Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Non-classical shocks to model oil-trapping
aa r X i v : . [ m a t h . A P ] N ov Asymptotic behavior of two-phase flows in heterogeneousporous media for capillarity depending only on space.II. Non-classical shocks to model oil-trapping
Cl´ement Canc`es ∗† November 12, 2018
Abstract
We consider a one-dimensional problem modeling two-phase flow in heterogeneous porous mediamade of two homogeneous subdomains, with discontinuous capillarity at the interface betweenthem. We suppose that the capillary forces vanish inside the domains, but not on the interface.Under the assumption that the gravity forces and the capillary forces are oriented in oppositedirections, we show that the limit, for vanishing diffusion, is not in general the optimal entropysolution of the hyperbolic scalar conservation law as in the first paper of the series [10]. A non-classical shock can occur at the interface, modeling oil-trapping. key words . scalar conservation laws with discontinuous flux, non-classical shock, two-phase flow, porousmedia, discontinuous capillarity AMS subject classification . The models of two-phase flows provide good first approximations to predict the motions of oil inthe subsoil. Although the theoretical knowledge concerning the question of the existence and theuniqueness of the solution to such models for homogeneous porous media [4, 15] and for media withregular enough variations [16] is quite complete, few results are available for discontinuous media, asfor example media made of several rock types [3, 7, 9, 11, 13, 18].One says that oil-trapping occurs when some oil can not pass through interfaces between differentrocks. Such a phenomenon plays an important role in the basin modeling, to predict the position ofeventual reservoirs where oil could be collected. As already explained in [7, 35], discontinuities of thecapillary pressure field can induce the so-called oil-trapping phenomenon.The effects of capillarity, which play a crucial role in oil-trapping, seem to play a less importantrole concerning the motion of oil in homogeneous porous media, and can sometime be neglected toprovide the so-called Buckley-Leverett equation.In this paper, we show that even if the dependence of the capillary pressure with respect to theoil-saturation of the fluid vanishes, the capillary pressure field still plays a crucial role to determinethe saturation profile. In order to carry out this study, we restrict our frame to the one-dimensionalcase. We will strongly use some recent results [9, 11, 13] obtained on flows in heterogeneous mediawith discontinuous capillary forces.We consider a one-dimensional porous medium, made of two different rocks, represented by Ω = R ⋆ − and Ω = R ⋆ + . Let π ( u, x ) be the capillary pressure, then it it is well known (see e.g. the ∗ UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France ( [email protected] ) † The author is partially supported by GNR MoMaS ∂ t u + ∂ x (cid:16) qc ( u, x ) + g ( u, x ) (1 − C∂ x π ( u, x )) (cid:17) = 0 , (1)where u is the oil saturation of the fluid, q is the total flow rate, supposed to be a nonnegative constant, C is a constant depending on the buoyancy forces and c ( u, x ) = c i ( u ) , g ( u, x ) = g i ( u ) , and π ( u, x ) = π i ( u ) if x ∈ Ω i . The functions c i are supposed to be increasing and Lipschitz continuous with c i (0) = 0 and c i (1) = 1,while g i are supposed to be Lipschitz continuous, strictly positive in (0 ,
1) satisfying g i (0) = g i (1) = 0and π i are increasing Lipschitz continuous functions.Physical experiments suggest that the dependence of π i with respect to u can be weak, at leastfor u far from 0 and 1. So we want to choose π ( u ) = P , and π ( u ) = P . The equation (1) turnsformally to the scalar conservation law with discontinuous flux function ∂ t u + ∂ x f ( u, x ) = 0 , (2)where f ( u, x ) (resp. f i ( u )) is equal to qc ( u, x ) − g ( u, x ) (resp. qc i ( u ) − g i ( u )).Such conservation laws have been widely studied in the last years. For a large overview on thistopic, we refer to the introduction of [8], or in a lesser extent to the associated paper [10]. In particular,it has been proven by Adimurthi, Mishra and Veerappa Gowda [2] that there might exist an infinitenumber of L -contraction semi-groups corresponding to the equation (2). Among them, in the casewhere the functions f i have at most a single extremum in (0,1), we mention the so-called optimalentropy solution which corresponds to the unique entropy solution in the case of a continuous fluxfunction f = f = f . We refer to [2] and to the first part of this communication [10] for a discussionon the so-called optimal entropy condition .In the sequel of this paper, we suppose that (H1) for i ∈ { , } , there exists a value u ⋆i ∈ [0 ,
1) such that f i ( u ⋆i ) = q , f i is increasing on [0 , u ⋆i ] and f i ( s ) > q for all s ∈ ( u i , ϕ i ( u ) = C Z u g i ( s ) ds. For technical reasons, we have to assume that (H2) there exist
R > α > m ∈ (0 ,
1) such that f ◦ ϕ − ( s ) ≥ q + R ( ϕ (1) − s ) m if s ∈ [ ϕ (1) − α, ϕ (1)] . (3)These assumptions are fulfilled by models widely used by the engineers, for which a classical choice of c i , g i is c i ( u ) = u a i u α i + ab (1 − u ) β i , g i ( u ) = K i u α i (1 − u ) β i bu a i + a (1 − u ) β i , where α i , β i ≥ a, b are given constants.The goal of this paper is to show that if the capillary forces at the level of the interface { x = 0 } are oriented in the opposite sense with respect to the gravity forces (in our case P < P ), then a2 u ⋆ u ⋆ f q Figure 1: example of functions f i satisfying Assumption (H1) . Note the we have note supposed, as itis done in [1, 8], that f i has a single local extremum in (0 , q . non classical stationary shock can occur at the interface. It was shown by Kaasschieter [24] that ifthe capillary pressure field is continuous at the interface (corresponding to the case P = P ), thenthe good notion of solution is the one of optimal entropy solution , computed by Adimurthi, Jaffr´e andVeerappa Gowda using a Godunov-type scheme [1]. We have pointed out in [10] that if the capillaryforces and the gravity forces are oriented in the same sense, the good notion of solution is also the oneof optimal entropy solution . If the assumptions stated above are fulfilled, if P < P and if the initialdata u is large enough to ensure that both phases move in opposite directions, i.e. u ⋆i ≤ u ( x ) ≤ i , (4)we will show that the limit is not the optimal entropy solution, but the entropy solution to the problem ∂ t u + ∂ x f i ( u ) = 0 ,u ( x = 0 − ) = 1 and u ( x = 0 + ) = u ⋆ ,u ( t = 0) = u . ( P lim )In the sequel, we denote by a + (resp. a − ) the positive (resp. negative) part of a , i.e. max(0 , a ) (resp.max(0 , − a )), and for i = 1 ,
2, for u, κ ∈ [0 , i + ( u, κ ) = (cid:26) f i ( u ) − f i ( κ ) if u ≥ κ, i − ( u, κ ) = (cid:26) f i ( κ ) − f i ( u ) if u ≤ κ, i ( u, κ ) = Φ i + ( u, κ ) + Φ i − ( u, κ ) = f i (max( u, κ )) − f i (min( u, κ )) . We can now define the notion of solution to ( P lim ), which is in fact an entropy in each subdomain Ω i ,with an internal boundary condition at the level of the interface. Definition 1.1 (solution to ( P lim )) Let u ∈ L ∞ ( R ) , u ⋆i ≤ u ( x ) ≤ a.e. in Ω i , A function u issaid to be a solution of ( P lim ) if it belongs to L ∞ ( R × R + ) , u ⋆i ≤ u ≤ a.e. in Ω i × (0 , T ) , and for = 1 , , for all ψ ∈ D + (Ω i × R + ) , for all κ ∈ [0 , , Z R + Z Ω i ( u ( x, t ) − κ ) ± ∂ t ψdxdt + Z Ω i ( u ( x ) − κ ) ± ψ ( x, dx + Z R + Z Ω i Φ i ± ( u ( x, t ) , κ ) ∂ x ψ ( x, t ) dxdt + M f i Z R + ( u i − κ ) ± ψ (0 , t ) dt ≥ , (5) where M f i is a Lipschitz constant of f i , and u = 1 , u = u ⋆ . For a given u in L ∞ ( R ), there exists a unique solution u to ( P lim ) in the sense of Definition 1.1, whichis in fact made on an apposition of two entropy solutions in R ± × R + . We refer to [27, 28] and [38] forproofs of existence and uniqueness to solutions to the problem ( P lim ). Moreover, thanks to [12], onecan suppose that u belongs to C ( R + ; L loc ( R )). Theorem 1.2
Let u ∈ L ∞ ( R ) with u ⋆i ≤ u ≤ a.e. in Ω i , then there exists a unique solution to ( P lim ) in the sense of Definition 1.1. Furthermore, if v is another solution to ( P lim ) corresponding to v ∈ L ∞ ( R ) with u ⋆i ≤ v ≤ a.e. in Ω i , then for all R > , for all t ∈ R + Z R − R ( u ( x, t ) − v ( x, t )) ± dx ≤ Z R + M f t − R − M f t ( u ( x ) − v ( x )) ± dx (6) where M f is a Lipschitz constant of both f i . Assume now that both phases move in the same direction:0 ≤ u ( x ) ≤ u ⋆i a.e. in Ω i , (7)then it will be shown that the relevant solution u to the problem is the unique entropy solution definedbelow. Definition 1.3
