ON A DEGENERATE PARABOLIC SYSTEM FOR COMPRESSIBLE, IMMISCIBLE…saad/cv/GaluSaadADEORIGINAL.pdf · Advances in Diﬀerential Equations November/December 2004 Volume 9, Numbers 11-12,

Advances in Differential Equations November/December 2004Volume 9, Numbers 11-12, Pages 1235–1278

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

POROUS MEDIA

Cedric Galusinski and Mazen SaadMathematiques Appliquees de Bordeaux UMR CNRS 5466

Universite Bordeaux 1, 351 cours de la Liberation, 33405 Talence, France

(Submitted by: Roger Temam)

Abstract. The aim of this paper is to analyze a model of a degener-ate nonlinear system arising from immiscible, compressible, two-phase,three-dimensional flows occurring in porous media. A degenerateweighted formulation is introduced to take into account the compress-ibility and the degeneracy. Two existence results of such a degenerateweak solution are introduced. The first result concerns the existenceof solutions under a reasonable assumption on the capillary pressure.This condition allows the degeneracy only where one of the two phasesis exclusively present. The second result establishes, for suitable initialdata, the existence of solutions when the degeneracy occurs where oneor the other phase is exclusively present. Nevertheless, for suitable ini-tial data, a classical weak solution is obtained when the degeneracy isnot too strong and occurs only where the injected phase is exclusivelypresent.

1. Introduction and model

We analyze a degenerate nonlinear system modeling a three-dimensionaldisplacement of two immiscible, compressible fluids in a porous medium.The equations describing the immiscible displacement of two compressiblefluids are given by the mass conservation of each phase:

φ(x)∂t(ρisi)(t, x)+div(ρiVi)(t, x)+ρisig(t, x) = ρisi f(t, x) i = 1, 2, (1.1)

where φ is the porosity of the medium, and ρi and si are respectively thedensity and the saturation of the ith fluid. The velocity of each fluid Vi isgiven 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.

1235

1236 Cedric Galusinski and Mazen Saad

where K is the permeability tensor of the porous medium, ki the relativepermeability of the ith phase, µi the i-phase’s viscosity (considered to beconstant) and pi the i-phase’s pressure. The effects of gravity are neglected.Here the functions f and g are respectively the injection and productionterms. Note that the saturation of the injected fluids is given (it appears inthe term ρis

i f in equation (1.1)), but the saturation of the produced fluid

is an unknown which appears in the terms ρisig.By definition of saturation, one has

s1(t, x) + s2(t, x) = 1. (1.3)

We consider s = s1 to be the saturation of the water phase or the gas phase,and s2 = 1 − s that 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 (dp12

ds (s) ≥ 0, for all s ∈[0, 1]). In this paper, the forced displacement of fluids is modeled. It isused in many enhanced recovery processes: a fluid such as water (or gas) isinjected into some wells in a reservoir while the resident hydrocarbons areproduced from other wells. Consequently, we consider the injected saturations1 = 1 and s

2 = 0.By these formulations, we can see that the unknown functions are the

saturation of one phase and one pressure. Let us denote as usualMi(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 1st phase,1 − ν(s) = M2(s)/M(s) the fractional flow of the 2nd phase,V = V1 + V2 the total velocity.

As in [5] and [16], we can express the total velocity in terms of p2 and p12.We have

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

defining a function p(s) such that dpds (s) = ν(s)dp12

ds (s), and setting p = 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)

where α(s) = M(s)ν(s)(1 − ν(s))dp12

ds (s) ≥ 0.

Degenerate parabolic system 1237

In [6] and [3] the authors consider an exponent state law to describe thedisplacement of one compressible fluid. We consider as in Aziz and Settari([4], p. 13) that it is possible to assume that the fluid compressibility isalso an exponent state law. The density of fluid depends on the pressure ofthe corresponding fluid, but taking advantage of the fact that this functionvaries slowly with capillary pressure (see Chavent et al. ([5], Chapter 4) formore details). Moreover, we assume that ρi = 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 twofluids. 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 consideringone fluid being injected (i.e., s

1 = 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 equa-tions 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]. Theequations (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 isconsistent with the incompressible model in the sense that the system is thelimiting form of the compressible system as the compressibility factor γ ofthe 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


and we consider no flux boundary conditions; in summary, we investigate thefollowing 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,

where n 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 mathemati-cal point of view. On the other hand, several papers are devoted to thenumerical [6] and mathematical ([1], [2], [3], [8], [11], [12]) studies of com-pressible, miscible fluids. Our model is close to that of [2]. Nevertheless, themodel arising from miscible fluids leads to a nondegenerate parabolic sys-tem. Furthermore, the authors consider a constant fluid viscosity, allowing amore regular pressure field. As a matter of fact, their model on the pressurereduces to

φ(x)a(u)∂tp − div(k(x)∇p) = qi − qs,

which allows one to exhibit a regular pressure field. In our model of immis-cible 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 thenlimited by an (L2(QT ))N regularity on the velocity field V = −KM(s)∇p.This makes the nonlinear terms such as s divV and γK(x)α(s)∇s ·∇p moredifficult to control in the equation governing the saturation s. 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 solu-tions are called degenerate weak solutions.

Models dealing with immiscible fluids (so with degenerate dissipativeterms) have already been studied from a mathematical point of view, butonly 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 termswe met in our model since divV = f − g and γ = 0. Classical weak solu-tions of parabolic problems can be defined for these models of incompressiblefluids.


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

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

(H1) ∃φ0 > 0, φ1 > 0 such that φ0 ≤ φ(x) ≤ φ1.(H2) ∃k0 > 0, k∞ > 0 such that (K(x)ξ, ξ) ≥ k0|ξ|2, for all ξ ∈ RN , and

almost every x ∈ Ω, ‖K‖(L∞(Ω))N×N ≤ k∞.(H3) M ∈ C0([0, 1]), ∃m0 > 0 such that M(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 statethe assumption on the function α:

(H7a) α ∈ C1([0, 1]), α(s) > 0 for 0 < s < 1, α(0) > 0, α(1) = 0,there exist α0 > 0, 0 < r2 ≤ 2, s1 < 1, m1 and M1 > 0 such thatα(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 exist r1 > 0 and m1, M1 > 0 such thatm1r1s

r1−1 ≤ α′(s) ≤ M1r1sr1−1 for all 0 ≤ s ≤ 1.

(H7c) α ∈ C1([0, 1]), α(s) > 0 for 0 < s < 1, α(0) = 0, α(1) = 0,there exist r1 > 0, r2 > 0, s1 < 1, m1 and M1 > 0 such thatm1r1s

r1−1 ≤ α′(s) ≤ M1r1sr1−1, for all s ∈ [0, s1].

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

1−ν(s)1−s = 0.

The assumption (H7a) indicates essentially that the function α does notdegenerate near s = 0 and behaves like the function (1 − s)r2 near s = 1.The assumption (H7b) indicates essentially that the function α behaves likethe function sr1 near s = 0 and is not vanishing at s = 1. These situationscorrespond to some models of capillary pressure which ensure these condi-tions. In [9], the authors assume a similar condition on the incompressiblecompositional model; see also [4]. The assumption (H7c) corresponds es-sentially to models on relative permeabilities. For example, if we consider


Corey’s model with relative permeabilities r′1 > 1 and r′2 > 1, we have

1 − ν(s) = (1 − s)r′2/(µ2

µ1sr′1 + (1 − s)r′2),

α(s) = sr′1(1 − s)r′2dp12

ds(s)/(µ2s

r′1 + µ1(1 − s)r′2).

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

In the next section we introduce first the existence of classical weak so-lutions under assumptions (H1)–(H6) and (H7a) for a particular choice ofinitial 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, wegive 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

2for 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 initialcondition 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 ,

γ〈φ∂tp, ψ〉 +∫

QT

K(x)M(s)∇p · ∇ψ dx dt =∫

QT

(f − g)ψ dx dt, (2.1)

−∫

QT

φ(x)s∂tχ dx dt −∫

Ωφ(x)s0(x)χ(0, x) dx +

∫QT

V · ∇(sχ) dx dt

−∫

QT

ν(s)V · ∇χ dx dt +∫

QT

Kα(s)∇s · ∇χ dx dt

− γ

∫QT

K(x)α(s)∇s · ∇pχ dx dt =∫

QT

(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.


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 thedegenerate system (1.15) in the sense of Definition 2.1.

Note that we can show that there exist c1, c2 > 0 such that

c1 ln(1 − s) ≤ L(s) ≤ c2 ln(1 − s), ∀s ∈ [0, 1).

2.2. Weak degenerate solutions. Denote by

β(s) = sr−1, h(s) =∫ s

0β(y)dy,

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

βθ(s) = sr−1+θ, hθ(s) =∫ s

0βθ(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)β′ 12 (s)∇s ∈ (L2(QT ))N ,

γ〈φ∂tp, ψ〉 +∫

QT


QT

(f − g)ψ dx dt, (2.3)

for all ψ ∈ L2(0, T ;H1(Ω)); we define

F (s, p, χ) = −∫

QT

φ(x)hθ(s)∂tχ dx dt −∫

Ωφ(x)hθ(s0(x))χ(0, x)dx

+∫

QT

V · ∇(sβθ(s)χ) dx dt −∫

QT

ν(s)V · ∇(βθ(s)χ) dx dt

+∫

QT

α(s)K∇s · ∇(βθ(s)χ) dx dt − γ

∫QT

K(x)α(s)∇s · ∇pβθ(s)χ dx dt

−∫

QT

(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 and χ ≥ 0, (2.5)


and furthermore,

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

([0, T ) × Ω

)\Qε.

(2.6)

Note that the properties on the function α12 (s)β′ 12 (s)∇s are a fortiori

valid with βθ instead of β. This is enough to give a sense to each term inthe above formulation.

The compressibility of fluids makes the system (1.15) highly nonlinear.Using classical energy estimates, these nonlinear terms are not controlledby the degenerate dissipative term in the equation governing the saturationvariable s. A classical weak formulation for parabolic problems cannot beestablished here. That is why we introduce a degenerate (at s = 0) weightβθ(s). We then call such solutions degenerate weak solutions. Remark thatthese solutions are not affected by this weight in regions where s = 0.

Theorem 2.2. Let (H1)–(H6) and (H7b) hold. For every γ > 0, there existsat least one degenerate weak solution to the degenerate system (1.15) in thesense of Definition 2.2.

Remark 2.1. In Definition 2.2, the neighborhood Qε could include a partof the boundary of (0, T ) × Ω, even for small ε. Consequently, the initialcondition and the boundary condition could be slightly violated. Even ifwe 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, theboundary condition, or the initial condition on a small (as small as wanted)ε neighborhood. Remark also that in this definition, the solution satisfies avariational inequality where the test function does not depend on ε.

