Clément Cancès
email: cances@ann.jussieu.fr

Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution

Keywords: entropy solution, scalar conservation law, discontinuous porous media, capillarity AMS subject classification. 35L65, 76S05

We consider an immiscible two-phase flow in a heterogeneous one-dimensional porous medium. We suppose particularly that the capillary pressure field is discontinuous with respect to the space variable. The dependence of the capillary pressure with respect to the oil saturation is supposed to be weak, at least for saturations which are not too close to 0 or 1. We study the asymptotic behavior when the capillary pressure tends to a function which does not depend on the saturation. In this paper, we show that if the capillary forces at the spacial discontinuities are oriented in the same direction that the gravity forces, or if the two phases move in the same direction, then the saturation profile with capillary diffusion converges toward the unique optimal entropy solution to the hyperbolic scalar conservation law with discontinuous flux functions.

the dead-oil problem in the one dimensional case

Suppose that R represents a one dimensional homogeneous porous medium, with porosity φ (which is supposed to be constant for the sake of simplicity). If u denotes the saturation of the water phase, and so (1u) the saturation of the oil phase thanks to the dead-oil approximation, writing the volume conservation of each phase leads to:

φ∂ t u + ∂ x V w = 0, (1)
-φ∂ t u + ∂ x V o = 0, (2)
where V o (resp. V w ) is the filtration speed of the oil phase (resp. water phase). Using the empirical diphasic Darcy law, we claim that

V β = -K k r,β (u) µ β (∂ x P β -ρ β g) , β = o, w, (3)
where K is the global permeability, only depending on the porous media, µ β , P β , ρ β are respectively the dynamical viscosity, the pressure and the density of the phase β, g represents the effect of gravity, k r,β denotes the relative permeability of the phase β. This last term comes from the interference of the two phases in the porous media.

There exists s ⋆ ∈ [0, 1) such that the function k r,w is non-decreasing, with k r,w (u) = 0 if 0 ≤ u ≤ s ⋆ < 1, and k r,w is increasing on [s ⋆ , 1]. The function k r,o is supposed to be nonincreasing, with k r,o (1) = 0. We suppose that there exists s ⋆ ∈ (s ⋆ , 1] such that k r,o (s) = 0 for s ∈ [s ⋆ , 1), and k r,o is decreasing on [0, s ⋆ ).

The pressures are supposed to be linked by the relation

P cap (u) = P w -P o , (4)
where P cap is a smooth non-decreasing function called capillary pressure.

Adding ( 1) and ( 2), and using ( 3) and (4) yields

-∂ x   β=o,w K k r,β (u) µ β (∂ x P β -ρ β g)   = 0,
and thus there exists q, called total flow-rate, only depending on time, such that

- β=o,w K k r,β (u) µ β (∂ x P β -ρ β g) = q. (5)
Using ( 3), ( 4) and ( 5), ( 1) can be rewritten

φ∂ t u + ∂ x qk r,w (u) k r,w (u) + µw µo k r,o (u) -K∂ x k r,w (u)k r,o (u) µ o k r,w (u) + µ w k r,o (u) (∂ x P cap (u) -(ρ w -ρ o )g) = 0. (6)
Supposing that the total flow rate q does not depend on times, and after a convenient rescaling, equation [START_REF] Aziz | Petroleum Reservoir Simulation[END_REF] becomes

∂ t u + ∂ x (f (u) -λ(u)∂ x π(u)) = 0, ( 7
)
where f is a Lipschitz continuous function, fulfilling f (0) = 0, f (1) = q, λ is a nonnegative Lipschitz continuous functions, with λ(0) = λ(1) = 0, and π is a non-decreasing function, also called capillary pressure. The effects of capillarity are often neglected, particularly in the case of reservoir simulation, and so [START_REF] Bachmann | Analysis of a scalar conservation law with a flux function with discontinuous coefficients[END_REF] turns to a nonlinear hyperbolic equation called Buckley-Leverett equation, and we have to consider the initial-value problem

∂ t u + ∂ x f (u) = 0, u(0) = u 0 . (BL)

discontinuous flux functions and optimal entropy solution

We now consider heterogeneous one dimensional porous media, i.e. an apposition of several homogeneous porous media with different physical properties. This leads to discontinuous functions with respect to the spatial variable. For the sake of simplicity, we assume that the heterogeneous porous medium is made of only two homogeneous porous media represented by the open subsets Ω 1 = R ⋆ -and Ω 2 = R ⋆ + . Keeping the notations of (6), φ, K, k r,β (u, •) and π(u, •) are now discontinuous functions, i.e. piecewise constant functions, denoted φ i , K i , k r,β,i and π i in Ω i . Thus the problem becomes   

∂ t u + ∂ x (f i (u) -λ i (u)∂ x π i (u)) = 0, u(0) = u 0 , + transmission condition at x = 0, (8)
where f i are Lipschitz continuous functions on [0, 1], and can be decomposed in the following way:

f i (u) = qr i (u) + λ i (u)(ρ w -ρ o )g, (9)
where r i is a non-decreasing Lipschitz continuous function fulfilling r i (0) = 0, r i (1) = 1, and λ i is a non-negative Lipschitz continuous function fulfilling λ i (0) = 0, λ i (1) = 0. We stress here the fact that q and (ρ wρ o )g neither depend on the subdomain i nor on time.

We now have to give more details on this transmission conditions at x = 0. First neglect the effects of capillarity, so that (8) becomes the apposition of two Buckley-Leverett equations, linked by a transmission condition.

   ∂ t u + ∂ x f i (u) = 0, u(0) = u 0 , + transmission condition at x = 0, (10)
We ask the conservation of mass at the interface between the two porous media, then we have to connect the flux. Denoting u i the trace (if it exists) of u |Ωi on {x = 0}, this means that the following Rankine-Hugoniot condition has to be fulfilled:

f 1 (u 1 ) = f 2 (u 2 ). ( 11
)
Some assumptions has to be done on the flux functions f i in order to carry out the study. Firstly, we suppose that the total flow-rate q is a non-negative constant. Dealing with non-positive q is also possible, since it suffices to change x by -x and u by (1u). Secondly, we suppose that each f i has a simple dynamic on [0, 1]. More precisely,

∃b i ∈ [0, 1) s.t. f i is decreasing on(0, b i ) and increasing on (b i , 1). ( 12
)
With Assumption ( 12), we particularly ensure that

q = f i (1) = max s∈[0,1] (f i (s)).
The physical meaning of ( 12) is that buoyancy works on the oil-phase in the sense of decaying x.

The case b i = 0 can also correspond to situations where the total flow rate q is sufficiently strong for ensuring that both phases always move in the same direction. Indeed, The oil-flux, given in Ω i by f i (u) has the same sign as the water-flux, given by qf i (u). Note that the assumption on the dynamic on f i is often fulfilled by the physical models, as it is stressed in [START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF] (see also [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF]).

Thirdly, we assume f 1 and f 2 are not linear on any non-degenerate interval of (0, 1). [START_REF] Brézis | Analyse Fonctionnelle: Théorie et applications[END_REF] This latter assumption allows us to claim, thanks to [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] (see also [START_REF] Vasseur | Strong traces for solutions of multidimensional scalar conservation laws[END_REF]) that a solution of q 0 1 b i Figure 1: example of f i fulfilling ( 12) and ( 13)

∂ t u + ∂ x f i (u) = 0 satisfying the entropy inequalities in Ω i × (0, T ): ∀κ ∈ [0, 1] ∂ t |u -κ| + ∂ x (sign(u -κ)(f i (u) -f i (κ))) ≤ 0 in D ′ (Ω i × [0, T )) (14)
admits a strong trace u i on {x = 0} × (0, T ).

Remark. 1.1 As it is proven in Section 3.1, if u 0 ∈ BV (R), the solution u we consider has strong traces on the interface without assuming [START_REF] Brézis | Analyse Fonctionnelle: Théorie et applications[END_REF]. The assumption [START_REF] Brézis | Analyse Fonctionnelle: Théorie et applications[END_REF] provides the existence of strong traces for initial data in L ∞ (R).

The problem ( 10)- [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF] has been widely studied recently (see e.g. [START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF][START_REF] Adimurthi | Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes[END_REF][START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF][START_REF] Bachmann | Analysis of a scalar conservation law with a flux function with 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 | Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients[END_REF][START_REF] Chen | Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems[END_REF][START_REF] Diehl | On scalar conservation laws with point source and discontinuous flux function[END_REF][START_REF] Diehl | Scalar conservation laws with discontinuous flux function. I. The viscous profile condition[END_REF][START_REF] Diehl | Scalar conservation laws with discontinuous flux function. II. On the stability of the viscous profiles[END_REF][START_REF] Garavello | Conservation laws with discontinuous flux[END_REF][START_REF] Jimenez | Some scalar conservation laws with discontinuous flux[END_REF][START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF][START_REF] Karlsen | On a nonlinear degenerate parabolic transportdiffusion 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][START_REF] Yu | Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux[END_REF][START_REF] Seguin | Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients[END_REF][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]). It has been particularly shown by Adimurthi, Mishra and Veerappa Gowda [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] that there are infinitely many solutions satisfying [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF]. Additional entropy conditions has to be considered at the interface {x = 0}. We refer to [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] and [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF] for a detailed discussion on the possible choices of entropy conditions at the interface. According to Kaasschieter [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] and Adimurthi, Jaffré and Veerappa Gowda [START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF], the relevant entropy condition at the interface for two-phase flows with continuous capillary pressure field is the so-called optimal entropy condition introduced in [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF]. Assuming that both f i are convex, the optimal entropy solution is characterized as follows: the discontinuity at the interface between u 1 and u 2 can not be undercompressive:

min {0, f ′ 1 (u 1 )} max {0, f ′ 2 (u 2 )} = 0. ( 15
)
Following the idea of Audusse and Perthame [START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF], the entropy condition at the interface can be derived by comparing the solution to steady states. Denoting by κ

