Peter Knabner
email: knabner@am.uni-erlangen.de

Jean E Roberts
email: jean.roberts@inria.fr

Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy-Forchheimer flow in the fracture

Keywords: flow in porous media, fractures, Darcy-Forchheimer flow, solvability, regularization, monotone operators écoulement en milieu poreux, fractures, écoulement de Darcy-Forchheimer, solvabilité, régularisation, opérateurs monotones

We consider a model for flow in a porous medium with a fracture in which the flow in the fracture is governed by the Darcy-Forchheimer law while that in the surrounding matrix is governed by Darcy's law. We give an appropriate mixed, variational formulation and show existence and uniqueness of the solution. To show existence we give an analogous formulation for the model in which the Darcy-Forchheimer law is the governing equation throughout the domain. We show existence and uniqueness of the solution and show that the solution for the model with Darcy's law in the matrix is the weak limit of solutions of the model with the Darcy-Forchheimer law in the entire domain when the Forchheimer coefficient in the matrix tends toward zero.

Introduction

Numerical modeling of fluid flow in a porous medium, even single-phase, incompressible fluid flow, is complicated because the permeability coefficient characterizing the medium may vary over several orders of magnitude within a region quite small in comparison to the dimensions of the domain. This is in particular the case when fractures are present in the medium. Fractures have at least one dimension that is very small, much smaller than a reasonable discretization parameter given the size of the domain, but are much more permeable (or possibly, due to crystalization , much less permeable) than the surrounding medium. They thus have a very significant influence on the fluid flow but adapting a standard finite element or finite volume mesh to handle flow in the fractures poses obvious problems. Many models have been developed to study fluid flow in porous media with fractures. Models may employ a continuum representation of fractures as in the double porosity models derived by homogenization or they may be discrete fracture models. Among the discrete fracture models are models of discrete fracture networks in which only the flow in the fractures is considered. The more complex discrete fracture models couple flow in the fractures or in fracture networks with flow in the surrounding medium. This later type model is the type considered here.

An alternative to the possibility of using a very fine grid in the fracture and a necessarily much coarser grid away from the fracture is the possibility of treating the fracture as an (n -1)-dimensional hypersurface in the n-dimensional porous medium. This is the idea that was developed in [START_REF] Alboin | Domain decomposition for flow in porous media with fractures[END_REF] for highly permeable fractures and in [START_REF] Martin | Modeling fractures and barriers as interfaces for flow in porous media[END_REF] for fractures that may be highly permeable or nearly impermeable. Similar models have also been studied in [START_REF] Faille | A new fault model in geological basin modelling, application to finite volume scheme and domain decomposition methods[END_REF][START_REF] Angot | Asymptotic and numerical modelling of flows in fractured porous media[END_REF][START_REF] Showalter | The narrow fracture approximation by channeled flow[END_REF]. These articles were all concerned with the case of single-phase, incompressible flow governed by Darcy's law and the law of mass conservation. In [START_REF] Frih | Modeling fractures as interfaces: a model for forchheimer fractures[END_REF] a model was derived in which Darcy's law was replaced by the Darcy-Forchheimer law for the flow in the fracture, while Darcy's law was maintained for flow in the rest of the medium. The model was approximated numerically with mixed finite elements and some numerical experiments were carried out.

The use of the linear Darcy law as the constitutive law for fluid flow in porous media, together with the continuity equation, is well established. For medium-ranged velocities it fits well with experiments [8, Chapter 5] and can be derived rigorously (on simpler periodic media) by homogenization starting from Stokes's equation [START_REF] Tartar | Convergence of the homogenization process[END_REF][START_REF] Allaire | Homogenization of the stokes flow in a connected porous medium[END_REF][START_REF] Allaire | One-phase newtonian flow[END_REF]. However, for high velocities experiments show deviations which indicate the need for a nonlinear correction term, [START_REF] Forchheimer | Wasserbewegung durch Boden[END_REF], [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF]Chapter 5]. The simplest proposed is a term quadratic in velocity, the Forchheimer correction. In fractured media, the permeability (or hydraulic conductivity) in the fractures is generally much greater than in the surrounding medium so that the total flow process in the limit is dominated by the fracture flow. This indicates that a modeling different from Darcy's model is necessary and leads us to investigate models combining Darcy and Darcy-Forchheimer flow.

In this paper we consider existence and uniqueness of the solution of corresponding stationery problems. Assumptions on coefficients should be weak so as not to prevent the use of the results in more complex real life situations. Therefore we aim at weak solutions of an appropriate variational formulation, where we prefer a mixed variational formulation, due to the structure of the problems and a further use of mixed finite element techniques. For a simple d-dimensional domain Ω and for the linear Darcy flow the results are well known (c.f. [START_REF] Brezzi | On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers[END_REF]) and rely on the coercivity of the operator A coming from Darcy's equation on the kernel of the divergence operator B coming from the continuity equation and the functional setting in H(div, Ω) for the flux and L 2 (Ω) for the pressure. For the nonlinear Darcy-Forchheimer flow the functional setting has to be changed to W 3 (div, Ω) (see Appendix A.1) for the flux so that A will remain (strictly) monotone and to L 3 2 (Ω) for the pressure. This makes it possible to extend the reasoning for the linear case to the homogeneous Darcy-Forchheimer problem and via regularization, using the Browder-Minty theorem for maximal monotone operators, also to prove unique existence in the inhomogeneous case. This work is carried out in the thesis [START_REF] Summ | Lösbarkeit un Diskretisierung der gemischten Formulierung für Darcy-Frochheimer-Fluss in porösen Medien[END_REF]; see also [START_REF] Knabner | Solvability of the mixed formulation for darcy-forchheimer flow in porous media[END_REF][START_REF] Fabrie | Regularity of the solution of darcy-forchheimer's equation[END_REF][START_REF] Amirat | Ecoulements en milieu poreux n'obeissant pas a la loi de darcy[END_REF] for related results. Here we extend this reasoning to the situation of two subdomains of the matrix separated by a fracture with various choices of the constitutive laws in domains and fractures. One would expect that the Darcy-Forchheimer law is more accurate than Darcy's law (and this will be partially made rigorous); therefore, (and for technical reasons) we start with a model having the Darcy-Forchheimer law throughout the domain (though with strongly variable coefficients) and extend the aforementioned reasoning for existence and uniqueness to this case, (Section 2). By its derivation, Darcy's law should be a limit case of the Darcy-Forchheimer law. This is made precise in Section 4 by showing that the solution of the Darcy model is a weak limit of solutions of the Darcy-Forchheimer model with the Forchheimer coefficient (multiplying the nonlinear term) going to 0. This was shown earlier in [START_REF] Amirat | Ecoulements en milieu poreux n'obeissant pas a la loi de darcy[END_REF] under slightly different assumptions, but we include it here for completeness. This opens up the possibility of treating various combinations of the constitutive laws. As rapid transport is more likely to take place in the fractures, we explicitly treat the case of Darcy's law in the matrix and the Darcy-Forchheimer law in fractures. By using the full Darcy-Forchheimer model as a regularization and deriving corresponding a priori bounds we can show the existence of a solution as a weak limit of the regularizing full models (Section 4). Uniqueness again follows as in all the other cases from the monotone structure of the problem (see Appendix A.2). Technical difficulties stem from the different functional settings for the linear case and the nonlinear case. It may be envisaged to extend this basic procedure in various directions. An obvious extension is to the case of a finite number of fractures and subdomains, as long as the fractures do not intersect, which is quite restrictive. But also a general case where d-dimensional subdomains are separated by (d-1)-dimensional fractures, which are separated by (d-2)-dimensional fractures, etc. may be attacked with this approach. Another extension could be the investigation of other nonlinear correction terms to Darcy's law: cf. [START_REF] Balhoff | Polynomial filtration laws for low reynolds number flows through porous media[END_REF].