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

jθ,λ(s) =

βθ(s) for 0 ≤ s ≤ s1

βθ(s1)(1 − s1)1−r′2−λ(1 − s)

r′2−1+λ for s1 ≤ s

, (2.7)

where r′ ≥ max(2, r2) (r2 is introduced in assumption (H7c)), and denoteby Jθ,λ its primitive

Jθ,λ(s) =∫ s

0jθ,λ(y)dy. (2.8)

To simplify notation, we denote J = J0,0 and j = j0,0.Let us also define µ and G by

µ(s) = β(s) for 0 ≤ s ≤ s1, (2.9)


where µ is continuous and

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

Also,

G(s) =∫ s

0µ(y)dy. (2.11)

According to the assumption (H7c), we have lims→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 and λ ≥ 7r2 +6− r′2 , let (H1)–(H6) and

(H7c) hold, and assume that the initial condition s0 satisfies G(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)µ′ 12 (s)∇s ∈ L2(QT ), J(s) ∈ L2(0, T ;H1(Ω)),

for all ψ ∈ L2(0, T ;H1(Ω)),

γ〈φ∂tp, ψ〉 +∫

QT


QT

(f − g)ψ dx dt; (2.14)

for all ε > 0, there exists Qε ⊂ QT , meas(Qε) < ε, such that, for allχ ∈ C1([0, T ) × Ω) with supp χ ⊂

([0, T ) × Ω

)\Qε,

−∫

QT

φ(x)Jθ,λ(s)∂tχ dx dt −∫

Ωφ(x)Jθ,λ(s0(x))χ(0, x)dx

+∫

QT

V · ∇(sjθ,λ(s)χ) dx dt −∫

QT

ν(s)V · ∇(jθ,λ(s)χ) dx dt

+∫

QT

Kα(s)∇s · ∇(jθ,λ(s)χ) dx dt − γ

∫QT

K(x)α(s)∇s · ∇pjθ,λ(s)χ dx dt

=∫

QT

(1 − s)gjθ,λ(s)χ dx dt. (2.15)


Note that each term in the formulas (2.14)–(2.15) is well defined. Inparticular, we give some details on the third and fourth integrals of theequality (2.15). From the definition of the function jθ,λ, all integrals aredefined in the region QT ∩ s < s1; we focus our attention on the integralsin the region QT ∩ s > s1. We have∫

QT∩s≥s1

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

)dx dt

=∫

QT∩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 thatfor s > 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 · ∇χ‖L1(QT ) + ‖V · ∇J(s)‖L1(QT )‖χ‖L∞(QT )

),

where c is a constant. In the same way, let us write∫QT∩s≥s1

Kα(s)∇s · ∇(jθ,λ(s)χ) dx dt

=∫

QT∩s≥s1Kα(s)j′θ,λ(s)∇s · ∇sχ dx dt +

∫QT∩s≥s1

Kα(s)∇Jθ,λ(s) · ∇χ dx dt,

and from the definitions of the function µ and jθ,λ, one obtains that thereexists a constant c(s1) such that α(s)j′θ,λ(s) ≤ c(s1)α(s)µ′(s) for all s > s1;then the above integrals make sense.

Remark that for r2 ≤ 2, which is relevant for most applications, the weakformulation (2.15) is not degenerated for s > s1 even though the functionα is degenerated around s = 1. This is due to the fact that the right-handside of the saturation equation of (1.15) is also degenerated at s = 1, sothat a test function (µ) blowing up at s = 1 can be used. A nondegeneratedissipative estimate follows around s = 1 α

12 (s)µ′ 12 (s)∇s ∈ L2(QT ). But this

test function obliges us to consider G(s0) ∈ L1(Ω). In order to overcome thisassumption, a weak degenerate solution at s = 1 could be tried, but the test


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

Theorem 2.3. Let (H1)–(H6) and (H7c) hold, and assume that the initialcondition s0 satisfies G(s0) ∈ L1(Ω). For every γ > 0, there exists at leastone degenerate weak solution to the degenerate system (1.15) in the sense ofDefinition 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 dominatedby C(1 − s)−1 for s ≥ s1 (see (2.13)). We remark also that this assumptionallows us to conserve, along the time, this property:

G(s(t)) ∈ L1(Ω) for almost every t ∈ (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 thedegeneracy of the function α, we introduce a modified problem where α isreplaced by αη(s) = α(s) + η only in the diffusion terms, with η > 0.

2.3. Weak solutions for the nondegenerate problem. We consider thenondegenerate 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 thefollowing assumption on α (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 orα(1) > 0.

Proposition 2.1. Let (H1)–(H6) and (H8) hold. For any η > 0, thereexists (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

γ〈φ∂tpη, ψ〉 +∫

QT

K(x)M(sη)∇pη · ∇ψ dx dt =∫

QT

(f − g)ψ dx dt (2.17)

1246 Cedric Galusinski and Mazen Saad∫Ω

φ(x)sη(T, x)χ(T, x)dx −∫

Ωφ(x)s0(x)χ(0, x)dx +

∫QT

sη∂tχ dx dt

+∫

QT

sηVη · ∇χ dx dt +∫

QT

Vη · ∇sηχ dx dt −∫

QT

ν(sη)Vη · ∇χ dx dt

+∫

QT

Kαη(sη)∇sη.∇χ dx dt − γ

∫QT

Kα(sη)∇sη · ∇pηχ dx dt

=∫

QT

(1 − sη)gχ dx dt (2.18)

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

The end of this paper is organized as follows: the next section is devotedto 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. Let X and Y be Banach spaces such that

X ⊂ Lr ⊂ Y with compact embedding X into Lr, r ≥ 1. (3.1)

Denote by W a set of functions and by φ a 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 Lq(0, T ;Lr) ∩ L1(0, T ;X),(2) φ∂tv is bounded in L1(0, T ;Y ), for v ∈ W .

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

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

∀η > 0, ∃M such that ‖v‖Lr ≤ η‖v‖X + M‖φv‖Y . (3.3)

Note that if v is bounded in X, then φv is bounded in Y , because

‖φv‖Y ≤ c1‖φv‖Lr ≤ c1φ1‖v‖Lr ≤ c2φ1‖v‖X .

Denote Em = v ∈ Lr; ‖v‖Lr < η+m‖φv‖Y . The sequence Em of open setsin Lr is increasing, and the union of Em covers Lr. Denote by S the unitsphere of X, S = v ∈ X; ‖v‖X = 1, which is relatively compact in Lr;then there exists a finite M such that S ⊂ EM , which yields

‖v‖Lr < η + M‖φv‖Y , ∀v ∈ S,

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


(i) W is bounded in Lq(0, T ;Lr) ∩ L1loc(0, T ;X),

(ii) ∀0 < t1 < t2 < T, ‖Thv − v‖L1(t1,t2;Lr) −→ 0 as h → 0 uniformly forv ∈ W ,

where Thv(t, x) = v(t + h, x).The first condition (i) is straightforward. To obtain the second condition

(ii), owing to (3.3), for every η > 0, there exists M such that

‖Thv − v‖L1(t1,t2;Lr) ≤ η‖Thv − v‖L1(t1,t2;X) + M‖φ(Thv − v)‖L1(t1,t2;Y ),

using now that the translations in time are continuous in L1(0, T ), andLemma 4 ([17]), we have

‖Thv − v‖L1(t1,t2;Lr) ≤ 2η‖v‖L1(0,T ;X) + Mh‖φ∂tv‖L1(0,T ;Y ) ≤ c3η + c4h.

Given ε > 0, for η = ε2c3

and h = ε2c4

it yields ‖Thv−v‖L1(t1,t2;Lr) ≤ ε, whichproves that W is relatively compact in Lp(0, T ;Lr), 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 isused.

Lemma 3.2. Let Ω be a bounded set in RN , and (fε)ε and (gε)ε be twosequences satisfying

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

(3.4)Then ∫

Ωfε(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 strongconvergence on (gε)ε in L1(Ω) for the first integral and by Lebesgue’s theoremfor the second.


4. Existence for the nondegenerate problem

The proof is carried out using the Schauder fixed-point theorem. We firstgive an outline. For physical relevance we should have 0 ≤ s(t, x) ≤ 1 foralmost every (t, x) ∈ QT . For this purpose, we introduce the following closedsubset K 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 parameter η.

Let s ∈ K be fixed, and let p be the solution of the parabolic equationγφ(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 solution s(t, x) to the equationφ(x)∂ts − sdiv(V) + div(ν(s)V)

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

(4.2)

For any s fixed in K, there exists a unique solution p of (4.1) in L2(0,T ;H1(Ω))∩C0(0, T ;L2(Ω)) satisfying

‖p‖L∞(0,T ;L2(Ω)) + ‖p‖L2(0,T ;H1(Ω)) ≤ C (4.3)

‖φ∂tp‖L2(0,T ;(H1(Ω))′) ≤ C (4.4)

‖V‖L2(0,T ;L2(Ω)) ≤ C, (4.5)

where C is a nonnegative constant which depends only on ‖f‖L2(QT ) +‖g‖L2(QT ). We omit here the proof of this classical result.

The existence of a solution to (2.16) is not obtained by applying theSchauder fixed-point theorem on the system (4.1), (4.2) because of the lack ofuniqueness on this system. Nevertheless, we establish in the next subsectiona fixed-point theorem for a regularized problem of (4.2) by introducing amap Tε such that Tε(s) = sε, where sε solves (4.10). Passing to the limitwith respect to ε, we obtain an existence result to (2.16).

4.1. Existence of a regularized solutions to (4.2). For (p,V) definedby (4.1), we prove that the solution of (4.2) exists and belongs to K. Forthat, we consider a regularization of p and V by convolution. First weextend p and V outside QT by functions still denoted p and V respectivelyin L2(0, T ;H1(RN )) ∩ C0(0, T ;L2(RN )) and (L2(0, T ;L2(RN )))N .

Next, we define pε = ρεp, Vε = ρεV, and gε = ρεg such that ρε(t, x) =ρ(t/ε, x/ε)/εN+1 and ρ ∈ C∞(RN+1) with ρ ≥ 0 and

∫RN+1 ρ(t, x) dx dt = 1.


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

pε −→ p strongly in L2(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 in L2(QT ). (4.8)

In the same way, we construct a regularized initial condition s0,ε such that

s0,ε −→ s0 strongly in L2(Ω). (4.9)

We introduce now the regularized problemφ(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 considers1ε and s2

ε, two solutions of (4.10). We set u = s1ε − s2

ε; then u satisfies

φ∂tu − udiv(Vε) + div((ν(s1ε) − ν(s2

ε))Vε) − div(Kαη(s)∇u)

− γKα(sε)∇u · ∇pε − γK(α(s1ε) − α(s2

ε))∇s1ε · ∇pε = −ugε (4.11)

withKαη(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

12

d

dt

∫Ω

φ|u|2 dx +∫

Ωαη(s)K∇u · ∇u dx

=∫

Ω|u|2divVε dx +

∫Ω(ν(s1

ε) − ν(s2ε))Vε · ∇u dx + γ

∫Ω

Kα(sε)∇u · ∇pεu

+ γ

∫Ω

K(α(s1ε) − α(s2

ε))∇s1ε · ∇pεu dx −

∫Ω

gε|u|2 dx.

We have the following estimates:∣∣∣ ∫Ω|u|2divVε dx

∣∣∣ ≤ ‖divVε‖L∞(Ω)‖u‖2L2(Ω);

using that fact that ν is a Lipschitz-continuous function, we have∣∣∣ ∫Ω(ν(s1

ε) − ν(s2ε))Vε · ∇u dx

∣∣∣ ≤ c‖Vε‖L∞(Ω)‖u‖L2(Ω)‖∇u‖L2(Ω)


≤ δ‖∇u‖2L2(Ω) + c‖Vε‖2

L∞(Ω)‖u‖2L2(Ω),

where δ is a given parameter. In the same way we have∣∣∣γ ∫Ω

Kα(sε)∇u · ∇pεu dx∣∣∣ ≤ δ‖∇u‖2

L2(Ω) + c‖K∇pε‖2L∞(Ω)‖u‖2

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(s1ε) − α(s2

ε))∇s1ε · ∇pεu dx

∣∣∣≤ c‖K∇pε‖L∞(Ω)‖∇s1

ε‖L2(Ω)‖α(s1ε) − α(s2

ε)‖L3(Ω)‖u‖L6(Ω)

≤ c‖K∇pε‖L∞(Ω)‖∇s1ε‖L2(Ω)‖u‖

12

L2(Ω)‖u‖

32

H1(Ω)

≤ c(δ)‖K∇pε‖4L∞(Ω)‖∇s1

ε‖4L2(Ω)‖u‖2

L2(Ω) + δ‖∇u‖2L2(Ω).