A function u is said to be an entropy solution if it belongs to L ∞ ( R × R + ) , ≤ u ≤ u ⋆i a.e. in Ω i × (0 , T ) , and for i = 1 , , for all ψ ∈ D + ( R × R + ) , for all κ ∈ [0 , , Z R + Z R | u ( x, t ) − κ | ∂ t ψdxdt + Z R | u ( x ) − κ | ψ ( x, dx + Z R + X i ∈{ , } Z Ω i Φ i ( u ( x, t ) , κ ) ∂ x ψ ( x, t ) dxdt + | f ( κ ) − f ( κ ) | Z R + ψ (0 , t ) dt ≥ . (8)Thanks to Assumption (H1) , there exist no χ ∈ [0 , max u ⋆i ] such that f ( χ ) = f ( χ ), f is decreasingand f is increasing on ( χ − δ, χ + δ ) for some δ >
0. Then the notion of entropy solution described by(8) introduced by Towers [33, 34] is equivalent to the notion of optimal entropy solution introduced in[2] (see also [8]). We take advantage of this by using the very simple algebraic relation (8).It has been proven that the entropy solution u exists and is unique for general flux functions f [6,Chapters 4 and 5]. In particular, the following comparison and L -contraction principle holds. Theorem 1.4
Let u ∈ L ∞ ( R ) with ≤ u ≤ u ⋆i a.e. in Ω i , then there exists a unique entropysolution in the sense of Definition 1.3. Furthermore, if v is another entropy solution corresponding to v ∈ L ∞ ( R ) with ≤ v ≤ u ⋆i a.e. in Ω i , then for all R > , for all t ∈ R + Z R − R ( u ( x, t ) − v ( x, t )) ± dx ≤ Z R + M f t − R − M f t ( u ( x ) − v ( x )) ± dx (9) where M f is a Lipschitz constant of both f i . .1 non classical shock at the interface As already mentioned, the optimal entropy solution can be seen as a extension to the case of discon-tinuous flux functions of the usual entropy solution [25] obtained for a regular flux function. We willnow illustrate that it is not the case with the solution to ( P lim ). Assume for the moment (it will beproved later) that in the case where u ( x ) ∈ ( u ⋆i ,
1) a.e. in Ω i , the corresponding solution u to ( P lim )admits u i as strong trace on the interface. One has the following Rankine-Hugoniot relation f ( u ) = f ( u ) = q, then u is a weak solution to (2), i.e. it satisfies for all ψ ∈ D ( R × R + ): Z R + Z R u∂ t ψ dxdt + Z R u ψ ( · , dx + Z R + Z R f ( u, · ) ∂ x ψ dxdt = 0 . (10)Firstly, suppose for the sake of simplicity that f ( u ) = f ( u ) = f ( u ), and that q = 0, then u ⋆i = 0for i ∈ { , } . The function u ( x ) = (cid:26) x < , x > P lim ) satisfying (5). However, since f (1) − f ( s )1 − s < s ∈ (0 , , the discontinuity at { x = 0 } does not fulfill the usual Oleinik entropy condition (see e.g. [31]). Thisdiscontinuity is thus said to be a non-classical shock .Suppose now that f ′ (1) < f ′ ( u ⋆ ) >
0, then the pair (1 , u ⋆ ) is a stationary under-compressible shock-wave , that are prohibited for optimal entropy solutions [2] as for classical entropysolutions in the case of regular flux functions. Remark. connection (
A, B ) , i.e. a stationaryundercompressible wave between the left state A and the right state B at the interface lead to another L -contraction semi-group (see [2, 8, 21]), which is so-called entropy solution of type ( A, B ) . However,we rather use the denomination non-classical shock for the connection between A and B since, asstressed above, the corresponding solution violates some fundamental properties of the classical entropysolutions. In this section, we assume that q = 0. Let u be the solution of the problem ( P lim ) corresponding tothe initial data u . Assume that u admits strong traces on the interface. The flow-rate of oil goingfrom Ω to Ω through the interface is given by f ( u ) = f ( u ) = 0 . Thus the oil cannot overcome the interface from Ω to Ω , thus if one supposes that u belongs to L ∞ ( R ), with 0 ≤ u ≤ x = − R ( R is an arbitrarypositive number) and x = 0 can only grow.Indeed, let t > t ≥
0, let ζ n ( x ) = min(1 , n ( x + R ) + , nx − ) and θ m ( t ) = min(1 , m ( t − t ) , m ( t − t )).Choosing ψ ( x, t ) = ζ n ( x ) θ m ( t ) in (10) for m, n ∈ N yields, using the positivity of f Z t t (cid:18)Z − R u ( x, t ) ζ n ( x ) dx (cid:19) ∂ t θ m ( t ) dt + Z t t θ m ( t ) n Z − /n f ( u ( x, t )) dx ! dt ≤ . u admits a strong trace on the interface,lim n →∞ n Z − /n f ( u ( x, t )) dx = f ( u ) = 0 . Then we obtain Z t t (cid:18)Z − R u ( x, t ) dx (cid:19) ∂ t θ m ( t ) dt ≤ . (11)The solution u belong to C ( R + ; L ( R )) thanks to [12], thus taking the limit as m → ∞ in (11) provides Z − R u ( x, t ) dx ≤ Z − R u ( x, t ) dx. Suppose now that q ≥
0. Thanks to what follows, we are able to solve the Riemann problem atthe interface for any initial data u ( x ) = (cid:26) u ℓ if x < ,u r if x > . The study of the Riemann problem is carried out in Section 5, leading to the following result. • If u ℓ > u ⋆ , then u = 1 and u = u ⋆ . We obtain the expected non-classical shock at the interface. • If u ℓ ≤ u ⋆ , then u = u ℓ and u is the unique value of [0 , u ⋆ ] such that f ( u ) = f ( u ℓ ).Using Assumption (H1) , this particularly implies that in both cases, the flux at the interface is givenby f ( u ) = f ( u ) = G ( u ℓ ,
1) (12)where G is the Godunov solver corresponding to the flux function f : G ( a, b ) = min s ∈ [ a,b ] f ( s ) if a ≤ b, max s ∈ [ b,a ] f ( s ) if a > b. This particularly yields that for any initial data u ∈ L ∞ ( R ) with 0 ≤ u ≤
1, the restriction u | Ω1 ofthe solution u to Ω is the unique entropy solution to ∂ t u + ∂ x f ( u ) = 0 in Ω × R + ,u ( · ,
0) = u in Ω ,u (0 , · ) = γ in R + (13)for γ = 1. Since the solution u to the problem (13) is a non-decreasing function of the prescribed trace γ on { x = 0 } , we can claim as in [10] that u | Ω1 = sup γ ∈ L ∞ ( R + )0 ≤ γ ≤ { v solution to (13) } . In particular, u is the unique weak solution (i.e. satisfying (10)) that is entropic in each subdomainand that minimizes the flux through the interface. We will introduce a family of approximate problems in Section 2, which takes into account the capil-larity, with small dependance ε of the capillary pressure with respect to the saturation. We use thetransmission conditions introduced in [9, 11, 13, 30] to connect the capillary pressure at the interface.For ε >
0, the problem ( P ε ) admits a unique solution u ε thanks to [11] and it is recalled that a6omparison principle holds for the solutions of the approximate problem ( P ε ). Particular sub- andsuper-solution are derived in order to show that if u ( x ) ≥ u ⋆I a.e. in Ω i , then the limit u of theapproximate solutions ( u ε ) ε> as ε tends to 0. An energy estimate is also derived.In Section 3, letting ε tend to 0, since no strong pre-compactness can be derived on ( u ε ) ε > L loc ( R × R + ) from the available estimates, we use the notion of process solution [20], which isequivalent to the notion of measure valued solution introduced by DiPerna [17] (see also [27, 32]). Theuniqueness of such a process solution allows us to claim that ( u ε ) converges strongly in L loc (( R × R + )towards the unique solution to ( P lim ).In Section 4, it is shown that if both phases move in the same direction, that is if 0 ≤ u ≤ u ⋆i a.e. in Ω i , then ( u ε ) converges towards the unique entropy solution to the problem in the sense ofDefinition 1.3.In Section 5, we complete the study of the Riemann problem at the interface. In this section, we take into account the effects of the capillarity, supposing that they are small. Wewill so build an approximate problem ( P ε ), whose unknown u ε will depend on a small parameter ε representing the dependance of the capillary pressure with respect to the saturation. We assume forthe sake of simplicity that the capillary pressure in Ω i is given by: π εi ( u ε ) = P i + εu ε . (14)It has been shown simultaneously in [9] and in [13] that a good way to connect the capillary pressuresat the interface is to require ˜ π ε ( u ε ) ∩ ˜ π ε ( u ε ) = ∅ , (15)where u ε and u ε are the traces of u ε on the interface, and where ˜ π εi is the monotonous graph given by˜ π εi ( s ) = π εi ( s ) if s ∈ (0 , , ( −∞ , P i ] if s = 0 , [ P i + ε, ∞ ) if s = 1 . We suppose that the capillary force is oriented in the sense of decreasing x , i.e. P < P (thecapillary force goes from the high capillary pressure to the low capillary pressure). Since ε is assumedto be a small parameter, we can suppose that 0 < ε < P − P , so that the relation (15) turns to u ε = 1 or u ε = 0 . (16)The flux function in Ω i is then given by: F εi ( x, t ) = f i ( u ε )( x, t ) − ε∂ x ϕ i ( u ε )( x, t ) . Because of the conservation of mass, we require the continuity of the flux functions at the interface.Thus the approximate problem becomes ∂ t u ε + ∂ x F εi = 0 ,u ε ( x = 0 − ) = 1 or u ε ( x = 0 + ) = 0 ,F ε (0 − ) = F ε (0 + ) ,u ( t = 0) = u . ( P ε )We are not able to prove the uniqueness of a weak solution of ( P lim ) if the flux F εi ”only” belongsto L (Ω i × R + ), and we will define the notion of prepared initial data, so that the flux belongs to L ∞ (Ω i × R + ). In this latter case, the uniqueness holds.7 .1 bounded flux solutions We define now the notion of bounded flux solution, that was introduced in this framework in [11, 13].