(x) = κ i if x ∈ Ω 1 , where f 1 (κ 1 ) = f 2 (κ 2 )
, then κ has to satisfy the relation

min {0, f ′ 1 (κ 1 )} max {0, f ′ 2 (κ 2 )} = 0.
We denote by κ opt (x) be the steady state corresponding to the optimal entropy connection appearing in the work of Adimurthi, Mishra and Veerappa Gowda [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF], described on Figure 2 and defined as follows:

• if f 1 (b 1 ) ≤ f 2 (b 2 ), then κ opt (x) = b 2 if x > 0, b 2 = min {ν | f 1 (ν) = f 2 (b 2 )} if x < 0; (16)
• if f 1 (b 1 ) ≥ f 2 (b 2 ), then κ opt (x) = b 1 if x < 0, b 1 = max {ν | f 2 (ν) = f 1 (b 1 )} if x > 0. (17)
It is easy to check that in both cases, the function (x, t) → κ opt (x) is a steady entropy solution in the sense of Definition 1.2. According to [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] and [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF], the condition (15) can be replaced by the case (a)

q b1 f1 f2 1 0 b2 b 2 case (b) f1 q f2 1 0 b1 b1 b2
Figure 2: We exhibit particular steady states, which are so called optimal connections in the work of Adimurthi, Mishra and Veerappa Gowda [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF]. The case (a) corresponds to the optimal connection (16) while the case (b) corresponds to the optimal connection [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF].

formulation: ∀ψ ∈ D + (R × [0, T )), T 0 R |u -κ opt |∂ t ψ dxdt + R |u 0 -κ opt |ψ(•, 0) dx + T 0 i=1,2 Ωi sign(u -κ opt ) (f i (u) -f i (κ opt )) ∂ x ψ dxdt ≥ 0. ( 18
)
This interface entropy condition does not require anymore that both f i are convex to provide a L 1 -contraction semi-group, as it will be stated in Theorem 1.3 and shown in [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF].

Definition 1.2 (entropy solution to (10)-( 11)) Let u 0 ∈ L ∞ (R) with 0 ≤ u 0 ≤ 1, and let T > 0. A function u is said to be an entropy solution to ( 10)- [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF] if

1. u ∈ L ∞ (R × (0, T )), 0 ≤ u ≤ 1 a.e.; 2. for all ψ ∈ D + (R × [0, T )), T 0 R u∂ t ψ dxdt + R u 0 ψ(•, 0)dx + i=1,2 T 0 Ωi f i (u)∂ x ψ dxdt = 0; ( 19
)
3. for i = 1, 2, for all ψ ∈ D + (Ω i × [0, T )), for all κ ∈ [0, 1], T 0 Ωi |u -κ|∂ t ψ dxdt + Ωi |u 0 -κ|ψ(•, 0)dx + T 0 Ωi sign(u -κ)(f i (u) -f i (κ))∂ x ψ dxdt ≥ 0; ( 20
)
4. the inequalities (18) hold.

In the following theorem, we claim the existence and the uniqueness of the entropy solution to the problem ( 10)- [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF].

Theorem 1.3 (Existence and uniqueness of the entropy solution) Let u 0 ∈ L ∞ (R) with 0 ≤ u 0 ≤ 1, and let T > 0, then there exists a unique entropy solution to ( 10)- [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF] in the sense of Definition 1.2. Furthermore, the function u can be supposed to belong to C([0, T ]; L 1 loc (R)), and if u, v are two entropy solutions associated to initial data u 0 , v 0 , then, for all R > 0, the following comparison principle holds:

∀t ∈ [0, T ], R -R (u(x, t) -v(x, t)) ± dx ≤ R+Ct -R-Ct (u 0 (x) -v 0 (x)) ± dx, where C = max i (Lip(f i )), with Lip(f i ) = sup s∈(0,1) |f ′ i (s)| .
The uniqueness and the L 1 -stability of the entropy solution stated above can be seen as a straightforward generalization of Theorem 3.1 in [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF] to the case where q = 0. The existence of such an entropy solution is provided in [START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF] by showing the convergence of the discrete solution corresponding to a Godunov-type scheme, while a modified Engquist-Osher scheme is considered in [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF].

In the particular case where [START_REF] Seguin | Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients[END_REF] that entropy solutions can be obtained as the limit for µ → 0 and δ → 0 of the solutions u µ,δ to the problem

f i (u) = k i u(1 -u) for k 1 , k 2 ∈ R + , then it is shown in
∂ t u µ,δ + ∂ x k δ (x)u µ,δ (1 -u µ,δ ) = µ∂ xx u µ,δ
where k δ is a smooth approximation of the piecewise constant function defined by k

(x) = k i if x ∈ Ω i .
It has been proven in [START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF] that the entropy solution can also be obtained as limit for ε tends to 0 of weak solutions to regularized problems

∂ t u ε + ∂ x (f i (u ε ) -ελ i (u ε )∂ x π i (u ε )) = 0
under the assumption that π 1 (0) = π 2 (0) and π 1 (1) = π 2 (1). This latter assumptions is relaxed in this paper and in [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].

The Godunov-type scheme proposed by Adimurthi, Jaffré and Veerappa Gowda [START_REF] Adimurthi | Godunov-type methods for conservation laws with a flux function discontinuous in space[END_REF] uses the fact that the fluxes are given by simple algebraic relations. Indeed, The flux G int (u 1 , u 2 ) = f 1 (u 1 ) = f 2 (u 2 ) at the interface corresponding to traces u 1 , u 2 is given by:

G int (u 1 , u 2 ) = max {G 1 (u 1 , 1), G 2 (0, u 2 )} , (21)
where G i is the Godunov solver corresponding to f i , that is

G i (u, v) =    min s∈[u,v] f i (s) if u ≤ v, max s∈[v,u] f i (s) if u ≥ v.
If the flux at the interface is given by

G int (u 1 , u 2 ) = f 1 (u 1 ) = G 1 (u 1 , 1), (22)
then the restriction u |Ω 1 of u to Ω 1 is the unique entropy solution to

     ∂ t u + ∂ x f 1 (u) = 0 in Ω 1 × (0, T ), u(0, •) = γ in (0, T ), u(•, 0) = u 0 in Ω 1 (23)
corresponding to γ = 1. Recall that the trace on {x = 0} has to be understood in a weak sense (see [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF][START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF]). Since the solution to [START_REF] Diehl | On scalar conservation laws with point source and discontinuous flux function[END_REF] depends in a non-decreasing way of the prescribed trace γ, we can claim that

u |Ω 1 = sup γ∈L ∞ ((0,T );[0,1]) {v solution to (23)} . (24)
Similarly, in the case where the flux at the interface is given by

G int (u 1 , u 2 ) = f 2 (u 2 ) = G 2 (0, u 2 ), (25)
the restriction u |Ω 2 of u to Ω 2 is characterized by

u |Ω 2 = inf γ∈L ∞ ((0,T );[0,1]) {v solution to (27)} , (26)
where

     ∂ t u + ∂ x f 2 (u) = 0 in Ω 2 × (0, T ), u(0, •) = γ in (0, T ), u(•, 0) = u 0 in Ω 2 (27)
Since the problem is conservative, in both cases the solution u is entropic in both Ω i × (0, T ), i.e. it satisfies [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF], and minimizes the flux through the interface {x = 0}. It is shown in [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] (see also [START_REF] Cancès | On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types[END_REF]) that this characterization still holds, but that the different physical assumptions lead to the selection of a solution to (10)- [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF] which is not the entropy solution in the sense of Definition 1.2.

heterogeneities involving discontinuous capillarities

Let us now come back to the problem [START_REF] Bachmann | Finite volume schemes for a non linear hyperbolic conservation law with a flux function involving discontinuous coefficients[END_REF]. Suppose for the sake of simplicity that the functions π i are smooth and increasing on [0, 1], and that λ i (u) > 0 if 0 < u < 1. The problem is then a spatial coupling of two parabolic problems, and we will need to ask two transmission conditions: one for the trace, and one for the flux. Concerning the latter, the conservation of mass yields a relation analogous to [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF], which can be written with rough notations:

f 1 (u 1 ) -λ 1 (u 1 )∂ x π 1 (u 1 ) = f 2 (u 2 ) -λ 2 (u 2 )∂ x π 2 (u 2 ). ( 28
)
Let us now focus on the trace condition at the interface. In the case of heterogeneous media, the capillary pressure can be discontinuous at the interface. Numerical schemes for simulating such flows has been proposed in [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF][START_REF] Enchéry | Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces[END_REF][START_REF] Ern | Discontinuous galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures[END_REF][START_REF] Ersland | Numerical methods for flow in a porous medium with internal boundaries[END_REF]. It has been shown independently in [START_REF] Buzzi | Interface conditions for degenerate two-phase flow equations in one space dimension[END_REF] and [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] (but see also [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF] and [START_REF] Enchéry | Numerical approximation of a two-phase flow in a porous medium with discontinuous capillary forces[END_REF]) that the connection of the capillary pressures π i (u i ) has to be done in a graphical sense, so that phenomena like oil trapping can appear. Thus we have to define the monotonous graphs πi .

πi (u) =    π i (u) if 0 < u < 1, (-∞, π i (0)] if u = 0, [π i (1), +∞) if u = 1.
It is shown in [START_REF] Buzzi | Interface conditions for degenerate two-phase flow equations in one space dimension[END_REF] and [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] that a natural way to connect the capillary pressures on the interface consists in asking:

π1 (u 1 ) ∩ π2 (u 2 ) = ∅. (29)
In order to state a convenient definition for the solution of ( 8)-( 28)-( 29), we introduce the Kirchhoff transformation

ϕ i (u) = u 0 λ i (s)π ′ i (s)ds. Definition 1.4 (bounded flux solution) Let u 0 ∈ L ∞ (R), 0 ≤ u 0 ≤ 1, and let T > 0.
A function u is said to be a bounded flux solution to ( 8)-( 28)-( 29) if it fulfills:

1. u ∈ L ∞ (R × (0, T )), 0 ≤ u ≤ 1 a.e., 2. ∂ x ϕ i (u) ∈ L ∞ (Ω i × (0, T )), 3. π1 (u 1 ) ∩ π2 (u 2 ) = ∅ for a.e. t ∈ (0, T ), 4. ∀ψ ∈ D(R × [0, T )), T 0 R u(x, t)∂ t ψ(x, t)dxdt + R 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 = 0. ( 30
)
The bounded flux solution are so called since the point 2 of Definition 1.4 ensures that the flux f i (u) -∂ x ϕ i (u) remains uniformly bounded. Such a condition will require assumptions on the initial data u 0 , as it will be stated in the following theorem.