The last term is simply estimated as∣∣∣ ∫Ω

gε|u|2 dx∣∣∣ ≤ ‖gε‖L∞(Ω)‖u‖2

L2(Ω).

Finally, we deduce from the coercivity of K that

12

d

dt

∫Ω

φ|u|2 dx + (k0η − 3δ)∫

Ω∇u · ∇u dx ≤ h(t)‖u‖2

L2(Ω),

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

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

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 in L2(0, T ;L2(Ω)).

Proof. The aim of the first part is to prove the physical relevance of solu-tions. Let us write precisely the extensions s, ν, and α, which are assumedto be Lipschitz continuous on [0, 1], of the functions s, ν, and α on R:

s(s) = s if 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,


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 = sε−|sε|2 and integrating over Ω, one has

12

d

dt

∫Ω

φ|s−ε |2 dx +∫

Ωs(sε)div(Vε)s−ε dx +

∫Ω

ν(sε)Vε · ∇s−ε dx

+∫

ΩK(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 ofthe 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 deduces ‖s−ε (·, t)‖L2(Ω) ≤ ‖s−0 ‖L2(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 have12

d

dt

∫Ω

φ|(sε − 1)+|2 dx −∫

Ωs(sε)div(Vε)(sε − 1)+ dx

−∫

Ων(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)+|2gε dx.

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

Ωdiv(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).


We then conclude thatd

dt

∫Ω

φ|(sε − 1)+|2 dx ≤ C2α2(1)η

||(sε − 1)+||2L2(Ω);

since s0 ≤ 1 in Ω , sε(·, t) ≤ 1 in Ω for all t ∈ (0, T ).For the second part, we multiply by sε, we use div(Vε(sε)2) = 2sε∇sε ·

Vε + (sε)2divVε, and we integrate over Ω:

12

d

dt

∫Ω

φ|sε|2 dx +∫

Ω2sε∇sε · Vε dx −

∫Ω

ν(sε)Vε · ∇sε dx

+∫

ΩK(x)αη(s)∇sε · ∇sε dx − γ

∫Ω

K(x)α(sε)∇sε · ∇pεsε dx

=∫

Ω(1 − sε)sεgε dx.

Using the facts that |sε| ≤ 1, |ν(sε)| ≤ 1, and K and α are bounded, wehave the following estimates:∣∣∣ ∫

Ω2sε∇sε · Vε dx +

∫Ω

ν(sε)Vε · ∇sε dx∣∣∣ ≤ 3‖Vε‖L2(Ω)‖∇sε‖L2(Ω)

≤ C‖Vε‖2L2(Ω) + δ‖∇sε‖2

L2(Ω);

similarly,∣∣∣γ ∫Ω

K(x)α(sε)∇sε · ∇pεsε dx∣∣∣ ≤ C‖∇pε‖L2(Ω)‖∇sε‖L2(Ω)

≤ C‖∇pε‖2L2(Ω) + δ‖∇sε‖2

L2(Ω)

and ∣∣∣ ∫Ω(1 − sε)sεgε dx

∣∣∣ ≤ 2|Ω| 12 ‖gε‖L2(Ω).

Using the coercivity of K we deduce that

12

d

dt

∫Ω

φ|sε|2 dx + (k0η − 2δ)∫

Ω|∇sε|2 dx ≤ C(‖gε‖2

L2(Ω) + ‖Vε‖2L2(Ω)).

Choosing now δ such that k0η − 2δ > 0 and integrating in time, we have

12

∫Ω

φ|sε(t, ·)|2 dx + (k0η − 2δ)∫ t

0

∫Ω|∇sε|2 dx

≤ ‖s0‖2L2(Ω) + C(‖gε‖2

L2(QT ) + ‖Vε‖2L2(QT ));

from (4.7) and (4.8) the right-hand side is independent of ε, which completesthe second part of this lemma.


For the third part, we prove that φ(x)∂tsε is uniformly bounded in L2(0, T ;(H1(Ω))′) + L1(QT ). From (4.2), one has for all χ ∈ L2(0, T ;H1(Ω)) ∩L∞(QT )

〈φ∂tsε, χ〉 −∫

QT

sεdiv(Vε)χ dx dt −∫

QT

ν(sε)Vε · ∇χ dx dt

+∫

QT

K(x)αη(s)∇sε · ∇χ dx dt −∫

QT

γK(x)α(sε)∇sε · ∇pεχ dx dt

=∫

QT

(1 − sε)gεχ dx dt.

We have sεdiv(Vε)χ = div(sεVεχ)−χVε ·∇sε−sεVε ·∇χ, and the functionssε, ν, α, and K are bounded; thus, a straightforward estimate leads to

|〈φ∂tsε, χ〉| ≤ C(‖Vε · ∇sε‖L1(QT ) + ‖Vε‖L2(QT ) + ‖∇sε‖L2(QT )

+ ‖∇pε‖L2(QT ) + ‖gε‖L2(QT ))(‖χ‖L2(0,T ;H1(Ω) + ‖χ‖L∞(QT);

then the result follows from the uniform bounds of ∇pε, Vε, and ∇sε in(L2(0, T ;L2(Ω)))N .

The third part (iii) of the lemma is a direct consequence of the embeddingof the Sobolev space W 1,q(Ω) ⊂ H1(Ω)∩L∞(Ω) for q > N , and consequently

L∞(0, T ;W 1,q(Ω)) ⊂ L2(0, T ;H1(Ω)) ∩ L∞(0, T ;L2(Ω)).

The fourth part is a consequence of Lemma 3.1. Indeed, from the parts (ii)and (iii), it suffices to take X = H1(Ω), r = 2, and Y = (W 1,q(Ω))′; then thesequence (sε)ε is relatively compact in Lp(0, T ;L2(Ω)) for all 1 ≤ p < ∞. 4.2. Fixed-point method for the regularized problem. The goal ofthis section is to prove the Schauder fixed-point theorem on Tε (see Step 3).

Let (sn)n ∈ K and s ∈ K such that sn −→ s in L2(QT ) as n −→ ∞.We first prove (see Step 1 and Step 2) the following, for a subsequenceTε(sn) → Tε(s).Step 1. Convergence of (Vn) towards V in L2(QT ). From estimates(4.3)–(4.4), the sequence

(pn)n is uniformly bounded in L2(0, T ;H1(Ω)) ∩ L∞(0, T ;L2(Ω))

and (φ∂tpn)n is uniformly bounded in L2(0, T ; (H1(Ω))′); thus, we can ex-tract a subsequence such that

pn −→ p strongly in L2(QT ), and a.e. in QT

pn −→ p weakly in L2(0, T ;H1(Ω)),φ∂tpn −→ φ∂tp weakly in L2(0, T ; (H1(Ω)′)).

(4.13)

The strong convergence of (Vn)n is a consequence of this lemma.


Lemma 4.3. The sequence (∇pn)n converges strongly to ∇p in L2(QT ).

Proof. Subtracting the relations satisfied by pn and p, we have

γφ∂t(pn − p) − div(KM(sn)(∇pn −∇p)) − div(K∇p(M(sn) − M(s))) = 0

KM(sn)∇pn · n = KM(s)∇p · n = 0 on ΣT ; (pn − p)(0, x) = 0 in Ω.

Taking pn − p as test function in the above equation,

γd

dt

∫Ω

φ|pn − p|2 dx +∫

ΩKM(sn)∇(pn − p) · ∇(pn − p)dx

= −∫

ΩK(M(sn) − M(s))∇p · ∇(pn − p)dx. (4.14)

Estimating the right-hand side by the Cauchy-Schwartz inequality and usingthe coercivity of K (see (H3)), we obtain after integration with respect to t,

k0m0

2

∫QT

|∇(pn − p)|2 dx dt ≤ C

∫QT

(M(sn) − M(s))|∇p|2 dx dt. (4.15)

We apply Lebesgue’s theorem to the right-hand side as n goes to ∞. Thiscompletes the proof of Lemma 4.3.

Finally, we have Vn = −KM(sn)∇pn, sn converges to s for almost every(t, x) ∈ QT , and the function M is continuous and bounded; then, thanks toLesbegue’s theorem, we deduce

Vn −→ V = −KM(s)∇p strongly in L2(QT ). (4.16)

Step 2. Convergence of (sn) towards s in C0(0, T ;L2(Ω)). Lemma 4.2remains valid for the sequence (sn)n; then

sn → s in L2(QT ) strongly, (4.17)

for a subsequence still denoted (sn)n.

Lemma 4.4. The sequence (sn)n is a Cauchy sequence in C0(0, T ;L2(Ω)).

Proof. We set wn,m = sn − sm and Vεn,m = Vε

n −Vεm = ρε Vn − ρε Vm,

where sn and sm are solutions to (4.10), and (pn,Vn) and (pm,Vm) solve(4.1). Subtracting the equations obtained respectively by sn and sm, weobtain

φ∂twn,m − wn,mdivVεn − smdivVε

n,m + div((ν(sn) − ν(sm))Vεn)

+ div(ν(sm)Vεn,m) − div(K(αη(sn) − αη(sm))∇sn)

− div(Kαη(sm)∇wn,m) − γK(α(sn)∇sn − α(sm)∇sm) · ∇pεn

− γKα(sm)∇sm · (∇pεn −∇pε

m) = −wn,mg,

with pεn = ρε pn and pε

m = ρε pm.


Multiplying by wn,m, which is a bounded function, and integrating overQt = (0, t) × Ω, we have∫

Ωφw2

n,m(t, x) dx +∫

Qt

Kαη(sm)∇wn,m · ∇wn,m dx dt =∫Qt

wn,mdivVεnwn,m dx dt +

∫Qt

smdivVεn,mwn,m dx dt

+∫

Qt

(ν(sn) − ν(sm))Vεn · ∇wn,m dx dt +

∫Qt

ν(sm)Vεn,m · ∇wn,m dx dt

−∫

Qt

K(αη(sn) − αη(sm))∇sn · ∇wn,m dx dt (4.18)

+ γ

∫Qt

K(α(sn)∇sn − α(sm)∇sm) · ∇pεnwn,m dx dt

+ γ

∫Qt

Kα(sm)∇sm · (∇pεn −∇pε

m)wn,m dx dt −∫

Qt

wn,mgεwn,m dx dt.

We denote by Ii, i =1,8, respectively the integrals on the right-hand side ofthe equality (4.18). The first integral I1 is estimated after an integration byparts:

|I1| ≤ 2∫

QT

|Vεn · ∇wn,mwn,m| dx dt

≤ k0η

8‖∇wn,m‖2

L2(QT ) + C‖wn,mVεn‖2

L2(QT ).

The term |I1| is then bounded by a term absorbed by the damping term of(4.18) and a term whose limit is zero as n and m go to infinity (by Lemma 3.2)as the wn,m go to zero almost everywhere according to (4.17).

