Clément Cancès
email: clement.cances@bretagne.ens-cachan.fr

Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities

Keywords: MSC subject classification. 35R05, 65M12 keywords. capillarity discontinuities, degenerate parabolic equation, finite volume scheme

We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existence-uniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model.

Introduction

The models of immiscible two-phase flows in porous media are widely used in petroleum engineering in order to predict the positions where oil could be collected. The discontinuities of the physical characteristics due to brutal change of lithology play a crucial role in the phenomenon of oil trapping, preventing the light hydrocarbons from reaching the surface. It seems that the discontinuities with respect to the space variable of a particular function, called the capillary pressure, are responsible of the phenomenon of oil-trapping [START_REF] Van Duijn | The effect of capillary forces on immiscible two-phase flows in heterogeneous porous media[END_REF][START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF].

In this paper, we consider one-dimensional two-phase flows in heterogeneous porous media, which are made of several homogeneous submedia. A simplified model of two-phase flow within this rock is described in the first section, leading to the definition of the weak solution. The transmission conditions at the interface between the different submedia are written using the graph formalism introduced in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] for the connection of the capillary pressures, which is simple to manipulate and allows to deal with any type of discontinuity of the domain, without any compatibility constraint, contrary to what occurs in [START_REF] Cancès | Nonlinear parabolic equations with spatial discontinuities[END_REF] and to a lesser extent in [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF][START_REF] Enchéry | Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces[END_REF].

The graph way to connect the capillary pressures at the interfaces is well suited to be discretized by a monotonous Finite Volume scheme. A discretization is proposed in the second section of the paper. Adapting the material from the book of Eymard, Gallouët & Herbin [START_REF] Eymard | Finite volume methods[END_REF] to our case, it is shown that the discrete solution provided by the scheme converges, up to a subsequence, to a weak solution as the step of the discretization tends to 0. The monotonicity of the transmission conditions is fundamental for proving the convergence of the scheme.

Unfortunately, we are not able to show the uniqueness of the weak solution to the problem, because of the lack of regularity. As it will be shown in the fourth section, supposing that the fluxes are uniformly bounded with regard to space and time is sufficient to claim the uniqueness of the solution. The uniqueness proof is an adaptation of the one given in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] to the case where the convection is not neglected. Here again, the monotonicity of the transmission conditions at the interfaces is strongly used.

The existence of a bounded flux solution is the topic of Section 3. It is shown that if the initial data is regular enough to ensure that the initial flux is bounded with respect to space, then the flux will remain bounded with respect to space and to time. Such a result has already been obtained in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF], where a parabolic regularization of the problem had been introduced. A maximum principle on the flux follows. We also quote [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF], in which a BV -estimate is shown on the flux. Since the monotonous schemes introduce some numerical diffusion, a strong analogy can be done between a uniformly parabolic regularization of the problem and the numerical approximation via a monotonous scheme. The convergence of the discrete solution to a bounded flux solution for regular enough initial data is thus naturally expected and stated in Theorem 3.1. The monotonicity of the transmission relations is essential during the proof.

We are able to prove the uniqueness of the bounded-flux solution to the problem using the doubling variable technique. This work performed in Section 4 is summarized in Theorem 4. [START_REF] Adimurthi | Conservation law with discontinuous flux[END_REF] In Section 5, a density argument allows to extend the existence and uniqueness frame to any initial data, using the notion of SOLA (Solution Obtained as Limit of Approximation). It is a more restrictive notion than the notion of weak solution, even if we are not able to prove the existence of a weak solution which is not a SOLA. The main result of the paper is given in Theorem 5.1, which claims that the whole sequence of discrete solutions built using the finite volume scheme introduced in Section 2 converges towards the unique SOLA to the problem. Finally, a numerical example is given in Section 6. This example gives an evidence of the entrapment of a certain quantity of oil under the interface.

Presentation of the problem

We consider a one-dimensional heterogeneous porous medium, which is an apposition of homogeneous porous media, representing the different geological layers. The physical properties of the medium only depend on the rock type and are piece-wise constant.

For the sake of simplicity, we only deal with two geological layers of same size. A generalization to an arbitrary finite number of geological layers would only lead to notation difficulties. In the sequel, we denote by Ω = (-1, 1) the heterogeneous porous medium, and by Ω 1 = (-1, 0), Ω 2 = (0, 1) the two homogeneous layers. The interface between the layers is thus {x = 0}. T is a positive real value.

We consider an incompressible and immiscible oil-water flow through Ω. Writing the conservation of each phase, and using Darcy's law leads to: for all (x, t) ∈ Ω i × (0, T ),

φ i ∂ t u -∂ x [µ o,i (u) (∂ x P o,i -ρ o g)] = 0, (1)
-φ i ∂ t u -∂ x [µ w,i (u) (∂ x P w,i -ρ w g)] = 0, (2)
where φ i ∈ (0, 1) is the porosity of the porous media Ω i , u is the oil-saturation (then (1u) is the watersaturation), µ β,i is the mobility of the phase β = w, o, where w stands for water, and o for oil. We denote by P β,i the pressure of the phase β, by ρ β its density, and by g the gravity.

Adding ( 1) and ( 2) shows that :

∂ x q = 0, where q = -µ w,i (u) (∂ x P w,i -ρ w g) -µ o,i (u) (∂ x P o,i -ρ o g) (3)
is the total flow-rate. For the sake of simplicity, we suppose that q does not depend on time, even if all the results presented below still hold for q ∈ BV (0, T ), as it is shown in [13, chapter 4]. Using (3) in ( 1) and (2) yields:

φ i ∂ t u + ∂ x µ o,i (u) µ o,i (u) + µ w,i (u) q + λ i (u)(ρ o -ρ w )g -λ i (u)∂ x (P o,i -P w,i ) = 0, (4)
where

λ i (u) = µ o,i (u)µ w,i (u) µ o,i (u) + µ w,i (u)
.

One assumes that the capillary pressure (P o,i -P w,i ) depends only on the saturation and of the rock type. More precisely, (P o,i -P w,i ) = π i (u), where π i (u) is supposed to be an increasing Lipschitz continuous function. The equation ( 4) becomes

φ i ∂ t u + ∂ x (f i (u) -λ i (u)∂ x π i (u)) = 0, (5)
where

f i (u) = µ o,i (u) µ o,i (u) + µ w,i (u) q(t) + λ i (u)(ρ o -ρ w )g.
We do the following assumptions on the functions appearing in the equation.

Assumptions 1 For i = 1, 2, one has:

1. π i is an increasing Lipschitz continuous function; 2. µ o,i is an increasing Lipschitz continuous function on [0, 1], with µ o,i (0) = 0; 3. µ w,i is a decreasing Lipschitz continuous function on [0, 1], with µ w,i (1) = 0.

Remark 1.1 It is often supposed for such problems that the functions µ β,i are monotonous in a large sense, and that there exist irreducible saturations s i , S i ∈ (0, 1), with s i < S i , such that

µ o,i (u) = 0 if u ∈ [0, s i ], µ w,i (u) = 0 if u ∈ [S i , 1].
If we assume that the functions µ β,i are strictly monotonous on their support, a convenient scaling would allow us to suppose that assumptions 1 are fulfilled.

We denote by ϕ i (s) = s 0 λ i (a)π ′ i (a)da, then (5) can be rewritten

φ i ∂ t u + ∂ x (f i (u) -∂ x ϕ i (u)) = 0. ( 6
)
Properties 1.1 It follows directly from assumptions 1 that for i = 1, 2 :

1. f i is Lipschitz continuous and f i (0) = 0, f i (1) = q;
2. λ i is Lipschitz continuous, and

λ i (0) = λ i (1) = 0, λ i (u) > 0 if u > 0; 3. ϕ i is an increasing Lipschitz continuous fulfilling ϕ i (0) = 0, ϕ ′ i (0) = ϕ ′ i (1) = 0.
We deduce from the properties 1.1 that ( 6) is a degenerated nonlinear parabolic equation.

Let us now focus on the transmission conditions through the interface {x = 0}. We denote by α i = lim s→0 π i (s) and β i = lim s→1 π i (s). We define the monotonous graphs πi by:

πi (s) =    π i (s) if s ∈ (0, 1), (-∞, α i ] if s = 0, [β i , +∞) if s = 1. (7)
Let u i denote the trace of u |Ωi on {x = 0} (which is supposed to exist for the moment). The trace on {x = 0} from Ω i of the pressure P β,i of the phase β is still denoted by P β,i . As it is exposed in [START_REF] Enchéry | Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces[END_REF] (see also [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF]), the pressure of the phase β can be discontinuous through the interface {x = 0} in the case where it is missing in the upstream side. This can be written

µ β,1 (u 1 )(P β,1 -P β,2 ) + -µ β,2 (u 2 )(P β,2 -P β,1 ) + = 0, β ∈ {o, w}. (8)
The conditions [START_REF] Bachmann | Finite volume schemes for a non linear hyperbolic conservation law with a flux function involving discontinuous coefficients[END_REF] have direct consequences on the connection of the capillary pressures through {x = 0}. Indeed, if 0 < u 1 , u 2 < 1, then the partial pressures P o and P w have both to be continuous, thus the connection of the capillary pressures π 1 (u 1 ) = π 2 (u 2 ) is satisfied. If u 1 = 0 and 0 < u 2 ≤ 1, then P o,1 ≥ P o,2 and P w,1 ≤ P w,2 , thus π 2 (u 2 ) ≤ π 1 (0). The same way, u 1 = 1 and 0 ≤ u 2 < 1 implies π 2 (u 2 ) ≥ π 1 [START_REF] Adimurthi | Conservation law with discontinuous flux[END_REF]. Checking that the definition of the graphs π1 and π2 implies π1 (0) ∩ π2 (0) = ∅, π1 (1) ∩ π2 (1) = ∅, we can claim that (8) implies:

π1 (u 1 ) ∩ π2 (u 2 ) = ∅. (9)
The conservation of each phase leads to the connection of the fluxes on {x = 0}. Denoting by F i the flux in Ω i , i.e. for all x ∈ Ω i ,

F i (x, t) = f i (u)(x, t) -∂ x ϕ i (u)(x, t),
the connection of the fluxes through the interface can be written

F 1 (0, •) = F 2 (0, •), (10)
where [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF] has to be understood in a weak sense. We now turn to the problem of the boundary conditions. Because of technical difficulties occurring during section 4, we want that the solution to the flow admits bounded fluxes, at least for regular initial data. This will force us to consider specific boundary conditions, which will involve bounded fluxes.

Let G i : (a, b) → G i (a, b) (i = 1,
2) be a function fulfilling the following properties :

• G i is Lipschitz continuous, non-decreasing w.r.t. its first argument, and non-increasing w.r.t. the second.

• for all a ∈ [0, 1], G i (a, a) = f i (a). F 1 (-1, t) = G 1 (u(t), u(-1, t)), F 2 (1, t) = G 2 (u(1, t), u(t)). (11)
The way in which we approximate the boundary condition shall be judiciously compared with the discretization of the boundary conditions for scalar hyperbolic conservation laws using monotonous Finite Volume schemes (see [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF]).

We consider an initial data u 0 ∈ L ∞ (Ω), with 0 ≤ u 0 ≤ 1, then we can write the initial-boundary-value problem:

               φ i ∂ t u + ∂ x [f i (u) -∂ x ϕ i (u)] = 0 in Ω i × (0, T ), F 1 (0, •) = F 2 (0, •) on (0, T ), π1 (u 1 ) ∩ π2 (u 2 ) = ∅ on (0, T ), u(t = 0) = u 0 in Ω, F 1 (-1, t) = G 1 (u(t), u(-1, t)) on (0, T ), F 2 (1, t) = G 2 (u(1, t), u(t))
on (0, T ).

(P)

We now define the notion of weak-solution Definition 1.1 A function u is said to be a weak solution to the problem (P) if it fulfills:

1. u ∈ L ∞ (Ω × (0, T )), with 0 ≤ u ≤ 1; 2. for i = 1, 2, ϕ i (u) ∈ L 2 (0, T ; H 1 (Ω i )); 3
φ i u(x, t)∂ t ψ(x, t)dxdt + i=1,2 Ωi φ i u 0 (x)ψ(x, 0)dx + T 0 i=1,2 Ωi [f i (u)(x, t) -∂ x ϕ i (u)(x, t)] ∂ x ψ(x, t)dxdt + T 0 G 1 (u(t), u(-1, t))ψ(-1, t)dt - T 0 G 2 (u(1, t), u(t))ψ(1, t)dt = 0. ( 12
)
2 The finite volume scheme

In this section, we build an implicit finite volume scheme in order to approximate a solution of (P). We will adapt the convergence proofs stated in [START_REF] Chainais-Hillairet | Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate[END_REF][START_REF] Eymard | Finite volume methods[END_REF][START_REF] Enchéry | Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces[END_REF], which are based on monotonicity properties of the scheme. This will allow us to claim the convergence in L p (Ω × (0, T )), up to a subsequence, of the discrete solutions built using the finite volume scheme towards a weak solution to the problem as step of the the discretization tends to 0.

The finite volume approximation

We first need to discretize all the data, so that we can define an approximate problem through the finite volume scheme.

Discretization of Ω: for the sake of simplicity, we will only deal with uniform spatial discretizations. Let N ∈ N ⋆ , one defines:

x j = j/N, ∀j ∈ [[-N, N]], x j+1/2 = j + 1/2 N , ∀j ∈ [[-N, N -1]].
One denotes by δx = 1/N . Discretization of (0, T ): once again, we will only deal with uniform discretizations. Let M ∈ N ⋆ , one defines: for all n ∈ [[0, M]], t n = nT /M . One denotes by δt = T /M . We denote by D the discretization of Ω × (0, T ) deduced of those of Ω and (0, T ).

