ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

UNIQUENESS FOR CROSS-DIFFUSION SYSTEMS ISSUING FROM SEAWATER INTRUSION PROBLEMS

CATHERINE CHOQUET, JI LI, CAROLE ROSIER

Abstract. We consider a model mixing sharp and diffuse interface approaches
for seawater intrusion phenomenons in confined and unconfined aquifers. More
precisely, a phase field model is introduced in the boundary conditions on
the virtual sharp interfaces. We thus include in the model the existence of
diffuse transition zones but we preserve the simplified structure allowing front
tracking. The three-dimensional problem then reduces to a two-dimensional
model involving a strongly coupled system of partial differential equations
of parabolic and elliptic type describing the evolution of the depth of the
interface between salt- and freshwater and the evolution of the freshwater
hydraulic head. Assuming a low hydraulic conductivity inside the aquifer, we
prove the uniqueness of a weak solution for the model completed with initial
and boundary conditions. Thanks to a generalization of a Meyer’s regularity
result, we establish that the gradient of the solution belongs to the spaceL^{r},
r > 2. This additional regularity combined with the Gagliardo-Nirenberg
inequality for r = 4 allows to handle the nonlinearity of the system in the
proof of uniqueness.

1. Introduction

Seawater intrusion in coastal aquifers is a major problem for water supply. The study of efficient and accurate models to simulate the displacement of a saltwater front in unsaturated porous media is motivated by the need of efficient tools for the optimal exploitation of fresh groundwater.

Observations show that, near the shoreline, fresh and salty underground water tend to separate into two distinct layers. It was the motivation for the derivation of seawater intrusion models treating salt- and freshwater as immiscible fluids. Points where the salty phase disappears may be viewed as a sharp interface. Nevertheless the explicit tracking of the interfaces remains unworkable to implement without further assumptions. An additional assumption, the so-called Dupuit approxima- tion, consists in considering that the hydraulic head is constant along each vertical direction. It allows to assume the existence of a smooth sharp interface. Classical sharp interface models are then obtained by vertical integration based on the as- sumption that no mass transfer occurs between the fresh and the salty area (see [4, 11] and even the Ghyben-Herzberg static approximation). This class of models allows direct tracking of the salt front. Nevertheless the conservative form of the

2010Mathematics Subject Classification. 35K51, 35K67, 35K57, 35A05, 76T05.

Key words and phrases. Uniqueness; cross-diffusion system; nonlinear parabolic equations;

seawater intrusion.

c

2017 Texas State University.

Submitted July 8, 2016. Published October 11, 2017.

1

equations is perturbed by the upscaling procedure. In particular the maximum principle does not apply. Of course fresh and salty water are two miscible fluids.

Following [6], we can mix the latter abrupt interface approach with a phase field approach (here an Allen–Cahn type model in fluid-fluid context see [1, 2, 5]) for re-including the existence of a diffuse interface between fresh and salt water where mass exchanges occur. We thus combine the advantage of respecting the physics of the problem and that of the computational efficiency.

From a theoretical point of view, an advantage resulting from the addition of the diffuse area compared to the sharp interface approximation is that the system now has a parabolic structure, so it is not necessary to introduce viscous terms in a preliminary fixed point for treating degeneracy as in the case of the sharp interface approach. Another important point is that we can demonstrate a more efficient and logical maximum principle from the point of view of physics, which is not possible in the case of classical sharp interface approximation. But the main point is that we can now show the uniqueness of the solution thanks to the parabolic structure of the system that yields more regularity for the solution.

This article is devoted to the study of the wellposedness of the sharp-diffuse in-
terface seawater intrusion model. We focus on confined aquifers. As already men-
tioned, the problem consists in a coupled system of quasi-linear parabolic-elliptic
equations. It belongs to the wide class of cross-diffusion systems for which the
equations are coupled in the highest derivatives terms and there is no general the-
ory for such a kind of problem. For dealing with the nonlinearity in the uniqueness
proof, we first prove aL^{r},r >2, regularity result for the gradient of the unknowns.

More precisely we generalize to the quasilinear case, the regularity result given by Meyers [10] in the elliptic case and extended to the parabolic case by Bensoussan, Lions and Papanicolaou, for any elliptic operatorA=−Pn

i,j=1∂jaij(x)∂i(see [3]).

The results assume that the operatorAsatisfies an uniform ellipticity assumption
and that its coefficients areL^{∞} functions. The hypothesis on A ensure the exis-
tence of an exponentr(A)>2 such that the gradient of the solution of the elliptic
equation (resp. of the parabolic equation) belongs to the space L^{r}with respect to
space (resp. L^{r} with respect to time and space). This additional regularity com-
bined with the Gagliardo-Nirenberg inequality let us handle the nonlinearity of the
system in the proof of uniqueness.

This article is organized as follows: First, in Section 2, we detail all the mathe- matical notations and we present some auxiliary results. In Section 3, we present a new proof of global in time existence for the problem. Sections 4 and 5 are devoted to the proofs of the regularity and uniqueness results.

2. Auxiliary results

We consider an open bounded domain Ω of R^{2} describing the projection of the
aquifer on the horizontal plane. The boundary of Ω, assumedC^{1}, is denoted by Γ.

The time interval of interest is (0, T), T being any nonnegative real number, and we set ΩT = (0, T)×Ω.

For the sake of brevity we shall writeH^{1}(Ω) =W^{1,2}(Ω) and
V =H_{0}^{1}(Ω), V^{0}=H^{−1}(Ω), H=L^{2}(Ω).

The embeddings V ⊂H =H^{0} ⊂V^{0} are dense and compact. For any T > 0, let
W(0, T) denote the space

W(0, T) :=

ω∈L^{2}(0, T;V), ∂tω∈L^{2}(0, T;V^{0})

endowed with the Hilbertian normk · k_{W}_{(0,T)}= k · k^{2}_{L}2(0,T;V)+k∂t· k^{2}_{L}2(0,T;V^{0})

^{1/2}
.
The following embeddings are continuous [9, Prop. 2.1 and Thm 3.1, chapter 1]

W(0, T)⊂ C([0, T]; [V, V^{0}]_{1/2}) =C([0, T];H)
while the embedding

W(0, T)⊂L^{2}(0, T;H) (2.1)

is compact (Aubin’s Lemma, see [12]). The following result by Mignot (see [8]) is used in the sequel.