The outline of this article is as follows: in Section 1 the model problem with Darcy flow in the matrix and Darcy-Forchheimer flow in the fracture as well as the problem with Darcy-Forchheimer flow in the matrix and in the fracture will be given. In Section 2 the existence and uniqueness of the solution to the problem with Darcy-Forchheimer flow in the matrix and in the fracture will be shown. Section 3 is concerned with showing that in a simple domain (one without a fracture) that the solution of the Darcy problem is obtained as the limit of the Darcy-Forchheimer problem when the Forchheimer coefficient tends to zero. Then Section 4 takes up the problem for extending the result of Section 3 to the case of a domain with a fracture in which it is shown that the problem with Darcy-Forchheimer flow in the fracture but with Darcy flow in the matrix is obtained as the limit of the problem with Darcy-Forchheimer flow everywhere as the Forchheimer coefficient in the matrix tends to zero.

1 Formulation of the problems Let Ω be a bounded domain in R d with boundary Γ, and let γ ⊂ Ω be a (d-1)-dimensional surface that separates Ω into two subdomains:

Ω ⊂ R d , Ω = Ω 1 ∪ γ ∪ Ω 2 , γ = (Ω 1 ∩ Ω 2 )
∩ Ω, Γ = ∂Ω, and Γ i = Γ ∩ ∂Ω i . We suppose for simplicity that γ is a subset of a hyperplane; i. e. that γ is flat. Taking the stratification of natural porous media into account this seems to be a feasible assumption covering a variety of situations. The extension to the case that γ is a smooth surface should not pose any major problems but would be considerably more complex as the curvature tensor would enter into the definitions of the tangential gradient and the tangential divergence. We consider the following problem, which was derived in [START_REF] Frih | Un modèle darcy-frochheimer pour un écoulement dans un milieu poreux fracturé[END_REF][START_REF] Frih | Modeling fractures as interfaces: a model for forchheimer fractures[END_REF]:

α i u i + ∇p i = 0 in Ω i div u i = q i in Ω i p i = p d,i on Γ i (1)
together with

(α γ + β γ |u γ |)u γ + ∇p γ = 0 on γ div u γ = q γ + [u 1 • n -u 2 • n] on γ p γ = p d,γ on ∂γ (2)
Inria and the interface condition

p i = p γ + (-1) i+1 κ(ξu i • n + ξu i+1 • n), on γ, i = 1, 2, (3)
where n is the unit normal vector on γ, directed outward from Ω 1 , κ is a coefficient function on γ related directly to the fracture width and inversely to the normal component of the permeability of the physical fracture, the parameter ξ is a constant greater than 1/2 and ξ = 1 -ξ, and for convenience of notation the index i of the subdomains is considered to be an element of Z 2 (so that if i = 2, then i + 1 = 1). The tensor coefficients α i , i = 1, 2, and α γ are related to the inverse of the permeability tensors on Ω i , i = 1, 2, and γ, respectively, and the coefficient β γ is the Forchheimer coefficient on γ, assumed to be scalar. We assume that the functions

α i : Ω i -→ R d,d , α γ : γ -→ R d-1,d-1
, are all symmetric and uniformly positive definite:

α i |x| 2 ≤ x • α i (y)x ≤ α i |x| 2 ∀y ∈ Ω i , x ∈ R d α γ |x| 2 ≤ x • α γ (y)x ≤ α γ |x| 2 ∀y ∈ γ, x ∈ R d-1 and β γ ≤ β γ (y) ≤ β γ ∀y ∈ γ, (4)
where α i , α γ , β γ > 0, and that the real valued coefficient function κ : γ -→ R is bounded above and below by positive constants:

0 < κ ≤ κ(y) ≤ κ ∀y ∈ γ. (5)
Note that only minimal assumptions concerning α i , i = 1, 2, α γ and β γ reflecting the structure of the problem are required and no further regularity, allowing for general heterogeneous media. However this means that the standard functional setting of the linear case has to be modified and thus also the regularity requirements concerning the source and boundary terms. We make the following assumptions concerning the data functions q and p d corresponding respectively to an external source term and to Dirichlet boundary data:

q = (q 1 , q 2 , q γ ) ∈ L 3 (Ω 1 ) × L 3 (Ω 2 ) × L 3 (γ) p d = (p d,1 , p d,2 , p d,γ ) ∈ (W 1 3 , 3 2 (Γ 1 ) ∩ H 1 2 (Γ 1 )) × (W 1 3 , 3 2 (Γ 2 ) ∩ H 1 2 (Γ 2 )) × H 1 2 (∂γ), (6)
where we have used the standard notation for the Lebesgue spaces L p , p ∈ R, p ≥ 1, and for the Sobolev spaces W k,p , k, p ∈ R, p ≥ 1; see [START_REF] Adams | Sobolev Spaces[END_REF]. Following standard practice we often write H k for the Sobolev space W k,2 , k ∈ R. We have required more regularity of the data functions than necessary for a weak formulation of problem (1), ( 2), (3) in order to use the same data functions for problem (1), ( 2), (3) and for problem [START_REF] Faille | A new fault model in geological basin modelling, application to finite volume scheme and domain decomposition methods[END_REF], [START_REF] Forchheimer | Wasserbewegung durch Boden[END_REF], [START_REF] Frih | Un modèle darcy-frochheimer pour un écoulement dans un milieu poreux fracturé[END_REF] given below.

To give a weak mixed formulation of problem (1), ( 2), (3), we introduce several spaces of functions:

M = {p = (p 1 , p 2 , p γ ) : p i ∈ L 2 (Ω i ), i = 1, 2, and p γ ∈ L 3/2 (γ)} p M = 2 i=1 p i 0,2,Ωi + p γ 0, 3 2 ,γ .
The space M being a product of reflexive Banach spaces is clearly a reflexive Banach space with the dual space

M ′ = {f = (f 1 , f 2 , f γ ) : f i ∈ L 2 (Ω i ), i = 1, 2, and f γ ∈ L 3 (γ)} f M ′ = 2 i=1 f i 0,2,Ωi + f γ 0,3,γ .
We also define

V = {v = (v 1 , v 2 , v γ ) : v i ∈ (L 2 (Ω i )) d , i = 1, 2, and v γ ∈ (L 3 (γ)) d-1 } v V = 2 i=1 v i 0,2,Ωi + v γ 0,3,γ RR n°8443
and its dual space

V ′ = {g = (g 1 , g 2 , g γ ) : g i ∈ (L 2 (Ω i )) d , i = 1, 2, and g γ ∈ (L 3 2 (γ)) d-1 } g V ′ = 2 i=1 g i 0,2,Ωi + g γ 0, 3 2 ,γ . Remark 1 For f = (f 1 , f 2 , f γ ) ∈ M ′ , respectively g = (g 1 , g 2 , g γ ) ∈ V ′ ,
we have used the ℓ 1 norm on R 3 to give the norm of f , respectively g, in terms of its three components f 1 , f 2 and f γ , respectively g 1 , g 2 and g γ , whereas the actual norm for the dual space would have used the ℓ ∞ or maximum norm. However these norms are equivalent since R 3 is of finite dimension and we have found it more convenient to use the ℓ 1 norms here.

For the domains

Ω 1 , Ω 2 in R d and γ in R d-1
, respectively, we need minimal regularity to make some of the expressions used below well defined. In particular, we need exterior normal vector fields on the boundaries. To assume that the domains are Lipschitzian will be sufficient, and this will be done henceforth. We will need in addition the space W defined by

W = {u = (u 1 , u 2 , u γ ) ∈ V : Div u := (divu 1 , divu 2 , divu γ -[u 1 • n -u 2 • n]) ∈ M ′ and u i • n ∈ L 2 (γ), i = 1, 2} u W = u V + Div u M ′ + 2 i=1 u i • n 0,2,γ ). (7)
One can show that W is also a reflexive Banach space and that

D = (D(Ω 1 )) d × (D(Ω 2 )) d × (D(γ)) d-1 (8)
is dense in W (see e.g. [START_REF] Summ | Lösbarkeit un Diskretisierung der gemischten Formulierung für Darcy-Frochheimer-Fluss in porösen Medien[END_REF]Lemma 3.13]), where by

D(O) is meant {ψ |O : ψ ∈ C ∞ (R n )}, for O a bounded domain in R n . We also have that for v ∈ W, v i ∈ H(div, Ω i ), i = 1, 2, and v γ ∈ H(div, γ) (since L 3 (γ) ⊂ L 2 (γ)) so that v i • n i ∈ H -1 2 (∂Ω i ) and v γ • n γ ∈ H -1 2 (∂γ),
where n i , i = 1, 2, and n γ are the exterior normal vectors on ∂Ω i , i = 1, 2, and on ∂γ, respectively.