Discretization of

u 0 : ∀j ∈ [[-N, N -1]], u 0,D (x j+1/2 ) = u 0 j+1/2 = 1 δx xj+1 xj u 0 (x)dx. ( 13
)
Discretization of the boundary conditions: ∀n ∈ [[0, M]],

u n+1 = 1 δt t n+1 t n u(t)dt, u n+1 = 1 δt t n+1 t n u(t)dt.
The Finite Volume scheme: the first equation of (P) can be rewritten:

φ i ∂ t u + ∂ x F i (x, t) = 0, in Ω × (0, T ) with F i (x, t) = f i (u) -∂ x ϕ i (u).
We consider the following implicit scheme:

∀j ∈ [[-N, N -1]], ∀n ∈ [[0, M -1]], φ i u n+1 j+1/2 -u n j+1/2 δt δx + F n+1 j+1 -F n+1 j = 0 ( 14
)
where F n+1 j is an approximation of the mean flux through x j on (t n , t n+1 ), and i is chosen such that (x j , x j+1 ) ⊂ Ω i . This notation will hold all along the paper. We choose a monotonous discretization of the flux: ∀j ∈

[[-N + 1, -1]] ∪ [[1, N -1]], ∀n ∈ [[0, M -1]], F n+1 j = G i (u n+1 j-1/2 , u n+1 j+1/2 ) - ϕ i (u n+1 j+1/2 ) -ϕ i (u n+1 j-1/2 ) δx , (15)
where G i is the same function as the one defined in [START_REF] Blanchard | Stefan problems with nonlinear diffusion and convection[END_REF]. We also define

F n+1 -N = G 1 (u n+1 , u -N +1/2 ), F n+1 N = G 2 (u N -1/2 , u n+1 ), (16)
F n+1 0 = G 1 (u n+1 -1/2 , u n+1 0,1 ) - 2(ϕ 1 (u n+1 0,1 ) -ϕ 1 (u n+1 -1/2 )) δx (17) = G 2 (u n+1 0,2 , u n+1 1/2 ) - 2(ϕ 2 (u n+1 1/2 ) -ϕ 2 (u n+1 0,2 )) δx , (18)
where

u n+1 0,1 , u n+1 0,2 moreover satisfy π1 (u n+1 0,1 ) ∩ π2 (u n+1 0,2 ) = ∅. (19)
Remark 2.1 The choice of the boundary conditions F n+1 ±N has been done in order to ensure

F n+1 -N ≤ G 1 ∞ < ∞, F n+1 N ≤ G 2 ∞ < ∞.
Thanks to the following lemma, such a couple (u n+1 0,1 , u n+1 0,2 ) is unique in [0, 1] 2 , thus the discrete transmission conditions system (17)-( 18)-( 19) is well posed. Lemma 2.1 For all (a, b) ∈ [0, 1] 2 , there exists a unique couple (c, d) ∈ [0, 1] 2 such that:

G 1 (a, c) - 2(ϕ 1 (c) -ϕ 1 (a)) δx = G 2 (d, b) - 2(ϕ 2 (b) -ϕ 2 (d)) δx , π1 (c) ∩ π2 (d) = ∅. (20

Proof

For i = 1, 2, π-1 i are continuous non-decreasing functions, increasing on [π i (0), π i (1)] and constant otherwise. Then we can build the continuous non-decreasing function Λ, defined by Λ :

R → R p → G 2 (π -1 2 (p), b) -G 1 (a, π-1 1 (p)) + 2 δx ϕ 1 • π-1 1 (p) -ϕ 1 (a) + ϕ 2 • π-1 2 (p) -ϕ 2 (b) .
For all p such that Λ(p) = 0, the couple (π -1 1 (p), π-1 2 (p)) is a solution to the discrete transmission conditions system (17)-( 18)- [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF]. It is easy to check, using the monotonicity of the functions G i that for all p ≤ min π i (0), Λ(p) ≤ 0. Symmetrically, for all p ≥ max π i (1), Λ(p) ≥ 0. Thus there exists p ⋆ such that Λ(p ⋆ ) = 0.

Suppose that there exists i such that p ⋆ ∈ (π i (0), π i (1)), then since ϕ i is increasing, Λ is increasing on a neighborhood of p ⋆ , and then the solution to the system (20) is unique.

Suppose now that p ⋆ / ∈ i (π i (0), π i (1)). Either p ⋆ ≤ min i π i (0), then c = d = 0, or p ⋆ ≥ max i π i (1), then c = d = 1, or p ⋆ ∈ [π k [START_REF] Adimurthi | Conservation law with discontinuous flux[END_REF], π l (0)] for k = l. We can suppose without any loss of generality that p ⋆ ∈ [π 1 (1), π 2 (0)], then the unique solution to the system (20) is given by c = 1, d = 0.

To conclude the proof of the lemma, it only remains to check that (a, b) → Λ is decreasing w.r.t. each one of its arguments, then the monotonicity of Λ and π-1

Existence and uniqueness of the discrete solution

We will now work on the implicit finite volume scheme given by ( 13)- [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] to show that this approximate problem is well-posed. Definition 2.1 Let N, M be two positive integers and D be the associated discretization of Ω × (0, T ). One defines:

X D,i = z ∈ L ∞ (Ω i × (0, T )) / ∀(x j , x j+1 ) ⊂ Ω i , ∀n ∈ [[0, M -1]], z |(xj,xj+1)×(t n ,t n+1 ] is a constant , and X D = z ∈ L ∞ (Ω × (0, T )) / ∀i = 1, 2, z |Ωi×(0,T ) ∈ X D,i .
One defines u D (x, t) ∈ X D , called discrete solution, given almost everywhere in (-1, 1) × (0, T ) by: for all

j ∈ [[-N, N -1]], for all n ∈ [[0, M -1]], u D (x, 0) = u 0,D (x) = u 0 j+1/2 if (x, t) ∈ (x j , x j+1 ), u D (x, t) = u n+1 j+1/2 if (x, t) ∈ (x j , x j+1 ) × (t n , t n+1 ],
where u n+1 j+1/2 j,n are given by the scheme [START_REF] Cancès | Nonlinear parabolic equations with spatial discontinuities[END_REF].

The monotonicity of the flux

F n+1 j w.r.t. u n+1 k+1/2 k
allows us to rewrite the scheme ( 14) under the form

H j+1/2 u n+1 j+1/2 , u n j+1/2 u n+1 k+1/2 k =j = 0, (21)
where H j+1/2 is continuous, increasing w.r.t. its first argument, and non-increasing w.r.t. all the others.

Definition 2.2 A function v D is said to be a discrete supersolution (resp. w D is a discrete subsolution) if it belongs to X (D), and if it satisfies: ∀j ∈ [[-N, N -1]], H j+1/2 v n+1 j+1/2 , v n j+1/2 v n+1 k+1/2 k =j ≥ 0, (resp. H j+1/2 w n+1 j+1/2 , w n j+1/2 w n+1 k+1/2 k =j ≤ 0 ).
Remark 2.2 A function u D is a discrete solution to the scheme if and only if it is both a supersolution and a subsolution.

Remark 2.3 It follows from the definition of the scheme, particularly from the definitions of the discrete boundary conditions [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only of the space. II. Occurrence of non-classical shocks to model oil-trapping[END_REF] and of the discrete fluxes at the interface (17)- [START_REF] Cancès | On the time continuity of entropy solutions[END_REF], that the constant function equal to 1 is a discrete supersolution, and that the constant function equal to 0 is a discrete subsolution.

We now focus on the existence and the uniqueness of the discrete solution to the scheme. In order to prove the existence of a discrete solution, we first need an a priori estimate on it.

Lemma 2.2 Let u D be a discrete solution to the scheme associated to the initial data u 0,D , let v D be a discrete supersolution associated to the initial data v 0,D , then for all t ∈ [0, T ],

i=1,2 Ωi φ i (u D (x, t) -v D (x, t)) + dx ≤ i=1,2 Ωi φ i (u 0,D (x) -v 0,D (x)) + dx. Symmetrically, if w D is a subsolution associated to the initial w 0,D , i=1,2 Ωi φ i (u D (x, t) -w D (x, t)) -dx ≤ i=1,2 Ωi φ i (u 0,D (x) -w 0,D (x)) -dx.

Proof

Denoting by a⊤b = max(a, b), and a⊥b = min(a, b), it follows from the monotonicity of the functions H j+1/2 implies that

H j+1/2 u n+1 j+1/2 , u n j+1/2 ⊤w n j+1/2 u n+1 k+1/2 ⊤w n+1 k+1/2 k =j ≤ 0, H j+1/2 w n+1 j+1/2 , u n j+1/2 ⊤w n j+1/2 u n+1 k+1/2 ⊤w n+1 k+1/2 k =j ≤ 0,
where w D is a subsolution. Since u n+1 j+1/2 ⊤w n+1 j+1/2 is either equal to u n+1 j+1/2 or to w n+1 j+1/2 ,

H j+1/2 u n+1 j+1/2 ⊤w n+1 j+1/2 , u n j+1/2 ⊤w n j+1/2 u n+1 k+1/2 ⊤w n+1 k+1/2 k =j ≤ 0. ( 22
)
Thanks to the conservativity of the scheme, subtracting ( 21) to [START_REF] Droniou | A density result in Sobolev spaces[END_REF], and summing on j

∈ [[-N, N -1]] yields i=1,2 Ωi φ i u D (x, t n+1 ) -w D (x, t n+1 ) -dx ≤ i=1,2 Ωi φ i (u D (x, t n ) -w D (x, t n )) -dx.

Since this inequality holds for any

n ∈ [[0, M -1]], it directly gives: ∀t ∈ [0, T ], i=1,2 Ωi φ i (u D (x, t) -w D (x, t)) -dx ≤ i=1,2 Ωi φ i (u D (x, 0) -w D (x, 0)) -dx. ( 23
)
The proof of the discrete comparison principle between a discrete solution and a discrete supersolution can be performed similarly.

Let us now state the existence and the uniqueness of the discrete solution.

Proposition 2.3 Let u 0 ∈ L ∞ (Ω), 0 ≤ u 0 ≤ 1 a.e.
, then there exists a unique discrete solution u D to the scheme, which furthermore fulfills 0 ≤ u D ≤ 1 a.e.. Moreover, if v 0 stands for another initial data, 0 ≤ v 0 ≤ 1, approximated by v 0,D following [START_REF] Cancès | Écoulements diphasiques en milieux poreux hétérogènes : modélisation et analyse des effets liés aux discontinuités de la pression capillaire[END_REF], and if we denote by v D the corresponding discrete solution, then the following L 1 -contraction principle holds;

i=1,2 Ωi φ i (u D (x, t) -v D (x, t)) ± dx ≤ i=1,2 Ωi φ i (u 0,D (x) -v 0,D (x)) ± dx, ∀t ∈ [0, T ].

Proof

It follows from Remark 2.3 and from Lemma 2.2 that the following L ∞ a priori estimate holds:

0 ≤ u D (x, t) ≤ 1, for all t ∈ [0, T ], for almost all x ∈ Ω.
Thanks to this estimate, mimicking the proof given in [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF], we can claim the existence of a discrete solution u D . Suppose that u D and v D are two solutions associated to the initial data u 0,D and v 0,D . As it was stressed in the remark 2.2, both u D and v D are both discrete sub-and supersolutions. Then, Lemma 2.2 ensures that the following L 1 -contraction principle holds:

i=1,2 Ωi φ i (u D (x, t) -v D (x, t)) ± dx ≤ i=1,2 Ωi φ i (u 0,D (x) -v 0,D (x)) ± dx, ∀t ∈ [0, T ].
The uniqueness of the discrete solution u D corresponding to the initial data u 0 follows.

The

L 2 ((0, T ); H 1 (Ω i )) estimates
The current subsection is devoted to the proof of the discrete energy estimate stated in Proposition 2.4. Since the discrete solutions are only piecewise constant, we need to introduce discrete semi-norms, which are discrete analogues to the L 2 ((0, T );

H 1 (Ω i )) semi-norms. Definition 2.3 Let i = 1, 2, one defines the discrete L 2 (0, T ; H 1 (Ω i )) semi-norms |•| 1,D,i on X D,i by: ∀z ∈ X D,i , |z| 2 1,D,i = M-1 n=0 δt j∈Jint,i δx z(x j+1/2 , t n+1 ) -z(x j-1/2 , t n+1 ) δx 2 , where J int,1 = [[-N + 1, -1]] and J int,2 = [[1, N -1]].
Proposition 2.4 For i = 1, 2, one defines the Lipschitz continuous increasing functions

ξ i : s → s 0 λ i (a)π ′ i (a)da.
There exists C > 0 only depending on π i , φ i , T, G i such that:

i=1,2 |ξ i (u D )| 2 1,D,i ≤ C.
This estimate is the discrete analogue to:

i=1,2 T 0 Ωi |∂ x ξ i (u)(x, t)| 2 dxdt ≤ C.
In order to prove Proposition 2.4, we will need the following technical lemma. To understand this lemma, first suppose that the total flow rate q is 0. Then, roughly speaking, it claims that, in the case where the capillary pressure is discontinuous at the interface, the discrete flux is oriented from the high capillary pressure to the low capillary pressure. Suppose now that q = 0. In order to respect the conservation of mass, some fluid will have to go through the interface, but we keep a control on the energy.

Lemma 2.5 Let (a, b) ∈ [0, 1] 2
, and let (c, d) ∈ [0, 1] 2 be the unique solution to the system [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF], as stated in Lemma 2.1, then the following inequality holds:

(π 1 (c) -π 2 (d)) G 1 (a, c) + ϕ 1 (a) -ϕ 1 (c) δx/2 = (π 1 (c) -π 2 (d)) G 2 (d, b) + ϕ 2 (d) -ϕ 1 (b) δx/2 ≥ -|q||π 1 (c) -π 2 (d)|.