Lemma 2.1. Let f :R→Rbe a continuous and nondecreasing function such that
lim sup_{|λ|→+∞}|f(λ)/λ|<+∞. Letω∈L^{2}(0, T;H)be such that∂tω∈L^{2}(0, T;V^{0})
andf(ω)∈L^{2}(0, T;V). Then

h∂tω, f(ω)iV^{0},V = d
dt

Z

Ω

Z ω(·,y)

0

f(r)dr

dy inD^{0}(0, T).

Hence for all0≤t1< t2≤T,
Z t_{2}

t1

h∂tω, f(ω)iV^{0},Vdt=
Z

Ω

Z ω(t_{2},y)

ω(t1,y)

f(r)dr dy.

We now present two preliminary lemma, which are consequences of the Meyers
regularity results [10], first for an elliptic equation, then for a parabolic one. The
adaptation of these results will be crucial for proving theL^{r}(0, T;W^{1,r}(Ω)),r >2,
regularity of the solutions.

•Elliptic case. We recall the following result (see Lions and Magenes [9]):

∀p: 1< p <∞,−∆ is an isomorphism fromW_{0}^{1,p}(Ω) toW^{−1,p}(Ω).

We setG= (−∆)^{−1}and g(p) =kGk_{L(W}−1,p(Ω);W_{0}^{1,p}(Ω)). We notice that g(2)=1.

Lemma 2.2. LetA∈(L^{∞}(Ω))^{n} be a symmetric tensor such that there existsα >0
satisfying

n

X

i,j=1

Ai,j(x)ξiξj≥α|ξ|^{2}, ∀x∈Ωandξ∈R^{n}.

We set β = max1≤i,j≤nkAi,jkL^{∞}(Ω). There exist r(α, β) >2, such that, for any
f ∈W^{−1,r}(Ω) and for anyg0∈W^{1,r}(Ω), the unique solutionuof the problem

∇ ·(A∇u) =f,∀x∈Ω
u∈H_{0}^{1}(Ω) +g0,

belongs toW^{1,r}(Ω). In addition, the following estimate holds:

kuk_{W}1,r(Ω)≤C(α, β, r)kf− ∇ ·(A∇g_{0})k_{W}−1,r(Ω), (2.2)
whereC(α, β, r)is a constant depending only onr and on constantsαandβ char-
acterizing the operatorA.

Remark 2.3. The proof given in [3] allows us to precise the constantC(α, β, r).

Letcbe a positive real number. We set µ= α+c

β+c, ν = c

β+c, (2.3)

wherecis introduced in order to ensureν < µ. Sinceg(2) = 1 and 0<1−µ+ν <1, by using the properties of the mapg, we can findr >2 such that

0< k(r) :=g(r)(1−µ+ν)<1. (2.4) Then, the smaller (1−µ+ν) is, the biggerrcan be. The determination ofrdepends on the constantsαandβ characterizing the elliptic operatorA.

Let us emphasize that Lemma 2.2 holds for all p such that 2 ≤ p ≤ r(α, β) thanks to the classical interpolation inequalities.

The limit case corresponds to the settings where the operatorAis proportional to the Laplacian: thenµ= 1,ν = 0 and (2.4) is satisfied for allr≥2. Taking into account the previous estimates, we can give an upper bound ofC(α, β, r) as follows

C(α, β, r)≤(1−g(r)(1−µ+ν))^{−1} g(r)

β+c = g(r)

(1−k(r))(β+c). (2.5)
Parabolic case. Let us give now a lemma for the parabolic context. We define
X_{p}=L^{p}(0, T;W_{0}^{1,p}(Ω)), endowed with the norm

k∇vk(L^{p}(Ω_{T}))^{n}=Z T
0

kv(t)k^{p}

W_{0}^{1,p}(Ω)dt^{1/p}
.

We introduceY_{p}=L^{p}(0, T;W^{−1,p}(Ω)) and we point out that the applicationv→
div_{x}v sends (L^{p}(Ω_{T}))^{n} into L^{p}(0, T;W^{−1,p}(Ω)). We endow Y_{p} with the norm
kfk_{Y}_{p}= inf_{div}_{x}_{g=f}kgk_{(L}p(Ω_{T}))^{n}. We can state the following Lemma (cf. [3]).

Lemma 2.4. Let A∈(L^{∞}(Ω))^{n} be a symmetric tensor defined as in Lemma 2.2.

Let f ∈L^{2}(0, T, H^{−1}(Ω)) andu^{0}∈H, there existsu∈L^{2}(0, T;H_{0}^{1}(Ω)) solution of

∂u

∂t +Au=f in Ω_{T}, u(0) =u^{0}.

Then, assuming that Γ is sufficiently regular, there exists r > 2, depending on
α, β and Ω such that if f ∈ L^{r}(0, T;W^{−1,r}(Ω)) and u^{0} ∈ W_{0}^{1,r}(Ω) then u ∈
L^{r}(0, T;W_{0}^{1,r}(Ω)). Furthermore, there existsC(α, β, r)ˆ >0 such that

kuk_{W}^{1,r}

0 (Ω)≤C(α, β, r)(kfˆ kL^{r}(0,T;W^{−1,r}(Ω))+ku^{0}k_{W}^{1,r}

0 (Ω)). (2.6)
Remark 2.5. As for lemma 2.2, it is possible to precise ˆC(α, β, r) (cf. [3]). We
setP =_{∂t}^{∂} −∆, the operator associated with the homogeneous Dirichlet boundary
conditions. We know that, being givenF ∈Y_{p}, there is a unique solution u∈X_{p}
such that

P u=F in Ω_{T}, u(0) =u^{0}.