Define the forms a :

W × W -→ R and b : W × M -→ R by a(u, v) = 2 i=1 Ωi α i u i • v i dx + γ (α γ + β γ |u γ |)u γ • v γ ds + 2 i=1 γ κ(ξu i • n + ξu i+1 • n)v i • n ds, b(u, r) = 2 i=1 Ωi divu i r i dx + γ (divu γ -[u 1 • n -u 2 • n] )r γ ds = < Div u, r > M ′ ,M .
Note that the form a is continuous and linear in its second variable while the form b is clearly continuous and bilinear. Define the continuous, linear forms g ∈ W ′ and f ∈ M ′ by

g : W -→ R g(v) = - 2 i=1 < p d,i , v i • n i > H 1 2 (Γi),H -1 2 (Γi) -< p d,γ , v γ • n γ > H 1 2 (∂γ),H -1 2 (∂γ)
and

f : M -→ R f (r) = 2 i=1 Ωi q i r i dx + γ q γ r γ ds. (9)
The weak mixed formulation of ( 1), ( 2) and ( 3) is

(P) Find u ∈ W and p ∈ M such that a(u, v) -b(v, p) = g(v) ∀v ∈ W b(u, r) = f (r) ∀r ∈ M.

Inria

Define also, for the moment only formally (see Lemma 2),

A : W -→ W ′ and B : W -→ M ′ < A(u), v > W ′ ,W = a(u, v) ∀v ∈ W < B(u), r > M ′ ,M = b(u, r) ∀r ∈ M
and note that B : W -→ M ′ is simply Div : W -→ M ′ so that for W, the kernel of B,

W = {v = (v 1 , v 2 , v γ ) ∈ W : Div v = B(v) = 0},
we have that

v W = v V + 2 i=1 v i • n 0,2,γ ∀v ∈ W.

Formulation with Darcy-Forchheimer flow in the matrix and in the fractures

With Ω, γ, Ω i , Γ i , n i , i = 1, 2, n γ and n as well as α i , α γ , β γ , κ, q i , q γ , p d,i , p d,γ , ξ, and ξ as in the preceding paragraph, and with β i : Ω i -→ R a function satisfying

β i ≤ β i (y) ≤ β i ∀ y ∈ Ω i , (10)
where β i , β i > 0, we now consider the following problem:

(α i + β i |u i |)u i + ∇p i = 0 in Ω i div u i = q i in Ω i p i = p d,i on Γ i (11)
together with

(α γ + β γ |u γ |)u γ + ∇p γ = 0 on γ div u γ = q γ + [u 1 • n -u 2 • n] on γ p γ = p d,γ
on ∂γ [START_REF] Forchheimer | Wasserbewegung durch Boden[END_REF] and the interface conditions

p i = p γ + (-1) i+1 κ(ξu i • n + ξu i+1 • n), i = 1, 2. (13)
Due to the Forchheimer regularization in the matrix equations, the spaces in the earlier definitions need to be replaced by spaces appropriate for the functional setting of the Forchheimer equations, i.e. L 2 (Ω i ) by L 3 2 (Ω i ), and consequently L 2 (Ω i ) by L 3 (Ω i ) for the dual spaces, thus obtaining a β-version of the earlier spaces, i. e. M β instead of M, etc. For the sake of clarity we state explicitly:

M β = {p = (p 1 , p 2 , p γ ) : p i ∈ L 3/2 (Ω i ), i = 1, 2, and p γ ∈ L 3/2 (γ)} p M β = 2 i=1 p i 0, 3 2 ,Ωi + p γ 0, 3 2 ,γ .
The space M β is clearly a reflexive Banach space with dual space

M ′ β = {f = (f 1 , f 2 , f γ ) : f i ∈ L 3 (Ω i ), i = 1, 2, and f γ ∈ L 3 (γ)} f M ′ β = 2 i=1 f i 0,3,Ωi + f γ 0,3,γ .
We also define

V β = {v = (v 1 , v 2 , v γ ) : v i ∈ (L 3 (Ω i )) d , i = 1, 2, and v γ ∈ (L 3 (γ)) d-1 } v V β = 2 i=1 v i 0,3,Ωi + v γ 0,3,γ , RR n°8443
which is similarly a reflexive Banach space, with its dual space

V ′ β = {g = (g 1 , g 2 , g γ ) : g i ∈ (L 3/2 (Ω i )) d , i = 1, 2, and g γ ∈ (L 3/2 (γ)) d-1 } g V ′ β = 2 i=1 g i 0, 3 2 ,Ωi + g γ 0, 3 2 ,γ .
Again we have used the equivalent ℓ 1 norm instead of the ℓ ∞ norm to construct the product space norm for M ′ β and W ′ β . We also need the space W β defined by

W β = {u = (u 1 , u 2 , u γ ) ∈ V β : Div u = (divu 1 , divu 2 , divu γ -[u 1 • n -u 2 • n]) ∈ M ′ β and u i • n ∈ L 2 (γ), i = 1, 2} u W β = u V β + Div u M ′ β + 2 i=1 u i • n 0,2,γ . (14)
One can show that W β is a reflexive Banach space, that D, given by ( 8), is dense in

W β , that for v ∈ W β , for i = 1, 2, v i ∈ H(div, Ω i ) and v γ ∈ H(div, γ) so that v i • n i ∈ H -1 2 (∂Ω i ) and v γ • n γ ∈ H -1 2 (∂γ). Further v i ∈ W 3 (div, Ω i ) (see Appendix A.1 ). Define the forms a β : W β × W β -→ R and b β : W β × M β -→ R by a β (u, v) = 2 i=1 Ωi (α i + β i |u i |)u i • v i dx + γ (α γ + β γ |u γ |)u γ • v γ ds + 2 i=1 γ κ(ξu i • n + ξu i+1 • n)v i • n ds, b β (u, r) = 2 i=1 Ωi divu i r i dx + γ (divu γ -[u 1 • n -u 2 • n] )r γ ds = < Div u, r > M ′ β ,M β .
Note that the form a β is continunous and linear in its second variable while b β is continuous and bilinear. Define the linear forms

g : W β -→ R and f : M β -→ R
as in (9) but with g ∈ W ′ β and f ∈ M ′ β which is valid with the regularity assumptions in [START_REF] Angot | Asymptotic and numerical modelling of flows in fractured porous media[END_REF]. The mixed weak formulation of ( 11), ( 12) and ( 13) is given by

(P β ) Find u ∈ W β and p ∈ M β such that a β (u, v) -b β (v, p) = g(v) ∀v ∈ W β b β (u, r) = f (r) ∀r ∈ M β .
Define again

A β : W β -→ W ′ β and B β : W β -→ M ′ β < A β (u), v > V ′ , V = a β (u, v) ∀v ∈ W β < B β (u), r > M ′ β , M β = b β (u, r
) ∀r ∈ M β for an equivalent operator equation and

W β = {u = (u 1 , u 2 , u γ ) ∈ W β : Div u := B β (u) = 0},
and note that

u W β = u V β + 2 i=1 u i • n 0,2,γ ∀u ∈ W β . ( 15
)
Remark 2 Note that none of the spaces W β , V β , W β , or M β and neither of the operators b β nor B β depends on the coefficient β. The index β is used simply to indicate that these are the spaces and operators used to define the problem (P β ).

Inria

To obtain some of the estimates that we will derive in the following sections we shall make use of the following technical lemma given in [15, lemmas 1.1 and 1.4].