Theorem 1.5 (existence of a bounded flux solution) Let f 1 , f 2 be Lipschitz continuous functions, and ϕ 1 , ϕ 2 be increasing Lipschitz continuous functions. Let

u 0 ∈ L 1 (R), 0 ≤ u 0 ≤ 1 fulfilling ∂ x ϕ i (u 0 ) ∈ L ∞ (Ω i )
, and π1 (u 0,1 ) ∩ π2 (u 0,2 ) = ∅, where u 0,i denotes the trace on {x = 0} of u 0|Ω i .

Then there exists a bounded flux solution. Moreover,

u belongs to C([0, T ]; L 1 (R)).
The first part of this theorem is a straightforward adaptation to the case of unbounded domains and non-monotonous f i of a result from [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF] and [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF] (see also [START_REF] Schweizer | Homogenization of degenerate two-phase flow equations with oil trapping[END_REF]). This is based on a maximum principle on the fluxes (f i (u) -∂ x ϕ i (u)). This particularly yields:

f i (u) -∂ x ϕ i (u) L ∞ (Ωi×(0,T )) ≤ max j=1,2 f j (u 0 ) -∂ x ϕ j (u 0 ) L ∞ (Ωj ) . (31)
If

u 0 ∈ L 1 (Ω), then choosing ψ = min(1, (1, R -|x|) + ) and letting R tend to ∞ gives u ∈ L ∞ ((0, T ); L 1 (R))
. Moreover, thanks to [START_REF] Cancès | On the time continuity of entropy solutions[END_REF], u can be supposed to belong to C([0, T ];

L 1 loc (R)). Then u belongs to C([0, T ]; L 1 (R)).
The choice of bounded flux solutions instead of more classical weak solution with ∂ x ϕ i (u) only belonging to L 2 ((0, T ); L 2 loc (Ω i )) has been motivated by the fact that it provides a comparison principle.

Proposition 1.6 Let u, v be two bounded flux solutions in the sense of Definition 1.4 associated to initial data u 0 , v 0 . Then, for all

ψ ∈ D + (R × [0, T )), T 0 R (u(x, t) -v(x, t)) ± ∂ t ψ(x, t)dxdt + R (u 0 (x) -v 0 (x)) ± ψ(x, 0)dx + T 0 i=1,2 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 i=1,2 Ωi ∂ x (ϕ i (u)(x, t) -ϕ i (v)(x, t)) ± ∂ x ψ(x, t)dxdt ≥ 0. ( 32
)
This proposition is not sufficient to claim the uniqueness, but it will be very useful in the sequel.

In order to obtain a uniqueness result, we have to ask furthermore that the initial data belongs to L 1 (R).

Theorem 1.7 (uniqueness of bounded flux solution)

Let u 0 ∈ L 1 (R), 0 ≤ u 0 ≤ 1 a.e., with ∂ x ϕ i (u 0 ) ∈ L ∞ (Ω i ) and π1 (u 0,1 ) ∩ π2 (u 0,2 ) = ∅.
Then there exists a unique bounded flux solution

u ∈ C([0, T ]; L 1 (R)) in the sense of definition 1.4.
This theorem is a straightforward consequence Proposition 1.6. Indeed, choosing ψ = min(1, (1, R-|x|) + ) in [START_REF] Jimenez | Some scalar conservation laws with discontinuous flux[END_REF], and letting R tend to +∞ gives the comparison principle:

∀t ∈ [0, T ], R (u(x, t) -v(x, t)) ± dx ≤ R (u 0 (x) -v 0 (x)) ± dx. (33)
The uniqueness result follows.

capillary pressure independent of the saturation

In some cases, the dependence of the capillary pressure π i with respect to the saturation seems to be weak, and some numerical simulation consider capillary pressures only depending on the porous medium, but not on the saturation. More precisely, we aim to consider graphs of capillary pressure on the form [START_REF] Karlsen | On a nonlinear degenerate parabolic transportdiffusion equation with a discontinuous coefficient[END_REF] so that the capillary pressure would roughly speaking not depend on u.

πi (u) =    P i if 0 < u < 1, (-∞, P i ] if u = 0, [P i , +∞) if u = 1,
If one considers an interface {x = 0} between two Ω i , where the πi are on the form [START_REF] Karlsen | On a nonlinear degenerate parabolic transportdiffusion equation with a discontinuous coefficient[END_REF], we can give an orientation to the interface: the interface is said to be positively oriented if P 1 > P 2 , and negatively oriented if P 1 < P 2 . A positively oriented interface involve positive capillary forces, and a negatively oriented involve positive capillary forces. The gravity effects are also oriented by the sign of (ρ wρ o )g in [START_REF] Bachmann | Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients[END_REF]. We have to make the assumption that "either the gravity effects and the interface are oriented in the same way, or the convective effects are larger than the gravity effects. "

Since we have supposed that gravity works in the sense of decaying x, we assume in the sequel that

P 1 < P 2 . ( 35
)
We build a family of approximate problems (P ε ) taking into account the capillary pressure: one suppose that π ε i (u) = P i + εu, where P i is a constant depending only on the homogeneous subdomain Ω i . In fact, any π ε i converging uniformly to P i on [0, 1] and such that u

→ u 0 λ i (s) (π ε i ) ′ (s)ds
converges uniformly toward 0 would fit.

Up to a smoothing of the initial data, we obtain a resulting sequence (u ε ) ε of bounded flux solutions for a problem of type ( 8)-( 28)-( 29). We will show that under Assumptions ( 12)-( 13), this sequence tends almost everywhere to the unique entropy solution to ( 10)- [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF].

This result has to be compared to the one presented in the associated paper [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], where it is shown that if Assumption [START_REF] Karlsen | Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient[END_REF] does not hold, non-classical shock can occur at the interface, representing oil-trapping.

organization of the paper

The paper is organized as follow: section 2 is devoted to the study of the approximate problem (P ε ). We first smooth the initial data in a convenient way, and then we give a L 2 ((0, T ); H 1 (Ω i ))-estimate on the approximate solutions that shows in particular that if the approximate solution u ε converges almost every where towards a function u, then u satisfies the points 1,2 and 3 of Definition 1.2. In order to prove that u ε converges almost everywhere, we derive a family BV -estimates. In order to check that the last point of Definition 1.2 is fulfilled by the limit u of the approximate solutions (u ε ) ε

The approximate problems

In this section we will define the approximate problem (P ε ), and its solution u ε . We will state a L 2 ((0, T ); H 1 loc (Ω i ))-estimate and a family of BV -estimates, which will be the key points of the proof of convergence of u ε toward a weak solution of the problem (P ε ).

In order to recover a family of entropy inequalities, we will build some steady solutions κ ε to the problem (P ε ), and study their limit as ε → 0. This last point will require strongly Assumption (12).

smoothing the initial data

As it has already been stressed in Theorem 1.5, we need to assume some regularity on the initial data to ensure the existence of a bounded flux solution to problems of the type ( 8)-( 28)- [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF]. Let u 0 belong to L ∞ (R), with 0 ≤ u 0 ≤ 1, we will build a family (u ε 0 ) ε of convenient approximate initial data.

Lemma 2.1 Let u 0 ∈ L ∞ (R), 0 ≤ u 0 ≤ 1, then there exists a family (u ε 0 ) ε of approximate initial data such that:

• u ε 0 ∈ C ∞ c (R ⋆ ), 0 ≤ u ε 0 ≤ 1, • u ε 0 → u 0 a.e. in R, ε∂ x u ε 0 ∞ → 0 as ε → 0, and ε∂ x u ε 0 ∞ ≤ 1 for all ε > 0, • If u 0 ∈ BV (R), then ∂ x u ε 0 L 1 (R) ≤ T V (u 0 ) + 4.
Proof: Let α > 0, and let ρ α be a mollifier with support in (-α, α).

Let v α = u 0 • χ α<|x|<1/α ⋆ρ α , then it is clear that v α ∈ C ∞ c (R ⋆ ), and that v α → u 0 a.e. in R as α → 0. Choosing ε = min α, min(1, √ α) ∂xv α ∞
, and u ε 0 = v α ends the proof of Lemma 2.1.

the problem (P ε )

Let P 1 , P 2 ∈ R, we define the functions π ε i by π ε i (u) = P i + εu, and

πε i (u) =    P i + εu if 0 < u < 1, (-∞, P i ] if u = 0, [P i + ε, +∞) if u = 1.
If ε is small, the intersection of the ranges of the functions π ε 1 and π ε 2 is empty, and then the graphical relation π1 (u 1 ) ∩ π2 (u 2 ) = ∅ connecting the capillary pressures at the interface becomes

P 1 P 2 πε 2 (u) πε 1 (u) ε ε saturation u 1 0 capillary pressure
(1 -u 1 )u 2 = 0. Let 0 ≤ u 0 ≤ 1, and let (u ε 0 ) ε be built as in Lemma 2.1, let ϕ i (u) = u 0 λ i (s)ds. The approximate problem is: find u ε s.t.        ∂ t u ε + ∂ x (f i (u ε ) -ε∂ x ϕ i (u ε )) = 0 in Ω i × (0, T ), πε 1 (u ε )(0 -, t) ∩ πε 2 (u ε )(0 + , t) = ∅ in (0, T ), f 1 (u ε )(0 -, t) -ε∂ x ϕ 1 (u ε )(0 -, t) = f 2 (u ε 2 )(0 + , t) -ε∂ x ϕ 2 (u ε )(0 + , t) in (0, T ), u ε (0) = u ε 0 in R. (P ε )
This problems is of type ( 8)-( 28)-( 29), then the notion of bounded flux solution is a good frame to solve it. Definition 2.2 (solution to (P ε )) A function u ε is said to be a (bounded flux) solution to (P ε ) if it fulfills