We estimate I2 in the same way as I1:

|I2| ≤∫

QT

|smVεn,m · ∇wn,m| dx dt +

∫QT

|wn,m∇sm · Vεn,m| dx dt

≤ ‖Vεn,m‖L2(QT )(‖∇wn,m‖L2(QT ) + ‖∇sm‖L2(QT )).

As Vεn,m goes to zero in L2(QT ) as n and m go to infinity, |I2| also goes to

zero as n and m go to infinity.

|I3| ≤ C‖(ν(sn) − ν(sm))Vεn‖2

L2(QT ) +k0η

8‖∇wn,m‖2

L2(QT ).

Using the facts that ν(sn)− ν(sm) goes to zero almost everywhere as n andm go to infinity, ν is bounded, and Vε

n converges strongly in L2(QT ), wethen can apply Lemma 3.2 to the first term of the right-hand side. The lastterm is smaller than the damping term of (4.18). For the fourth integral I4,


we conclude in the same way since Vεn,m goes to zero almost everywhere. It

is the same for I5 since αη(sn) − αη(sm) goes to zero almost everywhere asn and m go to infinity. The sixth integral is estimated below:

|I6| ≤k0η

8‖∇wn,m‖2

L2(QT ) + C‖∇pεnwn,m‖2

L2(QT );

by Lemma 3.2, the last term goes to zero since wn,m goes to zero almosteverywhere, |wn,m| ≤ 2, and (4.6) holds.

Also, I7 can be treated for example as I6. Finally, the last integral I8 isestimated as

|I8| ≤ 2‖g‖L2(QT )‖wn,m‖2L2(QT ) −→ 0, as n, m → ∞.

From the above estimates and the coercivity of K, we deduce from (4.18)that ∫

Ωw2

n,m(t, x)dx −→ 0 as n, m → +∞, for all t > 0,

which completes the proof. Step 3. The Schauder fixed-point theorem. Thanks to the uniquenessresult of (4.10) in Lemma 4.1, it is easy to show that the whole sequence sn

converges to s. This shows the continuity of Tε on K.From Lemma 4.2, Step 1 and Step 2, Tε is a continuous mapping from K

into itself with precompact image.From the Schauder fixed-point theorem, there exists sε ∈ K such that

Tε(sε) = sε.

4.3. Existence of a solution to (2.16). In the previous subsection, forall fixed ε > 0, we have shown the existence of a solution (pε,Vε, sε) to

γφ(x)∂tpε + divVε = f − g, Vε = −KM(sε)∇pε

φ(x)∂tsε − sεdiv(Vεε) + div(ν(sε)Vε

ε)−div(K(x)αη(sε)∇sε) − γK(x)α(sε)∇sε · ∇pε

ε = (1 − sε)gε

K(x)α(sε)∇sε · n = 0 on ΣT , sε(0, x) = s0(x) in Ω,

Vε · n = 0 on ΣT , pε(0, x) = p0(x) in Ω,

(4.19)

where Vεε = ρε Vε and pε

ε = ρε pε. According to Lemma 4.2, up to asubsequence, one can assume, as ε −→ 0,

sε −→ s strongly in L2(QT ), and a.e. in QT

sε −→ s weakly in L2(0, T ;H1(Ω)),0 ≤ s(t, x) ≤ 1 a.e. in QT ,

(4.20)

and using the convergences (4.6) and (4.7), we obtain that the limit s satisfies(4.2) in the sense of distribution.


Arguing as in Step 1 of the previous subsection, we also have the followingconvergence:

pε −→ p strongly in L2(0, T ;H1(Ω)), and a.e. in QT

φ∂tpε −→ φ∂tp weakly in L2(0, T ; (H1(Ω)′). (4.21)

Vε −→ V = −KM(s)∇p strongly in (L2(QT ))N . (4.22)

It is now possible to pass to the limit as ε goes to zero. The limit (p,V, s)satisfies the system (2.16). This completes the proof of Proposition 2.1.

5. Proof of Theorem 2.1

We have shown in Proposition 2.1 that the nondegenerate system admits asolution. Here we are going to obtain estimates on the solutions independentof the regularization η. First, it is easy to see that the estimates (4.3)–(4.5)are independent of η:

‖pη‖L∞(0,T ;L2(Ω)) + ‖pη‖L2(0,T ;H1(Ω)) ≤ C (5.1)

‖φ∂tpη‖L2(0,T ;(H1(Ω))′) ≤ C (5.2)

‖Vη‖L2(0,T ;L2(Ω)) ≤ C, (5.3)

where C is a nonnegative constant which depends only on ‖f‖L2(QT ) +‖g‖L2(QT ).

Let us define the function κ(s) = L′(s) ∈ C1[0, 1); that is, κ(s) = s for0 ≤ s ≤ s1,

κ′(s) = (ν(s) − s)−1κ(s), ∀s ∈ [s1, 1). (5.4)

Exactly as for µ in (2.13), we have

κ(s1)(1 − s1)(1 − s)−1 ≤ κ(s) ≤ k1κ(s1)(1 − s1)(1 − s)−1, ∀ s ≥ s1. (5.5)

Lemma 5.1. Assuming (H1)–(H6) and (H7a), and that L(s0) belongs toL1(Ω), the solutions to the saturation equation (2.16) satisfy

(i) 0 ≤ sη(t, x) ≤ 1, for almost every t, x in QT .(ii) The sequence (∇sη)η is uniformly bounded in (L2(QT ))N .(iii) The sequence (L(sη))η is uniformly bounded in L∞(0, T ;L1(Ω)).(iv) The sequence (

√ηκ′(sη)∇sη)η is uniformly bounded in (L2(QT ))N .

(v) The sequence (φ(x)∂tsη)η is uniformly bounded inL1(0, T ; (W 1,q(Ω))′) for q > N .

(vi) The sequence (sη)η is relatively compact in L2(QT ).


Proof. The first part, (i), is obtained in Lemma 4.2. Next, multiplying(2.16) by κ and integrating over Ω, one gets

d

dt

∫Ω

φL(sη) dx +∫

ΩKα(sη)κ′(sη)∇sη · ∇sη dx + η

∫Ω

Kκ′(sη)∇sη · ∇sη dx

= −∫

ΩsηVη · ∇κ(sη) dx +

∫Ω

ν(sη)Vη · ∇κ(sη) dx −∫

Ωκ(sη)Vη · ∇sη dx

+∫

ΩγKα(sη)∇sη · ∇pηκ(sη) dx +

∫Ω(1 − sη)κ(sη)g dx. (5.6)

The whole integral appearing in (5.6) can be split as∫Ω

=∫

Ω∩s<s1+

∫Ω∩s≥s1

,

so that the analysis developed in the previous section for a nondegeneratedissipative system is applied to the terms of the form

∫Ω∩s<s1. A new one

is applied to terms of the form∫Ω∩s>s1. We then obtain the estimate

d

dt

∫Ω

φL(sη) dx +k0α0

2

∫Ω∩s<s1

|∇sη|2 dx

+ k0

∫Ω∩s≥s1

α(sη)κ′(sη)|∇sη|2 dx + ηk0

∫Ω

κ′(sη)|∇sη|2 dx

≤ C(‖Vη‖2

L2(Ω) + ‖∇pη‖2L2(Ω) + ‖g‖L2(Ω)

)+

∫Ω∩s≥s1

((ν(sη) − sη)κ′(sη) − κ(sη))Vη · ∇sη dx

+∫

Ω∩s≥s1γKα(sη)∇sη · ∇pηκ(sη) dx +

∫Ω∩s≥s1

(1 − sη)κ(sη)g dx,

(5.7)

where C is a constant independent of η. Note that from the definition ofκ (5.4), the first integral on Ω ∩ s ≥ s1 of the right-hand side of (5.7)vanishes.

With the help of estimate (5.5), we have (1 − sη)κ(sη) ≤ k1κ(s1)(1 − s1)for s > s1, and there exist c1 > 0 and c2 > 0 such that

c1(1 − sη)−2 ≤ κ′(s) ≤ c2(1 − sη)−2, ∀s > s1.

Then,

d

dt

∫Ω

φL(sη) dx +k0α0

2

∫Ω∩s≤s1

|∇sη|2 dx


+ c1

∫Ω∩s≥s1

α(sη)(1 − sη)−2|∇sη|2 dx + ηk0

∫Ω


≤ c(‖Vη‖2L2(Ω) + ‖∇pη‖2

L2(Ω) + ‖g‖L2(Ω))

+ δ

∫Ω∩s≥s1

α(sη)κ2(sη)|∇sη|2 + c(δ)∫

Ω∩s≥s1|∇pη|2 dx

+ k1κ(s1)(1 − s1)∫

Ω∩s≥s1g dx. (5.8)

According to (5.5), choosing δ small enough so that

δκ2(s) ≤ c1

2(1 − s)−2,

the estimate (5.8) becomes

d

dt

∫Ω

φL(sη) dx +k0α0

2

∫Ω∩s≤s1

|∇sη|2 dx

+c1

2

∫Ω∩s≥s1

α(sη)(1 − sη)−2|∇sη|2 dx + ηk0

∫Ω


≤ c(‖Vη‖2L2(Ω) + ‖∇pη‖2

L2(Ω) + ‖g‖L2(Ω)).

Integrating over t between (0, T ), as α(s)(1 − s)−2 ≥ m1(1 − s1)r2−2 for alls ≥ s1 since r2 ≤ 2, we deduce (ii), (iii), and (iv) of this lemma.

For the fifth part, we prove that

φ(x)∂tsη is uniformly bounded in L2(0, T ; (H1(Ω))′) + L1(QT ).

Let χ ∈ L2(0, T ;H1(Ω)) ∩ L∞(QT ); we return to the saturation equation(2.16) to estimate

〈φ∂tsη, χ〉 = −∫

QT

Vη · ∇(sηχ) dx dt +∫

QT


−∫

QT

α(sη)K∇sη · ∇χ dx dt − η

∫QT

K∇sη · ∇χ dx dt

+∫

QT

γα(sη)K∇sη · ∇pηχ dx dt +∫

QT

(1 − sη)gχ dx dt.

(5.9)

By virtue of (ii) and the (L2(QT ))N bound on Vη, we have the desiredbound of all the terms of the above right-hand side; that is,

|〈φ∂tsη, χ〉| ≤ C(‖χ‖L2(0,T ;H1(Ω)) + ‖χ‖L∞(QT )).

Then part (v) of this lemma is a direct consequence of the Sobolev embeddingL∞(0, T ;W 1,q) ⊂ L2(0, T ;H1(Ω)) ∩ L∞(QT ), for q > N .

The last part, (vi), is a consequence of the compactness Lemma 3.1.


The proof of Lemma 5.1 is then complete. Remark that the same estimates as those of Lemma 4.3, based on the

strong convergence of the sequence (sη)η in L2(QT ), lead to the followingconvergence:

pη −→ p strongly in L2(0, T ;H1(Ω)), and a.e. in QT ,

Vη −→ V strongly in (L2(QT ))N .(5.10)

To complete the proof of Theorem 2.1, we deduce from (5.2) and Lemma 5.1that we can extract a subsequence such that

Vη −→ V strongly in (L2(QT ))N , and a.e. in QT ,

pη −→ p strongly in L2(0, T ;H1(Ω)),φ∂tpη −→ φ∂tp weakly in L2(0, T ; (H1(Ω)′),sη −→ s strongly in L2(QT ), and a.e. in QT ,

0 ≤ s(t, x) ≤ 1 a.e. in QT ,

∇sη −→ ∇s weakly in (L2(QT ))N .