Lemma 1 For x and y in R n , we have the following inequalities:

| |x|x -|y|y | ≤ (|x| + |y|) |x -y|, ( 16
) 1 2 |x -y| 3 ≤ (|x|x -|y|y) • (x -y), (17)
|x| -1 2 x -|y| -1 2 y ≤ √ 2 |x -y| 1 2 , ( 18
)
|x -y| 2 |x| + |y| ≤ x |x| - y |y| • (x -y). (19)
In [START_REF] Summ | Lösbarkeit un Diskretisierung der gemischten Formulierung für Darcy-Frochheimer-Fluss in porösen Medien[END_REF] and hereafter x → |x| -1 2 x on R n means the continuation of this function on R n \ {0} to R n obtained by defining |0| -1 2 0 := 0, which by ( 18) is indeed Hölder continuous with exponent 1 2 . Here we introduce some notation that we will use throughout the remainder of the article: for any positive integer n and any bounded domain O in R n , we know that L 3 (O) ֒→ L 2 (O) and that the inclusion map is continuous so that there is a constant C L,O depending on n and the measure of the space such that if

φ ∈ L 3 (O) then φ L 2 ≤ C L,O φ L 3 .
Here we shall assume that C L is a constant with C L,O ≤ C L for all of the spaces O that we deal with. (There are only a finite number for each problem.) Also we know that if s and t are such that 1 ≤ s ≤ t ≤ ∞ then the ℓ s and ℓ t norms on R n are equivalent (since all norms on finite dimensional spaces are equivalent), and we shall assume that there are positive real numbers C ℓ and c ℓ such that if x ∈ R n then c ℓ x ℓ t ≤ x ℓ s ≤ C ℓ x ℓ t for all dimensions n and all norms ℓ s and ℓ t with 1 ≤ s ≤ t ≤ ∞, that we encounter in the problems that follow. (Again there will only be a finite number.)

2 Existence and uniqueness of the solution of the problem (P β ) Darcy-Forchheimer flow in the fracture and in the subdomains

To show the existence and uniqueness of the solution of (P β ), following the argument of [15, Section 1] we show that the operator A β : W β -→ W ′ β is continuous and monotone and is uniformly monotone on W β to obtain a solution to the homogeneous problem with f = 0. (That

B β : W β -→ M ′
β satisfies the inf-sup condition follows just as in the linear case, cf. [START_REF] Martin | Modeling fractures and barriers as interfaces for flow in porous media[END_REF], however for completeness a demonstration is given in Appendix A.3). Then taking any solution to the second equation of (P β ) (whose existence is guaranteed by the inf-sup condition) an auxiliary homogeneous problem is constructed whose solution can be used to produce the solution of (P β ).

Lemma 2 The operator

A β : W β -→ W ′
β is continuous and strictly monotone and is furthermore uniformly monotone on W β .

Proof: To see that ∀u ∈ W β , A β (u) ∈ W ′ β i. e. that ∀u ∈ W β , A β (u) is bounded, suppose that u ∈ W β .
Then, using the equivalence of norms in a finite dimensional space and Hölder's inequality, we have, for each v in W β , Ωi

β i |u i |u i • v i dx ≤ β i C 3 ℓ Ωi |u i | 2 3 |v i | 3 dx ≤ β i C 3 ℓ Ωi (|u i | 2 3 ) 3 2 ds 2 3
Ωi

|v i | 3 3 ds 1 3 ≤ β i C 3 ℓ u i 2 0,3,Ωi v i 0,3,Ωi . (20)
We also have

Ωi α i u i • v i dx ≤ α i u i 0,2,Ωi v i 0,2,Ωi ≤ α i C 2 L u i 0,3,Ωi v i 0,3,Ωi . (21)
As we have similar inequalities for the norms on γ and as ξ ≥ ξ we conclude that for each v ∈ W β ,

A β (u) W ′ β = sup v∈W β v =0 | < A β (u), v > W ′ β , W β | v W β ≤ sup v∈W β v =0 2 i=1 α i C 2 L u i 0,3,Ωi v i 0,3,Ωi + β i C 3 ℓ u i 2 0,3,Ωi v i 0,3,Ωi + α γ C 2 L u γ 0,3,γ v γ 0,3,γ + β γ C 3 ℓ u γ 2 0,3,γ v γ 0,3,γ +ξ κ 2 i=1 u i • n 0,2,γ 2 i=1 v i • n 0,2,γ v W β ≤ 2 i=1 α i C 2 L u i 0,3,Ωi + β i C 3 ℓ u i 2 0,3,Ωi + α γ C 2 L u γ 0,3,γ + β γ C 3 ℓ u γ 2 0,3,γ + ξ κ 2 i=1 u i • n 0,2,γ ≤ αC 2 L u V β + βC 3 ℓ u 2 V β + ξ κ 2 i=1 u i • n 0,2,γ ≤ max{αC 2 L , ξ κ} + βC 3 ℓ u V β u W β , ( 22
)
where α is the max{α 1 , α 2 , α γ }, and similarly for β.

To see that A β : W β -→ W ′ β is continuous suppose that u and w are elements of W β . Using Hölder's inequality and then inequality [START_REF] Martin | Modeling fractures and barriers as interfaces for flow in porous media[END_REF] along with the equivalence of norms in finite dimensional spaces, we see that, for any v ∈ W β and for i = 1, 2,

Ωi β i (|u i |u i -|w i |w i ) • v i dx ≤ β i |u i |u i -|w i |w i 0, 3 2 ,Ωi v i 0,3,Ωi ≤ β i C 3 ℓ ( u i 0,3,Ωi + w i 0,3,Ωi ) u i -w i 0,3,Ωi v i 0,3,Ωi .
Then using the analogous inequality for the nonlinear term on γ we have

A β (u) -A β (w) W ′ β = sup v ∈ W β v = 0 < A β (u) -A β (w) , v > W ′ β , W β v W β ≤ αC 2 L u -w V β + βC 3 ℓ ( u V β + w V β ) u -w V β v V β + ξ κ 2 i=1 (u i -w i ) • n 0,2,γ 2 i=1 v i • n 0,2,γ v W β ≤ max{αC 2 L , ξ κ} + βC 3 ℓ ( u V β + w V β ) u -w W β .
To see that A β : W β -→ W ′ β is strictly monotone suppose again that u and w are elements of W β . Then using inequality [START_REF] Showalter | The narrow fracture approximation by channeled flow[END_REF], for i = 1, 2,

Ωi β i (|u i |u i -|w i |w i ) • (u i -w i ) dx ≥ β i c 3 ℓ 2 u i -w i 3 0,3,Ωi . (23)

Inria

We also note that if x, y ∈ R then ξ(x 2 + y 2 ) + 2 ξxy ≥ min{1, 2ξ -1}(x 2 + y 2 ). It follows that ∀u, w ∈ W β

< A β (u) -A β (w) , u -w > W ′ β , W β ≥ βc 3 ℓ 2 u -w 3 V β + κ min{1, 2ξ -1} 2 i=1 (u i -w i ) • n 2 0,2,γ ≥ C(β, κ, ξ) u -w 3 V β + 2 i=1 (u i -w i ) • n 2 0,2,γ ≥ 0, (24)
where β = min{β 1 , β 2 , β γ }, and where we have equality only if u = w.

To see that A β is uniformly monotone on W β it suffices to note that if u and w belong to

W β then < A β (u) -A β (w) , u -w > W ′ β , W β ≥ G( u -w W β ) u -w W β with G( u W β ) := C(β, κ, ξ) u 3 V β + 2 i=1 u i • n 2 0,2,γ u W β = C(β, κ, ξ) u 3 V β + 2 i=1 u i • n 2 0,2,γ u V β + 2 i=1 u i • n 0,2,γ -→ ∞ as u W β -→ ∞,
where we have used [START_REF] Knabner | Solvability of the mixed formulation for darcy-forchheimer flow in porous media[END_REF].

Lemma 3

The linear form b β : W β × M β -→ R satisfies the following inf-sup condition: there is a positive constant θ β such that ∀r ∈ M β

θ β r M β ≤ sup v∈W β b β (v, r) v W β . (25)
Proof: See Appendix A.3.

Proposition 1

The homogeneous problem

(P 0 β ) Find u 0 β ∈ W β and p 0 β ∈ M β such that a β (u 0 β , v) -b β (v, p 0 β ) = g(v) ∀v ∈ W β b β (u 0 β , r) = 0 ∀r ∈ M β
has a unique solution.

Proof: That there is a unique solution in W β to a β (u 0 β , v) = g(v), ∀v ∈ W β , i. e. to A β (u 0 β ) = g, now follows from the Browder-Minty theorem, [START_REF] Zeidler | Nonlinear function anaysis and its applications -Nonlinear monotone operators[END_REF]Theorem 26.A]. That there is a unique p 0 β ∈ M β such that (u 0 β , p 0 β ) is the unique solution of (P 0 β ) then follows as in the linear case as the operator B β is still linear.