Definition 2.1 (bounded flux solution to ( P ε )) Let u ∈ L ∞ ( R ) , ≤ u ≤ , a function u ε issaid to be a bounded flux solution if1. u ε ∈ L ∞ ( R × R + ) , ≤ u ≤ ;2. ∂ x ϕ i ( u ε ) ∈ L ∞ (Ω i × R + ) ∩ L loc ( R + ; L (Ω i )) ;3. u ε ( t ) (1 − u ε ( t )) = 0 for almost all t ≥ , where u εi denotes the trace of u ε | Ω i on { x = 0 } .4. ∀ ψ ∈ D ( R × R + ) , Z R + Z R u ε ( x, t ) ∂ t ψ ( x, t ) dxdt + Z R u ( x ) ψ ( x, dx + Z R + X i ∈{ , } Z Ω i [ f i ( u ε ) − ε∂ x ϕ i ( u ε )] ∂ x ψ ( x, t ) dxdt = 0 . (17) Remark. u ε solution belongs to C ( R + ; L ( R )) , in the sense that there exists ˜ u ε in C ( R + ; L ( R )) such that u ε ( t ) = ˜ u ε ( t ) for almost all t ≥ (see [12]). More precisely, all t ≥ is a Lebesgue point for u ε . So, the slight abuse of notation consisting in considering u ε ( t ) for all t ≥ will not lead to any confusion. Proposition 2.3
Let u and v be two bounded-flux solutions associated to initial data u , v , then forall ψ ∈ D + ( R × R + ) , Z R + Z R ( u − v ) ± ∂ t ψdxdt + Z R ( u − v ) ± ψ ( · , dx + Z R + X i Z Ω i (cid:16) Φ i ± ( u, v ) − ε∂ x ( ϕ i ( u ) − ϕ i ( v )) ± (cid:17) ∂ x ψdxdt ≥ . (18)We state now a theorem which is a generalization in the case of unbounded domains of Theorem 3.1and Theorem 4.1 stated in [11]. Theorem 2.4 (existence–uniqueness for bounded flux solutions)
Let u ∈ L ∞ ( R ) with ≤ u ≤ such that: • there exists a function ˆ u ∈ L ∞ ( R ) , with ≤ ˆ u ≤ a.e. in R , satisfying ∂ x ˆ u ∈ L ( R ) ∩ L ∞ ( R ) and such that ( u − ˆ u ) ∈ L ( R ) • ∂ x ϕ i ( u ) ∈ L ∞ (Ω i ) • lim x ր u ( x ) = 1 or lim x ց u ( x ) = 0 .Then there exists a unique bounded flux solution u ε to the problem ( P ε ) in the sense of Definition 2.1satisfying ( u ε − ˆ u ) ∈ L ( R ) . Furthermore, if u ε , v ε are two bounded flux solutions associated to initialdata u , v then u ( x ) ≥ v ( x ) a.e. in R ⇒ u ε ( x, t ) ≥ v ε ( x, t ) a.e. in R for all t ≥ . (19)8bviously, the existence of a bounded flux solution can not be extended to any initial data in L ( R ).Indeed, the initial data u has at least to involve bounded initial flux, i.e. ∂ x ϕ i ( u ) ∈ L ∞ ( R ). Anadditional natural assumption is needed to ensure the existence of such a bounded flux solution : theconnection in the graphical sense of the capillary pressures at the interface.If ( u − ˆ u ) and ( v − ˆ u ) belong to L for the same ˆ u , then (18) yields that the bounded flux solutions u ε and v ε corresponding to u and v satisfy the following contraction principle: ∀ t ∈ R + , Z R ( u ε ( x, t ) − v ε ( x, t )) ± dx ≤ Z R ( u ( x ) − v ( x )) ± dx, providing the uniqueness result stated in Theorem 2.4. We will study particular steady states of the approximate problem ( P ε ). We will consider steadybounded flux solutions s ε corresponding to a zero water flow rate, i.e. f i ( s ε ) − ε dd x ϕ i ( s ε ) = q in Ω i . (20)For ε >
0, there are infinitely many solutions s ε of the equation (20). We will construct some particularsolutions, that will permit us to show that the limit u as ε tends to 0 of bounded flux solutions u ε corresponding to large initial data admits the expected strong traces on the interface { x = 0 } .We will introduce now particular solutions of the ordinary differential equation y ′ = f i ◦ ϕ − i ( y ) − q. (21) Lemma 2.5
Let η > , there exists a solution y η to (21) for i = 1 which is nondecreasing on ( −∞ , − equal to ϕ (1) on [ − η, , satisfying y η ( x ) < ϕ (1) if x < − η and lim x →−∞ y η ( x ) = u ⋆ .Proof: Consider the problem (cid:26) w ′ ( x ) = R ( ϕ i (1) − w ( x )) m if x < − η,w ( − η ) = ϕ (1) , (22)where R and m are constants given by Assumption (H2) . The function w η ( x ) = ϕ i (1) − ( R (1 − m )( − x − η )) − m is a solution of (22). Because of (H2) , there exists a neighborhood ( − η − δ, − η ] of η such that w η isa super-solution of the problem (cid:26) y ′ ( x ) = f ◦ ϕ − ( y ) − q if x < − η,y ( − η ) = ϕ (1) . (23)Then there exists y η solution to (23) such that y η ( x ) = ϕ (1) if x ∈ ( − η,
0) and y η ( x ) ≤ w η ( x ) on ( − η − δ, − η ] . In particular, y η is not constant equal to 1. Thanks to (H1) , the function y η is increasing on the set { x ∈ Ω | y η ( x ) ∈ ( ϕ ( u ⋆ ) , ϕ (1)) } . Assume that there exists x ⋆ < − η such that y η ( x ⋆ ) = ϕ ( u ⋆ ),then one sets y η ( x ) = ϕ ( u ⋆ ) for all x ∈ ( −∞ , x ⋆ ]. If y η ( x ) > ϕ ( u ⋆ ) for all x <
0, then y η is increasingon ( −∞ , − η ). Thus it admits a limit as x tends to −∞ , and it is clear that the only possible limit is u ⋆ . (cid:3) emma 2.6 Let η > , then there exists a solution z η to (21) for i = 2 which is nondecreasing on R satisfying z η ( x ) ≤ ϕ (cid:16) u ⋆ (cid:17) if x ≤ η , z η ( x ) ≥ ϕ (cid:16) u ⋆ (cid:17) if x ≥ η and lim x →∞ z η ( x ) = ϕ (1) , lim x →−∞ z ( x ) = u ⋆ .Proof: The problem z ′ ( x ) = f ◦ ϕ − ( z ( x )) − q if x ∈ R ,z ( η ) = ϕ (cid:16) u ⋆ (cid:17) . admits a (unique) solution in C ( R , [0 , u ⋆ is a constant solution of (21) for i = 2, then onehas z ( x ) ≥ u ⋆ in R . Thanks to (H1) , the function z is nondecreasing. This implies that it admitslimits respectively in −∞ and in + ∞ . The only possible values for this limits are respectively u ⋆ and1. (cid:3) Proposition 2.7
Let η > , then there exists two families of steady bounded flux solutions ( s ε,η ) ε> and ( s ε,η ) ε> tending in L loc ( R ) as ε → respectively towards s η : x u ⋆ if x < − η, if x ∈ ( − η, ,u ⋆ if x > , and s η : x if x < ,u ⋆ if x ∈ (0 , η ) , if x > η. Proof:
We set s ε,η ( x ) = (cid:26) y η (cid:0) x + ηε − η (cid:1) if x < ,u ⋆ if x > , (24)and s ε,η ( x ) = (cid:26) x < ,z η (cid:0) x − ηε + η (cid:1) if x > , (25)where the functions y η and z η have been defined in Lemmas 2.5 and 2.6. Since the functions ϕ i ( s ε,η )and ϕ i ( s ε,η ) are monotone in Ω i , there derivatives dd x ϕ i ( s ε,η ) and dd x ϕ i ( s ε,η ) belong to L ( R ), and alsoto L ∞ ( R ) because s ε,η and s ε,η are solutions to (20). Thus they belong to L ( R ). Hence, for fixed ε , s ε,η and s ε,η are bounded flux solutions to the problem ( P ε ). The convergence as ε → s η and s η is a direct consequence of Lemmas 2.5 and 2.6. (cid:3) L ((0 , T ); H (Ω i )) estimate Our goal is now to derive an estimate which ensures that the effects of capillarity vanish almosteverywhere in Ω i × R + as ε tends to 0. Proposition 2.8
Let u ∈ L ∞ ( R ) with ≤ u ≤ a.e. satisfying the assumptions of Theorem 2.4and let u ε be the corresponding bounded flux solution. Then for all ε ∈ (0 , , for all T > , thereexists C depending only on u , g i , ϕ i , T such that √ ε k ∂ x ϕ i ( u ε ) k L (Ω i × (0 ,T )) ≤ C. (26) This particularly ensures that ε k ∂ x ϕ i ( u ε ) k L (Ω i × (0 ,T )) → as ε → . (27)10he idea of the proof of Proposition 2.8 is formally to choose ( u ε − ˆ u ) ψ as test function in (17) for afunction x ψ ( x ) compactly supported in Ω i . Using the fact that the flux F εi is uniformly boundedin L ∞ (Ω i × (0 , T )), we can let ψ tend towards χ Ω i , with χ Ω i ( x ) = 1 if x ∈ Ω i and 0 otherwise,and the estimate (26) follows. To obtain (27), it suffices to multiply (26) by √ ε . We refer to [10,Proposition 2.3] for a more details on the proof of Proposition 2.8. In order to ensure that the limit u of the approximate solutions u ε as ε → { x = 0 } , we will perturb the initial data u . Lemma 2.9
Let u ∈ L ∞ ( R ) satisfying (4) , then there exists ( u ε,η ) ε,η such that(a). s ε,η ( x ) ≤ u ε,η ( x ) ≤ s ε,η ( x ) a.e. in R , where the functions s ε,η and s ε,η are defined in (24) - (25) ,(b). ε k ∂ x ϕ i ( u ε,η ) k L ∞ (Ω i ) ≤ C where C depends neither on ε nor on η ,(c). u ε,η → u in L loc ( R ) as ( ε, η ) → (0 , .Proof: Let ( ρ n ) n ∈ N ⋆ be a sequence of mollifiers, then ρ n ∗ u is a smooth function tending u as n → ∞ . Then, for ε >
0, we choose n ∈ N ⋆ such thatmax n n, k ∂ x ϕ i ( u ∗ ρ n ) k L ∞ (Ω i ) o ≥ ε , (28)and we define u ε,η ( x ) = max { s ε,η ( x ) , min { s ε,η ( x ) , u ∗ ρ n ( x ) }} . (29)The point (a) is a direct consequence of (29). Letting ( ε, η ) → (0 ,
0) in (29) yieldslim ( ε,η ) → (0 , u ε,η ( x ) = max { u ⋆i , min { , u ( x ) }} . Since u is supposed to satisfy (4), this provideslim ( ε,η ) → (0 , u ε,η ( x ) = u ( x ) a.e. in R . The point (c) follows. In order to establish (b), it suffices to note that there exist an open subset ω of R such that u ε,η ( x ) is equal to u ∗ ρ n ( x ) for x ∈ ω , and such that u ε,η ( x ) is either equal to s ε,η ( x ) orto s ε,η ( x ) on ω c = R \ ω . It follows from (28) that ε k ∂ x ϕ i ( u ε,η ) k L ∞ (Ω i ∩ ω ) ≤ . One has f i ( u ε,η )( x ) − ε∂ x ϕ i ( u ε,η )( x ) = q a.e. in Ω i ∩ ω c , thus ε k ∂ x ϕ i ( u ε,η ) k L ∞ (Ω i ∩ ω c ) ≤ k q − f i k L ∞ ( u ⋆i , . This concludes the proof of Lemma 2.9. (cid:3)
Definition 2.10
A function u is said to be a prepared initial data if it satisfies (1 − u ) ∈ L ( R ) , ∂ x ϕ i ( u ) ∈ L ∞ (Ω i ) and s ε,η ( x ) ≤ u ( x ) ≤ s ε,η ( x ) a.e. in R (30) for some ε > , η > . ε, η ) s ε,η is decreasing with respect to both arguments and since the function( ε, η ) s ε,η is increasing with respect to both arguments, if u satisfies (30) for ε = ε and η = η ,then u satisfies (30) for all ( ε, η ) such that ε ≤ ε and η ≤ η . So the following Proposition is a directconsequence from (19). Proposition 2.11
Let u be a prepared initial data satisfying (30) for ε = ε and η = η , then for all ε ≤ ε , for all η ≤ η , the solution u ε to ( P ε ) satisfies s ε,η ( x ) ≤ u ε ( x, t ) ≤ s ε,η ( x ) for a.e. ( x, t ) ∈ R × R + . Since ( u ε ) ε is uniformly bounded between 0 and 1, there exists u ∈ L ∞ ( R × (0 , T )) such that u ε → u is the L ∞ weak-star sense. This is of course insufficient to pass in the limit in the nonlinear terms.Either greater estimates are needed, like for example a BV -estimate introduced in the work of Vol ′ pert[37] and in [10], or we have to use a weaker compactness result. This idea motivates the introductionof Young measures as in the papers of DiPerna [17] and Szepessy [32], or equivalently the notion ofnonlinear weak star convergence, introduced in [19] and [20], which leads to the notion of processsolution given in Definition 3.2. Theorem 3.1 (Nonlinear weak star convergence)
Let Q be a Borelian subset of R k , and ( u n ) bea bounded sequence in L ∞ ( Q ) . Then there exists u ∈ L ∞ ( Q × (0 , , such that up to a subsequence, u n tends to u ”in the non linear weak star sense” as n → ∞ , i.e.: ∀ g ∈ C ( R , R ) ,g ( u n ) → Z g ( u ( · , α )) dα for the weak star topology of L ∞ ( Q ) as n → ∞ . We refer to [17] and [20] for the proof of Theorem 3.1.
Because of the lack of compactness, we have to introduce the notion of process solution, inspired fromthe notion of measure valued solution introduced by DiPerna [17].