We set ˆg(p) =kP^{−1}k_{L(Y}_{p}_{;X}_{p}_{)}, we recall that ˆg(2) = 1. Using the properties of the
map ˆg(·), we claim that there existsr >2 such that

0<ˆk(r) := ˆg(r)(1−µˆ+ ˆν)<1, (2.7)

where the constants ˆµ, ˆν are defined by ˆ

µ=α+ ˆc

β+ ˆc and ˆν= cˆ

β+ ˆc, ∀ˆc >0. (2.8) Since ˆc > 0, we have ˆν < µ. Again, the smaller (1ˆ −µˆ+ ˆν), the bigger r, and the determination ofrwill depend on the constantsα, β characterizing the elliptic operatorA. The following condition is satisfied by the constant ˆC(α, β, r)

C(α, β, r)ˆ ≤(1−ˆg(r)(1−µˆ+ ˆν))^{−1} ˆg(r)

β+ ˆc = g(r)ˆ

(1−ˆk(r))(β+ ˆc). (2.9) 3. Global in time existence result

Mathematical setting. We consider that the confined aquifer is bounded by two layers, the lower surface corresponds to z = h2 and the upper surface z = h1. Quantityh2−h1is the thickness of the aquifer. We assume that depthsh1, h2are constant, such that h2 > δ1>0 and without lost of generality we can seth1= 0.

We introduce functionsTs andTf defined by

Ts(u) =h2−u ∀u∈(δ1, h2) and Tf(u) =u ∀u∈(δ1, h2).

Functions T_{s} and T_{f} are extended continuously and constantly outside (δ_{1}, h_{2}).

T_{s}(h) represents the thickness of the salt water zone in the reservoir, the previous
extension ofT_{s}forh≤δ_{1}enables us to ensure a thickness of freshwater zone always
greater than δ_{1} in the aquifer. We also emphasize that the function T_{f} only acts
on the source termQ_{f} for avoiding the pumping when the thickness of freshwater
zone is smaller thanδ1.

In the case of confined aquifer, the well adapted unknowns are the interface depthhand the freshwater hydraulic headf. The model reads (see [6]):

φ∂_{t}h− ∇ ·

KT_{s}(h)∇h

− ∇ · δ∇h

+∇ ·

KT_{s}(h)∇f

=−QsT_{s}(h), (3.1)

−∇ ·

h_{2}K∇f
+∇ ·

KT_{s}(h)∇h

=Q_{f}T_{f}(h) +Q_{s}T_{s}(h). (3.2)
The above system is complemented by the boundary and initial conditions

h=hD, f =fD in Γ×(0, T), (3.3)

h(0, x) =h0(x), in Ω, (3.4)

with the compatibility condition

h0(x) =hD(0, x), x∈Γ.

Let us now detail the mathematical assumptions. We begin with the characteris- tics of the porous structure. We assume the existence of two positive real numbers K−andK+such that the hydraulic conductivityKis a bounded symmetric elliptic and uniformly positive definite tensor

0< K−|ξ|^{2}≤ X

i,j=1,2

Ki,j(x)ξiξj≤K+|ξ|^{2}<∞ x∈Ω, ξ ∈R^{2}, ξ6= 0.

We assume that porosityφis constant in the aquifer. Indeed, in the field envisaged here, the effects due to variations in φ are negligible compared with those due to density contrasts. From a mathematical point of view, these assumptions do not change the complexity of the analysis but rather avoid cumbersome computations.

The parameterδrepresents the thickness of the diffuse interface. The source terms

Qf and Qs are given functions of L^{2}(0, T;H) and we assume that Qf ≥ 0 and
Qs≤0.

The functions hD and fD belong to L^{2}(0, T;H^{1}(Ω))∩H^{1}(0, T; (H^{1}(Ω))^{0})

×
L^{2}(0, T;H^{1}(Ω)) while functionh_{0} belongs toH^{1}(Ω). Finally, we assume that the
boundary and initial data satisfy conditions on the hierarchy of interfaces depths:

0< δ_{1}≤h_{D}≤h_{2}a.e. in Γ×(0, T), 0< δ_{1}≤h_{0}≤h_{2} a.e. in Ω.

Theorem 3.1 (Existence theorem). Assume a low spatial heterogeneity for the hydraulic conductivity tensor

K_{+}< h_{2}
h2−δ1

infr
δK_{−}

3h2

, K_{−}

. (3.5)

Then for anyT >0, problem (3.1)-(3.4)admits a weak solution(h, f)satisfying
(h−h_{D}, f−f_{D})∈W(0, T)×L^{2}(0, T;H_{0}^{1}(Ω)).

Furthermore the following maximum principle holds true :

0< δ_{1}≤h(t, x)≤h_{2} for a.e. x∈Ωand for anyt∈(0, T).

Remark 3.2. Assumption (3.5) (so as (4.4)) makes only sense when considering
low values forK. For the present application, this point is not restrictive since the
soil permeability typically ranges from 10^{−8}to 10^{−3} m/s.

With the additional diffuse interface, the system has a parabolic structure, it is
thus no longer necessary to introduce viscous terms in a preliminary fixed point step
for avoiding degeneracy. But we still need to impose a minimal freshwater thickness
strictly positive inside the aquifer to prove an uniform estimate in L^{2}(ΩT) of the
gradient off since the presence of the diffuse interface does not allow us to get this
estimate. Let us briefly sketch the strategy of the proof. First step consists in using
a Schauder fixed point theorem for proving an existence result for the problem.^{1}
Then we establish that the solution satisfies the maximum principles announced in
Theorem 3.1: First, we show thath≥h2 a.e. in ΩT; finally we prove thatδ1≤h
a.e. in ΩT under assumptionQf ≥0.