To handle a source term in the continuity equation we start from any solution to this equation and construct an auxiliary homogeneous problem whose solution is then combined with the solution to the (nonhomogeneous) continuity equation to produce the desired solution to the full problem. Proof: Since, according to Lemma 3, b β satisfies the inf-sup condition, the subproblem of (P β )

Find u ∈ W β such that b β (u, r) = f (r) ∀r ∈ M β
has a (non-unique) solution. Let u * ∈ W β denote one such. We consider the auxiliary problem

(P * β ) Find ũ ∈ W β and p ∈ M β such that a β (ũ + u * , v) -b β (v, p) = g(v) ∀v ∈ W β b β (ũ, r) = 0 ∀r ∈ M β .
Just as in Proposition 1, this problem has a unique solution, as one can show, just as in Lemma 2, that

a * β (u, v) := a β (u + u * , v)
defines a continuous operator, strictly monotone on W β and uniformly monotone on W β . Then, due to the bilinearity of b β , u := ũ + u * , together with p is a solution of (P β ).

To show uniqueness we refer to Lemma 7 in Appendix A.2.

Darcy as a limit of Darcy-Forchheimer -Simple Domain

Suppose here that O is a bounded domain in R d with boundary ∂O. The object of this section is to show that the solution of the Darcy problem

α O u = -∇p in O divu = q O in O p = p ∂,O
on ∂O may be obtained as the limit of a sequence of solutions of the Darcy-Forchheimer problems

α O u β + β O |u β |u β = -∇p β in O divu β = q O in O p β = p ∂,O
on ∂O, as β O → 0. As before we assume that the tensor coefficient function

α O : O -→ R d,d , is such that α O |x| 2 ≤ x • α O (y)x ≤ α O |x| 2 ∀y ∈ O, x ∈ R d , (26)
and the coefficient β O of the nonlinear term is assumed to be a positive real parameter as we are merely interested in obtaining the Darcy problem as a limit of Forchheimer problems. Let

W(O) = H(div, O) M(O) = L 2 (O) W β (O) = W 3 (div, O) M β (O) = L 3 2 (O),
and recall that the image of the normal trace map on W(O) is H -1 2 (∂O) while the image of the normal trace map on W β (O) is W -1 3 ,3 (∂O). Also as before the data functions are assumed to be such that q ∈ L 3 (O) and p ∂,O ∈ W

a O : W(O) × W(O) -→ R and b O : W(O) × M(O) -→ R (u, v) → O α O u • v dx (v, r) → O div(v) r dx,
Inria and the linear forms

g O ∈ W(O) ′ and f O ∈ M(O) ′ by g O : W(O) -→ R and f O : M(O) -→ R v → < p ∂,O , v • n > H 1 2 (∂O),H -1 2 (∂O) r → O q O r dx,
so that the problem (P Darcy ) can be written as

(P Darcy ) Find u ∈ W(O) and p ∈ M(O) such that a O (u, v) -b O (v, p) = g O (v) ∀v ∈ W(O) b O (u, r) = f O (r) ∀r ∈ M(O). Since a O is elliptic (coercive) on the subset W(O) = {v ∈ W(O) : b O (v, r) = 0, ∀r ∈ M(O)} and b O satisfies the inf-sup condition on W(O) × M(O): α O v W(O) ≤ a O (v, v) ∀v ∈ W(O) and θ O r M(O) ≤ sup v∈W(O) b O (v, r) v W(O) , ∀r ∈ M(O),
the Darcy problem (P Darcy ) has a unique solution [START_REF] Brezzi | On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers[END_REF].

(u O , p O ) ∈ W(O) × M(O),
To give the weak formulation of the Forchheimer problem note that since

W β (O) ⊂ W(O) the bilinear form a O is also defined on W β (O) × W β (O) and that the bilinear form b O is also defined on W β (O) × M β (O) (even though M β (O) ⊂ M(O)). Further b O also satisfies the analogous inf-sup condition onW β (O) × M β (O)
for some constant θ β,O ; see [START_REF] Summ | Lösbarkeit un Diskretisierung der gemischten Formulierung für Darcy-Frochheimer-Fluss in porösen Medien[END_REF] or the more general version in Lemma 3. Now define the mapping a β,O , linear in its second variable, by

a β,O : W β (O) × W β (O) -→ R (u, v) → O (α O + β O |u|)u • v dx,
and note that due to the regularity requirements on the data functions p ∂,O and q O that the linear forms g O and f O are defined and continuous on W β (O) and M β (O), respectively, (as well as on W(O) and M(O)), so that the problem (P F orch ) can be written as

(P F orch ) Find u β ∈ W β (O) and p β ∈ M β (O) such that a β,O (u β , v) -b O (v, p β ) = g O (v) ∀v ∈ W β (O) b O (u β , r) = f O (r) ∀r ∈ M β (O).
It is shown in [START_REF] Knabner | Solvability of the mixed formulation for darcy-forchheimer flow in porous media[END_REF] that the form a β,O is continuous, strictly monotone on W β (O), and coercive on The demonstration that the solutions of the problems (P F orch ) converge to the solution of (P Darcy ) is based on a priori bounds for u β,O and p β,O independent of the parameter β. In this section we will drop the spaces in the notation for the norms as only O or ∂O appears.

W β (O) = {v ∈ W β (O) : b O (v, r) = 0, ∀r ∈ M β (O)} [15,
Lemma 4 There is a constant C independent of β such that for β sufficiently small

p β,O M β (O) + u β,O W(O) + β 1 3 u β,O 0,3 ≤ C.
In addition, β u β,O 0,3 -→ 0, as β -→ 0.

Proof: Taking u β,O for the test function v in the first equation of (P F orch ) and noting that W β (O) ⊂ W(O), as in Section 2 (cf. estimate (24)) one obtains

α O u β,O 2 0,2 + c 3 ℓ 2 β u β,O 3 0,3 ≤ g O (u β,O ) + b(u β,O , p β,O ) ≤ g O W(O) ′ u β,O W(O) + div(u β,O ) 0,3 p β,O 0, 3 2 ≤ g O W(O) ′ u β,O 0,2 + div(u β,O ) 0,2 + div(u β,O ) 0,3 p β,O 0, 3 2 .
Next directly from the second equation of (P F orch ) (regarded as an equation in

M β (O) ′ = L 3 (O)), we obtain div(u β,O ) 0,3 = f O 0,3 , (27)
and, as there is a continuous embedding

M(O) ֒→ M β (O), i. e. L 2 (O) ֒→ L 3 2 (
O) so that the second equation of (P F orch ) holds for test functions in L 2 (O) as well as for those in

L 3 2 (O), we also have div(u β,O ) 0,2 = f O 0,2 . (28)
Combining these last three inequalities we obtain

α O u β,O 2 0,2 + c 3
ℓ 2 β u β,O 3 0,3 ≤ g O W(O) ′ ( u β,O 0,2 + f O 0,2 ) + f O 0,3 p β,O 0, 3 2 and 1 2 α O u β,O 2 0,2 + c 3
ℓ 2 β u β,O 3 0,3 ≤ 1 2α O g O 2 W(O) ′ + g O W(O) ′ f 0,2 + f O 0,3 p β,O 0, 3 2 ≤ D 1 + f O 0,3 p β,O 0, 3 2 , (29)
where D 1 is a constant depending only on the coefficient α O and the data functions determining g O and f O . Then with the first equation of (P F orch ), we obtain, ∀v ∈ W β (O),

|b O (v, p β,O )| ≤ |a β,O (u β,O , v)| + |g O (v)| ≤ α O u β,O 0,2 v 0,2 + C 3 ℓ β u β,O 2 0,3 v 0,3 + g O W(O) ′ v W β (O) ≤ α O C L u β,O 0,2 + C 3 ℓ β u β,O 2 0,3 + p ∂,O 1 3 , 3 2 v W β (O) ,
where we again use C L , respectively C ℓ , here specifically for the continuity constant for the embedding

L 3 (O) ֒→ L 2 (O), respectively ℓ 3 (R d ) ֒→ ℓ 2 (R d ). Using the inf-sup condition for b on W β (O) × M β (O) we have θ β,O r 0, 3 2 ≤ sup v∈W β (O) b O (v, r) v W β (O) .
and thus

θ β,O p β,O 0, 3 2 ≤ α O C L u β,O 0,2 + C 3 ℓ β u β,O 2 0,3 + p ∂,O 1 3 , 3 2 . ( 30
)
Plugging this estimate for p β into (29) we obtain

1 2 α O u β,O 2 0,2 + c 3 ℓ 2 β u β,O 3 0,3 ≤ D 1 + 1 θ β,O f O 0,3 α O C L u β,O 0,2 + C 3 ℓ β u β,O 2 0,3 + p ∂,O 1 3 , 3 2
.