Proof

In this proof, we suppose that π 1 (0) ≥ π 2 (0) and π 1 (1) ≥ π 2 (1), the other cases do not bring any other difficulties. One has π1 (c) ∩ π2 (d) = ∅, so there are three different cases:

• π 1 (c) = π 2 (d):
in this case, one has directly:

(π 1 (c) -π 2 (d)) G 1 (a, c) + ϕ 1 (a) -ϕ 1 (c) δx/2 = 0. • π 2 (d) < π 1 (0): the relation π1 (c) ∩ π2 (d) = ∅ ensures that c = 0, thus it follows from the monotonicity of ϕ 1 and G 1 that ϕ 1 (a) ≥ ϕ 1 (0) = 0, and G 1 (a, 0) ≥ G 1 (0, 0) = f 1 (0) = 0. This gives: (π 1 (c) -π 2 (d)) G 1 (a, c) + ϕ 1 (a) -ϕ 1 (c) δx/2 ≥ 0.
• π 1 (c) > π 2 (1): this implies d = 1. From the monotonicity of ϕ 2 and G 2 , we deduce that ϕ 2 (b) ≤ ϕ 2 (1) and

G 2 (1, b) ≥ G 2 (1, 1) = q. This yields (π 1 (c) -π 2 (d)) G 2 (d, b) + ϕ 2 (d) -ϕ 1 (b) δx/2 ≥ q|π 1 (c) -π 2 (d)|.
Proof of Proposition 2.4. First check that the scheme ( 14) can be rewritten

u n+1 j+1/2 -u n j+1/2 δt δx + F n+1 j+1 -f i (u n+1 j+1/2 ) -F n+1 j -f i (u n+1 j+1/2 ) = 0.
We multiply the previous equation by δtπ i (u n+1 j+1/2 ) and sum on j = -N, N -1. This leads to

A n+1 + B n+1 + C n+1 + D n+1 + E n+1 = 0, (24)
where

A n+1 = N -1 j=-N φ i π i (u n+1 j+1/2 ) u n+1 j+1/2 -u n j+1/2 δx ; B n+1 = j / ∈{-N,0,N } δt   π i (u n+1 j-1/2 ) G i (u n+1 j-1/2 , u n+1 j+1/2 ) -G i (u n+1 j-1/2 , u n+1 j-1/2 ) -π i (u n+1 j+1/2 ) G i (u n+1 j-1/2 , u n+1 j+1/2 ) -G i (u n+1 j+1/2 , u n+1 j+1/2 )   ; C n+1 = δt j / ∈{-N,0,N } π i (u n+1 j+1/2 ) -π i (u n+1 j-1/2 ) ϕ i (u n+1 j+1/2 ) -ϕ i (u n+1 j-1/2 ) δx ; D n+1 = δtF n+1 0 π 1 (u n+1 -1/2 ) -π 2 (u n+1 1/2 ) δx -δtπ 1 (u n+1 -1/2 )f 1 (u n+1 -1/2 ) + δtπ 2 (u n+1 1/2 )f 2 (u n+1 1/2 ) ; E n+1 = δtπ 1 (u n+1 -N +1/2 ) G 1 u n+1 , u n+1 -N +1/2 -G 1 u n+1 -N +1/2 , u n+1 -N +1/2 +δtπ 2 (u n+1 N -1/2 ) G 2 u n+1 N -1/2 , u n+1 -G 2 u n+1 N -1/2 , u n+1 N -1/2
.

Denoting by L G a Lipschitz constant of both G i ,

E n+1 ≥ -δtL G ( π 1 ∞ + π 2 ∞ ) . (25)
One has

F n+1 0 π 1 (u n+1 -1/2 ) -π 2 (u n+1 1/2 ) = F n+1 0 π 1 (u n+1 -1/2 ) -π 1 (u n+1 0,1 ) + F n+1 0 π 1 (u n+1 0,1 ) -π 2 (u n+1 0,2 ) + F n+1 0 π 2 (u n+1 0,2 ) -π 2 (u n+1 1/2 ) .
It has been proven in Lemma 2.5 that there exists C 1 depending only on q and π i such that

F n+1 0 π 1 (u n+1 0,1 ) -π 2 (u n+1 0,2 ) ≥ C 1 . (26)
Using the definition of F n+1 0 , it is then easy to check that there exists C 2 only depending on G i , q, π i ,

D n+1 ≥ δtC 2 + δt π 1 (u n+1 -1/2 ) -π 1 (u n+1 0,1 ) ϕ 1 (u n+1 -1/2 ) -ϕ 1 (u n+1 0,1 ) δx/2 +δt π 2 (u n+1 1/2 ) -π 2 (u n+1 0,2 ) ϕ 2 (u n+1 1/2 ) -ϕ 2 (u n+1 0,2 ) δx/2
.

Since π i is a non-decreasing function, G i : s → s 0 φ i π i (a)da is convex, then: ∀n ∈ [[0, M -1]], A n+1 ≥ N -1 j=-N G i (u n+1 j+1/2 ) -G i (u n j+1/2 ) δx. (27)
We denote by

Ψ i (s) = s 0 π i (τ )f ′ i (τ )dτ
, then an integration by parts leads to

Ψ i (b) -Ψ i (a) = π i (a)(G i (a, b) -f i (a)) -π i (b) (G i (a, b) -f i (b)) - b a π ′ i (s)(f i (s) -G i (a, b))ds.
Since f i (s) = G i (s, s), it follows from the monotonicity of G i and

π i that b a π ′ i (s)(f i (s) -G i (a, b))ds ≥ 0.
Thus

B n+1 ≥ δt j / ∈{-N,0,N } Ψ i (u n+1 j+1/2 ) -Ψ i (u n+1 j-1/2 ) ≥ δt Ψ 1 (u n+1 -1/2 ) -Ψ 1 (u n+1 -N +1/2 ) + Ψ 2 (u n+1 N -1/2 ) -Ψ 2 (u n+1 1/2 ) .
So there exists C 3 , only depending on π i , f i such that

B n+1 ≥ δtC 3 . (28)
Let

ξ i : s → s 0 λ i (a)π ′ i (a)da, Cauchy-Schwarz inequality yields: ∀(a, b) ∈ [0, 1] 2 , (π i (a) -π i (b))(ϕ i (a) -ϕ i (b)) ≥ (ξ i (a) -ξ i (b)) 2 .
This ensures that

C n+1 ≥ δt j / ∈{-N,0,N } ξ i (u n+1 j+1/2 ) -ξ i (u n+1 j-1/2 ) 2 δx ; ( 29
)
D n+1 ≥ δtC 2 + δt ξ 1 (u n+1 0,1 ) -ξ 1 (u n+1 -1/2 ) 2 δx/2 + δt ξ 2 (u n+1 1/2 ) -ξ 2 (u n+1 0,2 ) 2 δx/2 . ( 30
)
Summing [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF] on n ∈ [[0, M -1]], and taking into account ( 25), ( 27), ( 28), ( 29), [START_REF] Karlsen | L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients[END_REF], provides the existence of a quantity C, depending only on

T , π i , G i , φ i such that i=1,2 |ξ i (u D )| 2 1,D,i + M-1 n=0 δt    ξ 1 (u n+1 0,1 ) -ξ 1 (u n+1 -1/2 ) 2 δx/2 + ξ 2 (u n+1 1/2 ) -ξ 2 (u n+1 0,2 ) 2 δx/2    ≤ C. (31)
Remark 2. [START_REF] Adimurthi | Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes[END_REF] The estimate [START_REF] Mascia | Nonhomogeneous Dirichlet problems for degenerate parabolichyperbolic equations[END_REF] is stronger than the one stated in Proposition 2.4, since it lets also appear some contributions coming from the interface. They will be useful in the sequel. Indeed, if we denote by u D,i the trace of (u D ) |Ωi on the interface {x = 0}, and if we denote by γ D,i (t) = u n+1 0,i if t ∈ (nδt, (n + 1)δt], then it follows from [START_REF] Mascia | Nonhomogeneous Dirichlet problems for degenerate parabolichyperbolic equations[END_REF] that lim δt,δx→0

u D,i -γ D,i L p (0,T ) = 0, ∀p ∈ [1, ∞).
Suppose that u Di converges in L p (0, T ) towards a function u i , as it will be proven later. Then, we directly obtain that γ D,i also converges towards u i . Moreover, for all t > 0, π1 (γ Dd,1

(t)) ∩ π2 (γ Dd,2 (t)) = ∅. Since F = {(a, b) ∈ [0, 1] 2 | π1 (a) ∩ π2 (b) = ∅} is a closed set of [0, 1] 2 ,
we can claim that π1 (u 1 ) ∩ π2 (u 2 ) = ∅ a.e. in (0, T ).

Compactness of a family of approximate solutions

Let (M p ) p∈N , (N p ) p∈N be two sequences of positive integers tending to +∞. We denote D p the discretization of Ω × (0, T ) associated to M p , and N p . The L ∞ -estimate stated in Proposition 2.3 shows that there exists

u ∈ L ∞ (Ω × (0, T )), 0 ≤ u ≤ 1, such that, up to a subsequence, u Dp → u in the L ∞ (Ω × (0, T )) weak-⋆ sense as p → +∞.
We just need to prove that u Dp → u almost everywhere in Ω × (0, T ) to get the convergence of (u Dp ) towards u in L r (Ω × (0, T )) for any 1 ≤ r < +∞. To apply Kolmogorov criterion (see e.g. [START_REF] Brézis | Analyse Fonctionnelle: Théorie et applications[END_REF]) we need some estimates on the space and time translates of ξ i (u D ). Lemma 2.6 (space and time translates estimates) For all η ∈ R, for i = 1, 2, one denotes Ω i,η = {x ∈ Ω i / (x + η) ∈ Ω i }, then the following estimate holds:

ξ i (u D )(• + η, •) -ξ i (u D )(•, •) L 2 (Ωi,η ×(0,T )) ≤ |ξ i (u D )| 1,D,i |η|(|η| + 2δx). ( 32
)
One denotes w i,D the function defined almost everywhere by:

w i,D (x, t) = ξ i (u D )(x, t) in Ω i × (0, T ), 0 in R 2 \(Ω i × (0, T )).
There exists C 1 depending only on π i , φ i , T, G i and C 2 only depending on π i , φ i , T, λ i , G i such that:

∀η ∈ R, w i,D (• + η, •) -w i,D (•, •) L 2 (R 2 ) ≤ C 1 η, (33)
∀τ ∈ (0, T ),

w i,D (•, • + τ ) -w i,D (•, •) L 2 (Ωi×(0,T -τ )) ≤ C 2 τ. ( 34
)
The previous lemma is in fact a compilation of Lemmata 4.2, 4.3 and 4.6 of [START_REF] Eymard | Finite volume methods[END_REF] adapted to our framework. The estimates [START_REF] Michel | Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods[END_REF] and [START_REF] Otto | L 1 -contraction and uniqueness for quasilinear elliptic-parabolic equations[END_REF] allows us to use the Kolmogorov compactness criterion on the sequence (w i,Dp ) p∈N , and thus, there exists

w i ∈ L 2 (Ω i × (0, T )) such that for almost every (x, t) ∈ (Ω i × (0, T )), ξ i (u Dp )(x, t) → w i (x, t),
and then thanks to the

L ∞ -estimate 0 ≤ u Dp (x, t) ≤ 1, one can claim that ξ i (u Dp ) → w i in L r (Ω i × (0, T ))
, for all r ∈ [1, +∞[. Letting p tend to +∞ in [START_REF] Michel | Numerical comparison of invasion percolation models and finite volume methods for buoyancy driven migration of oil in discontinuous capillary pressure fields[END_REF] insures that w i belongs to L 2 (0, T ; H 1 (Ω i )). Since ξ -1 i is a continuous function, we can identify the limit:

w i = ξ i (u).
Thus ξ i (u) ∈ L 2 (0, T ; H 1 (Ω i )), and since ϕ i • ξ -1 i is a Lipschitz function, there exists C depending only on T, π i , φ i , G i , λ i such that:

ϕ i (u) L 2 (0,T ;H 1 (Ωi)) ≤ C, (35)
and that ξ i (u Dp ) → ξ i (u), up to a subsequence, in L r (Ω i × (0, T )) as p → +∞ for any r ∈ [1, +∞). Since ξ i , i = 1, 2, is an increasing function, one can claim that u Dp converges a.e. in Ω × (0, T ) towards u, and then:

u Dp → u in the L ∞ (Ω × (0, T ))-weak-⋆ sense, ( 36
) u Dp → u in L r (Ω i × (0, T )), ∀r ∈ [1, +∞[. ( 37
)
Roughly speaking, the approximation u D obtained via a monotonous finite volume scheme, which introduces numerical diffusion, is "close" to the approximation u ǫ obtained by adding additional diffusion -ǫ∆u ǫ to the problem. For such a continuous problem, we would have an estimate of type

T 0 Ωi (∂ x ξ i (u ǫ )) dxdt ≤ C ′ ,
which would lead to the relative compactness of the family (ξ i (u ǫ )) ǫ>0 in L 2 (Ω i × (0, T )). Then the family (ξ i (u ǫ )) ǫ>0 is also relatively compact in L 2 ((0, T ); H s (Ω i )) for all s ∈ (1/2, 1). This ensures that, up to a subsequence, the traces on the boundary and on the interface of (ξ i (u ǫ )) converge in L 2 (0, T ). The continuity of ξ -1 i , and the L ∞ -estimate ensure that the traces of u ǫ on the boundary and on the interface converge in L r (0, T ), for all r ∈ [1, ∞).

This sketch has to be modified in order to deal with discrete solutions, which do not belong to L 2 ((0, T ); H s (Ω i )) for s > 1/2. Nevertheless, a convenient estimate on the translates at the boundary, based on the discrete L 2 ((0, T ); H 1 (Ω i )) estimate stated in Proposition 2.4, leads to the following convergence result, which is proven in the multidimensional case in [17, Proposition 3.10]. Lemma 2.7 Let i = 1, 2, and let α ∈ ∂Ω i . We denote by u α,Dp,i the trace of (u Dp ) |Ωi on {x = α}. Then, one has:

for all r ∈ [1, ∞), u α,Dp,i → u |Ωi (α, •) in L r (0, T ) as p → +∞.
If we denote by u i (t) = u |Ωi (0, t), it follows from the remark 2.4 that π1 (u 1 ) ∩ π2 (u 2 ) = ∅ a.e. in (0, T ). We can summarize all the results of this subsection in the following proposition :

Proposition 2.8 Let (M p ) p , (N p ) p tend to ∞ as p → ∞, and let (D p ) p be the corresponding sequence of discretizations of Ω × (0, T ). Let u Dp p be the sequence of corresponding discrete solutions to the scheme, then, up to a subsequence (still denoted by u Dp p ), there exists u ∈ L ∞ (Ω × (0, T )), 0 ≤ u ≤ 1 a.e., with

ξ i (u) ∈ L 2 ((0, T ); H 1 (Ω i )) (i = 1, 2) such that: u Dp → u a.e. in Ω × (0, T ) as p → ∞.
Moreover, keeping the notations of Lemma 2.7,

u -1,Dp,1 (t) → u(-1, t) for a.e. t ∈ (0, T ) as p → ∞, u 1,Dp,2 (t) → u(1, t)
for a.e. t ∈ (0, T ) as p → ∞, u 0,Dp,i (t) → u |Ωi (0, t) = u i (t) for a.e. t ∈ (0, T ) as p → ∞, i = 1, 2, and π1 (u 1 ) ∩ π2 (u 2 ) = ∅ almost everywhere in (0, T ).