1. u ε ∈ L ∞ (R × (0, T )), 0 ≤ u ε ≤ 1 a.e., 2. ∂ x ϕ i (u ε ) ∈ L ∞ (Ω i × (0, T )), 3. ∀ψ ∈ D(R × [0, T )), T 0 R u ε (x, t)∂ t ψ(x, t)dxdt + R u ε 0 (x)ψ(x, 0)dx + T 0 i=1,2 Ωi (f i (u ε )(x, t) -ε∂ x ϕ i (u ε )(x, t)) ∂ x ψ(x, t)dxdt = 0. ( 36
)
We can use Theorem 1.5 to claim that there exists a family (u ε ) ε of bounded flux solution to (P ε ) in the sense of Definition 2.2. Moreover, this family of solution fulfills, thanks to (31) and Lemma 2.1: for all ε > 0,

ε∂ x ϕ i (u ε ) ∞ ≤ (max i (Lip(ϕ i )) + max i f i L ∞ (0,1) ). ( 37
)
Since u ε 0 belongs to L 1 (R), the solution u ε is furthermore unique in C([0, T ], L 1 (R)) thanks to Theorem 1.7.

a L

2 ((0, T ); H 1 loc (Ω i ))-estimate
All this subsection is devoted to prove the following estimate.

Proposition 2.3 Let K be a compact subset of Ω i , and let u ε be a solution of (P ε ) in the sense of Definition 2.2, then there exists C depending only on f i , K, T (and not on ε) such that

√ ε ϕ i (u ε ) L 2 ((0,T );H 1 (K)) ≤ C.
Particularly, this implies that ε∂ x ϕ i (u ε ) → 0 a.e. in Ω i × (0, T ) as ε → 0.

Proof: We fix ε > 0. Since the functions ϕ -1 i are not Lipschitz continuous, the problem (P ε ) is not strictly parabolic, and the function u ε is not a strong solution. In order to get more regularity on the approximate solution, we regularize the problem by adding an additional viscosity 1/n (n ≥ 1), so that the so built approximate solution u ε n is regular enough to perform the calculation below. Let n > 1, and let ϕ i,n (u) = ϕ i (u) + u/n, and let u ε n be a bounded flux solution of (P ε ) with ϕ i,n instead of ϕ i . From (37), we know that

∂ x ϕ i,n (u ε n ) is uniformly bounded in L ∞ (Ω i × (0, T )), and since ϕ -1 i,n is a Lipschitz continuous function, one has ∂ x u ε n ∈ L ∞ (Ω i × (0, T )). The following weak formulation holds: ∀ψ ∈ D(Ω × [0, T )), T 0 R u ε n (x, t)∂ t ψ(x, t)dxdt + R u ε 0 (x)ψ(x, 0)dx + T 0 i=1,2 Ωi (f i (u ε n )(x, t) -ε∂ x ϕ i,n (u ε n )(x, t)) ∂ x ψ(x, t)dxdt = 0. ( 38
) Let (a, b) ⊂ Ω i . Let ζ ∈ D + ((a, b)), we deduce from (38) that ∂ t u ε n | u ε n ζ 2 = T 0 b a f i (u ε n )∂ x (u ε n ζ 2 )dxdt -ε T 0 b a ∂ x ϕ i,n (u ε n )∂ x (u ε n ζ 2 )dxdt, (39)
where • | • is the duality bracket between L 2 ((0, T ); H -1 (a, b)) and L 2 ((0, T );

H 1 0 (a, b)). Since ϕ i,n is a Lipschitz continuous function with ( λ i ∞ + 1/n) as Lipschitz constant, one has T 0 b a ∂ x ϕ i,n (u ε n )∂ x (u ε n )ζ 2 dxdt ≥ 1 λ i ∞ + 1/n T 0 b a (∂ x ϕ i,n (u ε n )) 2 ζ 2 dxdt. (40)
Let Φ i be a primitive of f i , then:

T 0 b a f i (u ε n )∂ x (u ε n ζ 2 )dxdt = T 0 U ∂ x Φ i (u ε n )ζ 2 dxdt + T 0 b a f i (u ε n )u ε n ∂ x ζ 2 dxdt = T 0 b a [f i (u ε n )u ε n -Φ i (u ε n )] ∂ x ζ 2 dxdt. (41)
Admit for the moment Lemma 2.4, stated and proven below. We deduce from ( 39), ( 40), ( 41) and Lemma 2.4 that

ε λ i ∞ + 1 T 0 b a (∂ x ϕ i,n (u ε n )) 2 ζ 2 dxdt ≤ 1 2 ζ 2 ∞ |b -a| + T 0 b a |u ε n (f i (u ε n ) -ε∂ x ϕ i,n (u ε n )) -Φ i (u ε n )| |∂ x ζ 2 |dxdt.
Using now the fact that u ε n is a bounded flux solution, we deduce from (31

) that u ε n (f i (u ε n ) -ε∂ x ϕ i,n (u ε n )) -Φ i (u ε n
) is uniformly bounded independently of ε and n, and so there exists C only depending on f i , |b -a|, u 0 and λ i such that:

ε T 0 b a (∂ x ϕ i,n (u ε n )) 2 ζ 2 dxdt ≤ C ∂ x ζ 2 L 1 ((0,T );M(R)) + ζ 2 ∞ .
This estimate still holds for ζ(x, t) = χ (a,b) (x), for all (a, b) ∈ Ω i 2 , so we obtain

ε T 0 b a (∂ x ϕ i,n (u ε n )) 2 dxdt ≤ C(2T + 1). (42)
Classical compactness arguments provide the convergence, up to a subsequence, of (u ε n ) n to a solution u ε of (P ε ) in L p (R × (0, T )), 1 ≤ p < +∞. This ensures particularly that, up to a subsequence, lim n→+∞ u ε n = u ε a.e. in R × (0, T ).

Taking the limit w.r.t. n in (42) yields

ε T 0 b a (∂ x ϕ i (u ε )) 2 dxdt ≤ C(2T + 1).
Lemma 2.4 Let u ε n be an approximate solution of (P ε ) with ϕ i,n instead of ϕ i , and let ψ ∈ D + ((a, b)), then

∂ t u ε n | u ε n ζ 2 ≥ - 1 2 ζ 2 ∞ |b -a|. Proof: Since (u ε n ζ) ∈ L 2 ((0, T ); H 1 0 (a, b)), and ∂ t (u ε n ζ) ∈ L 2 ((0, T ); H -1 (a, b))
, and so, up to a negligible set, (u

ε n ζ) ∈ C([0, T ]; L 2 (a, b)), and
∂ t u ε n | u ε n ζ 2 = ∂ t u ε n ζ | u ε n ζ = 1 2 b a u ε n (x, T ) 2 ζ 2 (x)dx - 1 2 b a u 0 (x) 2 ζ 2 (x)dx ≥ - 1 2 ζ 2 ∞ |b -a|.

the BV -estimates

In this section we suppose that u 0 ∈ BV (R). In order to avoid heavy notations which would not lead to a good comprehension of the problem, the following proof will be formal. To establish the following estimates in a rigorous frame, one can introduce of a thin layer (-η, η) on which the pressure variates smoothly to replace the interface, and add some additional viscosity to obtain smooth strong solutions to the problem. This regularization of the problem has been performed in [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF] and in [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF].

For a, b ∈ [0, 1], we denote by

F i (a, b) = sign(a -b)(f i (a) -f i (b)).
Lemma 2.5 There exists C depending only on f i , T , u 0 such that

|∂ t F i (u ε , κ)| M b (Ωi×(0,T )) ≤ C.
Proof: Suppose in the sequel that u ε is a strong solution, i.e.

∂ t u ε + ∂ x [f i (u ε ) -ε∂ x ϕ i (u ε )] = 0 holds point-wise in Ω i × (0, T ). Let h > 0, and let t ∈ (0, T -h). Comparing u ε (•, • + h) and u ε with (33) yieds R |u ε (x, t + h) -u ε (x, t)|dx ≤ R |u ε (x, h) -u ε 0 (x)|dx.
Dividing by h and letting h tend to 0, one can claim using the fact that u ε is supposed to be a strong solution

R |∂ t u ε (x, t)|dx = i=1,2 Ωi |∂ x [f i (u ε )(x, t) -ε∂ x ϕ i (u ε )(x, t)] |dx ≤ R |∂ t u ε 0 (x)|dx ≤ i=1,2 Ωi |∂ x [f i (u ε 0 )(x) -ε∂ x ϕ i (u ε 0 )(x)] |dx
Lemma 2.1 then ensures that there exists C not depending on ε such that

T 0 R |∂ t u ε (x, t)|dxdt = T 0 i=1,2 Ωi |∂ x [f i (u ε )(x, t) -ε∂ x ϕ i (u ε )(x, t)] |dxdt ≤ C. ( 43
)
Thanks to the regularity of f i , this particularly ensures that, if we denote by M b (Ω i × (0, T )) the set of the bounded Radon measure on Ω i × (0, T ), i.e. the dual space of C c (Ω i × [0, T ), R) with the uniform norm, we obtain: ∀κ ∈ [0, 1],

|∂ t F i (u ε , κ)| M b (Ωi×(0,T )) ≤ C f ′ i ∞ .
Lemma 2.6 There exists C depending only on u 0 , f i and T such that

∂ x F i (u ε , κ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ)| M b (Ωi×(0,T )) ≤ C.
Proof: It follows from the work of Carillo (see e.g. [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF]) that for all κ ∈ [0, 1], for all