Now using the inequality

α O C L θ β,O f O 0,3 u β,O 0,2 ≤ 4 α O α O C L θ β,O 2 f O 2 0,3 + 1 4 α O u β,O 2 0,2
≤ D 2 + 1 4 α O u β,O 2 0,2

Inria

it is easy to see that

1 4 α O u β,O 2 0,2 + c 3 ℓ 2 β u β,O 3 0,3 ≤ D 1 + 4 α O α O C L θ β,O 2 f O 2 0,3 + 1 θ β,O f O 0,3 C 3 ℓ β u β,O 2 0,3
+ 1 θ β,O f O 0,3 p ∂,O 1 3 , 2 3 ≤ D 1 + D 2 + C 4 β u β,O 2
+ 1 q = 1 then ab ≤ a p p + b q q ) with a = (β s u β,O 0,3 ) 2 , b = 1, p = 3
2 and q = 3 one obtains

1 4 α O u β,O 2 0,2 + c 3 ℓ 2 β u β,O 3 0,3 ≤ D 1 + D 2 + D 3 + 2 3 C 4 β 1-2s (β s u β,O 0,3 ) 3 + 1 3 C 4 β 1-2s
and that

1 4 α O u β,O 2 0,2 + c 3 ℓ 2 β 1-3s - 2 3 C 4 β 1-2s (β s u β,O 0,3 ) 3 ≤ D 1 + D 2 + D 3 + 1 3 C 4 β 1-2s
or, in particular, that (with s = 1 2 )

1 4 α O u β,O 2 0,2 + c 3
ℓ 2 β -1 2 - 2 3 C 4 (β 1 2 u β,O 0,3 ) 3 ≤ D 1 + D 2 + D 3 + 1 3 C 4 := D 4 . (31)
Thus for β sufficiently small, we obtain an a priori bound on α

1 2 O u β,O 0,2 : α 1 2 O u β,O 0,2 ≤ 2 (D 4 ) 1 2 , (32)
and also that (β

1 2 u β,O 0,3 ) 3 ≤ c 3 ℓ 2 β -1 2 - 2 3 C 4 -1 D 4 so that β 1 2 u β,O 0,3 -→ 0 as β -→ 0. (33)
Rewriting (31) as

1 4 α O u β,O 2 0,2 + c 3 ℓ 2 β u β,O 3 0,3 ≤ D 4 + 2 3 C 4 (β 1 2 u β,O 0,3 ) 3 ,
we obtain in turn an a priori bound for β 1 3 u β,O 0,3 :

β 1 3 u β,O 0,3 ≤ (D 4 + ǫ) 1 3 , (34)
with ǫ > 0 arbitrarily small for β ≤ βǫ for some βǫ > 0. Now combining (30), (32) and (34) one obtains the following a priori bound on p β,O in L

p β,O 0, 3 2 ≤ 2α O C L θ β,O α 1 2 O (D 4 ) 1 2 + C 3 L θ β,O β 1 3 (D 4 + ǫ) 2 3 + 1 θ β,O p ∂,O 1 3 , 3 2 . ( 35
)
With ( 35), ( 32) and (28) the lemma is completed.

RR n°8443

From (32) and (35), we conclude that if {β j } is a sequence converging to 0 then there is a subsequence still denoted {β j } such that the sequences {u β j ,O } and {p β j ,O } are weakly convergent in (L 2 (O)) d and in L 3 2 (O), respectively:

u β j ,O ⇀ ũ in (L 2 (O)) d and p β j ,O ⇀ p in L 3 2 (O), (36)
i. e. explicitly

O αu β j ,O • v dx → O αũ • v dx ∀v ∈ (L 2 (O)) n and O p β j ,O q dx → O pq dx ∀q ∈ L 3 (O). (37)
Further, (33) implies that {β 2 (∂O). Thus each of the terms of (39) is well defined for v ∈ W(O), and we have since

1 2 j u β j ,O } converges strongly to 0 in L 3 (O). Thus O β j |u β j ,O |u β j ,O • v dx ≤ v 0,3 β j |u β j ,O |u β j ,O 0, 3 2 = v 0,3 O β 3 2 j |u β j ,O | 3 dx 2 3 = v 0,3 β 1 2 j u β j ,O 2
O α O ũ • v dx - O div(v)p dx = -< p ∂,O , v • n > H 1 2 (∂O),H -1 2 (∂O) ∀v ∈ W β (O). (39)
W β (O) is dense in W(O) that O α O ũ • v dx - O div(v)p dx = -< p ∂,O , v • n > H 1 2 (∂O),H -1 2 (∂O) ∀v ∈ W(O). (40)
Turning now to the second equation of (P Darcy ), we recall that f O = q O belongs to L 3 (O) = M β (O) ′ , and thus also to L 2 (O) = M(O) ′ . As we have seen, the second equation of (P F orch ) implies that for each β > 0, div(u β,O ) = f ∈ L 3 (O). This with (32) implies that u β,O is bounded in the W(O) norm and that for a subsequence {β ℓ } of {β j }, u β ℓ converges weakly to ũ in W(O).

It follows that

O div(ũ)r dx = O q O r ∀r ∈ M(O).
Thus the pair (ũ, p) in W(O) × M(O) is a solution of (P Darcy ) and (ũ, p) = (u O , p O ) by uniqueness.

4 Darcy as a limit of Darcy-Forchheimer -Domain with a Fracture

The object of this section is to obtain the original problem (P) (with Darcy flow in the subdomains Ω 1 and Ω 2 but Forchheimer flow in the fracture γ) as the limit of the problem (P β ) (with Forcheimer flow in the subdomains and in the fracture) studied in Section 2 when the Forchheimer coefficient Inria in the subdomains β decreases to 0. In this section, as in Section 3, for simplicity we shall assume that β i is the same constant, positive, real parameter for i = 1 and i = 2:

β 1 = β 2 = β > 0.
(The tensors β γ , α i and α γ (4), as well as κ (5), remain as in Section 2.) For each β sufficiently small, let (u β , p β ) ∈ W β × M β be the solution of (P β ). We will derive a priori bounds on (u β , p β ) which are independent of β, thus obtaining a limit function which we shall show is a solution to (P).

Lemma 6 There is a constant C independent of β, such that, for each β sufficiently small,

u β W + p β M β + β 1 3 2 i=1 u β,i 0,3,Ωi ≤ C.
In addition,

β 1 2 2 i=1
u β,i 0,3,Ωi -→ 0, as β -→ 0.