Step 1: Existence for the truncated system:

Definition of the map F = (F1,F2). For the fixed point strategy, we define
an application F: (W(0, T) +h_{D})×(L^{2}(0, T;H_{0}^{1}(Ω)) +f_{D})→(W(0, T) +h_{D})×
(L^{2}(0, T;H_{0}^{1}(Ω)) +f_{D}) by

F(¯h,f¯) = F1(¯h,f¯),F2(¯h,f¯)

= (h, f),

1More precisely, the present proof is based on the classical version of the Schauders fixed point
theorem applied to the initial problem. In [8], this fixed point theorem is applied to an auxiliary
truncated problem. The truncation is introduced to control theH^{1}-norm off. We thus had to
check that the mapFdefined below is sequentially continuous inL^{2}(0, T;H^{1}(Ω). Here we rather
choose working with the strong topologyL^{2}(0, T;L^{2}(Ω)), which is possible since the truncation
term has been dropped.

where the couple (h, f) is the solution of the following initial boundary value prob-
lem, for allw∈L^{2}(0, T;V):

Z T

0

φh∂th, wiV,V^{0}dt+
Z

Ω_{T}

(δ+Ts(¯h)K)∇h· ∇w dx dt +

Z

ΩT

QsTs(¯h

w dx dt− Z

ΩT

Ts(¯h)K∇f¯· ∇w dx dt= 0,

(3.6)

Z

Ω_{T}

h_{2}K∇f· ∇w dx dt−
Z

Ω_{T}

T_{s}(¯h)K∇¯h· ∇w dx dt

− Z

ΩT

(QsTs(¯h) +QfTf(¯h))w dx dt= 0.

(3.7)

We know from the classical theory of linear parabolic PDE’s that this variational linear system has a unique solution. The end of the present subsection is devoted to the proof of the existence of a fixed point ofF in some appropriate subset.

Sequential continuity of F_{1} in L^{2}(0, T;H) when F is restricted to any
bounded subset of W(0, T)×L^{2}(0, T;H^{1}(Ω)). Assume that given a bounded
sequence (¯h^{n},f¯^{n}) in (W(0, T)+hD)×(L^{2}(0, T;H_{0}^{1}(Ω))+fD) and a function (¯h,f¯)∈
(W(0, T) +hD)×(L^{2}(0, T;H_{0}^{1}(Ω)) +fD) such that

(h_{n}, f_{n})→(h, f) in (L^{2}(0, T;H))^{2}.
We thus have

(¯h^{n},f¯^{n})*(¯h,f¯) weakly inW(0, T)×L^{2}(0, T;H^{1}(Ω));

that is, ¯h^{n} * ¯h weakly in L^{2}(0, T, V) (the same for ¯f^{n} and ¯f) and ∂th^{n} * ∂th
weakly inL^{2}(0, T, V^{0}).

Set hn = F1(¯h^{n},f¯^{n}) and h =F1(¯h,f¯). We first intend to show that hn →h
weakly in W(0, T) and thus strongly in L^{2}(0, T;H) thanks to a classical result of
Aubin.

Pick a constantM >0, that we will precise later on, such that

k∇¯hnk(L^{2}(0,T;H))^{2} ≤M and k∇f¯nk(L^{2}(0,T;H))^{2}≤M. (3.8)
For alln∈N,hnsatisfies (3.6). Pick anyτ ∈[0, T] and takew= (hn−hD)χ(0,τ)(t)
in (3.6). It yields

φ Z τ

0

h∂t(hn−hD), hn−hDiV^{0},Vdt+
Z

Ωτ

(δ+KTs(¯h^{n}))∇hn· ∇hndx dt
+

Z

Ω_{T}

QsTs(¯hn

(hn−hD)dx dt− Z

Ω_{τ}

KTs(¯h^{n})∇f¯^{n}· ∇(hn−hD)dx dt

= Z

Ω_{τ}

(δ+KTs(¯h^{n}))∇hn· ∇hDdx dt−φ
Z τ

0

h∂thD, hn−hDiV^{0},V dt.

(3.9)

The functionshn−hDbelong toW(0, T) and hence toC([0, T];L^{2}(Ω)). Thanks to
Lemma 2.1, we can write

Z τ

0

h∂t(hn−hD), hn−hDiV^{0},Vdt= 1

2khn(·, τ)−hDk^{2}_{H}−1

2kh0−hD(·,0)k^{2}_{H}.
On the other hand, we have

Z

Ω_{τ}

δ+KT_{s}(¯h^{n})

∇h_{n}· ∇h_{n}dx dt≥δk∇h_{n}k^{2}_{L}2(0,τ;H)^{2}.

The real numberM >0 is such that sup_{n≥0}k∇f¯^{n}k(L^{2}(0,T ,H))^{2} ≤M. Using Cauchy-
Schwarz and Young inequalities, we obtain that, for anyη1>0,

Z

Ωτ

KTs(¯h^{n})∇f¯^{n}· ∇hndx dt

≤M K+lk∇hnk_{(L}2(0,τ;H))^{2}

≤ K_{+}^{2}M^{2}
η_{1} l^{2}+η1

4 k∇hnk^{2}_{(L}2(0,τ;H))^{2},
and

Z

ΩT

δ+KT_{s}(¯h^{n})

∇h_{n}· ∇h_{D}dx dt

≤η_{1}

4 k∇hnk^{2}_{(L}2(0,T;H))^{2}+(δ+K_{+}h_{2})^{2}
η1

k∇hDk^{2}_{(L}2(0,T;H))^{2}.
Since it depends onh_{D}, the next term is simply estimated by

Z

Ω_{T}

KTs(¯h^{n})∇f¯^{n}· ∇hDdx dt

≤M K+h2khDkL^{2}(0,T;H^{1}).
Finally we have

−

Z T

0

φh∂thD,(hn−hD)iV^{0},Vdt

≤φ^{2}

2δk∂thDk^{2}_{L}2(0,T;(H^{1}(Ω))^{0})+δ

2khnk^{2}_{L}2(0,T;H^{1})+δ

2khDk^{2}_{L}2(0,T;H^{1}),
and

−

Z

Ω_{T}

Q_{s}T_{s}(¯h^{n})(h_{n}−h_{D})dx dt

≤ kQsk^{2}_{L}2(0,T;H)

φ h^{2}_{2}+φ

4kh_{n}−h_{D}k^{2}_{L}2(0,T;H).
We chooseη1such thatδ−η1≥η0>0 for someη0>0. Using the above estimates
in (3.9), we obtain for allτ ∈[0, T]

φ

4k(h_{n}−h_{D})(·, τ)k^{2}_{H}+1

2(δ−η_{1})k∇u_{1,n}k^{2}_{(L}2(0,τ;H))^{2}

≤ K_{+}^{2}M^{2}
η1

l^{2}+φ

2kh0−h_{D}(·,0)k^{2}_{H}+(δ+K_{+}h_{2})^{2}
η1

k∇hDk^{2}_{(L}2(0,T;H))^{2}

+M K+h2khDk_{L}2(0,T;H^{1})+φ^{2}

2δk∂thDk^{2}_{L}2(0,T;(H^{1}(Ω))^{0})

+δ

2khDk^{2}_{L}2(0,T;H^{1})+kQ_{s}k^{2}_{L}2(0,T;H)

φ h^{2}_{2}.

(3.10)

We infer from (3.10) that there exist real numbersAM =AM(δ, K, h0, h2, hD, l, M) and BM = BM(δ, K, h0, h2, hD, l, M) depending only on the data of the problem such that

kh_{n}k_{L}∞(0,T;H)≤A_{M}, kh_{n}k_{L}2(0,T;V)≤B_{M}. (3.11)
Thus the sequence (hn)n is uniformly bounded inL^{∞}(0, T;H)∩L^{2}(0, T;V). Set

C_{M} = max(A_{M}, B_{M}).

We now prove that (∂t(hn −hD))n is bounded in L^{2}(0, T;V^{0}). Due to the
assumptionhD∈H^{1}(0, T; (H^{1}(Ω))^{0}), it will follow that (hn)nis uniformly bounded
inH^{1}(0, T;V^{0}). We have

k∂_{t}(h_{n}−h_{D})k_{L}2(0,T;V^{0})

= sup

kwk_{L2 (0,T;V}_{)}≤1

Z T

0

h∂t(h_{n}−h_{D}), wiV^{0},V dt

= sup

kwk_{L2 (0,T;V}_{)}≤1

Z T

0

−h∂_{t}h_{D}, wi_{V}^{0}_{,V}dt−1
φ

Z

Ω_{T}

δ+KT_{s}(¯h^{n})

∇h_{n}· ∇w dx dt

+ Z

Ω_{T}

KTs(¯h^{n})∇f¯^{n}· ∇w dx dt−
Z

Ω_{T}

QsTs(¯h^{n})w
dx dt

. Since

Z

ΩT

δ+KTs(¯h^{n})

∇hn· ∇w dx dt

≤ δ+K+h2

khnkL^{2}(0,T;H^{1}(Ω))kwkL^{2}(0,T;V),
and sincehn is uniformly bounded inL^{2}(0, T;H^{1}(Ω)), we write

Z

Ω_{T}

δ+KT_{s}(¯h^{n})

∇hn· ∇w dx dt

≤ δ+K_{+}h_{2}

C_{M}kwkL^{2}(0,T;V). (3.12)
Furthermore we have

Z

Ω_{T}

Ts(¯h^{n}))∇f¯^{n}· ∇w dx dt

≤M h2kwk_{L}2(0,T;V), (3.13)

Z

ΩT

Q_{s}T_{s}(¯h^{n})w dx dt

≤ kQ_{s}k_{L}2(0,T;H)h_{2}kwk_{L}2(0,T;V). (3.14)
Summing (3.12)–(3.14), we conclude that

k∂t(h_{n}−h_{D})kL^{2}(0,T;V^{0})≤D_{M}, (3.15)
where

DM =k∂thDk^{2}_{L}2(0,T;(H^{1}(Ω))^{0})+δCM+h2

φ(K+CM +M+kQsk_{L}2(0,T;H)).

We have proved that the sequence h_{n}

n is uniformly bounded in the space
W(0, T). Using Aubin-Lions’ lemma, we can extract a subsequence (hn_{k})k, con-
verging strongly inL^{2}(ΩT), almost everywhere in (0, T)×Ω, and weakly inW(0, T)
to some limit denoted byv. From the a.e. convergence in ΩT, we see that for all
w∈W(0, T),Tl(¯h^{n})∇w→Tl(¯h)∇wstrongly inL^{2}(ΩT) by dominated convergence.

It follows thatv solves (3.6) and (3.3)-(3.4). By uniqueness of the solution of that
system, we conclude that v =h and that the whole sequence hn →h weakly in
W(0, T) and strongly inL^{2}(0, T;H).

The sequential continuity ofF1in L^{2}(0, T;H) is established.

Sequential continuity of F2 in L^{2}(0, T;H) when F is restricted to any
bounded subset ofW(0, T)×L^{2}(0, T;H^{1}(Ω)) As above, we study the sequential
continuity of F2 by setting fn := F2(¯h^{n},f¯^{n}), f := F2(¯h,f¯), and showing first
that fn →f in L^{2}(0, T;H^{1}(Ω)) weakly. The key estimates are obtained using the
same type of arguments than those in the proof of the sequential continuity ofF1.
The details are omitted. We only point out that we can use the estimate (3.11)
previously derived forh_{n} to obtain the following estimates for f_{n}:

kfnkL^{∞}(0,T;H)≤E_{M} =E_{M} δ_{2}, K, f_{D}, h_{2}, l, M, C_{M}

, (3.16)

kf_{n}k_{L}2(0,T;V)≤F_{M} =F_{M} δ_{2}, K, f_{D}, h_{2}, l, M, C_{M}

. (3.17)

For proving the sequential compactness offn in L^{2}(0, T;H), we need some fur-
ther work since we can not use a Aubin’s compactness criterium in the elliptic
context characterizing fn. We actually get a stronger result: we claim and prove
thath_{2}f_{n}− Ts(¯h_{n}) converges inL^{2}(0, T;H^{1}(Ω)), whereTsis any function such that
T_{s}^{0} =T_{s}. Indeed, we recall that the variational formulations defining respectively
f_{n} andf are, for any w∈L^{2}(0, T;V),

Z

ΩT

h2K∇fn· ∇w dx dt− Z

ΩT

KTs(¯hn)∇¯hn· ∇w dx dt

− Z

Ω_{T}

(Q_{s}T_{s}(¯h_{n}) +Q_{f}T_{f}(¯h_{n}))w dxdt= 0,

(3.18)

Z

ΩT

h2K∇f· ∇w dx dt− Z

ΩT

KTs(¯h)∇¯h· ∇w dx dt

− Z

Ω_{T}

(Q_{s}T_{s}(¯h) +Q_{f}T_{f}(¯h))w dxdt= 0.

(3.19)

Choosingw=h2fn− Ts(¯hn)−h2fD+Ts(hD) in (3.18) we letn→ ∞. The already known convergence results let us pass to the limit in

n→∞lim Z

Ω_{T}

(Q_{s}T_{s}(¯h_{n}) +Q_{f}T_{f}(¯h_{n})) h_{2}f_{n}− Ts(¯h_{n})−h_{2}f_{D}+Ts(h_{D})
dxdt

= Z

ΩT

(QsTs(¯h) +QfTf(¯h)) h2f− Ts(¯h)−h2fD+Ts(hD) dx dt.

Using moreover (3.19) for the test functionw=h2f− Ts(¯h)−h2fD+Ts(hD), we conclude that

n→∞lim Z

ΩT

K∇ h2fn− Ts(¯hn)−h2fD+Ts(hD)

· ∇ h2fn− Ts(¯hn)−h2fD+Ts(hD) dx dt

= Z

ΩT

K∇ h2f− Ts(¯h)−h2fD+Ts(hD)

· ∇ h2f − Ts(¯h)−h2fD+Ts(hD) dx dt.

It follows that

n→∞lim Z

Ω_{T}

K∇(Fn−F)· ∇(Fn−F)dx dt= 0

ifFn =h2fn− Ts(¯hn)−h2fD+Ts(hD) andF=h2f− Ts(¯h)−h2fD+Ts(hD). Since
Kξ·ξ≥ K_{−}|ξ|^{2} for any ξ ∈R^{2} with K_{−} >0, the latter result and the Poincar´e
inequality let us ensure thatFn →F in L^{2}(0, T;V). SinceTs(¯hn)→ Ts(¯h) almost
everywhere in ΩT andh2>0, it follows in particular thatfn→f in L^{2}(0, T;H).

Existence of C ⊂ W(0, T)×L^{2}(0, T; (H^{1}(Ω)) such that F(C) ⊂ C. We aim
now to prove that there exists a nonempty bounded closed convex set ofW(0, T)×
L^{2}(0, T;H^{1}(Ω)), denoted byC, such thatF(C)⊂ C. We notice that this result will
imply that there exists a real numberM >0, depending only on the initial data,
such that for (h, f) =F(¯h,f¯)∈W, we have

k∇hk(L^{2}(0,T;H))^{2}≤M and k∇fk(L^{2}(0,T;H))^{2} ≤M. (3.20)

Takingw=h−hD ∈L^{2}(0, T;V) (resp. w=f −fD∈L^{2}(0, T;V)) in (3.6) (resp.

(3.7)) leads to φ

Z T

0

h∂_{t}h, h−h_{D}i_{V}^{0}_{,V}dt+
Z

ΩT

δ∇h· ∇(h−h_{D})dx dt
+

Z

Ω_{T}

KTs(¯h)∇h· ∇(h−hD)dx dt+ Z

Ω_{T}

QsTs(¯h

(h−hD)dx dt

− Z

ΩT

KTs(¯h)∇f¯· ∇(h−hD)dx dt= 0,

(3.21)

and Z

ΩT

h2K∇f· ∇(f−fD)dx dt− Z

ΩT

KTs(¯h)∇¯h· ∇(f−fD)dx dt

− Z

Ω_{T}

(Q_{s}T_{s}(¯h) +Q_{f}T_{f}(¯h))(f−f_{D})dxdt= 0.

(3.22)

We apply Lemma 2.1 to the functionf defined byf(u) =uforu∈Rto compute the first terms of (3.21). We obtain

Z T

0

h∂t(h−h_{D}),(h−hD)iV^{0},Vdt=1
2

Z

Ω

(h−hD)^{2}(T, x)dx−1
2

Z

Ω

(h−hD)^{2}(0, x)dx.

Summing equations (3.21) and (3.22), we obtain φ

2 Z

Ω

(h−hD)(T, x)^{2}dx+
Z

Ω_{T}

δ∇(h−hD)· ∇(h−hD)dx dt +

Z

ΩT

h2K∇(f −fD)· ∇(f−fD)dx dt +

Z

Ω_{T}

Ts(¯h)K∇(h−hD)· ∇(h−hD)dx dt

= Z

ΩT

Ts(¯h)K∇(¯h−hD)· ∇(f−fD) dx dt

| {z }

(1)

+ Z

Ω_{T}

KTs(¯h)∇( ¯f −fD)· ∇(h−hD)dx dt

| {z }

(2)

+φ 2

Z

Ω

(h−h_{D})(0, x)^{2}dx−
Z

Ω_{T}

δ∇hD· ∇(h−h_{D})dx dt

− Z

ΩT

h2K∇fD· ∇(f−fD)dx dt− Z

ΩT

Ts(¯h)K∇hD· ∇(h−hD)dx dt +

Z

Ω_{T}

T_{s}(¯h)K∇h_{D}· ∇(f−f_{D})dx dt+
Z

Ω_{T}

KT_{s}(¯h)∇f_{D}· ∇(h−h_{D})dx dt

− Z

Ω_{T}

QsTs(¯h)(h−hD)dx dt+ Z

Ω_{T}

QsTs(¯h) +QfTf(¯h)

(f−fD)dx dt.

We have

|(1)| ≤ Z

Ω_{T}

δ

3|∇(¯h−h_{D})|^{2}dx dt+3(h2−δ1)^{2}K_{+}^{2}
4δ

Z

Ω_{T}

|∇(f−f_{D})|^{2}dx dt,

|(2)| ≤ Z

ΩT

K_{−}h2

3 |∇( ¯f −f_{D})|^{2}dx dt
+3K_{+}^{2}(h_{2}−δ_{1})

4K_{−}h_{2}
Z

Ω_{T}

T_{s}(¯h)|∇(h−h_{D})|^{2}dx dt.

Also Z

Ω_{T}

δ∇h_{D}· ∇(h−h_{D})dx dt
≤ δ

2 Z

Ω_{T}

|∇(h−h_{D})|^{2}dx dt+δ
2

Z

Ω_{T}

|∇h_{D}|^{2}dx dt,

Z

Ω_{T}

h2K∇fD· ∇(f−fD)dx dt

≤h2K−

12 Z

Ω_{T}

|∇(f −fD)|^{2}dx dt+3K_{+}^{2}h2

K_{−}
Z

Ω_{T}

|∇fD|^{2}dx dt,

Z

ΩT

Ts(¯h)K∇hD· ∇(h−hD)dx dt

≤ K_{−}
8

Z

ΩT

T_{s}(¯h)|∇(h−h_{D})|^{2}dx dt+2K_{+}^{2}h_{2}
K_{−}

Z

ΩT

|∇h_{D}|^{2}dx dt,

Z

Ω_{T}

Ts(¯h)K∇hD· ∇(f−fD)dx dt

≤h2K−

12 Z

Ω_{T}

|∇(f−fD)|^{2}dx dt+3K_{+}^{2}h2

K_{−}
Z

Ω_{T}

|∇hD|^{2}dx dt,

Z

ΩT

Ts(¯h)K∇fD· ∇(h−hD)dx dt

≤ K_{−}
8

Z

ΩT

Ts(¯h)|∇(h−hD)|^{2}dx dt+2K_{+}^{2}h_{2}
K_{−}

Z

ΩT

|∇fD|^{2}dx dt,

Z

Ω_{T}

QsTs(¯h)(h−hD)dx dt

≤ δ 8 Z

ΩT

Ts(¯h)|∇(h−hD)|^{2}dx dt+2C_{P}^{2}
δ

Z

ΩT

Q^{2}_{s}T_{s}^{2}(¯h)dx dt,

Z

Ω_{T}

Q_{s}T_{s}(¯h) +Q_{f}T_{f}(¯h)

(f−f_{D})dxdt

≤h2K−

12 Z

Ω_{T}

|∇(f −fD)|^{2}dx dt+ 3C_{P}^{2}
h2K_{−}

Z

Ω_{T}

QsTs(¯h) +QfTf(¯h)2

dx dt,
where the Poincar´e’s constant is denoted by C_{P}. Gathering these estimates, we
conclude that

φ 2 Z

Ω

(h−hD)(T, x)^{2}dx+δ
2

Z

Ω_{T}

|∇(h−hD)|^{2}dx dt+h2K−

2 Z

Ω_{T}

|∇(f−fD)|^{2}dx dt
+h2K−

4 −3(h2−δ1)^{2}K_{+}^{2}
4δ

Z

Ω_{T}

|∇(f−fD)|^{2}dx dt
+3K−

4 −3K_{+}^{2}(h2−δ1)
4K_{−}h2

Z

Ω_{T}

Ts(¯h)|∇(h−hD)|^{2}dx dt

≤ δ 3 Z

Ω_{T}

|∇(¯h−h_{D})|^{2}dx dt+K_{−}h_{2}
3

Z

Ω_{T}

|∇( ¯f−f_{D})|^{2}dx dt

+φ 2 Z

Ω

(h−h_{D})(0, x)^{2}dx+δ
2
Z

ΩT

|∇h_{D}|^{2}dx dt

+5K_{+}^{2}h_{2}
K_{−}

Z

Ω_{T}

|∇f_{D}|^{2}dx dt+5K_{+}^{2}h_{2}
K_{−}

Z

Ω_{T}

|∇h_{D}|^{2}dx dt

+ 1 2φ

Z

Ω_{T}

Q^{2}_{s}T_{s}^{2}(¯h)dx dt+ 3C_{P}^{2}
h2K_{−}

Z

Ω_{T}

QsTs(¯h) +QfTf(¯h)2

dx dt. (3.23) Introducing the constant

C_{0}:=6φ
2
Z

Ω

(h−h_{D})(0, x)^{2}dx+δ
2

Z

Ω_{T}

|∇h_{D}|^{2}dx dt

+5K_{+}^{2}h_{2}
K_{−}

Z

Ω_{T}

|∇f_{D}|^{2}dx dt+5K_{+}^{2}h_{2}
K_{−}

Z

Ω_{T}

|∇h_{D}|^{2}dx dt

+ 1 2φ

Z

Ω_{T}

Q^{2}_{s}T_{s}^{2}(¯h)dx dt+ 3C_{P}^{2}
h2K_{−}

Z

Ω_{T}

QsTs(¯h) +QfTf(¯h)2

dx dt (3.24)

and recalling that the parameters satisfy (3.5), we infer that

δk∇(h−h_{D})k^{2}_{(L}2(ΩT))^{2}+h_{2}K_{−}k∇(f −f_{D})k^{2}_{(L}2(ΩT))^{2} ≤C_{0},
and

δk∇(¯h−h_{D})k^{2}_{(L}2(Ω_{T}))^{2}+h_{2}K_{−}k∇( ¯f −f_{D})k^{2}_{(L}2(Ω_{T}))^{2} ≤C_{0}.
Note that (3.23) yields

k∇(h−h_{D})k_{L}2(Ω_{T})≤p

C_{0}/δ, k∇(f−f_{D})k_{L}2(Ω_{T})≤p

C_{0}/h_{2}K_{−}

and 1

2 Z

Ω

(h−h_{D})(τ, x)^{2}dx≤C_{0}, for allτ≤T.

Conclusion. We introduce the set C:=

(h−hD, f−fD)∈W(0, T)×L^{2}(0, T; (H^{1}(Ω)) : (h(0,·),
f(0,·)) = (h0, f0), δk∇(h−hD)k^{2}_{L}2(Ω_{T})

+h2K−k∇(f−fD)k^{2}_{L}2(ΩT)≤C0, k∂thkL^{2}(0,T ,V^{0})≤DM.

(3.25)

where C0 is defined by (3.24) and M := max(p

C0/δ,p

C0/h2K_{−}). Then C is a
nonempty, closed, convex, bounded set in (L^{2}(0, T;H))^{2}, defined such thatF(C)⊂
C. Indeed, let us check that C is closed. Let (h_{n}, f_{n})_{n} be a sequence in C such
that (h_{n}, f_{n})→(h, f) inL^{2}(Ω_{T}). Since the sequence h_{n}

n is uniformly bounded
in the space W(0, T), we can extract a subsequence (hn_{k})k converging weakly in
W(0, T) to some limit denoted by ¯h. Then h = ¯h ∈ W(0, T) and khk_{W(0,T)} ≤
lim inf_{k→∞}kh_{n}_{k}k_{W}_{(0,T)}. Similarly, since the sequence f_{n}

n is uniformly bounded
in the space L^{2}(0, T;H^{1}(Ω)), there exists a subsequence such that ∇fn_{k} * ∇f
weakly inL^{2}(ΩT) andkfk_{L}2(0,T;H^{1}(Ω))≤lim inf_{k→∞}kfn_{k}k_{L}2(0,T;H^{1}(Ω)). SinceC is
also a bounded set inW(0, T))×L^{2}(0, T;H^{1}(Ω)), we also proved thatFrestricted
to C is sequentially continuous in (L^{2}(0, T;H))^{2}. For the fixed point strategy, it
remains to show the compactness ofF(C). Since we work in metric spaces, proving
its sequential compactness is sufficient. The compactness ofF1(C) is straightforward
due to the Aubin’s theorem. Let us further detail the proof forF2(C). Let{fn}be
a sequence in F2(C). It is associated with a sequence{(¯h_{n},f¯_{n}} inC. The Aubin’s
compactness theorem let us ensure that there exists a subsequence, not renamed

for convenience, and ¯h ∈ W(0, T) +hD such that ¯hn → ¯h in L^{2}(0, T;H) and
almost everywhere in ΩT. Thus we can follow the lines beginning just after (3.17)
for proving that fn → f in L^{2}(0, T;H). The sequential compactness of F2(C) in
L^{2}(0, T;H) is proven.

We now have the tools for using the Schauder’s fixed point theorem [13, Corollary
3.6]. There exists (h−h_{D}, f−f_{D})∈ Csuch that F(h, f) = (h, f). Then (h, f) is
a weak solution of problem (3.1)–(3.4).

Step 2: Maximum Principles. We are going to prove that for almost every x∈Ω and for allt∈(0, T),

δ1≤h(t, x)≤h2.

First show thath(t, x)≤h_{2} a.e. x∈Ω and for allt∈(0, T). We set
h_{m}= h−h_{2}^{+}

= sup(0, h−h_{2})∈L^{2}(0, T;V).

It satisfies ∇hm = χ_{{h>h}_{2}_{}}∇h and h_{m}(t, x) 6= 0 if and only if h(t, x) > h_{2},
where χ denotes the characteristic function. Let τ ∈ (0, T). Taking w(t, x) =
h_{m}(t, x)χ_{(0,τ)}(t) in (3.1) yields:

Z τ

0

φh∂th, h_{m}χ_{(0,τ}_{)}iV^{0},Vdt+
Z τ

0

Z

Ω

δ∇h· ∇hmdx dt +

Z τ

0

Z

Ω

KT_{s}(h)∇h· ∇h_{m}dx dt+
Z τ

0

Z

Ω

KT_{s}(h)∇f· ∇h_{m}dx dt
+

Z τ

0

Z

Ω

QsTs(h)hmdx dt= 0;

that is, Z τ

0

φh∂_{t}h, h_{m}i_{V}^{0}_{,V}dt+
Z τ

0

Z

Ω

δχ_{{h>h}_{2}_{}}|∇h|^{2}dx dt
+

Z τ

0

Z

Ω

KTs(h)χ_{{h>h}_{2}_{}}|∇h|^{2}dx dt+
Z τ

0

Z

Ω

KTs(h)∇f · ∇hm(x, t)dx dt +

Z τ

0

Z

Ω

QsTs(h)hm(x, t)dx dt= 0.

(3.26)

To evaluate the first term in the left-hand side of above equation, we apply Lemma
2.1 with functionf defined byf(λ) =λ−h2,λ∈R. We setW_{1}(0, T) :=W(0, T)×

L^{2}(0, T;H^{1}(Ω)). We write
Z τ

0

φh∂th, hmiV^{0},Vdt= φ
2
Z

Ω

h^{2}_{m}(τ, x)−h^{2}_{m}(0, x)
dx=φ

2 Z

Ω

h^{2}_{m}(τ, x)dx,
sincehm(0,·) = h0(·)−h2(·)+

= 0. Moreover sinceTs(h)χ_{{h>h}_{2}_{}}= 0 by definition
of Ts, the three last terms in the left-hand side of (3.26) are null. Hence (3.26)
becomes

φ 2 Z

Ω

h^{2}_{m}(τ, x)dx≤ −
Z τ

0

Z

Ω

δχ_{{h>h}_{2}_{}}|∇h|^{2}dx dt≤0.

Thenhm= 0 a.e. in ΩT.

Now we claim thatδ1≤h(t, x) a.e. x∈Ω and for all t∈(0, T). We set
h_{m}= h−δ_{1}−

∈L^{2}(0, T;V).

Letτ ∈(0, T). We recall thath_{m}(0,·) = 0 a.e. in Ω by the maximum principle satis-
fied by the initial datah_{0}. Moreover,∇(h−δ1)·∇hm=χ_{{δ}_{1}_{−h>0}}|∇(h−δ1)|^{2}. Thus,