Definition 3.2 (process solution to ( P lim )) A function u ∈ L ∞ ( R × R + × (0 , is said to be aprocess solution to ( P lim ) if ≤ u ≤ and for i = 1 , , ∀ ψ ∈ D + (Ω i × R + ) , ∀ κ ∈ [0 , , Z R + Z Ω i Z ( u ( x, t, α ) − κ ) ± ∂ t ψ ( x, t ) dαdxdt + Z Ω i ( u ( x ) − κ ) ± ψ ( x, dx + Z R + Z Ω i Z Φ i ± ( u ( x, t, α ) , κ ) ∂ x ψ ( x, t ) dαdxdt + M f i Z R + ( u i − κ ) ± ψ (0 , t ) dt ≥ , where M f i is any Lipschitz constant of f i , u = 1 and u = u ⋆ . Lemma 3.3
Let u be a η -prepared initial data in the sense of Definition 2.10 for some η > , andlet ( u ε ) ε be the corresponding family of approximate solutions. Then u ε ( x, t ) → for a.e. ( x, t ) ∈ ( − η, × R + , (31) u ε ( x, t ) → u ⋆ for a.e. ( x, t ) ∈ (0 , η ) × R + . (32)12 roof: Firstly, since u is a η -prepared initial data, there exists ε > s ε ,η ≤ u ≤ s ε ,η . Then it follows from Proposition 2.11 that for all ε ∈ (0 , ε ), for a.e. ( x, t ) ∈ R × R + s ε,η ( x ) ≤ u ε ( x, t ) ≤ s ε,η ( x ) . (33)This particularly shows that for all ε ∈ (0 , ε ), for a.e. ( x, t ) ∈ ( − η, × R + , u ε ( x, t ) = 1 , thus (31) holds. The assertion (32) can be obtained by using Proposition 2.7 and the dominatedconvergence theorem. (cid:3) Proposition 3.4 (convergence towards a process solution)
Let u be a prepared initial data inthe sense of Definition 2.10, and let ( u ε ) ε be the corresponding family of approximate solutions. Then,up to an extraction, u ε converges in the nonlinear weak-star sense towards a process solution u to theproblem ( P lim ).Proof: Since u ε is a weak solution of ( P ε ), which is a non-fully degenerate parabolic problem, i.e. ϕ − i is continuous, it follows from the work of Carrillo [14] that u ε is an entropy weak solution, i.e.: ∀ ψ ∈ D + (Ω i × R + ), ∀ κ ∈ [0 , Z R + Z Ω i ( u ε ( x, t ) − κ ) ± ∂ t ψ ( x, t ) dxdt + Z Ω i ( u ( x ) − κ ) ± ψ ( x, dx + Z R + Z Ω i (cid:2) Φ i ± ( u ε ( x, t ) , κ ) − ε∂ x ( ϕ i ( u ε )( x, t ) − ϕ i ( κ )) ± (cid:3) ∂ x ψ ( x, t ) dxdt ≥ . This family of inequalities is only available for non-negative functions ψ compactly supported in Ω i ,and so vanishing on the interface { x = 0 } . To overpass this difficulty, we use cut-off functions χ i,δ .Let δ >
0, we denote by χ i,δ a smooth non-negative function, with χ i,δ ( x ) = 0 if x / ∈ Ω i , and χ i,δ ( x ) = 1 if x ∈ Ω i , | x | ≥ δ . Let ψ ∈ D + (Ω × R + ), then ψχ i,δ ∈ D + (Ω i × R + ) can be used as testfunction in (34). This yields Z R + Z Ω i ( u ε − κ ) ± ∂ t ψχ i,δ dxdt + Z Ω i ( u − κ ) ± ψ ( · , χ i,δ dx + Z R + Z Ω i (cid:2) Φ i ± ( u ε , κ ) − ε∂ x ( ϕ i ( u ε ) − ϕ i ( κ )) ± (cid:3) ∂ x ψχ i,δ dxdt + Z R + Z Ω i (cid:2) Φ i ± ( u ε , κ ) − ε∂ x ( ϕ i ( u ε ) − ϕ i ( κ )) ± (cid:3) ψ∂ x χ i,δ dxdt ≥ . (34)We can now let ε tend to 0. Thanks to Theorem 3.1, there exists u ∈ L ∞ ( R × R + × (0 , ε → Z R + Z Ω i ( u ε ( x, t ) − κ ) ± ∂ t ψ ( x, t ) χ i,δ ( x ) dxdt = Z R + Z Ω i Z ( u ( x, t, α ) − κ ) ± ∂ t ψ ( x, t ) χ i,δ ( x ) dαdxdt, (35)lim ε → Z R + Z Ω i Φ i ± ( u ε ( x, t ) , κ ) ∂ x ψ ( x, t ) χ i,δ ( x ) dxdt = Z R + Z Ω i Z Φ i ± ( u ( x, t, α ) , κ ) ∂ x ψ ( x, t ) χ i,δ ( x ) dαdxdt. (36)13hanks to Proposition 2.8, one has ε∂ x ( ϕ i ( u ε ) − ϕ i ( κ )) ± tends to 0 a.e. in Ω i × (0 , T ) as ε → , then lim ε → Z R + Z Ω i ε∂ x ( ϕ i ( u ε )( x, t ) − ϕ i ( κ )) ± ∂ x ( ψ ( x, t ) χ i,δ ( x )) dxdt = 0 . (37)Since u is supposed to be a η -prepared initial data for some η >
0, we can claim thanks to Lemma 3.3that u ε ( x, t ) converges almost everywhere on ( − η, η ) × R + towards u i if x ∈ Ω i . Since for δ < η smallenough, the support of ∂ x χ ,δ is included in the set where u ε converges strongly, one haslim ε → Z R + Z Ω i Φ i ± ( u ε ( x, t ) , κ ) ψ ( x, t ) ∂ x χ i,δ ( x ) dxdt = Z R + Z Ω i Φ i ± ( u i , κ ) ψ ( x, t ) ∂ x χ i,δ ( x ) dxdt. (38)We let now δ tend to 0. Since χ i,δ ( x ) tends to 1 a.e. in Ω i , (35) and (36) respectively providelim δ → lim ε → Z R + Z Ω i ( u ε ( x, t ) − κ ) ± ∂ t ψ ( x, t ) χ i,δ ( x ) dxdt = Z R + Z Ω i Z ( u ( x, t, α ) − κ ) ± ∂ t ψ ( x, t ) dαdxdt, (39)lim δ → lim ε → Z R + Z Ω i Φ i ± ( u ε ( x, t ) , κ ) ∂ x ψ ( x, t ) χ i,δ ( x ) dxdt = Z R + Z Ω i Z Φ i ± ( u ( x, t, α ) , κ ) ∂ x ψ ( x, t ) dαdxdt. (40)One has also lim δ → Z Ω i ( u ( x ) − κ ) ± ψ ( x, χ i,δ ( x ) dx = Z Ω i ( u ( x ) − κ ) ± ψ ( x, dx. (41)One has | Φ i ± ( u i , κ ) | ≤ M f i ( u i − κ ) ± then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R + Z Ω i Φ i ± ( u i , κ ) ψ ( x, t ) ∂ x χ i,δ ( x ) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M f i ( u i − κ ) ± Z R + Z Ω i ψ ( x, t ) | ∂ x χ i,δ ( x ) | dxdt. Since | ∂ x χ i,δ | tends to δ x =0 in the M ( R )-weak star sense where D δ x =0 , ζ E M ( R ) , C ( R ) = ζ (0) , we obtain thatlim inf δ → lim ε → Z R + Z Ω i Φ i ± ( u ε ( x, t ) , κ ) ψ ( x, t ) ∂ x χ i,δ ( x ) dxdt ≥ M f i ( u i − κ ) ± Z R + ψ (0 , t ) dt. (42)Using (37),(39),(40),(41),(42) in (34) shows that u is a process solution in the sense of Definition 3.2. (cid:3) .3 uniqueness of the (process) solution It is clear that the notion of process solution is weaker than the one of solution given in Definition 1.1.We state here a theorem which claims the equivalence of the two notions, i.e. any process solution isa solution in the sense of Definition 1.1. Furthermore, such a solution is unique, and a L -contractionprinciple can be proven. Theorem 3.5 (uniqueness of the (process) solutions)