Proof: The proof follows closely the lines of the proof of Lemma 4. Taking for test function v = u β in the first equation of (P β ), noting that u β ∈ W and that g ∈ W ′ , and letting C ξ denote κ min(1, 2ξ -1) we obtain

2 i=1 α i u β,i 2 0,2 + c 3 ℓ 2 β u β,i 3
0,3 + α γ u β,γ 2 0,2 + c 3 ℓ 2 β γ u β,γ 3
0,3 + 2 i=1 C ξ u β,i • n 2 0,2 ≤ a β (u β , u β ) = g(u β ) + b β (u β , p β ) ≤ g W ′ u β W + Div u β M ′ β p β M β = g W ′ u β V + Div u β M ′ + 2 i=1 u β,i • n 0,2,γ + Div u β M ′ β p β M β ,
and from the second equation we have Div u β = f so that

Div u β M ′ β = f M ′ β Div u β M ′ = f M ′ . (41)
Combining these estimates, analogously to (29) we obtain

2 i=1 α i u β,i 2
0,2 + c 3 ℓ 2 β u β,i 3
0,3 + α γ u β,γ 2
0,2 + c 3 ℓ 2 β γ u β,γ 3
0,3 + 2 i=1 C ξ u β,i • n 2 0,2 ≤ g W ′ u β V + f M ′ + 2 i=1 u β,i • n 0,2,γ + f M ′ β p β M β ≤ g W ′ 2 i=1 u β,i 0,2 + u β,γ 0,3,γ + 2 i=1 u β,i • n 0,2,γ + f M ′ β p β M β + D 1 ,
where the constant term D 1 depends on g and on f . Then, using Young's inequality we have

2 i=1 1 2 α i u β,i 2
0,2 + c 3 ℓ 2 β u β,i 3
0,3 + α γ u β,γ 2 0,2 + 2 3 c 3 ℓ 2 β γ u β,γ 3
0,3 + 1 2 2 i=1 C ξ u β,i • n 2 0,2 ≤ f M ′ β p β M β + D 1 + D 2 ,
where D 2 depends on g, α i , β γ , and C ξ . The inf-sup condition for b β : W β × M β -→ R together with the first equation of (P β ) yields

θ β p β M β ≤ sup v∈W β b(v, p β ) v W β ≤ sup v∈W β a β (u β , v) -g(v) v W β ,
and using ( 20)

|a β (u β , v) -g(v)| ≤ 2 i=1 α i u β,i 0,2 v i 0,2 + C 3 ℓ β u β,i 2
0,3 v i 0,3 +κξ 2 i=1 u β,i • n 0,2,γ 2 i=1 v i • n 0,2,γ +α γ u β,γ 0,2,γ v γ 0,2,γ + C 2 ℓ β γ u β,γ 2
0,3,γ v γ 0,3,γ + g W ′ β v W β .
Then combining the last two estimates, analogously to (30) we have

θ β p β M β ≤ 2 i=1 C L α i u β,i 0,2 + C 3 ℓ β u β,i 2
0,3 + κξ u β,i • n 0,2,γ +C L α γ u β,γ 0,2,γ + C 2 ℓ β γ u β,γ 2
0,3,γ + g W ′ β . (42)
So

2 i=1 1 2 α i u β,i 2
0,2 + c 3 ℓ 2 β u β,i 3
0,3 + α γ u β,γ 2
0,2 + c 3 ℓ 3 β γ u β,γ 3
0,3 + 1 2 2 i=1 C ξ u β,i • n 2 0,2 ≤ D 1 + D 2 + D 3 + f M ′ β θ β 2 i=1 C L α i u β,i 0,2 + C 3 ℓ β u β,i 2
0,3 + κξ u β,i • n 0,2,γ +C L α γ u β,γ 0,2,γ + C 2 ℓ β γ u β,γ 2
0,3,γ ,
where D 3 depends on g, θ β , and f . Then using Young's inequality (three times with exponents 2 and 2 and twice with exponents 3 and 3 2 ) we obtain

2 i=1 1 4 α i u β,i 2
0,2 + c 3 ℓ 2 β u β,i 3
0,3 + 1 2 α γ u β,γ 2
0,2 + c 3 ℓ 3 β γ u β,γ 3
0,3 + 1 4 2 i=1 C ξ u β,i • n 2 0,2 ≤ D 1 + D 2 + D 3 + D 4 + 2 3 C 5 2 i=1 (β 1 2 u β,i 0,3 ) 3 , (43)
with D 4 depending on α i , α i , α γ , α γ , β γ , β γ , κ, κ, ξ, θ β , and f , and, analogously to (31),

2 i=1 1 4 α i u β,i 2 0,2 + c 3 ℓ 2 β -1 2 - 2 3 C 5 β 1 2 u β,i 0,3 3 + 1 4 C ξ u β,i • n 2 0,2 + 1 4 α γ u β,γ 2
0,2 + c 3 ℓ 3 β γ u β,γ 3
0,3 ≤ D 1 + D 2 + D 3 + D 4 .
(Recall that θ β does not depend on β.) Hence

β 1 2 u β,i 0,3 -→ 0 as β -→ 0, i = 1, 2,

Inria

For each β > 0, the first equation of (P β ) is

a β (u β , v) -b(v, p β ) = g(v), ∀v ∈ W β , 2 i=1 Ωi (α i + β|u β,i |)u β,i • v i dx + γ (α γ + β γ |u β,γ |)u β,γ • v γ ds + 2 i=1 γ 1 κ (ξu β,i • n + ξu β,i+1 • n)v i • n ds -b(v, p β ) = g(v), ∀v ∈ W β , or 2 i=1 Ωi β|u β,i |u β,i • v i dx + γ β γ |u β,γ |u β,γ • v γ ds = - 2 i=1 Ωi α i u β,i • v i dx - γ α γ u β,γ • v γ ds - 2 i=1 γ 1 κ (ξu β,i • n + ξu β,i+1 • n)v i • n ds + b(v, p β ) + g(v), ∀v ∈ W β .
Then taking the limit as β goes to 0 we have, due to [START_REF] Zeidler | Nonlinear function anaysis and its applications -Nonlinear monotone operators[END_REF] and Lemma 6 That the mapping C is monotone and continuous can be shown as in the proof of Lemma 2 where the monotonicity and continuity of A β are shown. Therefore C maps weakly convergent sequences to convergent sequences; see [START_REF] Zeidler | Nonlinear function anaysis and its applications -Nonlinear monotone operators[END_REF]. Thus, since u β,γ ⇀ ũγ , we have that C(u β,γ ) → C(ũ γ ) in L 3 (γ) which now yields that a(ũ, v) -b(v, p) = g(v), ∀v ∈ W.

lim β→0 γ β γ |u β,γ |u β,γ • v γ ds = - 2 i=1 Ωi α i ũi • v i dx - γ α γ ũγ • v γ ds - 2 i=1 γ 1 κ (ξ ũi • n + ξ ũi+1 • n)v i • n ds + b(v, p) + g(v), ∀v ∈ W β , and
Thus (ũ, p) ∈ W × M is a solution of (P).

As in (24) we see that A is strictly monotone on W. Thus we can refer to Lemma 7 for uniqueness.

Inria

A Appendix

In this appendix for the sake of completeness we give the definition and some basic properties of the spaces W p (div, Ω), and we include the demonstrations of some lemmas needed in the previous sections.

A.1 The spaces W p (div, O)

We recall the definition given in [START_REF] Knabner | Solvability of the mixed formulation for darcy-forchheimer flow in porous media[END_REF], [START_REF] Summ | Lösbarkeit un Diskretisierung der gemischten Formulierung für Darcy-Frochheimer-Fluss in porösen Medien[END_REF] As pointed out in [START_REF] Knabner | Solvability of the mixed formulation for darcy-forchheimer flow in porous media[END_REF] and in [START_REF] Summ | Lösbarkeit un Diskretisierung der gemischten Formulierung für Darcy-Frochheimer-Fluss in porösen Medien[END_REF] it suffices to note that W p (div, O) is a closed subset of (L p (O)) d to see that W p (div, O) is a reflexive Banach space. Further, normal traces of elements of W p (div, O) belong to W -1 p ,p (∂O).

A.2 A general uniqueness result

The object here is to show a uniqueness result that is used in the proofs of Theorems 1 and 2: that if A, B and B t are the operators associated with a mixed formulation and A is strictly monotone and B is surjective, then the mixed problem has no more than one solution; more precisely

A.3 Some inf-sup conditions

In this paragraph we give proofs of the fact that some of the bilinear operators considered in the text satisfy the inf-sup condition.

