker sense, to be made precise later, of solutions to system (1.15) under the conditions (H1)–(H6) and (H7b), and ﬁnally, for a particular choice of initial data, we give also a sense of solutions to system (1.15) under the conditions (H1)–(H6) and (H7c).

2. Main results

2.1. Classical weak solutions. Let us deﬁne a function L∈C²[0,1) by L(s) = s²

2 for 0≤s≤s1,

L(s) = (ν(s)−s)⁻¹L(s), ∀s∈[s1,1).

Deﬁnition 2.1. Let (H1)–(H6)and(H7a) hold, and assume that the initial condition s0 satisﬁes L(s0) ∈ L¹(Ω). Then (p, s) is a classical weak solu- tion of (1.15) if p ∈ L²(0, T;H¹(Ω))∩L^∞(0, T;L²(Ω)), V ∈ (L²(QT))^N, φ(x)∂tp ∈ L²(0, T; (H¹(Ω))), 0 ≤ s(t, x) ≤ 1, for almost every(t, x) ∈ (0, T)×Ω, L(s)∈L^∞(0, T;L¹(Ω)), ∇s∈(L²(QT))^N,

γφ∂tp, ψ+

Q_T

K(x)M(s)∇p· ∇ψ dx dt=

Q_T

(f −g)ψ dx dt, (2.1)

−

Q_T

φ(x)s∂tχ dx dt−

Ω

φ(x)s0(x)χ(0, x)dx+

Q_T

V· ∇(sχ)dx dt

−

Q_T

ν(s)V· ∇χ dx dt+

Q_T

Kα(s)∇s· ∇χ dx dt

−γ

Q_T

K(x)α(s)∇s· ∇pχ dx dt=

Q_T

(1−s)gχ dx dt, (2.2) for all ψ∈L²(0, T;H¹(Ω)), χ∈ C¹([0, T)×Ω) with supp χ⊂[0, T)×Ω.

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Theorem 2.1. Let (H1)–(H7a) hold, and let s0 satisfy ln(1−s0)∈L¹(Ω).

For every γ > 0, there exists at least one classical weak solution to the degenerate system (1.15)in the sense of Deﬁnition 2.1.

Note that we can show that there exist c1, c2 >0 such that c1ln(1−s)≤L(s)≤c2ln(1−s), ∀s∈[0,1).

2.2. Weak degenerate solutions. Denote by β(s) =s^r⁻¹, h(s) =

s 0

β(y)dy,

where r > 1 and r = r1 if r1 > 1 (r1 is introduced in assumption (H7b)), and r≤r1+ 2 ifr1 ≤1. Forθ≥0, we deﬁne

βθ(s) =s^r⁻^1+θ, hθ(s) = _s

0

βθ(y)dy.

Deﬁnition 2.2. Let θ ≥ 7r1+ 6−r, and let (H1)–(H6) and (H7b) hold.

Then (p, s) is a degenerate weak solution to (1.15) if and only if p∈L²(0, T;H¹(Ω))∩L^∞(0, T;L²(Ω)),

V∈(L²(QT))^N, φ(x)∂tp∈L²(0, T; (H¹(Ω))), 0≤s(t, x)≤1, a.e. in (t, x)∈(0, T)×Ω,

hθ(s)∈L²(0, T;H¹(Ω)), α¹²(s)β¹²(s)∇s∈(L²(QT))^N, γφ∂tp, ψ+

Q_T

K(x)M(s)∇p· ∇ψ dx dt=

Q_T

(f −g)ψ dx dt, (2.3) for all ψ∈L²(0, T;H¹(Ω)); we deﬁne

F(s, p, χ) =−

Q_T

φ(x)hθ(s)∂tχ dx dt−

Ω

φ(x)hθ(s0(x))χ(0, x)dx +

Q_T

V· ∇(sβθ(s)χ)dx dt−

Q_T

ν(s)V· ∇(βθ(s)χ)dx dt +

Q_T

α(s)K∇s· ∇(βθ(s)χ)dx dt−γ

Q_T

K(x)α(s)∇s· ∇pβθ(s)χ dx dt

−

Q_T

(1−s)gβθ(s)χ dx dt,

(2.4) with F satisfying

F(s, p, χ)≤0∀χ∈ C¹([0, T)×Ω), supp χ⊂[0, T)×Ω andχ≥0, (2.5)

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and furthermore,

∀ε >0,∃Q^ε⊂QT, meas(Q^ε)< ε, such that F(s, p, χ) = 0,∀χ∈ C¹([0, T)×Ω), supp χ⊂

[0, T)×Ω

\Q^ε. (2.6) Note that the properties on the function α¹²(s)β¹²(s)∇s are a fortiori valid with βθ instead of β. This is enough to give a sense to each term in the above formulation.

The compressibility of ﬂuids makes the system (1.15) highly nonlinear.

Using classical energy estimates, these nonlinear terms are not controlled by the degenerate dissipative term in the equation governing the saturation variable s. A classical weak formulation for parabolic problems cannot be established here. That is why we introduce a degenerate (at s= 0) weight βθ(s). We then call such solutions degenerate weak solutions. Remark that these solutions are not aﬀected by this weight in regions where s= 0.

Theorem 2.2. Let (H1)–(H6)and(H7b)hold. For everyγ >0, there exists at least one degenerate weak solution to the degenerate system (1.15)in the sense of Deﬁnition 2.2.

Remark 2.1. In Definition 2.2, the neighborhood Q^ε could include a part of the boundary of (0, T)×Ω, even for small ε. Consequently, the initial condition and the boundary condition could be slightly violated. Even if we have a control of the solution on the whole domain (0, T)×Ω and inde- pendently of ε, a “weak degenerate solution” can violate the equation, the boundary condition, or the initial condition on a small (as small as wanted) neighborhood. Remark also that in this definition, the solution satisfies a variational inequality where the test function does not depend onε.

Let us now give the definition of the weak solutions when the assumption (H7c) is satisfied. Forθ, λ≥0, letjθ,λ be the continuous function defined as

jθ,λ(s) =

βθ(s) for 0≤s≤s1

βθ(s1)(1−s1)¹⁻^r²⁻^λ(1−s)^r²⁻^1+λ fors1 ≤s , (2.7) where r ≥ max(2, r2) (r2 is introduced in assumption (H7c)), and denote by Jθ,λ its primitive

Jθ,λ(s) = _s

0

jθ,λ(y)dy. (2.8)

To simplify notation, we denote J =J0,0 and j=j0,0. Let us also deﬁneµ andG by

µ(s) =β(s) for 0≤s≤s1, (2.9)

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whereµ is continuous and

µ(s) = (ν(s)−s)⁻¹µ(s), ∀s≥s1. (2.10) Also,

G(s) = s

0

µ(y)dy. (2.11)

According to the assumption (H7c), we have lim

s→1⁻(1−ν(s))(1−s)⁻¹= 0⁺; it is then easy to obtain that

∀s≥s1, (1−s)⁻¹≤(ν(s)−s)⁻¹ ≤(1−s)⁻¹+ 2(1−ν(s))(1−s)⁻¹. (2.12) From the deﬁnition of µ, we have

µ(s) =µ(s1) exp(

_s

s₁

(ν(σ)−σ)⁻¹dσ), for all s≥s1, and consequently there exists k1 >0 such that

µ(s1)(1−s1)(1−s)⁻¹ ≤µ(s)≤k1µ(s1)(1−s1)(1−s)⁻¹, ∀s≥s1. (2.13) Deﬁnition 2.3. Let θ≥7r1+ 6−r andλ≥7r2+ 6−^r₂,let(H1)–(H6)and (H7c) hold, and assume that the initial conditions0 satisﬁesG(s0)∈L¹(Ω).

Then (p, s) is a degenerate weak solution of (1.15)if p∈L²(0, T;H¹(Ω))∩L^∞(0, T;L²(Ω)), V∈(L²(QT))^N,

φ(x)∂tp∈L²(0, T; (H¹(Ω))), 0≤s(t, x)≤1, for a.e. (t, x)∈(0, T)×Ω, G(s)∈L^∞(0, T;L¹(Ω)), α¹²(s)µ¹²(s)∇s∈L²(QT), J(s)∈L²(0, T;H¹(Ω)), for all ψ∈L²(0, T;H¹(Ω)),

γφ∂tp, ψ+

Q_T

K(x)M(s)∇p· ∇ψ dx dt=

Q_T

(f−g)ψ dx dt; (2.14) for all ε > 0, there exists Q^ε ⊂ QT, meas(Q^ε) < ε, such that, for all χ∈ C¹([0, T)×Ω) withsupp χ⊂

[0, T)×Ω

\Q^ε,

−

Q_T

φ(x)Jθ,λ(s)∂tχ dx dt−

Ω

φ(x)Jθ,λ(s0(x))χ(0, x)dx +

Q_T

V· ∇(sjθ,λ(s)χ)dx dt−

Q_T

ν(s)V· ∇(jθ,λ(s)χ)dx dt +

Q_T

Kα(s)∇s· ∇(jθ,λ(s)χ)dx dt−γ

Q_T

K(x)α(s)∇s· ∇pjθ,λ(s)χ dx dt

=

Q_T

(1−s)gjθ,λ(s)χ dx dt. (2.15)

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Note that each term in the formulas (2.14)–(2.15) is well defined. In particular, we give some details on the third and fourth integrals of the equality (2.15). From the definition of the function jθ,λ, all integrals are defined in the regionQT ∩ {s < s1}; we focus our attention on the integrals in the regionQT ∩ {s > s1}. We have

Q_T∩{s≥s₁}

ν(s)V· ∇(jθ,λ(s)χ)−V· ∇(sjθ,λ(s)χ) dx dt

=

Q_T∩{s≥s₁}

(ν(s)−s)jθ,λ(s)V· ∇χ

+ (ν(s)−s)j_θ,λ (s)V· ∇sχ−V· ∇Jθ,λ(s)χ

dx dt;

from (2.12) we have ν(s)−s≤1−s, for s > s1. Using (2.7), it yields that fors > s1,

(ν(s)−s)j_θ,λ (s)≤(r

2 −1 +λ)jθ,λ(s), and

(ν(s)−s)jθ,λ(s)≤βθ(s1)(1−s1)¹⁻^r

2−λ

(1−s)^r

2+λ ≤c(s1);

thus, the above integral can be bounded by c

V· ∇χL¹(Q_T)+V· ∇J(s)L¹(Q_T)χL^∞(Q_T)

, wherec is a constant. In the same way, let us write

Q_T∩{s≥s₁}Kα(s)∇s· ∇(jθ,λ(s)χ)dx dt

=

Q_T∩{s≥sKα(s)j1} _θ,λ (s)∇s· ∇sχ dx dt+

Q_T∩{s≥sKα(s)∇J1} θ,λ(s)· ∇χ dx dt, and from the deﬁnitions of the function µ and jθ,λ, one obtains that there exists a constantc(s1) such thatα(s)j_θ,λ (s)≤c(s1)α(s)µ(s) for all s > s1; then the above integrals make sense.

Remark that forr2 ≤2, which is relevant for most applications, the weak formulation (2.15) is not degenerated for s > s1 even though the function α is degenerated around s= 1. This is due to the fact that the right-hand side of the saturation equation of (1.15) is also degenerated at s = 1, so that a test function (µ) blowing up at s= 1 can be used. A nondegenerate dissipative estimate follows arounds= 1α¹²(s)µ¹²(s)∇s∈L²(QT).But this test function obliges us to considerG(s0)∈L¹(Ω). In order to overcome this assumption, a weak degenerate solution ats= 1 could be tried, but the test

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functionµ cannot degenerate simultaneously at s= 0 and s= 1 because µ has to be an increasing function to ensure a dissipative estimate.

Theorem 2.3. Let (H1)–(H6) and (H7c) hold, and assume that the initial condition s0 satisﬁes G(s0) ∈L¹(Ω). For every γ >0, there exists at least one degenerate weak solution to the degenerate system (1.15)in the sense of Deﬁnition 2.3.

The assumption on s0 is satisﬁed if the initial condition satisﬁes ln(1− s0) ∈ L¹(Ω). As a matter of fact, the function µ(s) = G(s) is dominated by C(1−s)⁻¹ fors≥s1 (see (2.13)). We remark also that this assumption allows us to conserve, along the time, this property:

G(s(t))∈L¹(Ω) for almost everyt∈(0, T).

In particular, no injected phase bag can appear along the time.

In order to prove Theorem 2.1, Theorem 2.2, and Theorem 2.3, one first proves the existence of solutions to a nondegenerate problem. To avoid the degeneracy of the function α, we introduce a modified problem whereα is replaced byαη(s) =α(s) +η only in the diffusion terms, withη >0.

2.3. Weak solutions for the nondegenerate problem. We consider the nondegenerate system

γφ(x)∂tpη+ divVη =f −g.

φ(x)∂tsη−sηdiv(Vη) + div(ν(sη)Vη)

−div(K(x)αη(sη)∇sη)−γK(x)α(sη)∇sη· ∇pη = (1−sη)g Vη =−K(x)M(sη)∇pη

Vη ·n= 0, K(x)αη(sη)∇sη·n= 0 on ΣT (2.16) pη(0, x) =p0(x), sη(0, x) =s0(x) in Ω.

For the existence of a solution of the nondegenerate system, we make the following assumption onα(which is weaker than assumptions (H7a), (H7b), or (H7c)):

(H8) α ∈ C⁰([0,1]), α(s) > 0 for 0 < s < 1, α(0) = 0, and α(1) = 0 or α(1)>0.

Proposition 2.1. Let (H1)–(H6) and (H8) hold. For any η > 0, there exists (pη, sη) a weak solution of (2.16) satisfying pη ∈ L²(0, T;H¹(Ω))∩ L^∞(0, T;L²(Ω)), Vη ∈ (L²(QT))^N, φ(x)∂tpη ∈ L²(0, T; (H¹(Ω))), sη ∈ L²(0, T;H¹(Ω))∩L^∞(0, T;L²(Ω)),0≤sη ≤1, sη ∈ C⁰(0, T;L²(Ω)), and

γφ∂tpη, ψ+

Q_T

K(x)M(sη)∇pη· ∇ψ dx dt=

Q_T

(f−g)ψ dx dt (2.17)

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Ω

φ(x)sη(T, x)χ(T, x)dx−

Ω

φ(x)s0(x)χ(0, x)dx+

Q_T

sη∂tχ dx dt +

Q_T

sηVη· ∇χ dx dt+

Q_T

Vη· ∇sηχ dx dt−

Q_T

ν(sη)Vη· ∇χ dx dt +

Q_T

Kαη(sη)∇sη.∇χ dx dt−γ

Q_T

Kα(sη)∇sη · ∇pηχ dx dt

=

Q_T

(1−sη)gχ dx dt (2.18)

for all ψ∈L²(0, T;H¹(Ω)), χ∈ C¹([0, T]×Ω), andq > N.

The end of this paper is organized as follows: the next section is devoted to a compactness result useful in proving Proposition 2.1, Theorem 2.1, Theorem 2.2, and Theorem 2.3 in Sections 4, 5, 6, and 7.

3. Preliminary result

3.1. Compactness result. LetX andY be Banach spaces such that X⊂L^r ⊂Y with compact embeddingX into L^r, r≥1. (3.1) Denote by W a set of functions and byφa function deﬁned on Ω such that 0< φ0 ≤φ(x)≤φ1 for a.e.x∈Ω. (3.2) Lemma 3.1. Assume (3.1), (3.2), 1< q≤ ∞, and

(1) W is bounded in L^q(0, T;L^r)∩L¹(0, T;X), (2) φ∂tv is bounded in L¹(0, T;Y), for v∈W.

Then, W is relatively compact in L^p(0, T;L^r), for all 1≤p < q.

Proof. Let us ﬁrst prove a result similar to Lemma 8 in [17]:

∀η >0, ∃M such that vL^r ≤ηvX+MφvY. (3.3) Note that ifv is bounded inX, thenφv is bounded in Y, because

φvY ≤c1φvL^r ≤c1φ1vL^r ≤c2φ1vX.

DenoteEm ={v∈L^r;vL^r < η+mφvY}.The sequenceEm of open sets in L^r is increasing, and the union of Em covers L^r. Denote by S the unit sphere of X, S = {v ∈ X; vX = 1}, which is relatively compact in L^r; then there exists a ﬁnite M such thatS ⊂EM, which yields

vL^r < η+MφvY, ∀v ∈S,

and the inequality (3.3) is reached for every w∈X by takingv=w/wX. Lemma 3.1 is a consequence of (Theorem 4, [17]) if we show

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(i) W is bounded inL^q(0, T;L^r)∩L¹_loc(0, T;X),

(ii) ∀0< t1 < t2 < T,Thv−vL¹(t₁,t₂;L^r) −→0 as h →0 uniformly for v ∈W,

whereThv(t, x) =v(t+h, x).

The ﬁrst condition (i) is straightforward. To obtain the second condition (ii), owing to (3.3), for everyη >0, there existsM such that

Thv−vL¹(t1,t2;L^r)≤ηThv−vL¹(t1,t2;X)+Mφ(Thv−v)L¹(t1,t2;Y), using now that the translations in time are continuous in L¹(0, T), and Lemma 4 ([17]), we have

Thv−vL¹(t1,t2;L^r)≤2ηvL¹(0,T;X)+M hφ∂tvL¹(0,T;Y)≤c3η+c4h.

Givenε >0, forη= _2c^ε

3 andh= _2c^ε

4 it yieldsThv−vL¹(t₁,t₂;L^r)≤ε, which proves thatW is relatively compact in L^p(0, T;L^r), 1≤p < q.

3.2. Convergence lemma. Along the way in this article we are often con- fronted with terms for which the following classical convergence lemma is used.

Lemma 3.2. Let Ω be a bounded set in R^N, and (fε)ε and (gε)ε be two sequences satisfying

fε(x)−→0 a.e. inΩ, |fε(x)| ≤C a.e. inΩ, C is a constant, and gε−→g strongly in L¹(Ω).

(3.4)

Then

Ω

fε(x)gε(x)dx−→0.

Proof. Let us write

Ω

|fε(x)gε(x)|dx≤

Ω

|fε(x)(gε(x)−g(x))|dx+

Ω

|fε(x)g(x)|dx

≤C

Ω|gε(x)−g(x)|dx+

Ω|fε(x)g(x)|dx;

the convergence to zero of the right-hand side is obtained by the strong convergence on (gε)εinL¹(Ω) for the ﬁrst integral and by Lebesgue’s theorem

for the second.

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4. Existence for the nondegenerate problem

The proof is carried out using the Schauder ﬁxed-point theorem. We ﬁrst give an outline. For physical relevance we should have 0 ≤ s(t, x) ≤ 1 for almost every (t, x)∈QT. For this purpose, we introduce the following closed subsetK of L²(QT):

K={u;u∈L²(QT),0≤u(t, x)≤1 for a.e. (t, x)∈QT}.

In this section we omit the dependence of solutions on the parameterη.

Lets∈ K be ﬁxed, and let p be the solution of the parabolic equation γφ(x)∂tp+ divV=f−g, V=−KM(s)∇p

V·n= 0 on ΣT, p(0, x) =p0(x) in Ω. (4.1) We associate with (p,V) a solutions(t, x) to the equation







φ(x)∂ts−sdiv(V) + div(ν(s)V)

−div(K(x)αη(s)∇s)−γK(x)α(s)∇s· ∇p= (1−s)g K(x)α(s)∇s·n= 0 on ΣT, s(0, x) =s0(x) in Ω.

(4.2) For anysﬁxed inK, there exists a unique solutionpof (4.1) inL²(0,T;H¹(Ω))

∩C⁰(0, T;L²(Ω)) satisfying

pL^∞(0,T;L²(Ω))+pL²(0,T;H¹(Ω))≤C (4.3) φ∂tpL²(0,T;(H¹(Ω))) ≤C (4.4) VL²(0,T;L²(Ω))≤C, (4.5) where C is a nonnegative constant which depends only on fL²(Q_T) + gL²(Q_T). We omit here the proof of this classical result.

The existence of a solution to (2.16) is not obtained by applying the Schauder ﬁxed-point theorem on the system (4.1), (4.2) because of the lack of uniqueness on this system. Nevertheless, we establish in the next subsection a ﬁxed-point theorem for a regularized problem of (4.2) by introducing a map Tε such that Tε(s) = sε, where sε solves (4.10). Passing to the limit with respect toε, we obtain an existence result to (2.16).

4.1. Existence of a regularized solutions to (4.2). For (p,V) deﬁned by (4.1), we prove that the solution of (4.2) exists and belongs to K. For that, we consider a regularization of p and V by convolution. First we extendp and V outsideQT by functions still denotedpand V respectively inL²(0, T;H¹(R^N))∩ C⁰(0, T;L²(R^N)) and (L²(0, T;L²(R^N)))^N.

Next, we deﬁnepε=ρεp,Vε =ρεV, andgε=ρεgsuch thatρε(t, x) = ρ(t/ε, x/ε)/ε^N⁺¹ and ρ∈ C^∞(R^N+1) withρ≥0 and

R^N+1ρ(t, x)dx dt= 1.

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We clearly have that pε (respectively Vε) belongs to C^∞(QT) (respectively (C^∞(QT))^N), and when εtends to zero, we have

pε−→p strongly inL²(0, T;H¹(Ω))∩L^∞(0, T;L²(Ω)), (4.6) Vε−→V strongly in (L²(QT))^N, (4.7) gε≥0, and gε−→g strongly inL²(QT). (4.8) In the same way, we construct a regularized initial condition s0,ε such that

s0,ε −→s0 strongly inL²(Ω). (4.9) We introduce now the regularized problem







φ(x)∂tsε−sεdiv(Vε) + div(ν(sε)Vε)

−div(K(x)αη(s)∇sε)−γK(x)α(sε)∇sε· ∇pε= (1−sε)gε

K(x)αη(s)∇sε·n= 0 on ΣT, sε(0, x) =s0,ε(x) in Ω.

(4.10) Lemma 4.1. For any ε > 0, there exists a unique solution sε in L²(0, T; H²(Ω))∩L^∞(0, T;H¹(Ω))to the regularized problem (4.10).

Proof. We refer to [14] for the existence parts. For uniqueness we consider s¹_ε and s²_ε, two solutions of (4.10). We setu=s¹_ε−s²_ε; thenu satisﬁes

φ∂tu−udiv(Vε) + div((ν(s¹_ε)−ν(s²_ε))Vε)−div(Kαη(s)∇u)

−γKα(sε)∇u· ∇pε−γK(α(s¹_ε)−α(s²_ε))∇s¹_ε· ∇pε=−ugε (4.11) with

Kαη(s)∇u·n= 0 on ΣT, u(0, x) = 0 in Ω.

Multiplying the equation (4.11) by u and integrating over Ω, we get 1

2 d dt

Ω

φ|u|²dx+

Ω

αη(s)K∇u· ∇u dx

=

Ω

|u|²divVεdx+

Ω

(ν(s¹_ε)−ν(s²_ε))Vε· ∇u dx+γ

Ω

Kα(sε)∇u· ∇pεu +γ

Ω

K(α(s¹_ε)−α(s²_ε))∇s¹ε· ∇pεu dx−

Ω

gε|u|²dx.

We have the following estimates:

Ω

|u|²divVεdx≤ divVεL^∞(Ω)u²_L2(Ω); using that fact thatν is a Lipschitz-continuous function, we have

Ω

(ν(s¹_ε)−ν(s²_ε))Vε· ∇u dx≤cVεL^∞(Ω)uL²(Ω)∇uL²(Ω)

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≤δ∇u²_L2(Ω)+cVε²_L∞(Ω)u²_L2(Ω), whereδ is a given parameter. In the same way we have

γ

Ω

Kα(sε)∇u· ∇pεu dx≤δ∇u²_L2(Ω)+cK∇pε²_L∞(Ω)u²_L2(Ω). We have H¹²(Ω)⊂L³(Ω) and H¹(Ω)⊂L⁶(Ω) for N = 3, and by an inter- polation argument we have

γ

Ω

K(α(s¹_ε)−α(s²_ε))∇s¹ε· ∇pεu dx

≤cK∇pεL^∞(Ω)∇s¹_εL²(Ω)α(s¹_ε)−α(s²_ε)L³(Ω)uL⁶(Ω)

≤cK∇pεL^∞(Ω)∇s¹εL²(Ω)u_L¹²2(Ω)u_H³²1(Ω)

≤c(δ)K∇pε⁴_L∞(Ω)∇s¹ε⁴_L2(Ω)u²_L2(Ω)+δ∇u²_L2(Ω). The last term is simply estimated as

Ω

gε|u|²dx≤ gεL^∞(Ω)u²_L2(Ω). Finally, we deduce from the coercivity of Kthat

1 2

d dt

Ω

φ|u|²dx+ (k0η−3δ)

Ω

∇u· ∇u dx≤h(t)u²_L2(Ω),

where h(t) ∈ L¹(0, T). Choose now δ such that k0η−3δ ≥ 0; Gronwall’s lemma allows us to conclude the Lipschitz property of (4.10), ensuring the

uniqueness result.

Now, we handle some estimates on the solution to (4.2) independent ofε.

Lemma 4.2. The solution to equation (4.2)satisﬁes (i) 0≤sε(t, x)≤1, for almost every t, x.

(ii) The sequence (sε)ε is uniformly bounded in L²(0, T;H¹(Ω))∩L^∞(0, T;L²(Ω)).

(iii) The sequence (φ(x)∂tsε)ε is uniformly bounded in L¹(0, T; (W^1,q(Ω))) for q > N.

(iv) The sequence (sε)ε is relatively compact inL²(0, T;L²(Ω)).

Proof. The aim of the ﬁrst part is to prove the physical relevance of solutions. Let us write precisely the extensions ˜s, ˜ν, and ˜α, which are assumed to be Lipschitz continuous on [0,1], of the functionss,ν, andα on R:

˜

s(s) =sif 0≤s≤1, s(s) = 0 if˜ s≤0, and ˜s(s) = 1 if s≥1,

˜

ν(s) =ν(s) if 0≤s≤1, ν(s) = 0 if˜ s≤0, and ˜ν(s) = 1 if s≥1,

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˜

α(s) =α(s) if 0≤s≤1, α(s) = 0 if˜ s≤0, and ˜α(s) =α(1) if 1≤s.

Thus, the equation (4.2) is replaced by

φ(x)∂tsε−˜s(sε)div(Vε) + div(˜ν(sε)Vε)

−div(K(x)αη(s)∇sε)−γK(x) ˜α(sε)∇sε· ∇pε= (1−sε)gε. (4.12) Multiplying this equation by −s⁻ε = ^s^ε^−|₂^s^ε^| and integrating over Ω, one has

1 2

d dt

Ωφ|s⁻ε|²dx+

Ω

˜

s(sε)div(Vε)s⁻_ε dx+

Ω

˜

ν(sε)Vε· ∇s⁻ε dx +

Ω

K(x)αη(s)∇s⁻ε · ∇s⁻ε dx+γ

Ω

K(x) ˜α(sε)∇sε· ∇pεs⁻_ε dx

=−

Ω

(1−sε)gεs⁻_ε dx.

Since ˜s(s) = ˜ν(s) = ˜α(s) = 0 for s ≤0, and according to the positivity of the fourth term of the left-hand side and the positivity of gε, one obtains

d dt

Ω

φ|s⁻ε|²dx≤0.

Integrating this inequality over (0, t) one deducess⁻ε(·, t)L²(Ω)≤ s⁻₀L²(Ω)

for all t∈(0, T); since s0 ≥0 in Ω, sε(·, t)≥0 in Ω for all t∈(0, T).

Multiplying again the equation (4.12) by (sε−1)⁺ and integrating over Ω we have

1 2

d dt

Ω

φ|(sε−1)⁺|²dx−

Ω

˜

s(sε)div(Vε)(sε−1)⁺dx

−

Ω

˜

ν(sε)Vε· ∇(sε−1)⁺dx+

Ω

K(x)αη(s)∇(sε−1)⁺· ∇(sε−1)⁺dx

−γ

Ω

K(x) ˜α(sε)∇sε· ∇pε(sε−1)⁺dx=−

Ω

|(sε−1)⁺|²gεdx.

Using now the fact that ˜s(s) = ˜ν(s) = 1 for s≥1 and Vε·n= 0 on∂Ω, we

have

Ω

div(Vε)(sε−1)⁺dx+

Ω

Vε· ∇(sε−1)⁺dx= 0;

since ˜α(s) =α(1) for s≥1, the ﬁfth term is bounded as follows:

γ

Ω

K(x) ˜α(sε)∇sε· ∇pε(sε−1)⁺dx

≤Cα(1)||∇(sε−1)⁺||L²(Ω)||(sε−1)⁺||L²(Ω).