Convergence of the scheme

We will now achieve the proof of the following result. Theorem 2.9 Let (M p ) p∈N , (N p ) p∈N be two sequences of positive integers tending to +∞, and (D p ) p∈N the associated sequence of discretizations of Ω × (0, T ). Then, up to a subsequence, the sequence u Dp p of the discrete solutions converges in L r (Ω × (0, T )) for all r ∈ [1, ∞) to a weak solution to the problem (P) in the sense of Definition 1.1.

Proof

As it has been seen in Proposition 2.8, the discrete solution u Dp converges, up to a subsequence, towards a function u fulfilling all the regularity criteria to be a weak solution. In order to prove the convergence of the subsequence to a weak solution, it only remains to show that the weak formulation (12) is satisfied by the limit u.

In order to simplify the proof of convergence of the scheme towards a weak solution, we will use a density result, which is a simple particular case of those stated in [START_REF] Droniou | A density result in Sobolev spaces[END_REF].

Lemma 2.10 Let a, b ∈ R, a < b, then: {ψ ∈ C ∞ c ([a, b])/ψ ′ ∈ C ∞ c ((a, b))} is dense in W 1,q (a, b), q ∈ [1,
+∞[. This lemma particularly allows us, thanks to a straightforward generalization, to restrict the set of test functions ψ for the weak formulation [START_REF] Brézis | Analyse Fonctionnelle: Théorie et applications[END_REF] to

T = {ψ ∈ D(Ω × [0, T ))/∂ x ψ ∈ D((∪ i=1,2 Ω i ) × [0, T ))}. Let ψ ∈ T . For j ∈ [[-N p , N p -1]], n ∈ [[0, M p -1]],
we denote by ψ n j+1/2 = ψ(x j+1/2 , t n ). Assume that p is large enough to ensure:

ψ n -1/2 = ψ n 1/2 , ∀n ∈ [[0, M p -1]], ( 38
) ∀n ∈ [[0, M p -1]], ψ n -N +1/2 = ψ(-1, t n ), ψ n N -1/2 = ψ(1, t n ). (39)
One has also

ψ Mp j+1/2 = 0, ∀j ∈ [[-N p , N p -1]]. (40) For j ∈ [[-N p , N p -1]], n ∈ [[0, M p -1]
], let us multiply equation ( 14) by ψ n j+1/2 δt, and sum on j ∈

[[-N p , N p -1]], n ∈ [[0, M p -1]], we get: Mp-1 n=0 Np-1 j=-Np φ i (u n+1 j+1/2 -u n j+1/2 )ψ n j+1/2 δx + Mp-1 n=0 δt Np-1 j=-Np (F n+1 j+1 -F n+1 j )ψ n j+1/2 = 0,
which can be rewritten thanks to [START_REF] Van Duijn | The effect of capillary forces on immiscible two-phase flows in heterogeneous porous media[END_REF], ( 39), (40):

Mp-1 n=0 Np-1 j=-Np φ i u n+1 j+1/2 (ψ n j+1/2 -ψ n+1 j+1/2 )δx - Np-1 j=-Np φ i u 0 j+1/2 ψ 0 j+1/2 δx + Mp-1 n=0 δt j / ∈{-Np,0,Np} F n+1 j (ψ n j-1/2 -ψ n j+1/2 ) - Mp-1 n=0 δtF n+1 -N ψ(-1, t n ) + Mp-1 n=0 δtF n+1 N ψ(1, t n ) = 0. ( 41
)
Using the definition of F n+1 j , we obtain

A p + B p + C p + D p + E p = 0, (42)
with, using the notation

ψ n -N -1/2 = ψ n -N +1/2 , and ψ n N +1/2 = ψ n N -1/2 , A p = Mp-1 n=0 Np-1 j=-Np φ i u n+1 j+1/2 (ψ n j+1/2 -ψ n+1 j+1/2 )δx ; B p = - Np-1 j=-Np φ i u 0 j+1/2 ψ 0 j+1/2 δx ; C p = Mp-1 n=0 δt j / ∈{-Np,0,Np} G i (u n+1 j-1/2 , u n+1 j+1/2 )(ψ n j-1/2 -ψ n j+1/2 ) ; D p = - Mp-1 n=0 δt Np-1 j=-Np ϕ i (u n+1 j+1/2 ) ψ n j+3/2 -2ψ n j+1/2 + ψ n j-1/2 δx ; E p = - Mp-1 n=0 δtG 1 (u n+1 , u n+1 -N +1/2 )ψ(-1, t n ) + Mp-1 n=0 δtG 2 (u n+1 N -1/2 , u n+1 )ψ(1, t n ) . Since h Ap : (x, t) → ψ n j+1/2 -ψ n+1 j+1/2 δt if (x, t) ∈ (x j , x j+1 )×(t n , t n+1
), converges uniformly towards -∂ t ψ as p → ∞, and since u D converges in L 1 (Ω × (0, T )) towards u,

lim p→∞ A p = - T 0 i=1,2 Ωi φ i u(x, t)∂ t ψ(x, t)dxdt. ( 43
)
Thanks to the convergence in L 1 (Ω) of u D (x, 0) towards u 0 , we have

lim p→∞ B p = - i=1,2 Ωi φ i u 0 (x)ψ(x, 0)dx. ( 44
)
We have to rewrite

C p = C 1 p + C 2 p , with C 1 p = Mp-1 n=0 δt j / ∈{-Np,0,Np} f i (u n+1 j-1/2 )(ψ n j-1/2 -ψ n j+1/2 ), C 2 p = Mp-1 n=0 δt j / ∈{-Np,0,Np} G i (u n+1 j-1/2 , u n+1 j+1/2 ) -f i (u n+1 j-1/2 ) (ψ n j-1/2 -ψ n j+1/2 ).
Thanks to Proposition 2.8, the quantity C 1 p converges towards -

T 0 i=1,2 Ωi f i (u)(x, t)∂ x ψ(x, t)dxdt as p → ∞. Concerning C 2 p , since G i (u n+1 j-1/2 , u n+1 j+1/2 ) -f i (u n+1 j-1/2 = G i (u n+1 j-1/2 , u n+1 j+1/2 ) -G i (u n+1 j-1/2 , u n+1 j-1/2 ) ≤ L G |u n+1 j+1/2 -u n+1 j-1/2 |,
and since (x, t) →

ψ n j-1/2 -ψ n j+1/2 δx on (x j-1/2 , x j+1/2 ) × (t n , t n+1 ) is uniformly bounded by ∂ x ψ ∞ , |C 2 p | ≤ L G ∂ x ψ ∞ T 0 Ω |δu Dp (x, t)|dxdt, where δu Dp (x, t) = u n+1 j+1/2 -u n+1 j-1/2 on (x j-1/2 , x j+1/2 ) × (t n , t n+1 ), (j ∈ [[-N p + 1, N p -1]]
), and 0 otherwise. It is easy to check, thanks to the discrete L 2 ((0, T ); H 1 (Ω i )) estimates stated in Proposition 2.4, that δu Dp tends to 0 in L 1 (Ω × (0, T )) as p → ∞. Then

lim p→∞ C p = - T 0 i=1,2 Ωi f i (u)(x, t)∂ x ψ(x, t)dxdt. ( 45
)
Since, using Proposition 2.8, ϕ i (u Dp ) tends to ϕ i (u) ∈ L 2 ((0, T ); H 1 (Ω i )) in the L 2 (Ω i × (0, T ))-topology, one has:

lim p→∞ D p = - T 0 i=1,2 Ωi ϕ i (u)(x, t)∂ 2 xx ψ(x, t)dxdt = T 0 i=1,2 Ωi ∂ x ϕ i (u)(x, t)∂ x ψ(x, t)dxdt. (46)
The strong convergence of the traces, stated in Proposition 2.8 allows us to claim that

lim p→∞ E p = - T 0 G 1 (u(t), u(-1, t))ψ(-1, t)dt + T 0 G 2 (u(1, t), u(t))ψ(1, t)dt. (47)
We can thus take the limit for p → ∞ in (42), and it follows from (43)-( 44)-( 45)-( 46)-(47) that u fulfills the weak formulation [START_REF] Brézis | Analyse Fonctionnelle: Théorie et applications[END_REF].

Uniform bound on the fluxes

In this section, we show that, under some regularity assumptions on the initial data, there exists a solution with bounded fluxes. This existence result is the consequence of some additional estimates on the discrete solution, and will be necessary to get the uniqueness result of Theorem 4.1.

Definition 3.1 A function u is said to be a bounded-flux solution to the problem (P) if:

1. u is a weak solution to the problem (P) in the sense of Definition 1.1;

2. ∂ x ϕ i (u) belongs to L ∞ (Ω i × (0, T )).
In order to get an existence result, we need more regularity on the initial data, as stated below.

Assumptions 2 We assume that:

1. ∂ x ϕ i (u 0 ) ∈ L ∞ (Ω i ), 0 ≤ u 0 ≤ 1;
2. π1 (u 0,1 ) ∩ π2 (u 0,2 ) = ∅, where u 0,i is the trace of u 0|Ω i on {x = 0}, Theorem 3.1 Suppose that assumptions 2 are fulfilled. Let (M p ) p∈N , (N p ) p∈N be two sequences of positive integers tending to +∞. Let (u Dp ) p∈N be the sequence of the associated discrete solutions obtained via the finite volume scheme [START_REF] Cancès | Nonlinear parabolic equations with spatial discontinuities[END_REF], and let u be an adherence value of the sequence (u Dp ) p∈N . Then u is a bounded flux solution to the problem (P) in the sense of Definition 3.1. This particularly ensures the existence of such a bounded-flux solution.

All the section 3 will be devoted to the proof of Theorem 3.1. We only need to verify the second point in Definition 3.1, because we have already proven in Theorem 2.9 that u is a weak solution. So the aim of this section is to get the uniform bound on the fluxes. Such an estimate can be found in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] in the case where the convection is neglected. It is obtained using a thin regular transition layer between Ω 1 and Ω 2 , and a regularization of the initial data u 0 . This technique was also used in [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF] to get a BV -estimate on the fluxes in the case of a non-bounded domain Ω, and for particular values of the data (which are supposed to be more regular). In this paper, we only deal with the discrete solution, which can be seen as a regularization of the solution to the continuous problem (P).

We extend the definitions of the discrete internal fluxes ( 15)-( 18) to the case n = -1, i.e. in the time t = 0. For all j ∈ [[-N + 1, N -1]], j = 0,

F 0 j = G i (u 0 j-1/2 , u 0 j+1/2 ) - ϕ i (u 0 j+1/2 ) -ϕ i (u 0 j-1/2 ) δx . ( 48
)
Thanks to Lemma 2.1, there exists a unique couple (u 0 0,1 , u 0 0,2 ) solution to the system

F 0 0 = G 1 (u 0 -1/2 , u 0 0,1 ) - ϕ 1 (u 0 0,1 ) -ϕ 1 (u 0 -1/2 ) δx/2 = G 2 (u 0 0,2 , u 0 1/2 ) - ϕ 2 (u 0 1/2 ) -ϕ 2 (u 0 0,2 ) δx/2 , ( 49
) π1 (u 0 0,1 ) ∩ π2 (u 0 0,2 ) = ∅. ( 50
)
Remark 3.1 u 0 0,1 and u 0 0,2 are given by Lemma 2.1, and so they are different of u 0,1 and u 0,2 .

Lemma 3.2 There exists C > 0 depending only on u 0 , ϕ i , q such that

max j∈[[-N+1,N-1]] |F 0 j | ≤ C.

Proof

Since ϕ i (u 0 ) is a Lipschitz continuous function, and ϕ -1 i is continuous, u 0|Ω i is a continuous function, and there exists y j+1/2 ∈ (x j , x j+1 ) such that u 0 j+1/2 = u 0 (y j+1/2 ). Then (48) can be rewritten

F 0 j = G i (u 0 (y j-1/2 ), u 0 (y j+1/2 )) - ϕ i (u 0 (y j+1/2 )) -ϕ i (u 0 (y j-1/2 )) δx .
Using the fact that

∂ x ϕ i (u 0 ) ∈ L ∞ (Ω i ) gives directly: ∀j ∈ [[-N + 1, N -1]] \ {0}, F 0 j ≤ max i=1,2 G i ∞ + 2 max i=1,2 ∂ x ϕ i (u 0 ) L ∞ (Ωi) . ( 51
)
The monotony of the transmission conditions π1 (u 0 0,1 ) ∩ π2 (u 0 0,2 ) = ∅ and π1 (u 0,1 ) ∩ π2 (u 0,2 ) = ∅ implies that either u 0 0,1 ≥ u 0,1 and u 0 0,2 ≥ u 0,2 , or u 0 0,1 ≤ u 0,1 and u 0 0,2 ≤ u 0,2 . Assume for example that u 0 0,1 ≥ u 0,1 and u 0 0,2 ≥ u 0,2 -the other case could be treated similarly-then one deduce from (49) that:

G 2 (u 0,2 , u 0 (y 1/2 )) - ϕ 2 (u 0 (y 1/2 )) -ϕ 2 (u 0,2 ) δx/2 ≤ F 0 0 ≤ G 1 (u 0 (y -1/2 ), u 0,1 ) - ϕ 1 (u 0,1 ) -ϕ 1 (u 0 (y -1/2 )) δx/2 ,
and so since ϕ i (u 0 ) is a Lipschitz continuous function,

|F 0 0 | ≤ max i=1,2 G i ∞ + 2 max i=1,2 ∂ x ϕ i (u 0 ) L ∞ (Ωi) . Proposition 3.3 There exists C > 0 depending only on u 0 , ϕ i , G i , such that max j∈[[-N+1,N-1]] max n∈[[0,M]] |F n j | ≤ C. Proof For all j ∈ [[-N + 1, N -1]] \ {0}, for all n ∈ [[0, M -1]], F n+1 j -F n j = G i (u n+1 j-1/2 , u n+1 j+1/2 ) -G i (u n j-1/2 , u n+1 j+1/2 ) + G i (u n j-1/2 , u n+1 j+1/2 ) -G i (u n j-1/2 , u n j+1/2 ) + ϕ i (u n+1 j-1/2 ) -ϕ i (u n j-1/2 ) δx - ϕ i (u n+1 j+1/2 ) -ϕ i (u n j+1/2 ) δx .
Thus, using ( 14) yields

F n+1 j -F n j = δt φ i δx G i (u n+1 j-1/2 , u n+1 j+1/2 ) -G i (u n j-1/2 , u n+1 j+1/2 ) u n+1 j-1/2 -u n j-1/2 F n+1 j -F n+1 j-1 + δt φ i δx G i (u n j-1/2 , u n+1 j+1/2 ) -G i (u n j-1/2 , u n j+1/2 ) u n+1 j+1/2 -u n j+1/2 F n+1 j+1 -F n j + δt φ i δx 2 ϕ i (u n+1 j-1/2 ) -ϕ i (u n j-1/2 ) u n+1 j-1/2 -u n j-1/2 F n+1 j -F n+1 j-1 - δt φ i δx 2 ϕ i (u n+1 j+1/2 ) -ϕ i (u n j-1/2 ) u n+1 j+1/2 -u n j+1/2 F n+1 j+1 -F n+1 j .
The monotonicity of the scheme is once again crucial, since it implies that there exist two non-negative values a n+1 j,j+1 , a n+1 j,j-1 such that

1 + a n+1 j,j-1 + a n+1 j,j+1 F n+1 j -a n+1 j,j-1 F n+1 j-1 -a n+1 j,j+1 F n+1 j+1 = F n j . (52)
The monotonicity of the graph transmission condition [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] ensures that either u n+1 0,1 ≥ u n 0,1 and u n+1 0,2 ≥ u n 0,2 , or u n+1 0,1 ≤ u n 0,1 and u n+1 0,2 ≤ u n 0,2 . Suppose for example that u n+1 0,1 ≥ u n 0,1 and u n+1 0,2 ≥ u n 0,2 , the other case being completely symmetrical.

F n+1 0 -F n 0 = G 1 (u n+1 -1/2 , u n+1 0,1 ) -G 1 (u n -1/2 , u n 0,1 ) + ϕ 1 (u n+1 -1/2 ) -ϕ 1 (u n -1/2 ) δx/2 - ϕ 1 (u n+1 0,1 ) -ϕ 1 (u n 0,1 ) δx/2 (53) = G 2 (u n+1 0,2 , u n+1 1/2 ) -G 2 (u n 0,2 , u n 1/2 ) + ϕ 2 (u n+1 0,2 ) -ϕ 2 (u n 1/2 ) δx/2 - ϕ 2 (u n+1 0,2 ) -ϕ 2 (u n 1/2 ) δx/2 . (54)
It follows from (53) and from the monotony of G 1 , ϕ 1 that

F n+1 0 -F n 0 ≤ G 1 (u n+1 -1/2 , u n 0,1 ) -G 1 (u n -1/2 , u n 0,1 ) + ϕ 1 (u n+1 -1/2 ) -ϕ 1 (u n -1/2 ) δx/2 .
Similar computations as those done to obtain (52) provide the existence of a non-negative value a n+1 0,-1 such that

1 + a n+1 0,-1 F n+1 0 -a n+1 0,-1 F n+1 -1 ≤ F n 0 . (55)
Considering (54) instead of (53) shows the existence of a non-negative value b n+1 0,1 such that

1 + b n+1 0,1 F n+1 0 -b n+1 0,1 F n+1 1 ≥ F n 0 . (56)
We denote by j n+1 max (resp. j n+1 min ) the integer such that

F n+1 j n+1 max = max j∈[[-N,N]] F n+1 j (resp. F n+1 j n+1 min = min j∈[[-N,N]] F n+1 j
).

Either j n+1 max ∈ {-N, N }, then it follows from the remark 2.

1 that max j∈[[-N,N]] F n+1 j ≤ max i=1,2 G i ∞ , or j n+1 max ∈ [[-N + 1, N -1]].
In the latter case, (52) and (55) imply

max j F n+1 j = F n+1 j n+1 max ≤ F n j n+1 max ≤ max j F n j .
Similarly, (52) and (55) yield min

j F n+1 j = F n+1 j n+1 min ≥ F n j n+1 min ≥ min j F n j .
We obtain a kind of discrete maximum principle on the discrete fluxes, which corresponds to the uniform bound on the continuous fluxes proven in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF]. It follows from Lemma 3.2 that

max n∈[[0,M]] max j∈[[-N+1,N-1]] F n+1 j ≤ max i=1,2 G i ∞ + 2 max i=1,2 ∂ x ϕ i (u 0 ) L ∞ (Ωi) .
Conclusion of proof of Theorem 3.1 Let (N p ) p∈N , (M p ) p∈N be two sequences of positive integers tending to +∞, and let (u Dp ) p∈N the sequences of associated discrete solutions. It has been seen in theorem 2.9 that (u Dp ) p tends to a weak solution u in L r (Ω × (0, T )), for all r ∈ [1, +∞). Let i = 1, 2, let (x, y) ∈ Ω i , let t ∈ (0, T ]. For p large enough,there exists j 0 , j 1 ∈ J int such that x j0 ≤ x ≤ x j0+1 and x j1 ≤ y ≤ x j1+1 , and there exists n such that t ∈ (t n , t n+1 ].

ϕ i (u Dp )(x, t) -ϕ i (u Dp )(y, t) = ϕ i (u n+1 j0+1/2 ) -ϕ i (u n+1 j1+1/2 ) = j0 j=j1+1 ϕ i (u n+1 j+1/2 ) -ϕ i (u n+1 j-1/2 ) ≤ j0 j=j1+1 ϕ i (u n+1 j+1/2 ) -ϕ i (u n+1 j-1/2 ) .
Using the definition of the discrete flux [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only of the space. I. Convergence to an entropy solution[END_REF]:

ϕ i (u Dp )(x, t) -ϕ i (u Dp )(y, t) ≤ j0 j=j1+1 δx F n+1 j -G i (u n+1 j-1/2 , u n+1 j+1/2 ) .
We deduce from Proposition 3.3 that there exists C > 0, depending only on u 0 , ϕ i , G i such that:

ϕ i (u Dp )(x, t) -ϕ i (u Dp )(y, t) ≤ j0 j=j1+1 δxC ≤ C(|x -y| + 2δx).
Letting p tend towards +∞, i.e. δx and δt towards 0 gives

|ϕ i (u)(x, t) -ϕ i (u)(y, t)| ≤ C|x -y|. (57)
So we deduce from (57

) that ∂ x ϕ i (u) ∈ L ∞ (Ω i × (0, T )).

Uniqueness of the bounded-flux solution

This section is devoted to the proof of Theorem 4.1, which is an adaptation of [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF]Theorem 5.1] to the case where the convection is taken into account. Ωi

φ i (u(x, t) -v(x, t)) ± dx ≤ Ωi φ i (u 0 (x) -v 0 (x)) ± dx.

This particularly implies the uniqueness of the bounded flux solution to the problem (P)

Obtaining a L 1 -contraction principle for a nonlinear parabolic equation is classical. We refer for example to [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF][START_REF] Gagneux | Unicité des solutions faibles d'équations de diffusion-convection[END_REF][START_REF] Otto | L 1 -contraction and uniqueness for quasilinear elliptic-parabolic equations[END_REF][START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF][START_REF] Mascia | Nonhomogeneous Dirichlet problems for degenerate parabolichyperbolic equations[END_REF][START_REF] Michel | Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods[END_REF][START_REF] Blanchard | Stefan problems with nonlinear diffusion and convection[END_REF] for the case of homogeneous domains, and for boundary conditions of Dirichlet or Neumann type. We have to adapt the proof of the L 1 -contraction principle to our problem, and thus particularly to the boundary conditions and to the transmission conditions at the interface. We need to introduce the cut-off functions ρ ε α ∈ C 0,1 (Ω, R + ) defined by

ρ ε α (x) = ε -|x -α| ε + . Lemma 4.2 For all θ ∈ D + ([0, T )), lim inf ε→0 T 0 θ(t) i=1,2 Ωi sign ± (u -v)(f i (u) -f i (v)) -∂ x (ϕ i (u) -ϕ i (v)) ± ∂ x ρ ε 0 (x)dxdt ≥ 0.

Proof

We define the subsets of (0, T )

E u>v = t ∈ (0, T ) u 1 (t) > v 1 (t) or u 2 (t) > v 2 (t) , E u≤v = (E u>v ) c = t ∈ (0, T ) u 1 (t) ≤ v 1 (t) and u 2 (t) ≤ v 2 (t) .
Since the trace on {x = 0} of the function sign

+ (u -v)(f i (u) -f i (v)
) is equal to 0 for all t ∈ E u≤v , it is easy to check that lim

ε→0 E u≤v θ i=1,2 Ωi sign + (u -v)(f i (u) -f i (v))∂ x ρ ε 0 dxdt = 0.
Thanks to the fact that the trace of (ϕ i (u)ϕ i (v))

+ is equal to 0 on the interface, one has also,

lim inf ε→0 E u≤v θ i=1,2 Ωi ∂ x (ϕ i (u) -ϕ i (v)) + ∂ x ρ ε 0 dxdt ≥ 0.
This particularly implies that lim inf

ε→0 E u≤v θ i=1,2 Ωi sign + (u -v)(f i (u) -f i (v)) -∂ x (ϕ i (u) -ϕ i (v)) + ∂ x ρ ε 0 dxdt ≥ 0. ( 58
)
Since u, v are two weak solutions, subtracting their corresponding weak formulation [START_REF] Brézis | Analyse Fonctionnelle: Théorie et applications[END_REF] for the test function ψ(x, t) = θ(t)ρ ε 0 (x) leads to

T 0 i=1,2 Ωi φ i (u -v)ρ ε 0 ∂ t θdxdt + i=1,2 Ωi φ i (u 0 -v 0 )ρ ε 0 θ(0)dx + T 0 θ i=1,2 Ωi ((f i (u) -f i (v)) -∂ x (ϕ i (u) -ϕ i (v))) ∂ x ρ ε 0 dxdt = 0.
Since ρ ε 0 tends to 0 in L 1 (Ω) as ε → 0, one has:

lim ε→0 T 0 i=1,2 Ωi φ i (u -v)ρ ε 0 ∂ t θdxdt + i=1,2 Ωi φ i (u 0 -v 0 )ρ ε 0 θ(0)dx = 0, thus lim ε→0 T 0 θ i=1,2 Ωi ((f i (u) -f i (v)) -∂ x (ϕ i (u) -ϕ i (v))) ∂ x ρ ε 0 dxdt = 0. (59)
Thanks to the L ∞ (Ω × (0, T )) bound on the fluxes, one has

T 0 θ i=1,2 Ωi ((f i (u) -f i (v)) -∂ x (ϕ i (u) -ϕ i (v))) ∂ x ρ ε 0 dxdt ≤ C θ L 1 (0,T ) ,
then, using a density argument, (59) holds for all θ ∈ L 1 (0, T ). Replacing θ by θχ Eu>v in (59), and splitting the positive and the negative parts a = a +a -, we obtain

Eu>v θ i=1,2 Ωi sign + (u -v)(f i (u) -f i (v)) -∂ x (ϕ i (u) -ϕ i (v)) + ∂ x ρ ε 0 dxdt = Eu>v θ i=1,2 Ωi sign -(u -v)(f i (u) -f i (v)) -∂ x (ϕ i (u) -ϕ i (v)) -∂ x ρ ε 0 dxdt +r(ε), (60)
with lim ε→0 r(ε) = 0.

For almost every t ∈ E u>v , it follows from the monotonicity of the graph relations for the capillary pressure

π1 (u 1 ) ∩ π2 (u 2 ) = ∅, π1 (v 1 ) ∩ π2 (v 2 ) = ∅, that t belongs to E u≥v = t ∈ (0, T ) u 1 (t) ≥ v 1 (t)
sign -(u -v)(f i (u) -f i (v)) -∂ x (ϕ i (u) -ϕ i (v)) -∂ x ρ ε 0 dxdt ≥ 0.
It follows directly from (60) that lim inf

ε→0 Eu>v θ i=1,2 Ωi sign + (u -v)(f i (u) -f i (v)) -∂ x (ϕ i (u) -ϕ i (v)) + ∂ x ρ ε 0 dxdt ≥ 0. (61)
Adding ( 58) and (61) achieves the proof of Lemma 4.2.

Lemma 4.3 For all θ ∈ D + ([0, T )), lim inf ε→0 T 0 θ Ω1 sign ± (u -v) (f 1 (u) -f 1 (v)) -∂ x (ϕ 1 (u) -ϕ 1 (v)) ± ∂ x ρ ε -1 dxdt ≥ 0, (62) lim inf ε→0 T 0 θ Ω2 sign ± (u -v) (f 2 (u) -f 2 (v)) -∂ x (ϕ 2 (u) -ϕ 2 (v)) ± ∂ x ρ ε 1 dxdt ≥ 0. (63)

Proof

For the sake of simplicity, we will only prove

lim inf ε→0 T 0 θ Ω2 sign + (u -v) (f 2 (u) -f 2 (v)) -∂ x (ϕ 2 (u) -ϕ 2 (v)) + ∂ x ρ ε 1 dxdt ≥ 0,
but all the steps of the proof can be extended to the other cases. We denote by F u>v and F u≥v the subsets of (0, T ) given by

F u>v = {t ∈ (0, T ) | u(1, t) > v(1, t)} , F u≤v = (F u>v ) c = {t ∈ (0, T ) | u(1, t) ≤ v(1, t)} .
Let ε > 0. For almost every t ∈ F u≤v , one has

Ω2 ∂ x (ϕ 2 (u)(x, t) -ϕ 2 (v)(x, t)) + ∂ x ρ ε 1 (x)dx ≤ 0.
Then, using the fact that for almost every t ∈ F u≤v , the trace of sign

+ (u(•, t) -v(•, t)) (f 2 (u)(•, t) -f 2 (v)(•, t)) on {x = 1} is equal to 0, lim inf ε→0 F u≤v θ Ω2 sign + (u -v) (f 2 (u) -f 2 (v)) -∂ x (ϕ 2 (u) -ϕ 2 (v)) + ∂ x ρ ε 1 dx ≥ 0. ( 64
)
We deduce from the weak formulation that for all θ ∈ D([0, T )),

lim ε→0 T 0 θ Ω2 (f 2 (u) -∂ x ϕ 2 (u)) ∂ x ρ ε 1 dx -G 2 (u(1, t), u(t)) dt = 0. ( 65
)
Since the fluxes f 2 (u) -∂ x ϕ 2 (u) and f 2 (v) -∂ x ϕ 2 (v) belong to L ∞ (Ω 2 × (0, T )), a density argument, which has already been used during the proof of Lemma 4.2, allows us to claim that (65) still holds for any θ ∈ L 1 (0, T ). So, it particularly holds if we replace θ by θχ Fu>v This leads to

lim ε→0 Fu<v θ Ω2 f 2 (u)-f 2 (v)-∂ x (ϕ 2 (u)-ϕ 2 (v)) ∂ x ρ ε 1 dxdt = Fu>v θ(t) (G 2 (u(1, t), u(t)) -G 2 (v(1, t), u(t))) dt.
It follows from the monotonicity of G 2 that

∀t ∈ F u>v , G 2 (u(1, t), u(t)) ≥ G 2 (v(1, t), u(t)), thus lim inf ε→0 Fu<v θ Ω2 f 2 (u) -f 2 (v) -∂ x (ϕ 2 (u) -ϕ 2 (v)) ∂ x ρ ε 1 dxdt ≥ 0. ( 66
)
In order to conclude the proof of Lemma 4.3, it only remains to check that lim inf

ε→0 Fu<v θ Ω2 f 2 (u) -f 2 (v) -∂ x (ϕ 2 (u) -ϕ 2 (v)) ∂ x ρ ε 1 dxdt = lim inf ε→0 Fu<v θ Ω2 sign + (u -v)(f 2 (u) -f 2 (v)) -∂ x (ϕ 2 (u) -ϕ 2 (v)) + ∂ x ρ ε 1 dxdt. ( 67
)
Since ϕ -1 2 is a continuous function, u(•, t) can be supposed to be continuous on Ω 2 for almost every t in (0, T ). Particularly, for almost every t ∈ F u>v , there exists a neighborhood V t of {x = 1} such that u(•, t) > v(•, t) for all x ∈ V t . On V t , one has

R(x, t) = f 2 (u) -f 2 (v) -∂ x (ϕ 2 (u) -ϕ 2 (v)) -sign + (u -v)(f 2 (u) -f 2 (v)) -∂ x (ϕ 2 (u) -ϕ 2 (v)) + = 0. Then, for almost every t ∈ F u>v , lim ε→0 Ω2 R(x, t)∂ x ρ ε 1 (x)dx = 0. Moreover, since the fluxes f 2 (u) -∂ x ϕ 2 (u) and f 2 (v) -∂ x ϕ 2 (v) belong to L ∞ (Ω 2 × (0, T )), there exists C > 0 not depending on ε such that for almost every t, Ω2 R(x, t)∂ x ρ ε 1 (x)dx ≤ C.
We deduce from the dominated convergence theorem that

lim ε→0 T 0 Ω2 R(x, t)∂ x ρ ε 1 (x)dx = 0.
This particularly implies that (67) holds. This achieves the proof of Lemma 4.3.

Proof of the Theorem 4.1. First, since u and v are weak solutions to a parabolic equation, they are also entropy solutions (see [START_REF] Gagneux | Unicité des solutions faibles d'équations de diffusion-convection[END_REF], [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF]), and it has been proven in [START_REF] Cancès | On the time continuity of entropy solutions[END_REF] that u and v belong to C([0, T ], L p (Ω)), in the sense that there exists ũ, ṽ ∈ C([0, T ], L p (Ω)) such that u = ũ, v = ṽ almost everywhere in Ω × (0, T ). Let u and v be two weak solutions, then some classical computations, based on the doubling variable technique applied on both the time and the space variable (see e.g. [START_REF] Gagneux | Unicité des solutions faibles d'équations de diffusion-convection[END_REF], [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF]) yields yields that for any

ψ ∈ D + (Ω i × [0, T )) T 0 Ωi φ i (u(x, t) -v(x, t)) ± ∂ t ψ(x, t)dxdt + Ωi φ i (u 0 (x) -v 0 (x)) ± ψ(x, 0)dx + T 0 Ωi sign ± (u(x, t) -v(x, t)) (f i (u)(x, t) -f i (v)(x, t)) ∂ x ψ(x, t)dxdt - T 0 Ωi ∂ x (ϕ i (u)(x, t) -ϕ i (v)(x, t)) ± ∂ x ψ(x, t)dxdt ≥ 0. ( 68
) Let θ ∈ D + ([0, T )), then summing (68) with respect to i = 1, 2, choosing ψ(x, t) = θ(t) 1 -ρ ε -1 (x) -ρ ε 0 (x) -ρ ε 1 (x)
as test function, and letting ε tend to 0 leads to, thanks to Lemmata 4.2 and 4.3 :

T 0 ∂ t θ(t) i=1,2 Ωi φ i (u(x, t) -v(x, t)) ± dxdt + i=1,2 Ωi φ i (u 0 (x) -v 0 (x)) ± θ(0)dx ≥ 0. ( 69
)
Since u, v belong to C([0, T ]; L 1 (Ω)), the relation (69) still holds for any θ ∈ BV (0, T ) with θ(T + ) = 0. Let t ∈ [0, T ], we choose θ = χ [0,t) in (69), obtaining this way the L 1 -contraction and comparison principle stated in the Theorem 4.1.

Solutions obtained as limit of approximations

We aim in this section to extend the existence-uniqueness result obtained in Theorems 3.1 and 4.1 for any initial data u 0 ∈ L ∞ (Ω), 0 ≤ u 0 ≤ 1 a.e.. We are unfortunately not able to prove the uniqueness of the weak solution to the problem (P) in such a general case, but we are able to prove the existence and the uniqueness of the solution obtained as limit of approximation by bounded flux solution. Moreover, this limit is the weak solution obtained via the convergence of the implicit scheme ( 14) studied previously.

Definition 5.1 A function u is said to be a SOLA (solution obtained as limit of approximation) to the problem (P) if it fulfils:

• u is a weak solution to the problem (P),

• there exists a sequence (u ν ) ν∈N of bounded flux solutions such that In order to illustrate this model, we use a test case developed by Anthony Michel [START_REF] Michel | Numerical comparison of invasion percolation models and finite volume methods for buoyancy driven migration of oil in discontinuous capillary pressure fields[END_REF]. The porous medium Ω = (0, 1) is made of sand for x ∈ (0, 0.5) ∪ (0.7, 1), with a layer of shale for x ∈ (0.5, 0.7).

u n → u in C([0, T ]; L 1 (Ω)), as n → +∞. Theorem 5.1 Let u 0 ∈ L ∞ (Ω), 0 ≤ u 0 ≤ 1 a.e.,
First case: The total flow rate is equal to 0, since f sand (1) = f shale (1) = 0, and the convection is the exclusive of the volume mass difference between the oil, which is lighter, and the water. The convection functions are given by:

f sand (u) = 100 * f shale (u) = 50 * u 2 (1 -u 2 ) 1 -2u + 2u 2 .
The capillary pressures are first given by π sand (u) = u 5 , π shale (u) = 0.5 + u 5 .

The function ϕ sand and ϕ shale , given by ϕ sand (u) = 10 * f i (s) otherwise.

A little quantity of oil enters the domain from the left boundary condition, and it moves forward in the first part made of sand. The discontinuity of the capillary pressure (figure 2) stops the migration of oil, which begins to collect at the left of the interface, as shown on the figure 1. One can check on the figure 3 that for t small enough, the oil-flux through the interface {x = 0.5} is equal to 0. The accumulation of oil at the left of {x = 0.5} implies an increase of the capillary pressure. As soon as the capillary pressure connects at {x = 0.5}, the oil can flow through the shale. The next discontinuity at {x = 0.7} does not impede the progression of the oil, since the capillary pressure force, oriented from the large pressure to the small pressure (here from the left to the right), works in the same direction that the buoyancy, which drives the migration of oil.

For t = 200, the presented solution is a steady solution, with constant flux (figure 3). Some oil remains blocked in the first subdomain (0, 0.5). Even if one puts u(t) = 0 for t ≥ 200, the main proportion of oil in This quantity is said to be trapped by the geology change. Further illustrations, and a scheme comparison will be given in [START_REF] Michel | Numerical comparison of invasion percolation models and finite volume methods for buoyancy driven migration of oil in discontinuous capillary pressure fields[END_REF].

Second case: We only change the values of the capillary pressure functions (and also the linked functions ϕ sand and ϕ shale ). The amplitude of the variation of each function is reduced from 1 to 0.2, i.e. The graph transmission condition for the capillary pressure turns to (1u sand )u shale = 0, where u sand (resp u shale ) denotes the trace of the oil saturation at the interfaces {x = 0.5} and {x = 0.7}. In this case, no oil can overpass the first interface, which is thus impermeable for oil. The only steady solution is u s (x) = 1 if x < 0.5, 0 if x > 0.5.

An asymptotic study for capillary pressures tending to functions depending only of space, and not on the saturation has been performed in [13, Chapter 5&6] (see also [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only of the space. I. Convergence to an entropy solution[END_REF][START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only of the space. II. Occurrence of non-classical shocks to model oil-trapping[END_REF]). It has been proven that either the limit solution for the saturation is an entropy solution for th e hyperbolic scalar conservation law with discontinuous fluxes in the sense of [START_REF] Towers | Convergence of a difference scheme for conservation laws with a discontinuous flux[END_REF][START_REF] Towers | A difference scheme for conservation laws with a discontinuous flux: the nonconvex case[END_REF][START_REF] Seguin | Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients[END_REF][START_REF] Adimurthi | Conservation law with discontinuous flux[END_REF][START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF][START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF][START_REF] Adimurthi | Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes[END_REF][START_REF] Bachmann | Analysis of a scalar conservation law with a flux function with discontinuous coefficients[END_REF][START_REF] Bachmann | Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients[END_REF][START_REF] Bachmann | Finite volume schemes for a non linear hyperbolic conservation law with a flux function involving discontinuous coefficients[END_REF][START_REF] Bachmann | Equations hyperboliques scalaires à flux discontinu[END_REF][START_REF] Jimenez | Some scalar conservation laws with discontinuous flux[END_REF] (see also [START_REF] Karlsen | On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient[END_REF][START_REF] Karlsen | Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient[END_REF][START_REF] Karlsen | L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients[END_REF]), mainly when the capillary forces at the interface are oriented in the same direction that the gravity forces, or that non-classical shocks can occur at the interfaces when the capillary forces and the gravity are oriented in opposite directions.

)

  Furthermore, (a, b) → c and (a, b) → d are continuous and nondecreasing w.r.t. each one of their arguments.

i

  ensures that (a, b) → c and (a, b) → d are non-decreasing.

Theorem 4 . 1

 41 If u,v are bounded-flux solutions in the sense of Definition 3.1 associated to the initial data u 0 , v 0 , then for all p ∈ [1, +∞[, u and v belong to C([0, T ]; L p (Ω)), and the following L 1 -contraction principle holds: ∀t ∈ [0, T ],

Figure 2 :

 2 Figure 2: Capillary pressure profiles for t = 20, t = 100, t = 200

u 0 s 2 ( 1 -s 2 ) 1 - 0 s 2 ( 1 -s 2 ) 1 -

 0212102121 2s + 2s 2 π ′ sand (s)ds, ϕ shale (u) = 0.1 * u 2s + 2s 2 π ′ shale (s)ds, are computed using an approximate integration formula. The initial data u 0 is equal to 0, and u = 0.001, u = 0. The convection is approximated by a Godunov scheme, defined byG i (a, b) =    min s∈[a,b] f i (s) if a ≤ b, max s∈[b,a]

Figure 3 :

 3 Figure 3: Oil-flux profiles for t = 20, t = 100, t = 200

π

  sand (u) = 0.2 * u 5 , π shale (u) = 0.5 + 0.2 * u 5 .

Figure 4 :Figure 5 :

 45 Figure 4: Saturation profiles for t = 100, t = 500, t = 900

Figure 6 :

 6 Figure 6: Oil-flux profiles for t = 100, t = 500, t = 900

  . for a.e. t ∈ (0, T ), π1 (u 1 (t)) ∩ π2 (u 2 (t)) = ∅, where u i denotes the trace of u |Ωi on {x = 0};

	4. for all ψ ∈ D(Ω × [0, T [), denoting by u(1, •) and u(-1, •) the traces of u on the boundary,
	T
	0 i=1,2 Ωi

Acknowledgements. The author would like to acknowledge the Professor Thierry Gallouët for his numerous recommendations and Anthony Michel from IFP for the fruitful discussions on the models. He also thanks Alice Pivan for her help with the English language.

The author is partially supported by the Groupement MoMaS

Proof

The set

is dense in {u 0 ∈ L ∞ (Ω) | 0 ≤ u 0 ≤ 1} for the L 1 (Ω)-topology. Then we can build a sequence (u 0,ν ) ν∈N such that lim ν→∞ u 0,νu 0 L 1 (Ω) = 0.

Let (u ν ) ν be the corresponding sequence of bounded flux solutions, then we deduce from the Theorem 4.1 that for all ν, µ ∈ N,

Then (u ν ) ν is a Cauchy sequence in C([0, T ]; L 1 (Ω)), thus it converges towards u ∈ C([0, T ]; L 1 (Ω)), and

Let us now check that u is a weak solution. Since ϕ i is continuous, and since 0 [START_REF] Seguin | Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients[END_REF] does not depend on u 0 , thus, up to a subsequence, (ϕ i (u ν )) ν converges weakly to ϕ i (u) in L 2 ((0, T ); H 1 (Ω i )). It also converges strongly in L 2 ((0, T ); H s (Ω i )) for all s ∈ (0, 1). This particularly ensures the strong convergence of the traces of (ϕ i (u ν )) ν on the interface. Since ϕ -1 i is continuous, we obtain the strong convergence of the traces of (u ν ) ν . Checking that the set

the limits u i fulfill π1 (u 1 ) ∩ π2 (u 2 ) = ∅, and so u is a weak solution, then it is a SOLA. If u and v are two SOLAs associated to the initial data u 0 and v 0 , we can easily prove, using the Theorem 4.

The uniqueness particularly follows. Let u 0 ∈ L ∞ (Ω), 0 ≤ u 0 ≤ 1, and let (u 0,ν ) ν ⊂ E a sequence of approximate initial data tending to u 0 in L 1 (Ω). We denote by u the unique SOLA associated to u 0 , and by (u ν ) ν the bounded flux solutions associated to (u 0,ν ) ν . Let (M p ) p∈N , (N p ) p∈N be two sequences of positive integers tending to +∞. Let p ∈ N, ν ∈ N, let u Dp the discrete solution corresponding to u 0 , and let u ν,Dp the discrete solution corresponding to u 0,ν .

From the discrete L 1 -contraction principle [START_REF] Enchéry | Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces[END_REF], and from the continuous one (71), we have

Letting p tend to ∞, it follows from the definition of (u 0,ν,D ) (adapted from ( 13)) that

We have proven in the Theorem 3.1 that the sequence of discrete solutions converges, under assumption on the initial data to the unique bounded flux solution, thus

Letting ν tend to ∞ provides lim p→∞ u Dpu L 1 (Ω×(0,T )) = 0.

The convergence occurs in L 1 (Ω × (0, T )), but the uniform bound on the sequence u Dp in L ∞ (Ω × (0, T )) ensures that the convergence also take place in all the L p (Ω × (0, T )), for p ∈ [1, ∞).


PharmD-PhD Canan G Nebigil

BS Pierre Hickel

PhD; Nadia Messaddeq

PhD Jean-Luc Vonesch

MD Marie P Douchet

MD-PhD Laurent Monassier

PhD Katalin György

PhD Rachel Matz

PhD Ramaroson Andriantsitohaina

MD; Philippe Manivet

PharmD-PhD Jean-Marie Launay

PhD Luc Maroteaux
email: lucm@igbmc.u-strasbg.fr

Ablation of serotonin 5-HT 2B receptors in mice leads to abnormal cardiac structure and function

Keywords: Cardiomyopathy, Cell adhesion molecules, Genetics, serotonin

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Cardiomyopathy is an important risk factor for subsequent cardiac morbidity and mortality. Relatively little is known about the molecular mechanism underlying cardiomyopathy and heart failure. The neurohormone, serotonin (5-Hydroxytryptamine, 5-HT) is involved in blood pressure regulation and cardiac function in adults. 5-HT plays an important role in hemodynamic stability. 5-HT specific reuptake inhibitors (by increasing the availability of 5-HT) produce arrhythmia including atrial fibrillation, bradycardia, and heart block [START_REF] Sheline | How safe are serotonin reuptake inhibitors for depression in patients with coronary heart disease?[END_REF] . The mitogenic action of 5-HT 2 triggers the valvular fibroplasia observed in carcinoid patients [START_REF] Robiolio | Carcinoid heart disease. Correlation of high serotonin levels with valvular abnormalities detected by cardiac catheterization and echocardiography[END_REF] and in obese people taking the 5-HT uptake inhibitor/5-HT 2B receptor ligand, fenfluramine, as an appetite suppressant [START_REF] Fitzgerald | Possible role of valvular serotonin 5-HT2B receptors in the cardiopathy associated with fenfluramine[END_REF][START_REF] Rothman | Aminorex, fenfluramine, and chlorphentermine are serotonin transporter substrates. Implications for primary pulmonary hypertension[END_REF] .

The various biological actions of 5-HT are mediated by numerous cognate receptors. There are at least 15 receptor subtypes that belong to four classes: 5-HT 1/5 , 5-HT 2 , 5-HT 3 , and 5-HT 4/6/7 [START_REF] Hoyer | IUPHAR classification of receptors for 5-Hydroxytryptamine (serotonin)[END_REF] . 5-HT binding to 5-HT 2A , 5-HT 2B or 5-HT 2C receptors activates phospholipase C, releases inositol trisphosphate and increases intracellular calcium levels. 5-HT 2B receptor is involved in 5-HT-induced mitogenesis in which c-Src is required for cell cycle progression via the mitogen activated protein kinase pathway [START_REF] Nebigil | 5-Hydroxytryptamine 2B receptor regulates cell-cycle progression: Cross talk with tyrosine kinase pathways[END_REF] . Stimulation of the 5-HT 2B receptor results in cross talk with the 5-HT 1B/1D receptor subtype via activation of phospholipase A 2 [START_REF] Tournois | Cross-talk between 5-hydroxytryptamine receptors in a serotonergic cell line. Involvement of arachidonic acid metabolism[END_REF] . The 5-HT 2B receptor also activates nitric oxide synthesis through a PDZ domain [START_REF] Manivet | PDZ-dependent activation of nitricoxide synthases by the serotonin 2B receptor[END_REF] .

To understand the specificity of each receptor subtypes, the genetic inactivation approach in mice has been used. Mutation of 5-HT receptors 5-HT 1A , 5-HT 1B , or 5-HT 2C in mice leads to behavioral abnormalities [START_REF] Murphy | Molecular manipulations as tools for enhancing our understanding of 5-HT neurotransmission[END_REF] . We have recently shown that 5-HT 2B receptor inactivation in mice leads to trabeculation defects in embryonic heart causing a 30% lethality at mid-gestation [START_REF] Nebigil | Serotonin 2B receptor is required for heart development[END_REF] . Now, we investigated cardiopathy in surviving 5-HT 2B receptor mutant mice. This study reveals that 5-HT via 5-HT 2B receptor is involved in the regulation of cardiomyocyte cytoarchitecture and function. 5-HT 2B receptor ablation in mice leads to cardiomyopathy, including left ventricular dysfunction without hypertrophy.

Methods

Generation of 5-HT 2B receptor knockout mice.

Targeted mutagenesis by homologous recombination has been previously described [START_REF] Nebigil | Serotonin 2B receptor is required for heart development[END_REF] . All animal experimentation was performed in accordance with institutional guidelines, and protocols were approved by the French Animal Care Committee in accordance with European regulations.

Morphological analysis of mouse embryos. Transmission electron microscopy (TEM)

and histological techniques were performed as previously described [START_REF] Choi | 5-HT2B receptor-mediated serotonin morphogenetic functions in mouse cranial neural crest and myocardiac cells[END_REF] . Immunohistochemistry was performed on heart cryosections with the anti-sarcomeric myosin heavy chain (MHC) antibody (MF-20). Anti-tropomyosine and N-cadherin antibody reactions were performed on paraffin sections as described [START_REF] Nebigil | Serotonin 2B receptor is required for heart development[END_REF] . Signal intensity was quantified using a fluoroimager (Typhoon, molecular Dynamics) and calculated as the product of averaged pixel intensity per area.

Cardiomyocyte density determination and confocal microscopic analysis.

Confocal microscope images of the sections were taken on a Leica TCS4D. Calculation of total number of nuclei per field was obtained by counting propidium iodide-stained nuclei. Nonmyocytes were tabulated by counting the number of nuclei not surrounded by cytoplasmic myosin and this number was used to calculate total myocytes as described [START_REF] Tardiff | A truncated cardiac troponin T molecule in transgenic mice suggests multiple cellular mechanisms for familial hypertrophic cardiomyopathy[END_REF] .

Analysis of hypertrophic cardiac genes by RT-PCR.

Semi-quantitative RT-PCR was performed on 1 µg of total RNA extracted from age matched control and knockout mice using the ribosomal elongation factor 1A as an internal control as previously described [START_REF] Nebigil | Serotonin 2B receptor is required for heart development[END_REF] .

The following primers were used: atrial natriuretic factor (ANF) 5'-ccaggccatattggagcaaa and 5'-gaagctgttgcagcctagtc; GATA4 5'-cactatgggcacagcagctcc and 5'ttggagctggcctgcgatgtc; alpha-MHC 5'-ctgctggagaggttattcctcg and 5'ggaagagtgagcggcgcatcaagg; and beta-MHC 5'-tgcaaaggctccaggtctgagggc and 5'gccaacaccaccctgtccaagttc. The PCR products were quantified using an image analyzer (Bio-Rad, GS-700) and calculated as arbitrary units.

Cardiomyocyte isolation and video imaging. Ventricular cardiomyocytes from newborn mice were isolated as previously described [START_REF] Adams | Enhanced Galphaq signaling: a common pathway mediates cardiac hypertrophy and apoptotic heart failure[END_REF] . Beating rate in response to dobutamine was determined by video recording of isolated cardiomyocytes. The analysis was performed on the stage of an inverted microscope (Leica DMRiB) using software developed by J.-L. V.

Echocardiography methods. Animals (19 week old mice) anesthetized with sodium pentobarbital (30 mg/kg intraperitoneally) were observed with two-dimensionally guided M-mode echocardiograms using a short focal length 12-MHz (Hewlett Packard Medical systems, USA) transducer. LV end systolic and diastolic diameters (respectively LVESD and LVEDD) were measured. The percentage of LV fraction shortening was then calculated.

Blood pressure measurements. Systolic arterial pressure and heart rate were recorded by the tail-cuff technique using the LE5002 Storage Pressure Meter (Letica, Spain) in awake 19 week old mutant and control mice. 19 week old mice, anesthetized with tribromoethanol (13 µl of 2.5% solution per g of body weight subcutaneously), were recorded with the four arms of ECG leads attached at the origin of each paw by uni and bipolar lead derivations. The signal was recorded by an electrocardiograph (EKG-Burdick, Siemens) connected to a data acquisition system (MP100 and Acknowledge Software, Biopac Systems Inc, USA).

Electrocardiogram (ECG).

Isolated perfused heart preparation. Hearts from mice (12-19 weeks, 23-25 g) anesthetized with sodium pentobarbital (60 mg/kg intraperitoneally), and heparinized (500 U/kg intraperitoneally) were cannulated and perfused according to Langendorff, at 37°C and pH 7.4, with modified Krebs-Henseleit solution containing (mM) NaCl 118, NaHCO 3 24, KCl 4.7, KH 2 PO 4 1.2, MgSO 4 1.2, CaCl 2 2.5, di-sodium EDTA 0.5 and glucose 10, gassed with 95 %O 2 / 5% CO 2 . Perfusion pressure was constant and equivalent to 1 meter H 2 O. The diastolic tension of the suture was adjusted to 1 g during the stabilization period of the heart.

Measurement of markers for cardiac failure and myocardial damage.

Enzymeimmunoassay for creatine kinase MB isoenzyme and for the cardiac troponin I was determined from samples of serum from adult mice [START_REF] Collinson | Troponin T or troponin I or CK-MB (or none?)[END_REF] .

Data analysis and statistics. All values represent average of independent experiments ± SEM (n=number of experiments as indicated in the text). Comparisons between groups

were performed using Student unpaired t test or ANOVA and a Fischer test. Significance was set at p<0.05.

Results

Heart morphology

5-HT 2B receptor inactivation leads to partial embryonic death due to trabecular defects in the heart leading to mid-gestation lethality (30% n= 120) [START_REF] Nebigil | Serotonin 2B receptor is required for heart development[END_REF] . The 5-HT 2B receptor mutant mice that reached birth displayed no obvious defects, although 30% (n=120) [START_REF] Nebigil | Serotonin 2B receptor is required for heart development[END_REF] of newborn mice developed signs of fatigue and dyspnea between postnatal days 2-5 and died within 24 h from the onset of these symptoms. A likely cause of neonatal death is inadequate cardiac output due to hypoplasia of the left ventricle, despite the lack of evidence for pulmonary edema. All 5-HT 2B receptor mutant mice surviving the first postnatal week developed to adulthood with cardiac problems. This variation in severity of the phenotype could not be attributed to variability in the genetic background of the mice (all the findings were obtained from 129/PAS pure background mice and similar mortality was also observed on a C57/Black6J-129/PAS mixed background).

Newborn 5-HT 2B receptor mutant hearts display a striking decrease in heart to body weight ratio (28%). This difference was 24% in 6-week-old mutants (Table 1). Histological analysis demonstrated that the decrease in heart mass was restricted to the ventricles (as shown in Figure 1A).

Cardiomyocyte number and size

The ratio of cardiomyocytes to total cells (stained with MF-20 antibody, myocyte specific MHC, and PI respectively) revealed 15% fewer cardiomyocytes in the newborn mutants as shown in Fig. 1B. Isolated mutant cardiomyocytes are 12% smaller in length compared to wild-type (n>15) (Fig. 1C). The decrease in ventricular mass observed in 5-HT 2B receptor mutant mice results, therefore, from decreases in both cell density and size of cardiomyocytes.

Hypertrophic gene expression in heart

To determine whether the loss of ventricular mass creates compensatory hypertrophic growth associated with altered expression of hypertrophic markers [START_REF] Chien | Transcriptional regulation during cardiac growth and development[END_REF] , ANF, alpha-MHC, beta-MHC and GATA4 expression was evaluated in 12 week old mutant hearts. Semiquantitative RT-PCR analysis of mutant heart mRNA demonstrated that none of these mRNAs showed significant variation in expression level (less than 5 % variation compared with control, n=5 different individuals). Similar results were obtained in newborn mutants (data not shown).

Cardiomyocyte function

To determine whether the cardiac phenotype of 5-HT 2B receptor mutant mice was cell intrinsic, the function of spontaneously beating isolated cardiomyocytes from newborn was investigated. Beta-adrenergic receptor agonist, dobutamine increased the beating rate of wild-type cardiomyocytes in a dose-dependent manner. However, mutant cardiomyocytes exhibited impaired response to dobutamine in the absence of sympathetic innervation (Fig. 2), indicating cell autonomous defects.

Ultrastructural analysis

A loss of myocardial organization, scattered area of degenerated cardiomyocytes and myofibrillar disarray were apparent in newborn mutant hearts. Wavy myofibrils were identified by anti-tropomyosine staining (Fig. 3A). In this area, myofilaments appeared misaligned, I bands were not detectable, abnormally wide Z bands were seen, and mitochondria were rounded and irregular (Fig. 3B). Sarcomeric length of mutants is 33% smaller compared to that of control mice (n=25). Notably, no evidence for myocardial apoptosis, fibrosis, or significant inflammatory cell infiltrates was found. Nearly identical histopathological findings were observed in all adult mutant hearts.

Furthermore, 5-HT 2B receptor mutant cardiomyocytes had reduced numbers of adherens junctions (Table 1) and the intercalated discs were consistently disorganized (Fig. 4A). Z line-associated protein expression was investigated. Vinculin staining in mutant newborn ventricles was unaltered (not shown). However, N-cadherin expression was reduced by 38.8% in mutant myocardium (Fig. 4B-Table 1).

Hemodynamic Measurements

Trans-thoracic echocardiograms (Fig. 5A, Table 2) show LV dilation and reduced systolic performance of the adult mutant mice. In male mutants, the LV end-diastolic diameter (LVEDD) was 25% higher than wild type. Extreme LV dilation (increased LVEDD) was observed, and the LV end-systolic diameter (LVESD) was increased by 50% in male 5-HT 2B receptor mutants (n>4). The percent of LV fraction shortening, as an indicator of systolic cardiac function, was significantly decreased in male (20%) (Fig. 5A, Table 2) but not in female mutants (not shown). However, when myocardial function was measured by Langendorff's heart preparation in vitro, the developed force in response to adrenergic stimuli (isoproterenol) was also significantly reduced in female mutants (Table 2, Figure 5B). A slight decrease in mutant female coronary flow was also observed, whereas no apparent change in basal blood pressure or heart rate was detected (Table 2).