Proof of Lemma 3: This proof is just as that for the problem with Darcy flow in the fracture and in the subdomain (see the proof of [START_REF] Martin | Modeling fractures and barriers as interfaces for flow in porous media[END_REF]Theorem 4.1]), only modified for the inf sup condition RR n°8443

1. 1

 1 Formulation with Darcy-Forchheimer flow in the fracture and Darcy flow in the matrix

Theorem 1

 1 The problem (P β ) admits a unique solution (u β , p β ) ∈ W β × M β .

1 3 , 3 2 1 2

 331 (∂O) ∩ W ,2 (∂O). Define the bilinear forms a O and b O by

0, 3 + D 3 ,

 33 with constant terms D 2 , which depends on f O , C L , α O , α O and θ β,O , and D 3 , which depends on f O , p ∂,O and θ β,O , and a constant coefficient C 4 , which depends on f O , θ β,O and C ℓ . Now using Young's inequality, (if p > 0 and 1 p

0, 3 →Lemma 5

 35 0 as β j → 0. Assume that the spatial dimension d satisfies d ≤ 6. Then the pair (ũ, p) defined by (36) is a solution to (P Darcy ) and hence is the unique solution of (P Darcy ): ũ = u O and p = p O .Proof: A priori, ũ ∈ W β (O) ⊂ W(O) and p ∈ M β (O) ⊂ M(O). It follows from (38) and (37) that

1 2 , 2 ( 1 3 , 3

 1213 However, if v ∈ D(O), then O α O ũ • v dx -O div(v)p dx = 0 so that ∇p = -α O ũ ∈ (L 3 (O)) d ,and thus p ∈ W 1, 3 2 (O). By the Sobolev embedding theorem,[START_REF] Adams | Sobolev Spaces[END_REF] , we have, if d ≤ 6, thenW 1, 3 2 (O) ⊂ L 2 (O) = M(O). Also we have supposed that p ∂,O belongs to W ∂O) as well as to W

2 i=1 2 i=1 2 i=1 3 2

 2223 in particular, for test functions v ∈ (D(Ω 1 )) d × (D(Ω 2 )) d × {0}, Ωi α i ũi • v i dx -Ωi divv i pi dx = 0.Thus ∇p i = -α i ũi ∈ L 2 (Ω i ) and therefore pi ∈ W 1,3 2 (Ω i ). From the Sobolev embedding theorem we then have pi ∈ L 2 (Ω i ) (for d ≤ 6) which means that p ∈ M. Now from the density of W β in W we conclude thatlim β→0 γ β γ |u β,γ |u β,γ • v γ ds = -Ωi α i ũi • v i dxγ α γ ũγ • v γ ds -• n + ξ ũi+1 • n)v i • n ds + b(v, p) + g(v), ∀v ∈ W.Now there remains to see thatlim β→0 γ β γ |u β,γ |u β,γ • v γ ds = γ β γ |ũ γ |ũ γ • v γ ds, ∀v γ ∈ L 3 (γ).Toward this end we define a mapping on L 3 (γ) × L 3 (γ) by(w γ , v γ ) → γ β γ |w γ |w γ • v γ dsand the associated mapping C : L 3 (γ) -→ L (γ).

  of the spaces W p (div, O) for O ⊂ R d a bounded domain in R d and p ∈ R a number with 1 ≤ p: W p (div, O) := {v ∈ (L p (O)) d : divv ∈ L p (O)} (46) with norm v W p (div,O) := v L p (O) + divv L p (O) .

Lemma 7

 7 Let X and Y be Hilbert spaces and let a be a form, linear in its second variable, on ×Y and b a bilinear form on Y × X and let A : Y -→ Y ′ and B : Y -→ X ′ be the associated linear operators defined by < A(v), w > Y ′ ,Y = a(v, w), ∀w ∈ Y and < B(v), r > X ′ ,X = b(v, r), ∀r ∈ X, respectively. Suppose further that f ∈ X ′ and g ∈ Y ′ . Then if A is strictly monotone and B is surjective, the problem(P) Find u ∈ Y and p ∈ X such that a(u, v) -b(v, p) = g(v) ∀v ∈ Y b(u, r) = f (r) ∀r ∈ X(47)has at most one solution.Proof: Suppose that (u, p) and (w, s) ∈ Y × X are solutions to (P). Thena(u, v) -a(w, v) -b(v, p -s) = 0, ∀v ∈ Y b(u -w, r) = 0, ∀r ∈ X,and taking as test functions v = u -w and r = p -s we obtain a(u, u -w) -a(w, u -w) = 0 or < A(u) -A(w), u -w > Y ′ ,Y = 0. Then, as A is strictly monotone we have u = w. To see that p = s we suppose the contrary and use the surjectivity of B to obtain an element v ∈ Y with < B(v), p -s > X ′ ,X = 0. However, as u = w we have 0 = a(u, v) -a(w, v) = b(v, p -s) = < B(v), p -s > X ′ ,X contradicting the choice of v.

  Proposition 1.2] and that the Forchheimer problem (P F orch ) has a unique solution (u β,O , p β,O ) ∈ W β (O)×M β (O), [15, Theorem 1.8]. Again see the more general vesion of this reasoning in Lemma 2.

RR n°8443

RR n°8443RESEARCH CENTRE PARIS -ROCQUENCOURTDomaine de Voluceau, -Rocquencourt B.P. 105 -78153 Le Chesnay Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Acknowledgement: We are grateful to the anonymous referee, who pointed out a considerable simplification of the original proof of Theorem 1.

This work was supported by the Franco-German cooporation program PHC Procope

as in (33), and, as in (32), (34) each of the terms u β,i 0,2 , u β,i • n 0,2,γ , u β,γ 0,2,γ , u β,γ 0,3,γ and β 1 3 u β,i 0,3 is bounded by a positive constant D 5 , depending on α i , α γ , β γ , κ, ξ, f and g but independent of β: u β,i 0,2 + u β,i • n 0,2,γ + u β,γ 0,2,γ + u β,γ 0,3,γ + β 1 3 u β,i 0,3 ≤ D 5 .

(44)

Combining ( 41) and (44) yields an a priori bound on u β,i in the H(div, Ω i )-norm. Equation (41) also gives an a priori bound on

To bound p β we recall (42)

and obtain for a positive constant D 6 , depending on α i , α γ , β γ , κ, ξ, f, g and θ β but independent of β :

which gives the a priori bound on p β M β .

Theorem 2 Suppose d ≤ 6. There exists a unique solution (u, p) ∈ W × M of problem (P), and (u, p) is a weak limit of solutions (u β , p β ) ∈ W × M in the sense made precise below.

Proof: The proof follows from the error bounds (44),(45) obtained in Lemma 6. As the spaces

2 (γ) and L 2 (γ) are reflexive Banach spaces they are sequentially weakly compact. Thus from (32) and (35), we conclude that if {β ℓ } is a sequence converging to 0 then there is a subsequence {β j } such that the sequences {u βj,i }, {p βj ,i }, {u βj,γ }, {p βj ,γ },

and β

We remark that since

We also note that divu βj ,γ ∈ L 2 (γ) so that u βj ,γ ∈ H(div, γ). Further, divu βj ,γ L 2 (γ) and thus u βj,γ H(div,γ) is bounded independently of β so that u βj ,γ converges weakly to ũγ in H(div, γ). Following the same lines of reasoning we conclude that

Thus with ũ = (ũ 1 , ũ2 , ũγ ) we have ũ ∈ W, and it is clear that b(ũ, r) =< Div ũ, r > M ′ , M = f (r), ∀r ∈ M, i. e. the second equation of (P) is satisfied by ũ.

in the W 3 (div) × L

β one solves the auxiliary problem ∆φ = ψ in Ω, φ = 0 on Γ, with right hand side ψ ∈ L 3 (Ω), the function that agrees with ψ i on Ω i . The solution φ is in W 2,3 (Ω) if Ω is sufficiently regular (otherwise we solve the same homogenous Dirichlet problem on a larger more regular domain and take the restriction of the solution to Ω), and φ 2,3,Ω ≤ C ψ 0,3,Ω with a constant C that depends only on Ω.

As

where n i is the exterieur unit normal vector on Ω i , and it follows that

To obtain the third component v γ , note that as v ∈ W 3 (div, Ω) we have

• n 2 = 0 on γ, and thus the problem in the fracture domain γ is decoupled from that in the subdomains Ω 1 and Ω 2 . One has only to define v γ := ∇φ γ where φ γ is the solution of ∆φ γ = ψ γ in γ, φ γ = 0 on ∂γ.

It is straightforward to verify now that

Proof: The proof of this lemma is just as that of Lemma 3 only the auxiliary problem in the subdomains is in the H(div) × L 2 setting. As the auxiliary problems in the subdomains and in the fracture domain decouple no difficulty arises from the fact that one of these involves H(div) × L 2 while the other involves W 3 (div) × L