(5.11)

We want to pass to the limit as η goes to zero in the weak formulation,


QT


QT

(f − g)ψ dx dt (5.12)

−∫

QT

φ(x)sη∂tχ dx dt −∫

Ωφ(x)s0(x)χ(0, x) dx

+∫

QT

Vη · ∇sηχ dx dt +∫

QT

Vη · ∇χsη dx dt −∫

QT


+ η

∫QT

K∇sη · ∇χ dx dt +∫

QT

α(sη)K∇sη · ∇χ dx dt

− γ

∫QT

K(x)α(sη)∇sη · ∇pηχ dx dt =∫

QT

(1 − sη)gχ dx dt (5.13)

for all ψ ∈ L2(0, T ;H1(Ω)) and χ ∈ C1([0, T )×Ω) with supp χ ⊂ [0, T )×Ω.In the above formulation the convergence results (5.11) allow us to passto the limit as η goes to zero for all the terms. This ends the proof ofTheorem 2.1.


We handle some estimates on the solutions of (4.2) independent of η.Recall that β is defined as β(s) = sr−1 and h is the primitive of β definedby h(s) =

∫ s0 β(y) dy.


Lemma 6.1. Let (H1)–(H6) and (H7b) hold. The solutions of the saturationequation (2.16) satisfy

(i) 0 ≤ sη(t, x) ≤ 1, for almost every (t, x) in QT .

(ii) The sequences (sr−22

η α12 (sη)∇sη)η, (sr−1

η ∇sη)η, and (α(sη)∇sη)η areuniformly bounded in (L2(QT ))N .

(iii) The sequence (h(sη))η is uniformly bounded in L∞(0, T ;L1(Ω)).

(iv) The sequence (η12 s

r−22

η ∇sη)η is uniformly bounded in (L2(QT ))N .(v) The sequence (φ(x)∂th(sη))η is uniformly bounded in

L1(0, T ; (W 1,q(Ω))′) for q > N .(vi) The sequence (h(sη))η (respectively (sη)η) is relatively compact in

L2(0, T ;L2(Ω)) (respectively in L2(0, T ;L2(Ω))).

Proof. The first part, (i), is obtained in Lemma 4.2. Next, multiplying theequation of saturation defined by (2.16) by β(sη) and integrating over Ω,one gets

d

dt

∫Ω

φh(sη) dx + (r − 1)∫

ΩKα(sη)sr−2

η ∇sη · ∇sη dx

+ η(r − 1)∫

ΩKsr−2

η ∇sη · ∇sη dx = −r

∫Ω

sr−1η Vη · ∇sη dx

+ (r − 1)∫

Ων(sη)sr−2

η Vη · ∇sη dx +∫

ΩγKα(sη)∇sη · ∇pηβ(sη) dx

+∫

Ω(1 − sη)β(sη)g dx. (6.1)

Using assumptions (H7b) and (H4), the two integrals on the right-hand sideare estimated as follows:∣∣∣ − r

∫Ω

sr−1η Vη · ∇sη dx + (r − 1)

∫Ω

ν(sη)sr−2Vη · ∇sη dx∣∣∣

≤ c‖Vη‖2L2(Ω) +

k0m1

4‖sr−1

η ∇sη‖2L2(Ω),

where k0 is defined in (H2) and m1 in (H7b).The third integral is estimated below,∣∣∣ ∫Ω

γKα(sη)∇sη · ∇pηβ(sη) dx∣∣∣ ≤ c‖∇pη‖2

L2(Ω) +k0

4‖α 1

2 (sη)sr−22

η ∇sη‖2L2(Ω),

and the fourth integral is bounded by∣∣∣ ∫Ω(1 − sη)β(sη)g dx

∣∣∣ ≤ c‖g‖L2(Ω).


Finally, from (6.1), the coercivity of K, and the above estimates, we deduce

d

dt

∫Ω

φh(sη) dx + k0

∫Ω

α(sη)sr−2η |∇sη|2 dx + ηk0

∫Ω

sr−2η |∇sη|2 dx

≤ c(‖Vη‖2L2(Ω) + ‖∇pη‖2

L2(Ω) + ‖g‖L2(Ω)) (6.2)

+k0

4‖α 1

2 (sη)sr−22

η ∇sη‖2L2(Ω) +

k0m1

4‖sr−1

η ∇sη‖2L2(Ω).

For 0 ≤ s ≤ 1, assumption (H7b) ensures that m1sr ≤ m1s

r1 ≤ α(s); then

k0m1

4

∫Ω

s2r−2η |∇sη|2 dx ≤ k0

4

∫Ω

α(sη)sr−2η |∇sη|2 dx.

Returning to equation (6.2), one gets

d

dt

∫Ω

φh(sη) dx +k0

2

∫Ω

α(sη)sr−2η |∇sη|2 dx + ηk0

∫Ω

sr−2η |∇sη|2 dx

≤ c(‖Vη‖2L2(Ω) + ‖∇pη‖2

L2(Ω) + ‖g‖L2(Ω)).

Integrating over t between (0, T ), we deduce parts (ii), (iii), and (iv) of thislemma, except the estimate on the term∫

QT

|α(sη)∇sη|2 dx dt ≤ M1

∫QT

α(sη)sr1η |∇sη|2 dxdt

≤ M1

∫QT

α(sη)sr−2η |∇sη|2 dx dt,

since r1 ≥ r − 2.For the fifth part, we prove that

φ(x)∂th(sη) is uniformly bounded in L2(0, T ; (H1(Ω))′) + L1(QT ).

Let χ ∈ L2(0, T ;H1(Ω)) ∩ L∞(QT ); multiplying the saturation equation byβ(sη)χ and integrating over QT , one has

〈φ∂th(sη), χ〉 = −∫

QT

Vη · ∇(sηβ(sη)χ) dx dt

+∫

QT

ν(sη)Vη · ∇(β(sη)χ) dx dt −∫

QT

α(sη)K∇sη · ∇(β(sη)χ) dx dt

− η

∫QT

K∇sη · ∇(β(sη)χ) dx dt +∫

QT

γα(sη)K∇sη · ∇pηβ(sη)χ dx dt

+∫

QT

(1 − sη)gβ(sη)χ dx dt. (6.3)


We are going to give estimates on each term on the right-hand side:∫QT

Vη · ∇(sηβ(sη)χ) dx dt

= r

∫QT

sr−1η Vη · ∇sηχ dx dt +

∫QT

srηVη · ∇χ dx dt

≤ r‖Vη‖L2(QT )‖sr−1η ∇sη‖L2(QT )‖χ‖L∞(QT ) + C‖Vη‖L2(QT )‖∇χ‖L2(QT ).

The second integral on the right-hand side of (6.3) is estimated as followsby using (i) of Lemma 6.1 and from assumption (H4) (ν(s) ≤ Cs):∣∣∣ ∫

QT

ν(sη)Vη · ∇(β(sη)χ) dx dt∣∣∣

≤ C(‖sr−1

η ∇sη‖L2(QT ) + ‖Vη‖L2(QT )

)(‖χ‖L∞(QT ) + ‖∇χ‖L2(QT )

).

Similarly, we have∣∣∣ ∫QT

α(sη)K∇sη · ∇(β(sη)χ) dx dt∣∣∣

≤C(‖α 1

2 (sη)sr−22

η ∇sη‖2L2(Ω)+ ‖sr−1

η ∇sη‖L2(QT )

)(‖χ‖L∞(QT )+ ‖∇χ‖L2(QT )

).

We estimate the fourth integral on the right-hand side of (6.3):∣∣∣η ∫QT

K∇sη · ∇(β(sη)χ) dx dt∣∣∣

≤ C(‖η 1

2 sr−22

η ∇sη‖2L2(QT ) + η‖sr−1

η ∇sη‖L2(QT )

)(‖χ‖L∞(QT ) + ‖∇χ‖L2(QT )

).

Finally, the two last terms in (6.3) are estimated as∣∣∣∣∫QT

γα(sη)K∇sη · ∇pηβ(sη)χ dx dt +∫

QT

(1 − sη)gβ(sη)χ dx dt

∣∣∣∣≤ C

(‖sr−1

η ∇sη‖L2(QT )‖∇pη‖L2(QT ) + ‖g‖L1(QT )

)‖χ‖L∞(QT ).

We plug these estimates into (6.3); according to (ii) and (iv) of this lemma,we obtain

|〈φ∂th(sη), χ〉| ≤ C(‖χ‖L2(0,T ;H1(Ω)) + ‖χ‖L∞(QT ));

then part (v) of this lemma is a direct consequence of the Sobolev embedding

L∞(0, T ;W 1,q(Ω)) ⊂ L2(0, T ;H1(Ω)) ∩ L∞(QT ), for q > N.

The last part, (vi), is a consequence of the compactness Lemma 3.1. Indeed,we have

‖∇h(sη)‖L2(QT ) = r‖sr−1η ∇sη‖L2(QT );


using part (ii) we deduce that the sequence (h(sη))η is uniformly boundedin L2(0, T ;H1(Ω)). The function h is bounded on [0, 1]; consequently, thesequence (h(sη))η is uniformly bounded in L∞(QT ). Moreover, (h(sη))η sat-isfies part (v) of Lemma 6.1, so Lemma 3.1 allows us to conclude that (h(sη))η

is relatively compact in L2(0, T ;L2(Ω)). Since the function h is increasingand h−1 is continuous, (sη)η is relatively compact in L2(0, T ;L2(Ω)). Theproof of Lemma 6.1 is complete.

The strong convergence on velocity is also valid under assumption (H7b):

pη −→ p strongly in L2(0, T ;H1(Ω))), and a.e. in QT ,

Vη −→ V strongly in (L2(QT ))N .(6.4)

Now, we are concerned with an almost-everywhere convergence of the gra-dient of the saturation weighted by a degenerate function of the saturation.

Lemma 6.2. Let q = 3r1 +2, where r1 is defined in assumption (H7b). Thesequence ((sq

ηα(sη)∇sη))η is a Cauchy sequence in measure.

Proof. First of all, we will derive an essential estimate to the proof. LetA be a primitive of α. Define the function B(s) = A2(s), and denote byb(s) = 2A(s)α(s) its derivative. Let µ > 0; we set the cut function Tµ

defined by Tµ(s) = min(µ,max(−µ, s)) for all s ∈ R, and we denote Θµ(s) =∫ s0 Tµ(τ) dτ .Consider the sequences (sη)η and (sη′)η′ satisfying equation (2.16); let

Aη,η′ = A(sη) − A(sη′) and Bη,η′ = B(sη) − B(sη′).Subtracting the saturation equations satisfied by sη and sη′ multiplied by

b(sη)Tµ(Bη,η′) and b(sη′)Tµ(Bη,η′) respectively, we obtain∫Ω

φ(x)Θµ(Bη,η′(t, x)) dx (6.5)

+∫

Qt

(K∇A(sη) · ∇(b(sη)Tµ(Bη,η′)) − K∇A(sη′) · ∇(b(sη′)Tµ(Bη,η′))

)dx dt

= −∫

Qt

(Vη · ∇

(sηb(sη)Tµ(Bη,η′)

)− Vη′ · ∇

(sη′b(sη′)Tµ(Bη,η′)

))dx dt

+∫

Qt

(ν(sη)Vη · ∇

(b(sη)Tµ(Bη,η′)

)− ν(sη′)Vη′ · ∇

(b(sη′)Tµ(Bη,η′)

))dx dt

− η

∫Qt

K∇sη · ∇(b(sη)Tµ(Bη,η′)) dx dt + η′∫

Qt

K∇sη′ · ∇(b(sη′)Tµ(Bη,η′)) dx dt

+∫

Qt

(γK∇A(sη) · ∇pηb(sη)Tµ(Bη,η′) − γK∇A(sη′) · ∇pη′b(sη′)Tµ(Bη,η′)

)dx dt

+∫

Qt

((1 − sη)gb(sη) − (1 − sη′)gb(sη′)

)Tµ(Bη,η′) dx dt.


According to Lemma 6.1 the sequences

(∇A(sη))η, (∇B(sη))η, (∇b(sη))η are uniformly bounded in (L2(QT ))N .(6.6)

Indeed, we have ∇A(sη) = α(sη)∇sη; the result is straightforward from part(ii) of Lemma 6.1. Next, ∇B(sη) = 2A(sη)∇A(sη), and we have |A(sη)| ≤M1. Finally, ∇b(sη) = 2α(sη)∇A(sη) + 2α′(sη)A(sη)∇sη, we have |α(sη)| ≤M1 and α′(sη)A(sη) ≤ M2

1 s2r1η ≤ M2

1m1

α(sη), and consequently |∇b(sη)| ≤2(M1 + M2

1m1

)|∇A(sη)|, which establishes (6.6).We denote by Ii, i = 1, 6, the integrals which appear on the right-hand

side of (6.5). To estimate the first integral I1, we compute

Vη · ∇(sηb(sη)Tµ(Bη,η′)) − Vη′ · ∇(sη′b(sη′)Tµ(Bη,η′)) =

(Vη · ∇B(sη) + Vη · ∇b(sη)sη − Vη′ · ∇B(sη′) − Vη′ · ∇b(sη′)sη′)Tµ(Bη,η′)

+ (Vη − Vη′)sηb(sη) · ∇Tµ(Bη,η′) + (sηb(sη) − sη′b(sη′))Vη′ · ∇Tµ(Bη,η′),

and consequently we have∣∣∣ ∫Qt

(Vη · ∇(sηb(sη)Tµ(Bη,η′)) − Vη′ · ∇(sη′b(sη′)Tµ(Bη,η′)) dx dt∣∣∣

≤ ‖(Vη · ∇B(sη) + Vη′ · ∇B(sη′))Tµ(Bη,η′)‖L1(QT )

+ ‖(sηVη · ∇b(sη) + sη′Vη′ · ∇b(sη′))Tµ(Bη,η′)‖L1(QT )

+ ‖b‖L∞(R)‖(Vη − Vη′)‖L2(QT )‖∇Tµ(Bη,η′)‖L2(QT )

+ ‖(sηb(sη) − sη′b(sη′))Vη′ · ∇Tµ(Bη,η′)‖L1(QT );

from (6.4) the sequence (Vη)η converges strongly in (L2(QT ))N ; from (6.6),|Tµ(Bη,η′(t, x))| ≤ µ for almost every (t, x) ∈ (0, T ) × Ω and Bη,η′ convergesto zero almost everywhere in (0, T ) × Ω. Lemma 3.2 ensures that the firstand second terms on the right-hand side of the above estimate go to zeroas η and η′ tend to zero. The same lemma can be applied to the last term,arguing that (sηb(sη)−sη′b(sη′)) is bounded and goes to zero for almost every(t, x) ∈ (0, T ) × Ω. Obviously, from (6.4) the third term of the right-handside goes to zero.

The same arguments based on Lemma 3.2, the strong convergence estab-lished in (6.4), statement (6.6), and the bounds obtained in Lemma 6.1 allowus to handle the integrals I2, I5, and I6. Now, we will show that there existsa constant C independent of η and η′ such that |I3| ≤ Cη. For that, fromthe definition of the function b, we have∫

Qt

K∇sη · ∇(b(sη)Tµ(Bη,η′)) dx dt = 2∫

Qt

K∇A(sη) · ∇A(sη)Tµ(Bη,η′) dx dt


+ 2∫

Qt

A(sη)K∇A(sη) · ∇Tµ(Bη,η′) dx dt

+ 2∫

Qt

A(sη)α′(sη)K∇sη · ∇sηTµ(Bη,η′) dx dt;

the two first integrals of the right-hand side are obviously uniformly boundedindependent of η and η′. This is also true for the third one, because A(sη)·α′(sη) ≤ M2

1

m21α2(sη). In the same manner, we have |I4| ≤ Cη′.

Set Wµ(η, η′) to be the right-hand side of (6.5); then Wµ(η, η′) goes tozero when η and η′ go to zero, for all µ > 0. Denote by V (µ) the first termon the left-hand side of (6.5); we have |V (µ)| ≤ Cφ1meas(Ω)µ, which goesto zero as µ goes to zero uniformly on η and η′.

For instance, from (6.5) we have shown the essential result∫QT


)dx dt

= Wµ(η, η′) + V (µ). (6.7)

For a given function f from (0, T )×Ω to R and a real r, one denotes by f ≥r the set (t, x) ∈ (0, T )×Ω; f(t, x) ≥ r. Denote u(s) =

∫ s0 b(z)A(z)α(z)dz.

A second step consists of establishing that for δ > 0,

meas|∇u(sη) −∇u(sη′)| ≥ δ −→ 0,

as η and η′ go to zero. We remark that |∇u(sη) − ∇u(sη′)| ≥ δ ⊂ A1 ∪A2 ∪ A3 ∪ A4, where A1 = |∇A(sη)| ≥ k, A2 = |∇A(sη′)| ≥ k, A3 =|Bη,η′ | ≥ µ, and A4 = |∇u(sη) − ∇u(sη′)| ≥ δ ∩ |∇A(sη)| ≤ k ∩|∇A(s′η)| ≤ k ∩ |Bη,η′ | ≤ µ. From (6.6), we conclude easily for the firsttwo sets. Indeed, one has

meas(A1) ≤1k‖∇A(sη‖L1(Q)

≤ C

k,

and an analogous estimate holds for A2. Hence, by choosing k large enough,meas(A1) + meas(A2) is arbitrarily small. Similarly, one gets

meas(A3) ≤1µ‖Bη,η′‖

L1(QT ),

which, for µ > 0 fixed, tends to 0 when η, η′ → 0. Then, it remains to handlethe set A4. We have

δmeas(A4) ≤∫A4

|∇u(sη) −∇u(sη′)|2 dx dt


≤ 2∫A4

|b(sη)A(sη)∇Aη,η′ |2 dx dt

+ 2∫A4

|b(sη)A(sη) − b(sη′)A(sη′)|2|∇Aη′ |2 dx dt

≤ 2M3

1

k0

∫A4

b(sη)A(sη)K∇Aη,η′ · ∇Aη,η′ dx dt

+ 2k2

∫QT

|b(sη)A(sη) − b(sη′)A(sη′)|2 dx dt.

The parameter k is chosen large enough and fixed; then last term of the aboveinequality goes to zero as η, η′ → 0. We denote this term by Wk(η, η′), and

δmeas(A4) ≤ (6.8)

2M31

k0

∫QT

b(sη)(A(sη) + A(sη′))K∇Aη,η′ · ∇Aη,η′1|Bη,η′ |≤µ dx dt + Wk(η, η′).

Next, from the definition of the function B we have

K∇Aη,η′ · ∇(b(sη)Tµ(Bη,η′))

= b(sη)K∇Aη,η′ · ∇Tµ(Bη,η′) + K∇Aη,η′ · ∇b(sη)Tµ(Bη,η′)

= b(sη)(A(sη) + A(sη′))K∇Aη,η′ · ∇Aη,η′1|Bη,η′ |≤µ

+ b(sη)Aη,η′K∇Aη,η′ · ∇(A(sη) + A(sη′))1|Bη,η′ |≤µ

+ K∇Aη,η′ · ∇b(sη)Tµ(Bη,η′). (6.9)

Note that |Aη,η′ |1|Bη,η′ |≤µ ≤ √µ and |Tµ(Bη,η′)| ≤ µ; then the last two

terms tend to zero as µ goes to zero in L1(QT ) and therefore can be includedin the function V (µ).

Now, we are concerned with the left-hand side of (6.9). We have∫Qt

K∇Aη,η′ · ∇(b(sη)Tµ(Bη,η′)) dx dt =∫Qt


)dx dt

−∫

Qt

K(b(sη) − b(sη′))∇A(sη′) · ∇Bη,η′1|Bη,η′ |≤µ dx dt

−∫

Qt

K∇A(sη′) · ∇(b(sη) − b(sη′))Tµ(Bη,η′) dx dt. (6.10)

Note that |(b(sη) − b(sη′))1|Bη,η′ |≤µ| ≤ 2M1√

µ and |Tµ(Bη,η′)| ≤ µ; thenthe last two terms tend to zero as µ goes to zero uniformly with respect to η


and η′, and therefore can be included in the function V (µ). Finally, estimate(6.8) can be estimated from (6.10), (6.9), and (6.7) as

δmeas(A4) ≤ Wµ(η, η′) + V (µ) + Wk(η, η′);

for all ε > 0, there exists k0 such that for all k ≥ k0

meas(A1) + meas(A2) ≤ε

2,

there exists µ0 such that for all µ ≤ µ0 we have

|V (µ)| ≤ δε

8,

and there exists η0 such that for all η, η′ ≤ η0 we have

|Wµ0(η, η′)| + |Wk0(η, η′)| ≤ δε

8and

‖Bη,η′‖L1(QT )

≤ µ0ε

4.

Then, for all ε > 0, for all δ > 0, there exists η0 such that for all η, η′ ≤ η0,

meas|∇u(sη) −∇u(sη′)| ≥ δ ≤ ε. (6.11)

That is, the sequence (b(sη)α(sη)A(sη)∇sη)η is a Cauchy sequence in mea-sure.

Let us show, for all ε′ > 0, for all δ′ > 0, there exists η1 such that for allη, η′ ≤ η1,

meas|sqη∇A(sη) − sq

η′∇A(sη′)| ≥ δ′ ≤ ε′.

We have

sqη∇A(sη) − sq

η′∇A(sη′) = sqη∇Aη,η′ + (sq

η − sqη′)∇A(sη′).

The parameter q is chosen so that q = 3r1+2; then sq = s3r1+2 ≤ c1b(s)A(s),where c1 = r2

1

2m31

and

|sqη∇Aη,η′ | ≤ c1b(sη)A(sη)|∇Aη,η′ |

≤ c1|∇u(sη) −∇u(sη′)| + c1|(b(sη)A(sη) − b(s′η)A(sη′))∇A(sη′)|,so that

|sqη∇A(sη) − sq

η′∇A(sη′)| ≤ |(sqη − sq

η′)∇A(sη′)| + c1|∇u(sη) −∇u(sη′)|+ c1|(b(sη)A(sη) − b(sη′)A(sη′))∇A(sη′)|. (6.12)

The first and last terms of the right-hand side go to zero in L1(QT ) as η andη′ go to zero. Also, from (6.11), one proves Lemma 6.2.