ψ ∈ D + (Ω i × [0, T )), T 0 Ωi |u ε -κ|∂ t ψdxdt + Ωi |u ε -κ|ψ(0)dx T 0 Ωi F i (u ε , κ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ)| ∂ x ψdxdt ≥ 0. ( 44
)
Let η > 0, we denote by ω η (x) = (1 -|x|/η) + , and suppose now that ψ belongs to D + (Ω i × [0, T )), i.e. ψ does not vanish on the interface {x = 0}. Estimate [START_REF] Vasseur | Strong traces for solutions of multidimensional scalar conservation laws[END_REF] still holds when we consider

ψ η = ψ(1 -ω η ) as test function. T 0 Ωi |u ε -κ|(1 -ω η )∂ t ψdxdt + Ωi |u ε -κ|ψ(0)(1 -ω η )dx + T 0 Ωi F i (u ε , κ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ)| (1 -ω η )∂ x ψdxdt ≥ T 0 Ωi F i (u ε , κ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ)| ψ∂ x ω η dxdt. ( 45
)
The fact that the flux induced by u ε is uniformly bounded w.r.t. ε thanks to [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF] implies that there exists C depending only on u 0 , f i such that

F i (u ε , κ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ)| L ∞ (Ωi×(0,T )) ≤ C. ( 46
)
Then we obtain the following estimate on the right-hand-side in (45):

T 0 Ωi F i (u ε , κ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ)| ψ∂ x ω η dxdt ≤ CT ψ ∞ .
Letting η tend to 0 in inequality (45) gives:

∀ψ ∈ D + (Ω i × [0, T )), ∀κ ∈ [0, 1], T 0 Ωi |u ε -κ|∂ t ψdxdt + Ωi |u ε -κ|ψ(0)dx + T 0 Ωi F i (u ε , κ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ)| ∂ x ψdxdt ≥ -CT ψ ∞ (47)
We introduce now a monotonous function χ ψ ∈ D + (Ω i ) equal to 1 on the support of ψ(•, t) for all t ∈ [0, T ] (so that ∂ x χ ψ L 1 (Ωi) = 1), then ψ ∞ χ ψ ≥ ψ. Choosing ψ ∞ χ ψψ and ψ ∞ χ ψ + ψ as test function in (47) yields, using (46) once again

∂ t |u ε -κ| -∂ x F i (u ε , κ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ)| M b (Ωi×(0,T )) ≤ 2CT. ( 48
)
Thanks to [START_REF] Towers | A difference scheme for conservation laws with a discontinuous flux: the nonconvex case[END_REF], there exists C ′ depending only on u 0 , f i and T such that

|∂ t |u ε -κ|| M b (Ωi×(0,T )) ≤ C ′ . ( 49
)
Lemma 2.6 is so a consequence of ( 48) and (49).

Proposition 2.7 Let u 0 ∈ BV (R) and let K = [a, b] ⊂ Ω i . We introduce z ε K (x, t) defined on the whole space R 2 given by:

z ε K,κ (x, t) = F i (u ε , κ)(x, t) if (x, t) ∈ K × (0, T ), 0 otherwise.
There exists C depending only on u 0 , f i , T, K and a uniformly bounded function r K,κ , with r K,κ (ε) tends uniformly to 0 with respect to κ as ε → 0, such that, for all

(ξ, h) ∈ R 2 , R 2 z ε K,κ (x + ξ, t + h) -z ε K,κ (x, t) dxdt ≤ C(|ξ| + |h|) + r K,κ (ε)
.

Proof: Let h ∈ R, one has R 2 z ε K,κ (x, t + h) -z ε K,κ (x, t) dxdt ≤ ∂ t z ε K,κ M b (R 2 ) |h| ≤ |∂ t F i (u ε , κ)| M b (Ωi×(0,T )) + 2 F i (u ε , κ) L ∞ (T + b -a) |h|
It follows from Lemma 2.5 that there exists C 1 depending only on u 0 , f i , K, T, ϕ i such that

R 2 z ε K,κ (x, t + h) -z ε K,κ (x, t) dxdt ≤ C 1 |h|. (50)
We also define

q ε K,κ (x, t) = ε∂ x |ϕ i (u ε )(x, t) -ϕ i (κ)| if (x, t) ∈ K × (0, T ), 0 otherwise.
Proposition 2.3 ensures that q ε K,κ (x, t) converges to 0 almost everywhere in R 2 , and the estimate (46) ensures us that q ε K,κ (x, t) stays uniformly bounded in L ∞ (R 2 ) with respect to ε. Let ξ ∈ R, then we have:

R 2 z ε K,κ (x + ξ, t) -q ε K,κ (x + ξ, t) -z ε K,κ (x, t) -q ε K,κ (x, t) dxdt ≤ |∂ x (F i (u ε , κ) -ε∂ x |ϕ i (u ε )(x, t) -ϕ i (κ)|)| M b (Ωi×(0,T )) +2(T + b -a) ( F i (u ε , κ) -ε∂ x |ϕ i (u ε )(x, t) -ϕ i (κ)| L ∞ ) |ξ|. (51)
Using (46) and Lemma 2.6 in (51) yields that there exists C 2 depending only on u 0 , f i , K, T, ϕ i such that

R 2 z ε K,κ (x + ξ, t) -q ε K,κ (x + ξ, t) -z ε K,κ (x, t) -q ε K,κ (x, t) dxdt ≤ C 2 |ξ|.
This particularly ensures that:

R 2 z ε K,κ (x + ξ, t) -z ε K,κ (x, t) dxdt ≤ C 2 |ξ| + 2 q ε K,κ (x, t) L 1 (R 2 ) . (52)
By choosing C = max(C 1 , C 2 ), one we deduce from (50) and (52) that

R 2 z ε K,κ (x + ξ, t + h) -z ε K,κ (x, t) dxdt ≤ C(|h| + |ξ|) + 2 q ε K,κ (x, t) L 1 (R 2 ) .
We conclude the proof of Proposition 2.7 by checking that q ε K,κ (x, t) L 1 (R 2 ) converges uniformly to 0 with respect to κ as ε tends to 0.

some steady solutions

The spatial discontinuities of the saturation and capillary pressure allow us to consider Kružkov entropies |u -κ| only for non-negative test functions vanishing on the interface, i.e. in D + (R ⋆ × [0, T )). This is not enough to obtain the convergence of u ε toward an entropy solution u in the sense of Definition 1.2, and a relation has also to be derived at of the interface.

In order to deal with general test functions belonging to D + (R × [0, T )), we will so have to introduce some approximate Kružkov entropies | • -κ ε (x)|, where κ ε are steady solutions of (P ε ). Letting ε → 0, those steady solutions converge to piecewise constant function κj defined below. The functions | • -κ j (x)| correspond to the so called partially adapted entropies introduced by Audusse and Perthame [START_REF] Audusse | Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies[END_REF]. We will then be able to compare the limit u of approximate solutions u ε to this limit κj , and then to prove that u is the unique entropy solution.

The building of convenient κ ε strongly uses Assumption [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF]. It will be shown in [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] that if [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF] fails, non classical shocks can occur at the interface, and the limit ũ of the approximate solutions u ε is thus not an entropy solution.

Recall that q ≥ 0, P 1 < P 2 , and suppose that 0 < ε < P 2 -P 1 . Some simple adaptations can be done to cover the case q < 0. The transmission condition (29) can be summarized as follow: either u 1 = 1, or u 2 = 0.

We have to introduce the following sets:

E 1 = {κ 1 /∃κ 2 with f 1 (κ 1 ) = f 2 (κ 2 )},
3 Convergence toward the entropy solution

almost everywhere convergence

We state now a lemma which is an adaptation of Helly's selection theorem criterion, which states that if (v ε ) ε is a family of measurable functions on an open subset U of R k (k ≥ 1), uniformly bounded in L ∞ loc (U) and BV loc (U), one can extract a subfamily still denoted by (v ε ) ε that converges almost everywhere in U, and the limit belongs to BV loc (U). Lemma 3.1 Let U be an open subset of R k (k ≥ 1). Let (v ε ) ε be a family of functions uniformly bounded in L ∞ (U) with respect to ε. Let K be a compact subset of U. For ζ ∈ R k , we denote by

K ζ = { x ∈ K | x + ζ ∈ K }.
One assumes that there exists C > 0 depending only on K (and thus not on ε) and a function r fulfilling lim ε→0 r(ε) = 0 , such that for all ζ ∈ R k ,

K ζ |v ε (x + ζ) -v ε (x)|dx ≤ C|ζ| + r(ε). ( 58
)
Then, there exists a sequence (ε n ) n tending to 0, and v ∈ BV loc (U) such that lim n→+∞ v εn = v a.e. in U.

Proof: Let K be a compact subset of U. Estimate (58) says, roughly speaking, that v ε is almost a BV -function on K, i.e. as close to BV (K) as wanted, provided that ε is supposed to be small enough. So we will build a family (w ε ) ε of BV -functions, which will be close to the family (v ε ) ε , at least for small ε, and we will show that (w ε ) ε admits an adherence value v in BV (K) for the L 1 (K)topology, and that this v is also an adherence value of (v ε ) ε . Another proof for the a.e. convergence toward a function v can be derived directly from Kolmogorov compactness criterion (see e.g. [START_REF] Brézis | Analyse Fonctionnelle: Théorie et applications[END_REF]). But the advantage of the following method is that it provides directly some regularity on the limit v ∈ BV loc (U).

Let (ρ ε ) ε be a sequence of mollifiers, i.e. smooth, non negative and compactly supported functions with support included in the ball of center 0 and radius ε, and fulfilling ρ ε L 1 (R k ) = 1 for all ε > 0. We define the smooth functions

w ε = ṽε ⋆ ρ ε , where ṽε (x) = v ε (x) if x ∈ U and ṽε (x) = 0 if x ∈ U c .
Thanks to the regularity of w ε ,

K ζ |w ε (x + ζ) -w ε (x)|dx ≤ ∇w ε (L 1 (K)) k |ζ|. (59)
Suppose that ε < d(K, ∂U) (with the convention d(K, ∅) = ∞). Thanks to (58), we have also

K ζ |w ε (x + ζ) -w ε (x)|dx ≤ {K ζ +ε} |v ε (x + ζ) -v ε (x)|dx ≤ C|ζ| + r(ε), (60)
where

{K ζ + ε} = { x ∈ U | d(x, K) ≤ ε}.
Since r(ε) tends to 0 as ε → 0, this particularly ensures lim sup

ε→0 ∇w ε (L 1 (R k )) k ≤ C.
The family (w ε ) ε is thus bounded in BV (K ζ ) in the neighborhood of ε = 0, and thus, thanks to Helly's selection criterion, there exist v ∈ BV (K ζ ), and (ε n ) n tending to 0 such that w εn → v a.e. in K as n → ∞.