There exists a unique process solution u to the problem ( P lim ), and furthermore this solution does not depend on α , i.e. u is a solution to theproblem ( P lim ) in the sense of definition 1.1. Furthermore, if u , v are two initial data in L ∞ ( R ) satisfying (4) and let u and v be two solutions associated to those initial data, then for all t ∈ [0 , T ) , Z R − R ( u ( x, t ) − v ( x, t )) ± dx ≤ Z R + M f t − R − M f t ( u ( x ) − v ( x )) ± dx. (43)This theorem is a consequence of [38, Theorem 2]. Let u ( x, t, α ) and v ( x, t, β ) be two process solutionscorresponding to initial data u and v . Classical Kato inequalities can be derived in each Ω i × R + byusing the doubling variable technique: ∀ ψ ∈ D + (Ω i × R + ), Z R + Z Ω i Z Z ( u ( x, t, α ) − v ( x, t, β )) ± ∂ t ψ ( x, t ) dαdβdxdt + Z Ω i ( u ( x ) − v ( x )) ± ψ ( x, dx + Z R + Z Ω i Z Z Φ i ± ( u ( x, t, α ) , v ( x, t, β )) ∂ x ψ ( x, t ) dαdβdxdt ≥ . The treatment of the boundary condition at the interface is an adaptation to the case of processsolution to the work of Otto summarized in [28] and detailed in [27] leading to (see [38, Lemma 2]): ∀ ψ ∈ D + (Ω i × R + ), Z R + Z Ω i Z Z ( u ( x, t, α ) − v ( x, t, β )) ± ∂ t ψ ( x, t ) dαdβdxdt + Z Ω i ( u ( x ) − v ( x )) ± ψ ( x, dx + Z R + Z Ω i Z Z Φ i ± ( u ( x, t, α ) , v ( x, t, β )) ∂ x ψ ( x, t ) dαdβdxdt ≥ . (44)Choosing ψ ε ( x, s ) = | x | ≤ R + M f s,R + M f s + ε − | x | ε if R + M f t ≤ | x | ≤ R + M f s + ε | x | ≥ R + M f s + ε if s ≤ t and ψ ε ( x, s ) = 0 if s > t as test function in (44) and letting ε tend to 0 provide the expected L -contraction principle (43).Finally, if u and ˜ u are two process solutions associated to the same initial data u , we obtain a L -contraction principle of the following form: for a.e. t ∈ R + , Z R Z Z ( u ( x, t, α ) − ˜ u ( x, t, β )) ± dαdβdx ≤ , thus u ( x, t, α ) = ˜ u ( x, t, β ) a.e. in R × R + × (0 , × (0 , u does not depend on the processvariable α . 15 heorem 3.6 Let u be a prepared initial data in the sense of Definition 2.10, and let u ε be thecorresponding solution to the approximate problem ( P ε ). Then u ε converges to the unique solution u to ( P lim ) associated to initial data u in the L p ((0 , T ); L q ( R )) -sense, for all p, q ∈ [1 , ∞ ) .Proof: We have seen in Proposition 3.4 that u ε converges up to an extraction towards a process solu-tion. The family ( u ε ) ε admits so a unique adherence value, which is a solution thanks to Theorem 3.5,thus the whole family converges towards this unique limit u .Let K denotes a compact subset of R × [0 , T ], then one has Z Z K ( u ε − u ) dxdt = Z Z K ( u ε ) dx − Z Z K u ε udx + Z Z K u dx. Since u ε converges in the nonlinear weak star sense towards u ,lim ε → Z Z K ( u ε ) dx = Z Z K u dx. Moreover, u ε converges in the L ∞ weak star topology towards u , thenlim ε → Z Z K u ε udx = Z Z K u dx. Thus we obtain lim ε → Z Z K ( u ε − u ) dxdt = 0 . One concludes using the fact the | u ε − u | ≤ ε > (cid:3) L ∞ ( R ) In this section, we extend the result of Theorem 3.6 to any initial data in L ∞ ( R ) satisfying (4) thanksto density argument. Theorem 3.7
Let u ∈ L ∞ ( R ) satisfying (4) , and let ( u ,n ) n ∈ N ⋆ be a sequence of prepared initial datatending to u in L loc ( R ) . Then the sequence ( u n ) n of solutions to ( P lim ) corresponding to the sequence ( u ,n ) of initial data converges in C ( R + ; L loc ( R )) towards the unique solution to ( P lim ) correspondingto solution the initial data u .Proof: First, note that for all u ∈ L ∞ ( R ) satisfying (4), there exists a sequence ( u ,n ) n ∈ N ⋆ of preparedinitial data tending to u in L loc ( R ) thanks to Lemma 2.9.Thanks to (43), one has for n, m ∈ N ⋆ , for all t ∈ R + Z R − R ( u n ( x, t ) − u m ( x, t )) ± dx ≤ Z R + M f t − R − M f t ( u ,n ( x ) − u ,m ( x )) ± dx, then ( u n ) n is a Cauchy sequence in C ( R + ; L loc ( R )). In particular, there exists u such thatlim n →∞ u n = u in C ( R + ; L loc ( R )) . It is then easy to check that u is the unique solution to ( P lim ). (cid:3) Entropy solution for small initial data
In this section, we suppose that the initial data u belongs to L ( R ), and that0 ≤ u ≤ u ⋆i a.e. in Ω i . (45)This initial data can be smoothed using following lemma whose proof is almost the same as the proofof Lemma 2.9. Lemma 4.1
There exists ( u ε ) ε> ⊂ L ( R ) such that • ∂ x ϕ i ( u ε ) ∈ L ∞ (Ω i ) , • ess lim x ր u ε ( x ) = 1 , • lim ε → u ε = u in L loc ( R ) . For all ε >
0, there exists a unique bounded flux solution u ε to ( P ε ) corresponding to u ε thanksto Theorem 2.4. The following theorem claims that as ε tends to 0, u ε tends to the unique entropysolution in the sense of Definition 1.3. Theorem 4.2 (convergence towards the entropy solution)
Let u ∈ L ∞ ( R ) satisfying (45) andlet ( u ε ) ε be a family of approximate initial data built in Lemma 4.1. Let u ε be the bounded flux solutionto ( P ε ) corresponding to u ε , then u ε converges to u in L loc ( R × R + ) as ε tends to where u is theunique entropy solution in the sense of Definition 1.3.Proof: Using the technics introduced in [10, Proposition 2.8], we can show that for all λ ∈ [0 , q ] thereexists a steady solution κ ελ to the problem ( P ε ), corresponding to a constant flux f i ( κ ελ ) − ε∂ϕ i ( κ ελ ) = λ, and such that this solution converges uniformly on each compact subset of R ⋆ as ε tends to 0 towards κ λ ( x ) = min κ { f ( κ, x ) = λ } . Following the idea of Audusse and Perthame [5], we will now compare the limit u of u ε as ε to 0 withthe steady state κ λ . Let λ ∈ [0 , q ]. Since u ε and κ ελ are both bounded flux solutions, it follows fromProposition 2.3 that for all ψ ∈ D + ( R × R + ), Z R + Z R ( u ε − κ ελ ) ± ∂ t ψdxdt + Z R ( u ε − κ ελ ) ± ψ ( · , dx + Z R + X i Z Ω i (cid:16) Φ i ± ( u ε , κ ελ ) − ε∂ x ( ϕ i ( u ε ) − ϕ i ( κ ελ )) + (cid:17) ∂ x ψdxdt ≥ . (46)Choosing λ = q and ψ ( x, t ) = ( T − t ) + ξ ( x ) for some arbitrary T > ξ ∈ D + ( R ) yields Z T Z Ω ( u ε − κ εq ) + ξdxdt ≤ Z T ( T − t ) X i =1 , Z Ω i ε∂ x (cid:0) ϕ i ( u ε ) − ϕ i ( κ εq ) (cid:1) + ∂ x ξdxdt. (47)Since u ε is bounded between 0 and 1, it converges in the nonlinear weak star sense, thanks to Theo-rem 3.1 towards a function u ∈ L ∞ ( R × R + × (0 , ≤ u ≤ u ≤ κ q = u ⋆i a.e. in Ω i × R + × (0 , . (48)17et λ ∈ [0 , q ], then taking the limit for ε → Z R + Z R Z | u − κ λ | ∂ t ψdαdxdt + Z R | u − κ λ | ψ ( · , dx + Z R + X i Z Ω i Z Φ i ( u, κ λ ) ∂ x ψdαdxdt ≥ . (49)Suppose that u ⋆ ≥ u ⋆ . Let κ ∈ [0 , u ⋆ ], we denote by ˜ κ = f − ( f ( κ )) ∩ [0 , u ⋆ ]. Then choosing λ = f ( κ )in (49), and letting ε tend to 0 gives: ∀ κ ∈ [0 , u ⋆ ], ∀ ψ ∈ D + ( R × R + ), Z T Z Ω Z | u − ˜ κ | ∂ t ψdαdxdt + Z Ω | u − ˜ κ | ψ ( · , dx + Z R + Z Ω Z | u − κ | ∂ t ψdαdxdt + Z Ω | u − κ | ψ ( · , dx + Z R + Z (cid:18)Z Ω Φ ( u, ˜ κ ) ∂ x ψdx + Z Ω Φ ( u, κ ) ∂ x ψdx (cid:19) dαdt ≥ . (50)It follows from the work of Jose Carrillo [14] that the following entropy inequalities hold for testfunctions compactly supported in Ω : ∀ κ ∈ [0 , ∀ ψ ∈ D + (Ω × R + ), Z R + Z Ω | u ε − κ | ∂ t ψdxdt + Z Ω | u ε − κ | ψ ( · , dx + Z R + Z Ω (Φ ( u ε , κ ) − ε∂ x | ϕ ( u ε ) − ϕ ( κ ) | ) ∂ x ψdxdt ≥ . (51)Thus letting ε tend to 0 in (51) provides: ∀ ψ ∈ D + (Ω × R + ), ∀ κ ∈ [0 , Z R + Z Ω Z | u − κ | ∂ t ψdαdxdt + Z Ω | u − κ | ψ ( · , dx + Z R + Z Ω Z Φ ( u, κ ) ∂ x ψdαdxdt ≥ . (52)Let δ >
0, and let ψ ∈ D + ( R × R + ), we define ψ ,δ ( x, t ) = ψ ( x, t ) χ ,δ ( x ) , ψ ,δ = ψ − ψ ,δ , where χ ,δ is the cut-off function introduced in section 3.2. Then using ψ ,δ as test function in (52)and ψ ,δ in (50) leads to: Z R + Z R Z | u − κ | ∂ x ψdαdxdt + Z R | u − κ | ψ ( · , dx + Z R + X i Z Ω i Z Φ i ( u, κ ) ∂ x ψdαdxdt + Z R + Z Ω Z (Φ ( u, κ ) − Φ ( u, ˜ κ )) ψ∂ x χ ,δ dαdxdt ≥ R ( κ, ψ, δ ) , (53)where lim δ → R ( κ, ψ, δ ) = 0 . Since f is increasing on [0 , u ⋆ ] and f ([ u ⋆ , ⊂ [ q, ∞ ), either κ ≤ u ⋆ ,or f ( κ ) ≥ f ( u ⋆ ) . This ensures thatΦ ( u, κ ) = | f ( u ) − f ( κ ) | , ∀ u ∈ [0 , u ⋆ ] , ∀ k ∈ [0 , u ⋆ ] . This yields | Φ ( u, κ ) − Φ ( u, ˜ κ ) | = (cid:12)(cid:12) | f ( u ) − f ( κ ) | − | f ( u ) − f (˜ κ ) | (cid:12)(cid:12) ≤ | f ( κ ) − f (˜ κ ) | = | f ( κ ) − f ( κ ) | . (54)18aking the inequality (54) into account in (53), and letting δ → ∀ κ ∈ [0 , k u k ∞ ], ∀ ψ ∈ D + ( R × R + ), Z R + Z R Z | u − κ | ∂ x ψdαdxdt + Z R | u − κ | ψ ( · , dx + Z R + X i Z Ω i Z Φ i ( u, κ ) ∂ x ψdαdxdt + | f ( κ ) − f ( κ ) | Z T ψ (0 , · ) dt ≥ . Using the work of Florence Bachmann [6, Theorem 4.3], we can claim that u is the unique entropysolution to the problem. Particularly, u does not depend on α (introduced for the nonlinear weak starconvergence). As proven in the proof of Theorem 3.6, this implies that u ε converges in L loc ( R × R + )towards u . (cid:3) In this section, we complete the resolution of the Riemann problem at the interface { x = 0 } , whoseresult has been given in section 1.2. Consider the initial data u ( x ) = (cid:26) u ℓ if x < ,u r if x > . We aim to determine the traces ( u , u ) at the interface of the solution u ( x, t ) corresponding to u .This resolution has already been performed in the following cases.(a). u ⋆ < u ℓ ≤ u ⋆ ≤ u r <
1: it has been seen that u = 1 and u = u ⋆ .(b). 0 ≤ u ℓ ≤ u ⋆ and 0 ≤ u r ≤ u ⋆ : Since u is the unique optimal entropy solution studied in [2, 24],then u = u ℓ and u is the unique value in [0 , u ⋆ ] such that f ( u ℓ ) = f ( u ).In the cases(c). u ⋆ < u ℓ ≤ u r = 1,(d). u ℓ = u ⋆ and u r = 1,it is possible to approach the solution u by bounded flux solutions u ε that are constant equal to 1 inΩ × R + . Then one obtains u = u = 1 for the case (c) and u = u ⋆ and u = 1 for the case (d).The last points we have to consider are(e). u ⋆ < u ℓ ≤ ≤ u r < u ⋆ ,(f). 0 ≤ u ℓ ≤ u ⋆ and u ⋆ < u r ≤ { x = 0 } . Lemma 5.1
Let u r ∈ [0 , u ⋆ ) . For all ε > , there exists a function v ε solution to the problem ∂ t v ε + ∂ x (cid:0) f ( v ε ) − ε∂ x ϕ ( v ε ) (cid:1) = 0 if x > , t > ,f ( v ε ) − ε∂ x ϕ ( v ε ) = f ( u ⋆ ) if x = 0 , t > ,v ε = u r if x > , t = 0 , (55) satisfying furthermore u r ≤ v ε ≤ u ⋆ and ∂ x ϕ ( v ε ) ∈ L ∞ ( R + × R + ) . emma 5.2 Let u ℓ ∈ [0 , u ⋆ ] , u r ∈ ( u ⋆ , and let u be the unique value of [0 , u ⋆ ] such that f ( u ) = f ( u ℓ ) . For all ε > there exists a function w ε solution to the problem ∂ t w ε + ∂ x (cid:0) f ( w ε ) − ε∂ x ϕ ( w ε ) (cid:1) = 0 if x > , t > ,f ( w ε ) − ε∂ x ϕ ( w ε ) = f ( u ) = f ( u ℓ ) if x = 0 , t > ,w ε = u r if x > , t = 0 , (56) satisfying furthermore u ≤ w ε ≤ u r and ∂ x ϕ ( w ε ) ∈ L ∞ ( R + × R + ) . The case (e).
Assume that u ℓ > u ⋆ and u r < u ⋆ . Let ( u η ) η be a family of initial data such that ∂ x ϕ i ( u η ) ∈ L ∞ (Ω i ), u η ( x ) = 1 for x ∈ ( − η, u η ( x ) ∈ [ u ℓ ,
1] for a.e. x ∈ Ω , u η ( x ) = u r a.e. in Ω and such that k u η − u ℓ k L (Ω ) ≤ η. Then thanks to Theorem 2.4, there exists a unique bounded flux solution u ε,η to the problem ( P ε )corresponding to the initial data u η . It is easy to check that the solution defined in Ω × R + by thefunction v ε introduced in Lemma 5.1 and coinciding in Ω × R + with the unique bounded flux solutioncorresponding to the initial data ˜ u η ( x ) = ( u η ( x ) if x < , x > . In particular, as ε tends to 0, it follows from arguments similar to those developed in the previoussections that u ε,η converges in L loc (Ω i × R + ) towards the unique entropy solution to the problemproblem ∂ t u η + ∂ x f ( u η ) = 0 if x < , t > ,u η = 1 if x = 0 , t > ,u η = u η if x < , t = 0 . (57) ∂ t u + ∂ x f ( u ) = 0 if x > , t > ,u = u ⋆ if x = 0 , t > ,u = u r if x > , t = 0 . (58)Note that the trace condition on the interface { x = 0 } in (58) is fulfilled in a strong sense since u r ≤ u ( x, t ) ≤ u ⋆ a.e. in Ω × R + and f is increasing on [ u r , u ⋆ ].The solution to (57) depends continuously on the initial data in L loc . Hence, letting η tend to 0in (57) provides that the limit u of u η is the unique entropy solution to the problem ∂ t u + ∂ x f ( u ) = 0 if x < , t > ,u = 1 if x = 0 , t > ,u = u ℓ if x < , t = 0 . Note that since u ⋆ < u ℓ ≤ u ≤ s ∈ [ u, f ( s ) = f (1) = q , the trace prescribed on the interface { x = 0 } is fulfilled in a strong sense. This particularly yields that in the case (e), the solution to theRiemann problem is given by u = 1 , u = u ⋆ . The case (f ).