To complete the proof of Theorem 2.2, we deduce from estimates (5.1)–(5.3), Lemma 6.1, and Lemma 6.2 that we can extract a subsequence suchthat

Vη −→ V strongly in (L2(QT ))N , and a.e. in QT ,

pη −→ p strongly in L2(0, T ;H1(Ω)),

φ∂tpη −→ φ∂tp weakly in L2(0, T ; (H1(Ω)′)),

sη −→ s strongly in L2(QT ), and a.e. in QT ,

0 ≤ s(t, x) ≤ 1 a.e. in QT , (6.13)

hθ(sη) −→ hθ(s) strongly in L2(QT ),

sr−22

η α(sη)12∇sη −→ s

r−22 α(s)

12∇s weakly in (L2(QT ))N ,

sqηα(sη)∇sη −→ sqα(s)∇s a.e. in (0, T ) × Ω, with q = 3r1 + 2.

We want to pass to the limit as η goes to zero in the weak formulation


QT


QT

(f − g)ψ dx dt (6.14)

−∫

QT

φ(x)hθ(sη)∂tχ dx dt −∫

Ωφ(x)hθ(s0(x))χ(0, x) dx

+∫

QT

Vη · ∇sηβθ(sη)χ dx dt +∫

QT

Vη · ∇βθ(sη)sηχ dx dt

+∫

QT

Vη · ∇χsηβθ(sη) dx dt −∫

QT

ν(sη)Vη · ∇βθ(sη)χ dx dt

−∫

QT

ν(sη)Vη · ∇χβθ(sη) dx dt + η

∫QT

K∇sη · ∇βθ(sη)χ dx dt

+ η

∫QT

K∇sη · ∇χβθ(sη) dx dt +∫

QT

α(sη)K∇sη · ∇βθ(sη)χ dx dt

+∫

QT

α(sη)K∇sη ·∇χβθ(sη) dx dt−γ

∫QT

K(x)α(sη)∇sη ·∇pηβθ(sη)χ dx dt

=∫

QT

(1 − sη)gβθ(sη)χ dx dt (6.15)

for all ψ ∈ L2(0, T ;H1(Ω)), and χ ∈ C1([0, T )×Ω) with suppχ ⊂ [0, T )×Ω.In the above formulation, for θ ≥ 0, the convergence results (6.13) allow usto pass to the limit as η goes to zero, up to a subsequence, for all termsexcept the eighth and tenth. For that, for θ > 0, the eighth integral passes


to the limit zero in the following way:

η

∣∣∣∣∫QT

K∇sη · ∇βθ(sη)χ dx dt

∣∣∣∣ ≤ Cη

∫QT

sr−2+θη |∇sη|2|χ| dx dt (6.16)

≤ Cη

∫QT∩sη≤η

12r

sr−2+θη |∇sη|2 dx dt ‖χ‖L∞(QT )

+ Cη

∫QT∩sη>η

12r

sr−2+θη |∇sη|2 dx dt ‖χ‖L∞(QT )

≤ηθ2r C‖η 1

2 s(r−2)/2η ∇sη‖L2(QT )‖χ‖L∞(QT )+Cη

12 ‖sr−1

η ∇sη‖L2(QT )‖χ‖L∞(QT ).

It is now an obvious conclusion that the eighth term in the equation (6.15)goes to zero. Now, we are concerned with tenth term. The sequence(α(sη)K∇sη · ∇βθ(sη))η is nonnegative and

α(sη)K∇sη · ∇βθ(sη) = (r − 1 + θ)sr−2+θη α(sη)K∇sη · ∇sη,

which converges almost everywhere, up to a subsequence, to α(s)K∇s ·∇βθ(s), since r − 2 + θ − 2q − r1 ≥ 0, which gives θ ≥ 7r1 + 6 − r.

Consider a nonnegative test function (χ ≥ 0); then Fatou’s lemma ensures

lim infη→0

∫QT

α(sη)K∇sη · ∇βθ(sη)χ dx dt ≥∫

QT

α(s)K∇s · ∇βθ(s)χ dx dt,

and the limit solution (p, s) is then obtained to satisfy the inequality (2.5)in Definition 2.2.

The proof of (2.6) is a consequence of Egorov’s theorem. In fact, we have

∀ε > 0,∃Qε ⊂ QT measurable such that meas(Qε) < ε andα(sη)K∇sη · ∇βθ(sη) −→ α(s)K∇s · ∇βθ(s) uniformly in QT \Qε.

Consider a test function χ such that supp χ ⊂([0, T ) × Ω

)\Qε; then∫

QT \Qε

α(sη)K∇sη · ∇βθ(sη)χ dx dt −→∫

QT \Qε

α(s)K∇s · ∇βθ(s)χ dx dt,

as η goes to zero. The proof of Theorem 2.2 is then complete. Remark that if we had a smoother solution, sη for example, and ∇A(sη)

uniformly bounded in (L2+ε(QT ))N , we would have Qε = ∅ in our definitionof degenerate weak solution.


The sketch of the proof of Theorem 2.3 is very close to that of Theorem 2.2.Compared to Theorem 2.3, a specific analysis has to be added to overcomethe difficulty of the degenerate dissipation close to s = 1.


We start with estimates on solutions to the η-regularized problem. Wecomplete the uniform (with respect to η) estimates (5.1), (5.2), (5.3) byLemma 7.1. Recall that the functions µ, jθ,λ, Jθ,λ, and G are respectivelydefined by (2.9)–(2.10), (2.7), (2.8), and (2.11), and j and J are consideredto be j = j0,0 and J = J0,0.

Lemma 7.1. Assuming (H1)–(H6) and (H7c), and that G(s0) belongs toL1(Ω), the solutions to the saturation equation (2.16) satisfy

(i) 0 ≤ sη(t, x) ≤ 1, for almost every t, x in QT .(ii) The sequences (α

12 (sη)µ′ 1

2 (sη)∇sη)η, (∇J(sη))η, and (α(sη)∇sη)η

are uniformly bounded in L2(QT ).(iii) The sequence (G(sη))η is uniformly bounded in L∞(0, T ;L1(Ω)).(iv) The sequence (

√ηµ′(sη)∇sη)η is uniformly bounded in (L2(QT ))N .

(v) The sequence (φ(x)∂tJ(sη))η is uniformly bounded inL1(0, T ; (W 1,q(Ω))′) for q > N .

(vi) The sequences (J(sη))η and (sη)η are relatively compact in L2(QT ).

Proof. The first part, (i), is obtained in Lemma 4.2. Next, multiplying(2.16) by µ (defined in (2.9)–(2.10)) and integrating over Ω, one getsd

dt

∫Ω

φG(sη) dx +∫

ΩKα(sη)µ′(sη)∇sη · ∇sη dx + η

∫ΩKµ′(sη)∇sη · ∇sη dx

= −∫

ΩsηVη · ∇µ(sη) dx +

∫Ω

ν(sη)Vη · ∇µ(sη) dx (7.1)

+∫

ΩγKα(sη)∇sη ·∇pηµ(sη) dx +

∫Ω(1 − sη)µ(sη)g dx −

∫Ωµ(sη)Vη.∇sη dx.

The whole integral appearing in (7.1) can be split as∫Ω

=∫

Ω∩s<s1+

∫Ω∩s≥s1

,

so that the same analysis as the one developed in the previous section isapplied to the terms of the form

∫Ω∩s<s1. We then obtain the estimate

d

dt

∫Ω

φG(sη) dx +k0

2

∫Ω∩s<s1

α(sη)µ′(sη)|∇sη|2 dx

+ k0

∫Ω∩s≥s1

α(sη)µ′(sη)|∇sη|2dx + ηk0

∫Ω

µ′(sη)|∇sη|2 dx

≤ c(‖Vη‖2L2(Ω) + ‖∇pη‖2

L2(Ω) + ‖g‖L2(Ω))

+∫

Ω∩s≥s1((ν(sη) − sη)µ′(sη) − µ(sη))Vη · ∇sη dx (7.2)


+∫

Ω∩s≥s1γKα(sη)∇sη · ∇pηµ(sη) dx +

∫Ω∩s≥s1

(1 − sη)µ(sη)g dx.

By definition of the function µ, the second term of the right-hand side van-ishes.

With help of the estimate (2.13), we have (1− sη)µ(sη) ≤ k1µ(s1)(1− s1)for s > s1; then

d

dt

∫Ω

φG(sη) dx +k0

2

∫Ω

α(sη)µ′(sη)|∇sη|2 dx + ηk0

∫Ω


≤ c(‖Vη‖2L2(Ω) + ‖∇pη‖2

L2(Ω) + ‖g‖L2(Ω))

+ δ

∫Ω∩s≥s1

α(sη)µ2(sη)|∇sη|2 + c(δ)∫

Ω∩s≥s1|∇pη|2 dx

+ k1µ(s1)(1 − s1)∫

Ω∩s≥s1g dx. (7.3)

From (2.10), (2.12), and (2.13), we can choose δ such that δµ2(s) ≤ k04 µ′(s);

the estimate (7.3) becomes

d

dt

∫Ω

φG(sη) dx +k0

4

∫Ω

α(sη)µ′(sη)|∇sη|2 dx + ηk0

∫Ω


≤ c(‖Vη‖2L2(Ω) + ‖∇pη‖2

L2(Ω) + ‖g‖L2(Ω)).

Integrating over t between (0, T ) we deduce the first part of (ii), (iii), and(iv) of this lemma. Furthermore, the second part of (ii) follows from theupper bound

α(s)µ′(s) ≥ m1(1 − s)r2µ(s)(ν(s) − s)−1 ≥ m1µ(s1)(1 − s1)(1 − s)r2−2

≥ cj2(s), ∀s ≥ s1.

The third part is already proved for sη ≤ s1 and is trivial for sη ≥ s1,since

1sη≥s1α(sη) ≤ c(s1)1sη≥s1µ′(sη).

For the fifth part, we prove that

φ(x)∂tJ(sη) is uniformly bounded in L2(0, T ; (H1(Ω))′) + L1(QT ).

Let χ ∈ L2(0, T ;H1(Ω)) ∩ L∞(QT ); we return to (6.3):

〈φ∂tJ(sη), χ〉 = −∫

QT

(Vη · ∇(sηj(sη)χ) − ν(sη)Vη · ∇(j(sη)χ)

)dx dt

−∫

QT

α(sη)K∇sη · ∇(j(sη)χ) dx dt − η

∫QT

K∇sη · ∇(j(sη)χ) dx dt


+∫

QT

γα(sη)K∇sη · ∇pηj(sη)χ dx dt +∫

QT

(1 − sη)gj(sη)χ dx dt. (7.4)

We are going to give estimates for each term on the right-hand side. Theestimates of the proof of Theorem 2.2 are valid for the integral term of theform

∫QT∩s<s1. We then estimate only the terms of the form

∫QT∩s≥s1.

From (2.12) and the definition of the function j (see (2.7)), we have thefollowing properties:

ν(s) − s ≤ 1 − s for all s > s1,

(1 − s)j′(s) = ( r′2 − 1)j(s) for all s > s1,

(1 − s)j(s) = βθ(s1)(1 − s1)1−r′2 (1 − s)

r′2 ≤ c(s1) for all s > s1.

The first integral of the right-hand side is estimated as follows:∫QT∩s≥s1

(ν(sη)Vη · ∇(j(sη)χ) − Vη · ∇(sηj(sη)χ)

)dx dt =∫

QT∩s≥s1(ν(sη)−sη)j(sη)Vη·∇χ+((ν(sη)−sη)j′(sη)+j(sη))Vη·∇sηχ dx dt

≤ C‖Vη‖L2(QT )‖∇χ‖L2(QT )

+ ‖((1 − sη)j′(sη) + j(sη))∇sη‖L2(QT )‖Vη‖L2(QT )‖χ‖L∞(QT )

≤ C‖Vη‖L2(QT )‖∇χ‖L2(QT ) + ‖∇J(sη)‖L2(QT )‖Vη‖L2(QT )‖χ‖L∞(QT ),

and by virtue of (ii), we have the desired bound. The bound on the secondterm of (7.4) is estimated classically by∫

QT∩s≥s1Kα(sη)∇sη · ∇(j(sη)χ) dx dt ≤

C‖√

α(sη)j′(sη)∇sη|2L2(QT )‖χ‖L∞(QT )+C‖α(sη)∇J(sη)‖L2(QT )‖∇χ‖L2(QT ).

From the definitions of the functions µ and j, one obtains that there existsa constant c(s1) such that

α(sη)j′(sη) ≤ c(s1)α(sη)µ′(sη) for all s > s1,

and from the part (ii) of this lemma, we have the desired bound. Similarly,we can estimate the other integrals of the right-hand side of (7.4); we obtain

|〈φ∂tJ(sη), χ〉| ≤ C(‖χ‖L2(0,T ;H1(Ω)) + ‖χ‖L∞(QT )).

Then part (v) of this lemma is a direct consequence of the Sobolev embeddingL∞(0, T ;W 1,q) ⊂ L2(0, T ;H1(Ω)) ∩ L∞(QT ), for q > N .


The last part, (vi), is a consequence of the compactness Lemma 3.1, andthe proof of Lemma 6.1 obtained in the previous section is also valid with Jinstead of h. The proof of Lemma 7.1 is then complete.

The strong convergence on velocity is also valid under assumption (H7c):

pη −→ p strongly in L2(0, T ;H1(Ω))), and a.e. in QT ,Vη −→ V strongly in (L2(QT ))N .

(7.5)

Lemma 7.2. Let q1 = 3r1 + 2 and q2 = 3r2 + 2, where r1 and r2 aredefined in assumption (H7c). The sequences ((1sη≤s1s

q1η α(sη)∇sη))η and

((1sη≥s1(1 − sη)q2α(sη)∇sη))η are Cauchy sequences in measure.

Proof. The proof is exactly the same as that of Lemma 6.2, arguing that(α(sη)∇sη)η is always uniformly bounded in (L2(QT ))N as in the case ofassumption (H7b). It is also true for the sequences (∇b(sη))η and (∇B(sη))η

appearing in the proof of Lemma 6.2. In addition, taking into account(7.5) and Lemma 7.1, the convergence result (6.11) is also valid under theassumption (H7c); that is,

for all ε > 0, for all δ > 0, there exists η0 such that for all η, η′ ≤ η0,

meas|∇u(sη) −∇u(sη′)| ≥ δ ≤ ε, (7.6)

where ∇u(s) = b(s)A(s)∇A(s). Now, we have

1sη≤s1sq1η ∇A(sη) − 1sη′≤s1s

q1

η′∇A(sη′)

= 1sη≤s1sq1η ∇Aη,η′ +

(1sη≤s1s

q1η − 1sη′≤s1s

q1

η′)∇A(sη′);

the last term goes to zero as η and η′ go to zero in L1(QT ), and

1sη≥s1(1 − sη)q2∇A(sη) − 1sη′≥s1(1 − sη′)q2∇A(sη′) =

1sη≥s1(1−sη)q2∇Aη,η′ +(1sη≥s1(1−sη)q2−1sη′≥s1(1−sη′)q2

)∇A(sη′).

In the same way, the last term goes to zero as η and η′ go to zero in L1(QT ).As q1 = 3r1 + 2, we have

1sη≤s1sq1η ≤ 1sη≤s1s

3r1+2η ≤ c11sη≤s1b(sη)A(sη),

where c1 = r21

2m31, so that

|1sη≤s1sq1η ∇A(sη) − 1sη′≤s1s

q1

η′∇A(sη′)|≤ |(1sη≤s1s

q1η − 1sη′≤s1s

q1

η′ )∇A(sη′)| + c1|∇u(sη) −∇u(sη′)|+ c1|(b(sη)A(sη) − b(sη′)A(sη′))∇A(sη′)|. (7.7)


The last term goes to zero as η and η′ go to zero in L1(QT ), and we concludeas at the end of Lemma 6.2 that the sequence (1sη≤s1s

q1η ∇A(sη))η is a

Cauchy sequence in measure.In the same way, we have

1sη≥s1(1 − sη)q2 ≤ 1sη≥s1(1 − sη)3r2+2 ≤ c21sη≥s1b(sη)A(sη),

where c2 is a constant independent of η; then we get

|1sη≥s1(1 − sη)q2∇A(sη) − 1sη′≥s1(1 − sη′)q2∇A(sη′)|≤ |

(1sη≥s1(1 − sη)q2 − 1sη′≥s1(1 − sη′)q2

)∇A(sη′)|

+ c2|∇u(sη) −∇u(sη′)| + c2|(b(sη)A(sη) − b(sη′)A(sη′))∇A(sη′)|. (7.8)

The last term goes to zero as η and η′ go to zero in L1(QT ), and we concludeas the end of Lemma 6.2 that the sequence (1sη≥s1(1− sη)q2∇A(sη))η is aCauchy sequence in measure, which completes the proof of this lemma.

To conclude the proof of Theorem 2.3, we deduce from estimates (5.1)–(5.3), Lemma 7.1, and Lemma 7.2 that we can extract a subsequence suchthat

pη −→ p strongly in L2(0, T ;H1(Ω)), and a.e. in QT ,

Vη −→ V strongly in (L2(QT ))N ,

φ∂tpη −→ φ∂tp weakly in L2(0, T ; (H1(Ω)′)),

sη −→ s strongly in L2(QT ), and a.e. in QT ,

0 ≤ s(t, x) ≤ 1 a.e. in QT ,

J(sη) −→ J(s) strongly in L2(QT ), (7.9)

J(sη) −→ J(s) weakly in L2(0, T ;H1(Ω)),

α12 (sη)µ′ 1

2 (sη)∇sη −→ α12 (s)µ′ 1

2 (s)∇s weakly in (L2(QT ))N ,

j′(sη)(sη − ν(sη))∇sη −→ j′(s)(s − ν(s))∇s weakly in (L2(QT ))N ,

1sη≤s1sq1η α(sη)∇sη −→ 1s≤s1s

q1α(s)∇s a.e. in QT with q1 = 3r1 + 2,

1sη≥s1(1 − sη)q2α(sη)∇sη −→ 1s≥s1(1 − s)q2α(s)∇s a.e. in QT

with q2 = 3r2 + 2.

We want to pass to the limit as η goes to zero in the weak formulation


QT


QT

(f − g)ψ dx dt (7.10)


−∫

QT

φ(x)Jθ,λ(sη)∂tχ dx dt −∫

Ωφ(x)Jθ,λ(s0(x))χ(0, x) dx (7.11)

+∫

QT

Vη · ∇sηjθ,λ(sη)χ dx dt +∫

QT

Vη · ∇jθ,λ(sη)(sη − ν(sη))χ dx dt

+∫

QT

Vη · ∇χsηjθ,λ(sη) dx dt −∫

QT

ν(sη)Vη · ∇χjθ,λ(sη) dx dt

+ η

∫QT

K∇sη · ∇jθ,λ(sη)χ dx dt + η

∫QT

K∇sη · ∇χjθ,λ(sη) dx dt

+∫

QT

α(sη)K∇sη · ∇jθ,λ(sη)χ dx dt +∫

QT

α(sη)K∇sη · ∇χjθ,λ(sη) dx dt

− γ

∫QT

α(sη)∇sη · ∇pηjθ,λ(sη)χ dx dt =∫

QT

(1 − sη)gjθ,λ(sη)χ dx dt

for all ψ ∈ L2(0, T ;H1(Ω)) and χ ∈ C1([0, T ) × Ω) with supp χ ⊂ ([0, T ) ×Ω)\Qε, for a set Qε ⊂ QT , meas(Qε) < ε.

In the above formulation, for θ ≥ 0 and λ ≥ 0, the convergence results(7.9) allow us to pass to the limit as η goes to zero, up to a subsequence,for all the terms except for the seventh and ninth. For that, for θ > 0 andλ ≥ 0, we give only some details on the convergence of the seventh integral.Let us write

η

∫QT

K∇sη · ∇jθ,λ(sη)χ dx dt

= η

∫sη<s1

K∇sη · ∇jθ,λ(sη)χ dx dt + η

∫sη>s1

K∇sη · ∇jθ,λ(sη)χ dx dt.

From (6.16), the first integral of the right-hand side goes to zero as η goesto zero, and we have

η∣∣∣ ∫

sη>s1K∇sη · ∇jθ,λ(sη)χ dx dt

∣∣∣ ≤ Cη

∫sη>s1

(1 − sη)r′2−2|∇sη|2|χ| dx dt

≤ Cη

∫sη>s1∩(1−sη)≤η

1r′

(1 − sη)r′2−2|∇sη|2 dxdt‖χ‖L∞(QT )

+ Cη

∫sη>s1∩(1−sη)>η

1r′

(1 − sη)r′2−2|∇sη|2 dx dt‖χ‖L∞(QT )

≤ Cη12 ‖

√ηµ′(sη)∇sη‖L2(QT )‖χ‖L∞(QT ) + Cη

12 ‖∇J(sη)‖L2(QT )‖χ‖L∞(QT ).

According to Lemma 7.1 ((ii), (iv)), the right-hand side goes to zero as ηgoes to zero.


Now we are concerned with the ninth term. Let us write∫QT

α(sη)K∇sη · ∇jθ,λ(sη)χ dx dt

=∫

QT

1sη≤s1α(sη)K∇sη · ∇βθ(sη)χ dx dt

+∫

QT

1sη≥s1Kα(sη)j′θ,λ(sη)∇sη · ∇sηχ dx dt.

The first term on the right-hand side has already been treated in the previoussection, where we have seen that, for θ ≥ 7r1 + 6 − r,

1sη≤s1α(sη)K∇sη · ∇βθ(sη) −→ 1s≤s1α(s)K∇s · ∇βθ(s) a.e. in QT .

To treat the second term, we choose λ large enough so that the second termintegrated on the right-hand side converges almost everywhere; that is,

r2 +r′

2− 2 + λ ≥ 2q2 + 2r2;

that is, λ ≥ 2q2 + r2 + 2− r′2 = 7r2 + 6− r′

2 . Finally, for θ ≥ 7r1 + 6− r andλ ≥ 7r2 + 6 − r′

2 , we have

α(sη)K∇sη · ∇jθ,λ(sη) −→ α(s)K∇s · ∇jθ,λ(s) a.e. in QT .

By Egorov’s theorem,

∀ε > 0,∃Qε ⊂ QT measurable such that meas(Qε) < ε andα(sη)K∇sη · ∇jθ,λ(sη) −→ α(s)K∇s · ∇jθ,λ(s) uniformly in QT \Qε.

Consider a test function χ such that supp χ ⊂([0, T ) × Ω

)\Qε; then∫

QT \Qε

α(sη)K∇sη · ∇jθ,λ(sη)χ dx dt −→∫

QT \Qε

α(s)K∇s · ∇jθ,λ(s)χ dx dt

uniformly in QT \Qε as η goes to zero, which completes the proof of Theo-rem 2.3. Acknowledgment. We thank the referee for the care she/he brought tothe reading of this paper and for many remarks which helped improve thearticle.

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