Furthermore, for all n ∈ N ⋆ ,

w εn -v εn L 1 (K) ≤ K B(0,εn)
|v εn (xy)v εn (x)|ρ εn (y)dydx

≤ Cε n + r(ε n ).
This ensures that v εn tends also almost everywhere toward v as n tends to +∞.

Lemma 3.1 will be used to prove the following convergence assertion.

Proposition 3.2 Suppose that u 0 ∈ BV (R), and let u ε be a solution to (P ε ). Up to an extraction, there exists u ∈ L ∞ (R × (0, T )), 0 ≤ u ≤ 1 a.e. such that u ε → u a.e. in R × (0, T ).

Furthermore, there exists u 1 , u 2 ∈ L ∞ (0, T ), such that

lim η→0 1 η T 0 0 -η |u(x, t) -u 1 (t)|dxdt = 0, lim η→0 1 η T 0 η 0 |u(x, t) -u 2 (t)|dxdt = 0.
Proof: Let K be a compact subset of Ω i . We define the function

H i : [0, 1] → R by H i (u) = 1 0 (F i (u, σ) -f i (σ)) dσ, (61)
so that, thanks to Proposition 2.7, there exists C depending on u 0 , f i , T, K, and a function r tending to 0 as ε tends to 0 such that for all ξ ∈ R, h ∈ (0, T ),

T -h 0 K |H i (u ε )(x + ξ, t + h) -H i (u ε )(x, t)| dxdt ≤ C(|ξ| + |h|) + r(ε). ( 62
)
An integration by parts in (61) yields: ∀u ∈ [0, 1]

H i (u) = - 1 0 (σ -b i )∂ σ (F i (u, σ) -f i (σ))dσ + (2b i -1)f i (u) = 2 u 0 (σ -b i )f ′ i (σ)dσ + (2b i -1)f i (u) (63)
where, thanks to [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF], f i is decreasing on [0, b i ] and increasing on [b i , 1]. Using Proposition 2.7 with κ = 0,

T -h 0 K |f i (u ε )(x + ξ, t + h) -f i (u ε )(x, t)| dxdt ≤ C(|ξ| + |h|) + r(ε), (64)
where C and r have been updated. Denoting by

A i (u) = u 0 (σ -b i )f ′ i (σ)dσ,
we obtain from (63) and ( 64)

T -h 0 K |A i (u ε )(x + ξ, t + h) -A i (u ε )(x, t)| dxdt ≤ C(|ξ| + |h|) + r(ε), (65)
with a new update for C and r. Thus we deduce from Proposition 3.1 that, up to an extraction, A i (u ε ) converges almost everywhere toward A i ∈ BV (K × (0, T )). It follows from ( 12) that A i is an increasing function, and so we obtain the convergence almost everywhere in K × (0, T ) of u ε toward a measurable function v.

Since for all ε > 0, 0 ≤ u ε ≤ 1, there exists u ∈ L ∞ (R × (0, T )), 0 ≤ u ≤ 1 such that u ε converges to u in the L ∞ (R × (0, T ))-⋆-weak sense, we thus have, up to an extraction, u ε → u a.e. in K × (0, T ).

(66) Since (66) holds for any compact subset K of Ω i , for i = 1, 2, we can claim that, up to an extraction, u ε → u a.e. in R × (0, T ).

Moreover, since A i (u) belongs to BV (R × (0, T )), we can claim that A i (u) admits a strong trace on {x = 0} × (0, T ) (see for instance [START_REF] Attouch | Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization[END_REF]). Using once again the fact that A -1 i is a continuous function, we can claim that u admits also a strong trace on each side of the interface.

convergence toward the entropy solution

In this section, it is proven that the limit value u for the family (u ε ) ε exhibited previously is the entropy solution described in Definition 1.2. Proposition 3.3 Let u 0 ∈ BV (R), 0 ≤ u 0 ≤ 1 a.e., and let (u ε 0 ) ε a family of approximation of u 0 given by Lemma 2.1, and let (u ε ) ε be the induced sequence of bounded flux solution of ( 8)-( 28)- [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF]. Then under Assumption ( 12)

lim ε→0 u ε = u in L p loc (R × [0, T ]), ∀p ∈ [1, ∞)
where u is the unique entropy solution to ( 10)-( 11) associated to initial data u 0 .

Proof: Thanks to Proposition 3.2, we can suppose that there exists u ∈ L ∞ (R × (0, T )) such that, up to a subsequence,

u ε → u in L 1 loc (R × (0, T )) as ε → 0, (67)
then f i (u ε ) → f i (u) in L 1 loc (Ω i × (0, T )) as ε → 0. Furthermore, thanks to Proposition 2.3, one has ε∂ x ϕ i (u ε ) → 0 in L 1 loc (Ω i × (0, T )) as ε → 0. ( 68
)
As a consequence, letting ε tend to 0 in [START_REF] Karlsen | L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients[END_REF] provides (recall that u ε 0 tends to u 0 in L 1 loc (R)) that u is a weak solution to (10)- [START_REF] Bear | Dynamic of Fluids in Porous Media[END_REF], i.e. that it satisfies [START_REF] Cancès | On the time continuity of entropy solutions[END_REF].

Since for ε > 0, (εϕ i )

-1 is a continuous function, it follows from the work of Carrillo [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF] that it fulfills the following entropy inequalities: ∀κ ∈

[0, 1], ∀ψ ∈ D + (Ω i × [0, T )) T 0 Ωi |u ε -κ|∂ t ψ dxdt + Ωi |u ε 0 -κ|ψ(•, 0) dx + T 0 Ωi (F i (u ε , κ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ)|) ∂ x ψ dxdt ≥ 0. (69)
The assertion (67) particularly yields that for i = 1, 2 and for all κ ∈ [0, 1],

F i (u ε , κ) → F i (u, κ) in L 1 loc (R × (0, T )) as ε → 0. ( 70
)
On the other hand, it follows from (68) that

ε∂ x |ϕ i (u ε ) -ϕ i (κ)| → 0 in L 1 loc (R × (0, T )) as ε → 0. (71)
Taking (67), (70) and (71) into account in (69) provides the inequality [START_REF] Cancès | Two-phase flows involving capillary barriers in heterogeneous porous media[END_REF].

The last point remaining to check is that the interface entropy condition (18) holds. Let (κ ε ) ε be a family of steady states to (P ε ) converging in L 1 loc (R) as ε tends to 0 to κ opt defined in ( 16)- [START_REF] Cancès | Finite volume scheme for two-phase flow in heterogeneous porous media involving capillary pressure discontinuities[END_REF]. Recall that such a family exists thanks to Proposition 2.9. Then for all fixed ε > 0, κ ε is a steady bounded flux solution. Hence, it follows from (32) that

T 0 R |u ε -κ ε |∂ t ψ dxdt + R |u ε 0 -κ ε | ψ(•, 0) dxdt + T 0 i=1,2 Ωi (F i (u ε , κ ε ) -ε∂ x |ϕ i (u ε ) -ϕ i (κ ε )|) ∂ x ψ dxdt ≥ 0. ( 72
)
It follows from Proposition 2.3 that

ε∂ x |ϕ i (u ε ) -ϕ i (κ ε )| → 0 in L 1 loc (Ω i × (0, T )) as ε → 0.
Letting ε tend to 0 in (72) provides directly the fourth point [START_REF] Cancès | On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types[END_REF] the fourth point in Definition 1.2.

We state now the main result, which is in fact the extension of Proposition 3.3 to a larger class of initial data. Theorem 3.4 (main result) Let u 0 ∈ L ∞ (R), 0 ≤ u 0 ≤ 1, and let (u ε 0 ) ε a family of approximation of u 0 given by Lemma 2.1, and let (u ε ) ε be the induced sequence of bounded flux solution of (P ε ). Then under Assumption [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF],

lim ε→0 u ε = u in L p loc (R × [0, T ]), ∀p ∈ [1, ∞)
where u is the unique entropy solution to ( 10)-( 11) associated to initial data u 0 .

Proof: Let u 0 ∈ L ∞ (R), 0 ≤ u 0 ≤ 1, and let ν > 0. There exists u 0,ν in BV (R), 0 ≤ u 0,ν ≤ 1 such that for all R > 0, u 0,νu 0 L 1 (-R,R) ≤ C(R)ν.

(73)

If one regularizes u 0,ν into u ε 0,ν using Lemma 2.1, and if one denotes by u ε ν the associated unique bounded flux solution, we have seen that u ε ν converges almost everywhere to u ν as ε tends to 0. As previously, we denote by u ε 0 the regularization of u 0 obtained via Lemma 2.1, and u ε the unique associated bounded flux solution.

Let R > 0, then we have

T 0 R -R |u ε -u|dxdt ≤ T 0 R -R |u ε -u ε ν |dxdt + T 0 R -R |u ε ν -u ν |dxdt + T 0 R -R |u ν -u|dxdt. (74)
The contraction principle stated in Theorem 1.