Following the technique used in [10] and in Section 4, there exists a unique function u εℓ solution to the problem: f ( u εℓ ) − ε dd x ϕ ( u εℓ ) = f ( u ℓ ) if x < ,u εℓ (0) = 1 if x = 0 . u ε be the function defined by u ε ( x, t ) = ( u εℓ ( x ) if x < , t ≥ ,w ε ( x, t ) if x > , t ≥ , where w ε is the function introduced in Lemma 5.2. Then u ε is a bounded flux solution to the prob-lem ( P ε ) in the sense of Definition 2.1.One has u εℓ → u ℓ in L loc (Ω ) as ε → , and w ε → w in L loc (Ω × R + ) as ε → w is the unique solution to ∂ t w + ∂ x f ( w ) = 0 if x > , t > ,w = u = f − ◦ f ( u ℓ ) if x = 0 , t > ,w = u r if x > , t = 0 . Since w ( x, t ) ∈ [ u , u r ] a.e. in Ω × R + and since min s ∈ [ u ,w ] f ( s ) = f ( u ) = f ( u ℓ ), the trace w = u is satisfied in a strong sense on { x = 0 } . This yields that the solution to the Riemann problem in thecase (f) is given by u = u ℓ , u = f − ◦ f ( u ℓ ) . The model presented here shows that for two-phase flows in heterogeneous porous media with negligibledependance of the capillary pressure with respect to the saturation, the good notion of solution is notalways the entropy solution presented for example in [1, 6], and particular care as to be taken withrespect to the orientation of the gravity forces. Indeed, some non classical shock can appear at thediscontinuities of the capillary pressure field, leading to the phenomenon of oil trapping. We stressthe fact that the non classical shocks appearing in our case have a different origin, and a differentbehavior of those suggested in the recent paper [36] (see also [26]). Indeed, in this latter paper, thislack of entropy was caused by the introduction of the dynamical capillary pressure [22, 23, 29], i.e. thecapillary pressure is supposed to depend also on ∂ t u . In our problem, the lack of entropy comes onlyfrom the discontinuity of the porous medium.In order to conclude this paper, we just want to stress that this model of piecewise constant capillarypressure curves can not lead to some interesting phenomenon. Indeed, if the capillary pressure functions π i are such that π ((0 , ∩ π ((0 , = ∅ , it appears in [11, Section 6] (see also [7]) that some oil canoverpass the boundary, and that only a finite quantity of oil can be definitely trapped. Moreover, thisquantity is determined only by the capillary pressure curves and the difference between the volumemass of both phases, and does not depend on u . The model presented here, with total flow-rate q equal to zero, do not allow this phenomenon, and all the oil present in Ω at the initial time remainstrapped in Ω for all t ≥ References [1] Adimurthi, J´erˆome Jaffr´e, and G. D. Veerappa Gowda. Godunov-type methods for conserva-tion laws with a flux function discontinuous in space.
SIAM J. Numer. Anal. , 42(1):179–208(electronic), 2004.[2] Adimurthi, Siddhartha Mishra, and G. D. Veerappa Gowda. Optimal entropy solutions for con-servation laws with discontinuous flux-functions.
J. Hyperbolic Differ. Equ. , 2(4):783–837, 2005.213] B. Amaziane, A. Bourgeat, and H. Elamri. Existence of solutions to various rock types model oftwo-phase flow in porous media.
Applicable Analysis , 60:121–132, 1996.[4] S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov.
Boundary value problems in mechanics ofnonhomogeneous fluids , volume 22 of
Studies in Mathematics and its Applications . North-HollandPublishing Co., Amsterdam, 1990. Translated from the Russian.[5] E. Audusse and B. Perthame. Uniqueness for scalar conservation laws with discontinuous flux viaadapted entropies.
Proc. Roy. Soc. Edinburgh Sect. A , 135(2):253–265, 2005.[6] F. Bachmann.
Equations hyperboliques scalaires `a flux discontinu . PhD thesis, Universit´e Aix-Marseille I, 2005.[7] M. Bertsch, R. Dal Passo, and C. J. van Duijn. Analysis of oil trapping in porous media flow.
SIAM J. Math. Anal. , 35(1):245–267 (electronic), 2003.[8] R. B¨urger, K. H. Karlsen, and J. D. Towers. An Engquist-Osher-type scheme for conservation lawswith discontinuous flux adapted to flux connections.
SIAM J. Numer. Anal. , 47(3):1684–1712,2009.[9] F. Buzzi, M. Lenzinger, and B. Schweizer. Interface conditions for degenerate two-phase flowequations in one space dimension.
Analysis , 29(3):299–316, 2009.[10] C. Canc`es. Asymptotic behavior of two-phase flows in heterogeneous porous media for capil-larity depending only of the space. I. Convergence to the optimal entropy solution. submitted,arXiv:0902.1877, 2009.[11] C. Canc`es. Finite volume scheme for two-phase flow in heterogeneous porous media involvingcapillary pressure discontinuities.
M2AN Math. Model. Numer. Anal. , 43:973–1001, 2009.[12] C. Canc`es and T. Gallou¨et. On the time continuity of entropy solutions. arXiv:0812.4765v1, 2008.[13] C. Canc`es, T. Gallou¨et, and A. Porretta. Two-phase flows involving capillary barriers in hetero-geneous porous media.
Interfaces Free Bound. , 11(2):239–258, 2009.[14] J. Carrillo. Entropy solutions for nonlinear degenerate problems.
Arch. Ration. Mech. Anal. ,147(4):269–361, 1999.[15] G. Chavent and J. Jaffr´e.
Mathematical Models and Finite Elements for Reservoir Simulation ,volume vol.17. North-Holland, Amsterdam, stud. math. appl. edition, 1986.[16] Z. Chen. Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of aweak solution.
J. Differential Equations , 171(2):203–232, 2001.[17] R. J. DiPerna. Measure-valued solutions to conservation laws.
Arch. Rational Mech. Anal. ,88(3):223–270, 1985.[18] G. Ench´ery, R. Eymard, and A. Michel. Numerical approximation of a two-phase flow in a porousmedium with discontinuous capillary forces.
SIAM J. Numer. Anal. , 43(6):2402–2422, 2006.[19] R. Eymard, T. Gallou¨et, M. Ghilani, and R. Herbin. Error estimates for the approximate solu-tions of a nonlinear hyperbolic equation given by finite volume schemes.
IMA J. Numer. Anal. ,18(4):563–594, 1998.[20] R. Eymard, T. Gallou¨et, and R. Herbin. Finite volume methods. Ciarlet, P. G. (ed.) et al., inHandbook of numerical analysis. North-Holland, Amsterdam, pp. 713–1020, 2000.[21] M. Garavello, R. Natalini, B. Piccoli, and A. Terracina. Conservation laws with discontinuousflux.
Netw. Heterog. Media , 2(1):159–179 (electronic), 2007.2222] S.M. Hassanizadeh and W.G. Gray. Mechanics and thermodynamics of multiphase flow in porousmedia including interphase boundaries.
Adv. Water Resour. , 13:169–186, 1990.[23] S.M. Hassanizadeh and W.G. Gray. Thermodynamic basis of capillary pressure in porous media.
Water Resources Res. , 29:3389–3405, 1993.[24] E. F. Kaasschieter. Solving the Buckley-Leverett equation with gravity in a heterogeneous porousmedium.
Comput. Geosci. , 3(1):23–48, 1999.[25] S. N. Kruˇzkov. First order quasilinear equations with several independent variables.
Mat. Sb.(N.S.) , 81 (123):228–255, 1970.[26] P. G. LeFloch.
Hyperbolic systems of conservation laws . Lectures in Mathematics ETH Z¨urich.Birkh¨auser Verlag, Basel, 2002. The theory of classical and nonclassical shock waves.[27] J. M´alek, J. Neˇcas, M. Rokyta, and M. R˚uˇziˇcka.
Weak and measure-valued solutions to evolution-ary PDEs , volume 13 of
Applied Mathematics and Mathematical Computation . Chapman & Hall,London, 1996.[28] F. Otto. Initial-boundary value problem for a scalar conservation law.
C. R. Acad. Sci. ParisS´er. I Math. , 322(8):729–734, 1996.[29] D. Pavone. Macroscopic equations derived from space averaging for immiscible two-phase flow inporous media.
Revue de l’Institut fran¸cais du p´etrole , 44(1):29–41, janvier-fvrier 1989.[30] B. Schweizer. Homogenization of degenerate two-phase flow equations with oil trapping.
SIAMJ. Math. Anal. , 39(6):1740–1763, 2008.[31] J. Smoller.
Shock waves and reaction-diffusion equations , volume 258 of
Fundamental Principlesof Mathematical Sciences . Springer-Verlag, New York, second edition, 1994.[32] A. Szepessy. Convergence of a streamline diffusion finite element method for scalar conservationlaws with boundary conditions.
RAIRO Mod´el. Math. Anal. Num´er. , 25(6):749–782, 1991.[33] J. D. Towers. Convergence of a difference scheme for conservation laws with a discontinuous flux.
SIAM J. Numer. Anal. , 38(2):681–698 (electronic), 2000.[34] J. D. Towers. A difference scheme for conservation laws with a discontinuous flux: the nonconvexcase.
SIAM J. Numer. Anal. , 39(4):1197–1218 (electronic), 2001.[35] C. J. van Duijn, J. Molenaar, and M. J. de Neef. The effect of capillary forces on immiscibletwo-phase flows in heterogeneous porous media.
Transport in Porous Media , 21:71–93, 1995.[36] C. J. van Duijn, L. A. Peletier, and I. S. Pop. A new class of entropy solutions of the Buckley-Leverett equation.
SIAM J. Math. Anal. , 39(2):507–536 (electronic), 2007.[37] A. I. Vol ′ pert. Spaces BV and quasilinear equations. Mat. Sb. (N.S.) , 73 (115):255–302, 1967.[38] J. Vovelle. Convergence of finite volume monotone schemes for scalar conservation laws on boundeddomains.