ECG analysis

ECG analysis in mutants revealed neither atrio-ventricular nor intraventricular conduction defects (similar PR intervals, QRS duration and amplitude). The resting heart rate was significantly decreased in the anesthetized female mutants. The P duration, but not P amplitude, was significantly increased in female (47%) and to a lesser extent in male (17%) mutants. The most striking difference between wild type and 5-HT 2B receptor mutants (both females and males) was dramatically elevated T wave amplitude, which is an indicator of abnormalities in repolarization of ventricles (Fig. 5C, Table ). However, serum potassium levels were not altered (not shown).

Biochemical Markers of heart failure.

Clinical indications of human acute myocardial infarction and injury are revealed by serum levels of cardiac specific biochemical markers, troponin-I and creatine kinase-MB [START_REF] Collinson | Troponin T or troponin I or CK-MB (or none?)[END_REF] . Striking elevated markers were observed in the serum of 5-HT 2B receptor mutants (6 week old), (Fig. 5D). Interestingly, male 5-HT 2B receptor mutants exhibited more pronounced biological changes than females.

Discussion

In this study, we provide the first evidence that Gq-coupled 5-HT 2B receptor ablation in mice leads to cardiomyopathy with LV dysfunction, dilation and an abnormal structure within Z band correlated with a deficiency in N-cadherin expression.

5-HT 2B receptor mutant mice exhibit thinning of the ventricular wall and the reduction in ventricular mass that appears due to two complementary mechanisms: loss of myocardial cells and decrease in cell size. The primary loss of myocardial cells could be due to apoptosis and/or impaired proliferation of cardiomyocytes. No apoptotic bodies were observed by TEM yet mitogen activated protein kinase (MAPK/ERK) activation in response to 5-HT was strongly reduced in newborn mutant hearts (unpublished observation). Together these data suggest that ventricular hypoplasia is mainly due to impaired proliferation but not to apoptosis. Mutation of the thin filament protein, troponin T in mice, also results in cardiomyopathy, due to a primary loss of cardiomyocytes and decrease in cell size [START_REF] Tardiff | A truncated cardiac troponin T molecule in transgenic mice suggests multiple cellular mechanisms for familial hypertrophic cardiomyopathy[END_REF] . Myofibril loss is the most obvious structural change in human cardiomyopathy [START_REF] Mann | Cellular versus myocardial basis for the contractile dysfunction of hypertrophied myocardium[END_REF] and sarcomeric disarray is characteristic of failing hearts [START_REF] Schaper | Impairment of the myocardial ultrastructure and changes of the cytoskeleton in dilated cardiomyopathy[END_REF] . Actually, the decrease in cardiomyocyte size could be due to impaired growth during postnatal development. The loss of ventricular mass creates biomechanical stress on the remaining viable heart muscle, which typically triggers a hypertrophic response by inducing embryonic gene re-expression. However, in the 5-HT 2B receptor mutant heart, despite increased preload conditions (increased LVESD), the expression of hypertrophic markers was not elevated and there were no morphological signs of hypertrophy. Unlike the 5-HT 2B receptor mutant mice, alpha-MHC and myf5 mutant mice develop a hypertrophy and interstitial fibrosis accompanied cardiomyopathy [START_REF] Mckinsey | Cardiac hypertrophy: sorting out the circuitry[END_REF] . Why 5-HT 2B receptor mutants fail to have a hypertrophic response remains to be investigated. Combined myofibrillar breakdown and inhibited myofibrilogenesis may account for loss of ventricular mass without substantial hypertrophy. Mice overexpressing tropomodulin [START_REF] Sussman | Myofibril degeneration caused by tropomodulin overexpression leads to dilated cardiomyopathy in juvenile mice[END_REF] or mutated troponin T are models of dilated cardiomyopathy with inhibited myofibrilogenesis without a hypertrophic response [START_REF] Tardiff | A truncated cardiac troponin T molecule in transgenic mice suggests multiple cellular mechanisms for familial hypertrophic cardiomyopathy[END_REF] .

Other neurotransmitters and hormones that use Gq protein signaling are also involved in cardiomyopathies. In vitro and in vivo studies have indicated a role for hormones such as angiotensin II, bradykinin B 2 [START_REF] Emanueli | Dilated and failing cardiomyopathy in bradykinin B(2) receptor knockout mice[END_REF] , endothelin 1, norepinephrine and prostaglandin F2A, not only in stimulation of cardiac hypertrophy, but also in decompensation of the hypertrophied heart through induction of cardiomyocyte apoptosis [START_REF] Dorn | Gq signaling in cardiac adaptation and maladaptation[END_REF] . Targeted expression of the carboxy-terminus of the alpha subunit of Gq or overexpression of the Gq protein in the heart causes cardiomyopathy. However, the regulation of cardiomyocyte cytoarchitecture through Gq-coupled pathway is poorly understood.

Our data suggest that alteration in cardiomyocyte cytoarchitecture results from 5-HT 2B receptor mutation. How does 5-HT 2B receptor affect the organization of myofibrils and related cardiomyocytes cytoarchitecture? 5-HT 2B receptor mutant cardiomyocytes exhibit abnormal organization of contractile elements, including Z-stripe enlargement (Fig. 5). Interestingly, most of the mutations leading to dilated cardiomyopathy in humans affect structural proteins involved in cytoskeleton-extracellular matrix interaction at the Z-stripe [START_REF] Beltrami | The cellular basis of dilated cardiomyopathy in humans[END_REF] . The altered intercalated disc structures observed in the 5-HT 2B receptor mutant mice heart could be a molecular mechanism leading to impaired contractility and myofibrillar degeneration. Z line-associated structures are responsible for the lateral alignment of myofibrils and their anchorage is at N-cadherin and vinculin containing costameres along the cell membrane. The 5-HT 2B receptor mutant mice exhibit decreased N-cadherin levels. N-cadherin plays an important role in maintaining myofibril integrity [START_REF] Soler | N-cadherin involvement in cardiac myocyte interaction and myofibrillogenesis[END_REF] , in cardiomyocyte interaction and in myofibrilogenesis [START_REF] Radice | Developmental defects in mouse embryos lacking N-cadherin[END_REF] . Down regulation of N-cadherin and disruption of intercellular adhesion have also been reported in failing guinea pig hearts [START_REF] Wang | Chronic pressure overload cardiac hypertrophy and failure in guinea pigs: III. Intercalated disc remodeling[END_REF] . Addition of antibodies against N-cadherin to cardiomyocyte cultures also induces myofibrillar and cytosolic disorganization [START_REF] Goncharova | The involvement of adherens junction components in myofibrillogenesis in cultured cardiac myocytes[END_REF] . Furthermore, mutation of the Drosophila 5-HT2Dro receptor (pharmacological ortholog to 5-HT 2B receptor) results in embryos which do not gastrulate properly due to changes in E-cadherin-dependent cell adhesiveness [START_REF] Colas | Serotonin synchronises convergent extension of ectoderm with morphogenetic gastrulation movements in Drosophila[END_REF] . Our data suggest that 5-HT 2B receptor, in mammals, is required for proper myofibril integrity and myofibrilogenesis by regulating N-cadherin expression.

The 5-HT 2B receptor mutant mice phenotype has similarity with the natural history of patients with dilated cardiopathy. LV dilatation and depressed LV systolic performance in the mutant mice are typical features used to diagnose dilated cardiomyopathy in human. Moreover, serum biochemical indicators of myocardial infarction are increased in the 5-HT 2B receptor mutant mice. No apparent changes in basal blood pressure and heart rate are detected (Table ), suggesting that either 5-HT 2B receptor is not involved in basal blood pressure control or that systemic vascular flow redistribution compensates at least partially this impaired contractility.

5-HT 2B receptor mutant mice exhibit gender differences: Consistent with the idea that the morphological lesions detected in male mutant mice underlie abnormal functions, female mutant mice with less severe histopathological findings did not reveal significant functional changes under the steady-state condition. Similar gender differences occur in other cardiomyopathy models such as in the beta-MHC mutant mouse [START_REF] Berul | Familial hypertrophic cardiomyopathy mice display gender differences in electrophysiological abnormalities[END_REF] . In X-linked cardiomyopathy in humans, heart failure occurs rapidly after onset of symptoms in male, but is delayed in its onset and progression in females [START_REF] Towbin | The role of cytoskeletal proteins in cardiomyopathies[END_REF] . Cardioprotective effects in females have been attributed to estrogen action.

5-HT 2B receptor specific agonist, norfenfluramine, ergot drugs, and 5-HT released from carcinoid tumors contribute to valvular fibroplasia in humans [START_REF] Fitzgerald | Possible role of valvular serotonin 5-HT2B receptors in the cardiopathy associated with fenfluramine[END_REF][START_REF] Rothman | Aminorex, fenfluramine, and chlorphentermine are serotonin transporter substrates. Implications for primary pulmonary hypertension[END_REF] . However, the lack of detectable valvular defects in mutant mice, indicates that the 5-HT 2B receptor is not required for heart valve development.

Mutation of a non-cytoskeletal molecule, the 5-HT 2B receptor, provides the first genetic evidence that 5-HT via this receptor regulates cardiomyocyte function and structure. These findings should facilitate a genetic approach and new avenues of drug design in fighting cardiovascular disease.

Figure 1 .

 1 Figure 1. 5-HT 2B receptor mutant mice exhibit decreased ventricular mass due to decreased cell number and size. A) Representative sagittal section from paraffinembedded adult hearts (12 week old). lv: left ventricle, rv: right ventricle. B) The cardiomyocytes (MHC -positive cells) and total cells (PI stained nucleus) per field were counted (n>100 from four independent staining). Numbers are expressed as mean±SD for n = 8 (sections). C) Ventricular myocytes from newborn mice were isolated and their size was measured. Values are expressed as mean ± SEM, n>100. Bars: A 200 µm, B 50 µm, C 100 µm. D) Markers known to be expressed in hypertrophic growth (ANF, MHC, and GATA4) were analyzed by RT-PCR in adult mice (12 weeks). 5-HT 2B receptor mutant (-/-) and wild-type (+/+) mice.

Figure 2 .

 2 Figure 2. Cardiac phenotype in 5-HT 2B receptor mutant is intrinsic to cardiomyocytes. A) The beating rate of isolated cardiomyocytes from newborn hearts in response to different concentrations of dobutamine is shown. B) The beating rate in isolated single cardiomyocytes was deduced from these digital video recordings. The beating rate is expressed as mean myocyte contractions per minute ± SEM, * p<0.05: difference between mutant (-/-, n=5) and wild-type (+/+, n=3) mice.

Figure 3 .

 3 Figure 3. 5-HT 2B receptor mutant mice display dramatic disruption of cardiomyocyte cytoarchitecture with myofibrillar disarrays. A) Representative sections from paraffin-embedded adult hearts stained with tropomyosine antibody. B) High magnification of sarcomeres from LVs in hearts. A-bands (A), M-lines (M) and I-bands (I) Z-bands (Z), desmosomes (d), mitochondria (m), nexus (n);. Bars: A 50 µm, and B 1 µm. 5-HT 2B receptor mutant (-/-), and control (+/+) mice.

Figure 4 .

 4 Figure 4. Intracellular junctional structures are perturbed in 5-HT 2B receptor mutant mice. A) High magnification of intercalated disc (ID) structures from LVs of newborn mice. Actin fibers (F); Desmosomes (D); Mitochondria (M); Nexus (N); Z-bands (Z). B) Immunohistochemistry for the intercalated disc protein N-cadherin in newborn hearts. The DAPI staining shows the distribution of cells (right). Bars: B 50 µm, and A 0.5µm. 5-HT 2B receptor mutant (-/-), and control (+/+) mice.

Figure 5 .

 5 Figure 5. Functional analysis reveals LV dysfunction in 5-HT 2B receptor mutant mice. A) Ventricular diameters (left) and LV shortening fraction (right) of adult heart were obtained by two-dimensional echocardiogram. Values are expressed as mean ± SEM, n=5. FS (LV fraction shortening). B) Langendorff preparation of isolated hearts from adult female perfused and stimulated with adrenergic agonist. Values are given ± SEM (n=3) . C) ECG from wild (left panel) and mutant (right panel) adult male, in the DI two arms bipolar derivation. Letters indicate the P, Q, R S and T waves. Arrows indicate all P waves. D) Classical markers of human heart failure, creatine kinase MB isoenzyme (CK-MB mass measurement) and for the cardiac troponin I (cTnI) were detected in serum of mice by enzyme-immunoassay. Values in ng/ml are given ± SEM (n>10) and stars indicate difference between mutant (-/-) and wild-type (+/+) mice (P<0.05).

Table 1

 1 Morphometry of 5-HT 2B receptor mutant mice cardiac parameters Heart to body weight ratio in % (n>5 per group). ‡ Sarcomeric length in µm (n=25 per group, 2 individual each); § N-cadherin expression in arbitrary units (n=10 per group, 4 individual each); II Intercalated disc size in µm assessed by direct measurement on electron micrograph per unit picture (n=5 per group, 2 individual each); Values are expressed as mean ± SEM, * p<0.05: difference between mutant (-/-) and wild-type (+/+) mice.

	Genotype	+/+	-/-
		Heart to body weight ratio †
	Newborn	1.06±0.12	0.76*±0.07
	6 weeks	0.67±0.40	0.51*±0.03
		Sarcomeric length ‡
	Newborn	2.04±0.02	1.38*±0.05
		N-cadherin expression §
	Newborn	73.4±12.4	44*.0±8.2
		Intercalated disc size II
	Newborn	2.54±0.08	0.64*±0.09
	†

Acknowledgements

This work has been supported by funds from the Centre National de la Recherche Scientifique, the Institut National de la Santé et de la Recherche Médicale, the Hôpital Universitaire de Strasbourg, the Université Louis Pasteur, and by grants from the Fondation de France, the Fondation pour la Recherche Médicale, from the Association pour la Recherche contre le Cancer (# 9503, 7389), the Ligue Nationale contre le Cancer and a Lilly-University L. Pasteur Fellowship (C.G.N.).