Figure 3 :

 3 Figure 3: capillary pressures graphs πε i .

T 0 i=1, 2 T 0 i=1, 2

 22 ν (x, t)u(x, t)|dxdt ≤ T R+MT -R-MT |u 0,ν (x)u 0 (x)|dx (75)whereM ≥ max i Lip(f i ). We denote by ζ(x, t) = min(1, (R + 1 + M (Tt) -|x|) + ). One has ζ = 1 on (-R, R), ζ ≥ 0 and ζ ∈ L 1 (R) thus Proposition 1.6 that for all ψ ∈ W 1,1 (R × (0, T )) with ψ ≥ 0 a.e. and ψ(•, T ) = 0, one has T 0 R |u ε (x, t)u ε ν (x, t)|∂ t ψ(x, t)dxdt + R |u ε 0 (x)u ε 0,ν (x)|ψ(x, 0)dx + Ωi sign(u ε (x, t)u ε ν (x, t)) (f i (u ε )(x, t)f i (u ε ν )(x, t)) ∂ x ψ(x, t)dxdt -Ωi ε∂ x |ϕ i (u ε )(x, t)ϕ i (u ε ν )(x, t)|∂ x ψ(x, t)dxdt ≥ 0. (77)We denote by Λ1 (t) = [-R -1 -M (Tt), -R -M (Tt)], Λ 2 (t) = [R + M (Tt), R + 1 + M (Tt)], ϑ(x) = 1 if x < 0

i=1, 2 i=1, 2 i=1, 2

 222 Λi(t) εϑ(x)∂ x |ϕ i (u ε )(x, t)ϕ i (u ε ν (x, t))|dxdt. (78)We deduce, using (75) (76) and (78) in (74), thatT 0 R -R |u ε (x, t)u(x, t)|dxdt ≤ T R |u ε 0 (x)u ε 0,ν (x)|ζ(x, 0)dx -Λi(t) εϑ(x)∂ x |ϕ i (u ε )(x, t)ϕ i (u ε ν (x, t))|dxdt +T R+MT -R-MT |u 0,ν (x)u 0 (x)|dx + t)u ν (x, t)|dxdt.(79)We can now let ε tend to 0. Thanks to Proposition 2.3, we can claim that lim Λi(t)εϑ(x)∂ x |ϕ i (u ε )(x, t)ϕ i (u ε ν (x, t))|dxdt = 0.We also deduce from Proposition 3t)u ν (x, t)|dxdt = 0.Since ζ(x, 0) is compactly supported, it follows from Lemma 2.1 and the dominated convergence theorem thatlim ε→0 R |u ε 0 (x)u ε 0,ν (x)|ζ(x, 0)dx = R |u 0 (x)u 0,ν (x)|ζ(x, 0)dx. ε (x, t)u(x, t)|dxdt ≤ T R+MT -R-MT |u 0,ν (x)u 0 (x)|dx + R |u 0 (x)u 0,ν (x)|ζ(x, 0)dx. (80)The inequality (80) holds for any ν > 0, and letting ν tend to 0 leads to lim ε (x, t)u(x, t)|dxdt = 0, then u ε tends to u in L 1 loc (R × [0, T ]) as ε → 0. Since 0 ≤ u ε ≤ 1 a.e., Hölder inequality gives the convergence in L p loc (R × [0, T ]) for all finite p.

  Λi(t) (x). Taking ψ(x, t) = (Tt)ζ(x, t) in (77) yields : Λi(t) sign(u ε (x, t)u ε ν (x, t)) (f i (u ε )(x, t)f i (u ε ν )(x, t)) dxdt Λi(t) εϑ(x)∂ x |ϕ i (u ε )(x, t)ϕ i (u ε ν (x, t))|dxdt ≥ 0.Since M ≥ max i Lip(f i ), one has

							,
							-1 if x > 0,
	so that
		∂ x ζ(x, t) = χ -i=1,2 ϑ(x)χ Λi (t) (x), ∂ t ζ(x, t) = -M i=1,2 T 0 R |u ε (x, t) -u ε ν (x, t)|ζ(x, t)dxdt + T R |u ε 0 (x) -u ε 0,ν (x)|ζ(x, 0)dx
			-M	0	T	(T -t))	i=1,2 Λi(t)	|u ε (x, t) -u ε ν (x, t)|dxdt
		T
	+	0 i=1,2 T (T -t) T i=1,2 -0 (T -t)
	0	(T -t)	i=1,2 Λi(t)

sign(u ε (x, t)u ε ν (x, t)) (f i (u ε )(x, t)f i (u ε ν )(x, t)) dxdt ≤ M T 0 (Tt) i=1,2 Λi(t) |u ε (x, t)u ε ν (x, t)|dxdt

and thus

T 0 R |u ε (x, t)u ε ν (x, t)|ζ(x, t)dxdt ≤ T

It follows from Assumption [START_REF] Bertsch | Analysis of oil trapping in porous media flow[END_REF] that either E 1 = [0, 1], or E 2 = [0, 1], and so we are ensured that κ ∈ [0, 1] belongs either to E 1 , or to E 2 (or of course to both). Check also that for all z ∈ (0, q], it follows from ( 12) that there exists a unique κ i (z) such that f i (κ i (z)) = z. On the contrary, if z ∈ E i with z ≤ 0, z has two antecedents through f i . Let κ belong to E j , one denotes

and

Definition 2.8 (reachable steady state) A function κ(x) is said to be a reachable steady state if there exists a steady entropy solution κ ε (x) to the problem (P ε ) in the sense of Definition 2.2 converging to κ(x) in L 1 (R) as ε tends to 0.

This section is devoted to establish the following proposition, that exhibits all the reachable steady states. In particular, all the steady states that are not undercompressible are reachable.

Proposition 2.9 For all κ ∈ E j , there exists a family of steady solutions (κ ε ) ε to the problem

where κj (x) can be chosen between:

In particular, κ opt defined in ( 16)-( 17) is a reachable steady state.

Proof: Let κ ∈ E j .

• If κ = 1, the three limits κj are identically equal to 1, which is a steady solution fulfilling Proposition 2.9.

• We suppose now that κ < 1, and f j (κ) > 0. Even in this case, the three reachable limit are the same. Thus we only have to build one sequence of converging steady solutions. Let y be a solution of:

The solution y(x) converges to κ j 1 as x → +∞. The family (κ ε ) ε defined by: ∀ε

fulfills so the conclusion of Proposition 2.9.

• Suppose now f j (κ) = 0. The solutions κ ε (x) built with (56)-(57) converges toward the two reachable steady states i) and ii). One can also choose κ ε (x) = 0, which is of course a steady solution.

• It remains the case f j (κ) < 0. The solutions κ ε (x) built with (56)-(57) still converges toward the two reachable steady states i) and ii).

Let w be a solution of    d dx ϕ 2 (w) = f 2 (w)f j (κ), for x > 0, w(0) = 0. and κ ε (x) = κ j 1 if x < 0, w(x/ε) if x > 0, then κ ε converges toward the third reachable steady state.


domain: math nlin

text:
Ramen Ghosh

Mahmoud Tahmasebi

Senior Member, IEEE Marion Mcafee

Dynamic Mode Decomposition and Koopman Operator for Iterated Function Systems

Keywords: Koopman Operator, Dynamic Mode Decomposition, Data Driven Dynamical System, Iterated Function System, Non-linear Stochastic Dynamics

The article aims to demonstrate the effectiveness of using the Koopman operator and dynamic mode decomposition (DMD) for iterated function systems (IFS). Specifically, we show how these tools can be used to analyze and predict the behaviour of stochastic nonlinear dynamical systems represented by discrete-time Markov chains on a compact state space. In particular, we focus on an ergodic nonlinear IFS, which has not been studied with these approaches before. Our paper presents the first application of dynamic mode decomposition to this type of system.

I. INTRODUCTION

This article introduces dynamic mode decomposition (DMD), and Koopman operator theory for iterated function systems (IFS) [START_REF] Ghosh | Iterated function systems: A comprehensive survey[END_REF]. IFS is a category of stochastic dynamical systems modelled as a discrete-time Markov chain on a general state space. Under some regularity assumption, it admits an invariant measure through the associated Markov operator [START_REF] Ghosh | Iterated function systems: A comprehensive survey[END_REF]. The governing equations for several complex systems and processes cannot be derived from first principles, and their resulting models are too complicated to be of practical use or both. Using a state-space model for applications such as control and prediction is troublesome due to the difficulty or impossibility of measuring the actual state of the system when applied to such a system, which further complicates the issue. As an alternative, just a subset of measurements, also known as observables, will be exposed. For such a complicated system, the Koopman operator is one of the modelling tools that may be utilized [START_REF] Budišic | The koopman operator in systems and control: Concepts, methodologies, and applications[END_REF]. Since the Koopman operator represents a system in a high-dimensional linear space, spectral techniques may be used to investigate the system. Koopman Mode Decomposition, which decomposes the system based on the eigenfunctions of the Koopman operator, was initially presented in [START_REF] Koopman | Hamiltonian systems and transformation in Hilbert space[END_REF]. Since then, the Koopman Operator has gained popularity for its utility as a data-driven method through various forms of Koopman Mode Decomposition [START_REF] Mezić | Spectral properties of dynamical systems, model reduction and decompositions[END_REF]. Dynamic Mode Decomposition (DMD) is a well-established ATU Sligo supported all the authors R. Ghosh is with the Centre for Mathematical Modelling and Intelligent Systems for Health and Environment (MISHE), Atlantic Technological University, ATU Sligo, Ash Lane, F91 YW50 Sligo, Ireland.

ramen.ghosh@atu.ie marion.mcafee@atu.ie data-driven technique first described in [START_REF] Schmid | Dynamic mode decomposition of numerical and experimental data[END_REF]. In [START_REF] Rowley | Spectral analysis of nonlinear flows[END_REF], it was shown that the dynamic mode decomposition is related to the Koopman mode decomposition (KMD). The purpose of DMD methods is to locate a matrix close to approximating a finite approximation of the Koopman operator [START_REF] Mezić | On numerical approximations of the Koopman operator[END_REF]. DMD may be constructed in various forms and is suitable for usage in various contexts. Despite their widespread use, many DMD methods have a significant shortcoming: they are susceptible to failure if the data in question includes noise or unpredictability. Koopman operator technique has been used in random dynamical systems, especially in homogeneous Markov processes, stochastic differential equations driven by Gaussian white noise and stochastic systems generated by i.i.d random maps, see [START_REF] Mezić | Spectral properties of dynamical systems, model reduction and decompositions[END_REF][START_REF] Wanner | Robust approximation of the stochastic Koopman operator[END_REF]. To the author's knowledge, this work is the first to formally introduce the dynamic mode decomposition and Koopman operator for iterated function systems and guarantees the approximation of ergodic non-linear IFS.

II. MATHEMATICAL PRELIMINARIES

Definition 1 (Markov Chain on Metric Space). Let (Ω, E , P) be a probability space. A Markov chain on a metric space X is a sequence of X valued random variables {X k } k≥0 whose interdependence satisfies the Markov property, i.e, for all A ∈ B(X )

P [X k+1 ∈ A | X 0 , X 1 , . . . , X k ] = P [X k+1 ∈ A | X k = x] . (1)
Definition 2 (IFS with state-independent probability with finite number of maps). Let (X , ρ) be a compact metric space. Let {w i } m i=1 be continuous self-transformations on X and µ be a probability measure on [m] := {1, 2, . . . , m}. Let σ 0 , σ 1 , . . . be i.i.d discrete random variables taking values in [m] and

P (σ i = j) = µ (j) j ∈ [m] and i = 0, 1, 2, . . . , m. (2)
An IFS with state-independent probability is a Markov process that is realized by the recursion

X k+1 := w σ k (X k ), k = 0, 1, 2, . . . , (3)
where X 0 ∼ ν ∈ M p (X ) is assumed to be independent of σ k ∀k.

We describe below how the iteration progress in Algorithm 1.

A.

X k+1 = w σ k (X k )
Step 3: Return {X k } k∈N current observables to the expected value of future observables

Kg(x) = ν(x, dy)g(y) = E [g(X k+1 ) | X k = x] = E [(g • w ω k ) (X k ) | X k = x] (4)
We will denote the one-step Koopman evolution for a discretetime map as K 1 = K. The operator K τ is also called the transition operator for the associated Markov process. The Definition 3 will be useful once the family of Koopman operators {K τ } satisfy the following semigroup property:

K τ1+τ2 g = (K τ2 • K τ1 ) g for τ 1 , τ 2 ≥ 0.
For deterministic Koopman operator, this is true, provided the system is autonomous and solutions exist and are unique since, in this case, the flow forms a semigroup. For the stochastic Koopman family of operators on an i.i.d. system, the semigroup property is guaranteed by the independence of X τ1 (ω) and X τ2 (ω) since

K τ1+τ2 g(x) = E P [g (X τ1+τ2 (w))] = E P [(g (X τ2 )(X τ1 )(w))] = E P [K τ2 g (X τ1 (w))] = K τ1 K τ2 g.
We wish to find a matrix M which represents K on a finitedimensional subspace. We use dynamic mode decomposition [START_REF] Schmid | Dynamic mode decomposition of numerical and experimental data[END_REF] to find that invariant subspace and the matrix M as presented in the Algorithm 2.

The accuracy of the approximation M depends on the observable g 1 , g 2 , . . . , g n spanning an invariant subspace V of K.

Assume that rank M 1 = n;

(5)

X 0 = 1 n M 1 M ⋆ 1 ; (6)
X 1 = 1 n M 2 M ⋆ 1 . (7)
Then we have,

M = M 2 M † 1 = (M 2 M ⋆ 1 ) (M 1 M ⋆ 1 ) -1 = X 1 X -1 0 (8)
More precisely

[X 0 ] ij = 1 n n-1 t=0 g i (t)g j (t) (9)
But due to the assumption that the IFS possesses a unique invariant measure, we have,

lim n→∞ 1 n n-1 t=0 g i (t)g j (t) → ⟨g i , g j ⟩. (10)
Similarly,

[X 1 ] ij = 1 n n-1 t=0 g i (t + 1)g j (t), (11)
and,

lim n→∞ 1 n n-1 t=0 g i (t + 1)g j (t) → ⟨Kg i , g j ⟩. ( 12
)
Algorithm 2: Koopman-DMD Algorithm Input: Let X 0 , X 1 , X 2 , . . . , X n be a trajectory of the IFS system, and

ĝ =      g 1 g 2 . . . g d      : R → R d (13)
be a vector-valued observable, let

ĝ(k) = ĝ(X k ) (14)
be the vector containing the data from a sample. Output: the eigenvalues, left and right eigenvectors (λ i , u i , v i ), i = 1, 2, . . . , d of M. Then the dynamic eigenvalues are λ i , the dynamic modes are v i , and the numerical eigenfunctions.

Step 1: construct the matrices

M 1 = ĝ(0) ĝ(1) • • • ĝ(n -1) n×n ( 15
)
M 2 = ĝ(1) ĝ(2) • • • ĝ(n) n×n (16)
Step 2: Form the matrix

M = M 2 M † 1 ( 17
)
where M † 1 is the Moore-Penrose pseudoinverse.

Step 3: Compute the eigenvalues, left and right eigenvectors (λ i , u i , v i ), i = 1, 2, . . . , d of M. Then the dynamic eigenvalues are λ i , the dynamic modes are v i , and the numerical eigenfunctions are given by

ϕ i = u ⊤ i M 1 (18)
In the following result, we show the convergence of dynamic eigenvalues to the eigenvalues of the Koopman operator K under some regularity assumptions.

Theorem 1. Let ω 0 , ω 1 , ω 2 , . . . are i.i.d discrete random variables taking values in [m]. Let ν * be the unique invariant measure for the IFS generated by the self-maps {w 1 , w 2 , . . . , w m }. Let V be the K-invariant subspace of the linear space formed by the span of the observables {g 1 , g 2 , . . . , g n }. Let λ j,n be the dynamic eigenvalues and v j,n be the dynamic modes produced by Algorithm 2 generated by the trajectory X 0 , X 1 , . . . , X n .

Then the dynamic eigenvalues converge to the eigenvalues of K restricted to V almost surely.

III. A NUMERICAL ILLUSTRATION

Iterated function system generated by logistic maps and their asymptotic properties has been studied considerably and are still active research area. We consider an iterated function system generated by the following logistic maps on [0, 1] and probabilities as follows w 1 (x) = 0.8x(1 -x) with probability 1 4 ;

w 2 (x) = 2.8x(1 -x) with probability 1 2 ;

w 3 (x) = 3.8x(1 -x) with probability 1 4 .
Using the Algorithm 3, we approximated the above non-linear IFS and Figure 1234shows the numerical illustration. The code is available in https://github.com/M2219/ifsdmd.

Algorithm 3: IFS-DMD Algorithm

Input: Initial value X 0 , 3 functions {w k } 3 k=1 , probability measure µ over [START_REF] Koopman | Hamiltonian systems and transformation in Hilbert space[END_REF]; Output: A sequence of random variables X k , k = 100 with Markov property, dynamic mode spectrum λ i and associated dynamic modes.

Step 1: Set X 0

Step 2: Repeat the following for 100 iterations as Algorithm 1;

Step 3: Return {X k } k∈N as data snapshots.

Step 4:

M 1 = X 1 X 2 . . . X k-1 , M 2 = X 2 X 3 . . . X k ;
Step 5: compute SVD of M 1 , M 1 = U ΣV * , where U k×r , V k×r , are unitary, Σ r×r diagonal with singular values, r is the dimension of interest;

Step 6: define the matrix à = U * M 2 V Σ -1 ;

Step 7: compute eigenvalues and eigenvectors of Ã, Ãy = λy;

Step 8: the DMD mode corresponding to the DMD eigenvalue λ is given by Φ = U y. Error between actual values from IFS and predicted value via DMD

IV. CONCLUSION AND FUTURE WORK

In this article, the Koopman operator is defined for IFS, and an ergodic non-linear IFS has been approximated using Error between actual values from IFS and predicted value via DMD the DMD algorithm; the Koopman operator, dynamic mode decomposition (DMD), and data-driven dynamical systems are active fields of study in mathematics, engineering, and physics. Future research aims to develop efficient algorithms for approximating the Koopman operator, understand its relationship with other linear operators, and create more robust DMD variants. There are connections between Koopman-DMD and neural networks, as both offer data-driven approaches to analyze and model dynamics, extract features, and reduce dimensionality. Koopman-DMD captures nonlinear system dynamics without explicit assumptions, similar to how neural networks model complex nonlinear relationships. Both methods involve feature extraction and dimensionality reduction. Efforts have been made to combine Koopman-DMD with neural networks, leveraging their strengths to capture linear and nonlinear dynamics. Exploring these connections can lead to novel techniques for improved modelling and analysis of complex systems.

Figure 1 .

 1 Figure 1. Approximation of IFS via DMD algorithm, initial value X 0 = 0.2

Figure 2 .

 2 Figure 2.Error between actual values from IFS and predicted value via DMD

Figure 3 .

 3 Figure 3. Approximation of IFS via DMD algorithm, initial value X 0 = 0.5

Figure 4 .

 4 Figure 4.Error between actual values from IFS and predicted value via DMD

  Tahmasebi, the Centre for Mathematical Modelling and Intelligent Systems for Health and Environment (MISHE), Atlantic Technological University, ATU Sligo, Ash Lane, F91 YW50 Sligo, Ireland.

	mahmoud.tahmasebi@research.atu.ie
	M. McAfee the Centre for Mathematical Modelling and Intelli-
	gent Systems for Health and Environment (MISHE), Atlantic Techno-
	logical University, ATU Sligo, Ash Lane, F91 YW50 Sligo, Ireland.

M.

  Initial value X 0 , m functions {w k } m k=1 , probability measure µ over [m] Output: A sequence of random variables X k with Markov property (see Definition 1) Step 1: Set k = 0 Step 2: Repeat the following: Step 2.1: Draw (σ n ) from µ Step 2.2:

Koopman Operator and Dynamic Mode Decomposition for IFS Definition 3. Let O be the space of observables, the stochastic Koopman operator K for IFS as defined in equation (3), maps Algorithm 1: State-Independent IFS Iteration Input: