Generalized Buckley-Leverett system Wladimir Neves UFRJUniversidade - - PowerPoint PPT Presentation

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Generalized Buckley-Leverett system

Wladimir Neves UFRJ–Universidade Federal do Rio de Janeiro Rio de Janeiro - Brazil wladimir@im.ufrj.br Joint work with Nikolai Chemetov–CMAF-Lisbon HYP-2012, June 28, 2012

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Introduction

Let us consider a pisto-like displacement of fluids in porous media. A displacement of oil by water. Piston-like means: The process of motion in porous media, always has two parts. One of oil (only) and other of water (only), i.e. immiscible. The liquids are also assumed incompressible.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Non-linear porous-media theory

The mathematical model is described by: (t, x) ∈ R × Rd− points in the time-space domain. m(x)− porosity. vi(t, x) ∈ Rd, (i = 1, 2)− seepage velocity field. si(t, x) ∈ R, (i = 1, 2)− saturation of each ith component, 0 ≤ s1, s2 ≤ 1, s1 + s2 = 1. (1) Conservation of Mass (Continuity equation): m ∂t(ρi si) + divx(ρi vi) = 0, (i = 1, 2) (2) where ρi is mass density of the ith-phase of liquids.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

From the incompressibility assumption, we have ∂t(si) + divx(vi) = 0, (i = 1, 2) (3) where we took m ≡ 1 without loss of generality. Conservation of Linear Momenta (Darcy’s law equation): µi k0 kri(s1)vi = −∇xpi + ρi g h, (i = 1, 2) (4) where for each component i = 1, 2, pi(t, x) is the pressure, µi is the dynamic viscosity and kri is the relative permeability. Moreover, k0(x) is the absolute permeability of the porous medium and ρ g h is the external gravitational force, which is used dropped.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

From the above considerations and, denoting λi = µi/(k0 kri(s1)) (i = 1, 2), we have λi vi = −∇xpi. (5) Then, from equations (3) and (5), we are in condition to formulate the Muskat problem. One remarks that, in each part of the medium, where just exist one component, the saturation si is obviously constant and equals one.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Muskat problem (original formulation): λo vo = −∇xpo, divx(vo) = 0, in Qo, λi vw = −∇xpi, divx(vw) = 0, in Qw, po = pw, vw · nt = vo · nt,

• n Γt,

(6) where the under-scrip o, w stand respectively for oil and water. Further, we have assumed that, pc = 0, i.e. the capillarity pressure is zero on Γt. Qo, Qw– regions of oil and water respectively. Γt– free boundary between Qw and Qo. nt– unitary normal field to Γt. Therefore, Muskat problem (original formulation) is a time-dependent elliptic-diffraction problem with a free-boundary.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Muskat problem (weak formulation): First, we define on Qw ∪ Γt ∪ Qo: u(t, x) := sw(t, x), hence so(t, x) = 1 − u(t, x). v(t, x) := vw(t, x) + vo(t, x) – total velocity. Moreover, we define the pressure p on Qw ∪ Γt ∪ Qo as p(t, x) :=

• pw

in Qw ∪ Γt, po in Qo. (7) After some algebraic computation, we have the following system, (also) called Buckley-Leverett system: ∂tu + divx(v g(u)) = 0, divx(v) = 0, h(u) v = −∇xp. (8)

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Remarks: The Muskat problem (original formulation) was introduced in 1934 by Morris Muskat; Two fluid systems in porous media. The encroachment of water into oil sand, Physics Vol. 5. The Muskat problem (weak formulation) appears for the first time into the paper: Weak formulation of a multidimensional Muskat problem, written by L. Jiang and Z. Chen, 1990. On that paper are proved without functional formalism that, both formulations are equivalent. The Muskat is an open problem! Let us look closer to the weak formalism.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

From equation (8), it follows that, we have to deal with a scalar non-homogeneous conservation law ∂tu + divxϕ(t, x, u) = 0, with ϕ(t, x, p) = v(t, x) g(p), (9) and v is expected to be just in L2! Do not mind w.r.t. g and h regularities, used to be good enough... The best existence (and pre-compactness) result for this type of equation is given be E. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Rational Mech. Anal., 195 (2010), 643–673.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

One of the hypothesis on that important paper is maxλ∈[a,b] |ϕ(t, x, λ)| ∈ Lp (p > 2), which is not our case. (The regularity assumption is related to Murat’s Interpolation Lemma). Moreover, even if we have enough regularity, the paper uses an extension of the H-measure introduced by Luc Tartar, and also, the same idea of the localization property. In this way, ϕ should be non-degenerated, that is

• ∀a, b ∈ R, a < b

∀(τ, ξ) ∈ Rd+1 \ {0}

• λ ∈ (a, b) =

⇒ (τ, ξ) ·

• 1, ∂λϕ(t, x, λ)
• = 0 for a.a. (t, x)
• .

The most simple and one of the most important examples of non-homogeneous flux functions is given by (9) and, it does not satisfy the above condition! Indeed, let us give a simple example.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Example: We consider the 2-dimensional case and denote v = (V1, V2). Therefore, we must have for L3-a.e. (t, x) ∈ R × R2

• 1, v(t, x) g′(λ)
• ·(τ, ξ) = τ +g′(λ)
• V1(t, x) ξ1+V2(t, x) ξ2
• = 0,

for all (τ, ξ1, ξ2) ∈ R3, (τ 2 + ξ2

1 + ξ2 2 = 0), and

λ ∈ (a, b) ⊂ R, a < b, which is false once one takes τ = 0, ξ1 = ±V2, ξ2 = ∓V1. Therefore, we could not attack the existence of solution to Muskat problem on that way!

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Some ideas have been proposed to by-pass this difficult, but most of them focused in the saturation equation.

• N. Chemetov and WN considered a new idea. They proposed

a generalized Darcy’s law equation, in fact a regularization of the standard one, i.e. τ ∂tv − ν ∆v + h(u) v = −∇p, (10) where τ, ν are small positive parameters. They introduced a Generalized Buckley-Leverett system. The new formulation proposed brings enough regularity of the seepage velocity field. They showed solvability of the generalized system using the idea of Kinetic Theory.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Functional notation

Let T > 0 be a real number, Ω ⊂ Rd (with d = 1, 2 or 3) an open and bounded domain having a C 2− smooth boundary Γ. We define ΩT := (0, T) × Ω, ΓT := (0, T) × Γ. We use standard notations for the Lebesgue function space Lp(Ω), the Sobolev spaces W s,p(Ω) and Hs(Ω) ≡ W s,2(Ω). The vector counterparts of these spaces are denoted by L2(Ω) = (L2(Ω))d and Hs(Ω) := (Hs(Ω))d. For any u ∈ L2(Ω), satisfying div (u) = 0 in D

′(Ω),

the normal component of u, i.e. un:= u · n, exists and belongs to H−1/2(Γ).

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

We will also use the following divergence free spaces Vs(Ω) : = {u ∈ Hs(Ω) : div (u) = 0 in D

′(Ω),

• Γ

u · n dx = 0}, Vs(Γ) : = {u ∈ Hs(Γ) :

• Γ

u · n dx = 0}, V−s(Γ) : = (Vs(Γ))′ . We define by uτ := u − unn the tangent component for some u defined on Γ, G1(ΓT) : =

• u ∈ L2(0, T; V1/2(Γ)) : uτ ∈ H1/2(0, T; V−1/2(Γ),

un ∈ H3/4(0, T; V−1(Γ)

• ,

S1(ΓT) :=

• u ∈ L2(0, T; V1/2(Γ)) ∩ H1(0, T; V−1/2(Γ)
• .

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Statement of the Buckley-Leverett problem

We are concerned with the following initial-boundary value problem, denoted as IBVP: Find a pair (u, v) = (u(t, x), v(t, x)) : ΩT → R × Rd solution to the generalized Buckley-Leverett system in the domain ΩT ∂tu + div

• v g(u)
• =

0, (11) τ ∂tv − ν∆v + h(u)v = −∇p, div(v) = 0, (12) satisfying the boundary conditions (u, v) = (ub, b)

• n ΓT,

(13) and the initial conditions (u, v) = (u0, v0) in Ω. (14)

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

We assume that our data satisfies the following regularity properties g, h ∈ W 1,∞

loc (R),

such that 0 < h0 ≤ h(u), (15) ub ∈ L∞(ΓT), such that 0 ≤ ub ≤ 1

• n ΓT,

u0 ∈ L∞(Ω), such that 0 ≤ u0 ≤ 1 in Ω, (16) and v0 ∈ V0(Ω) and b ∈ G1(ΓT), such that b(0) · n = v0 · n in H−1/2(Γ). (17)

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Now, since equation (11) is a hyperbolic scalar conservation law, the saturation function u may admit shocks. Therefore, in order to select the more correct physical solution, we need the entropy concept as given at the following Definition A pair F(u) := (η(u), q(u)) is called an entropy pair for (11), if η : R → R is a continuous differentiable and also convex function and q : R → R satisfies q′(u) = η′(u) g′(u) for all u ∈ R. (18) Moreover, we say that F(u) is a generalized convex entropy pair if it is a uniform limit of a family of convex entropy pairs over compact sets.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Here, we consider the following parameterized family of entropy pairs for (11) F(u, v) =

• |u − v|, sgn(u − v)
• g(u) − g(v)
• (19)

for each v ∈ R. Another two examples of parameterized family of entropy pairs for (11), which will be useful in the Kinetic formulation to be used latter are F±(u, v) =

• (u − v)±, sgn
• (u − v)±

g(u) − g(v)

• (20)

for each v ∈ R, where (u − v)+ := max{(u − v), 0}, (u − v)− := min{(u − v), 0}.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Definition A pair (u, v) ∈ L∞(ΩT) × L2(0, T; V1(Ω)) is called a generalized solution to the IBVP, if the pair (u, v) satisfies, for all test functions φ, ψ

• QT
• |u − v| φt + sgn(u − v)(g(u) − g(v)) v · ∇φ
• dx dt

+ M

• ΓT

|ub(r) − v|φ(r) dHn(r) +

• Ω

|u0 − v|φ(0) dx ≥ 0, (21)

• ΩT

{τ v·ψt −ν ∇v : ∇ψ−h(u) v·ψ} dxdt +τ

• Ω

v0·ψ(0) dx = 0. (22)

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

The above definition is motivated by Theorem 4.1 in Chen and Frid [1]. Then, we have the following Theorem If the data g, h, ub, u0, v0, b fulfills the regularity properties (16)-(17), then the initial-boundary value problem IBVP: (11)-(14) has a generalized solution (u, v), satisfying

• u 1

a . a. in ΩT, v ∈ C(0, T; V0(Ω)) ∩ L2(0, T; V1(Ω)) ∩ H1(0, T; V−1(Ω)).

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Parabolic approximation

In order to show existence of a generalized solution to (11)–(14), we study first the following approximated parabolic associated system ∂tuε + div

• vε g(uε)
• =

ε ∆uε, (23) τ ∂tvε − ν∆vε + h(uε)vε = −∇pε, div (vε) = 0. (24) joint with the (uε

b, bε) and (uε 0, vε 0) respectively regularized

boundary and initial data satisfying suitable compatibility conditions on Γ and at t = 0.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

From well-known theory for parabolic and Navier-Stokes type equations, see Ladyzhenskaya et all [2], using Schauder’s fixed point argument, based on the maximum principle for uε, i.e. 0 ≤ uε ≤ 1 a.e. in ΩT, we get (as now a standard procedure) the solvability of the approximated system (23)–(23) and establish the following

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Proposition For each ε > 0, there exists a unique solution (uε, vε) of the system (23)–(23), which satisfies the following properties uε ∈ L∞(0, T; H1(Ω)) ∩ L2(0, T; H2(Ω)), such that √ε ∇uεL2(ΩT ) ≤ C, 0 ≤ uε ≤ 1, a.e. on ΩT, (25) and vε ∈ L2(0, T; V1(Ω)) ∩ H1(0, T; V−1(Ω)), such that √τ vC([0,T];V0(Ω)) + vL2(0,T;V1(Ω)) (26) +√τ vH1(0,T;V−1(Ω)) C, (27) where C is a positive constant independent of ε and τ.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

The limit transition on ε → 0+

The limit transition on ε → 0+ is obtained from the Kinetic Theory. One remarks that, the standard W 1,1 or BV estimates seems to be not possible to derive. Special attention should be done on the values of the kinetic function on the boundary of ΓT.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

The main idea. Sketch of the proof Let (η(u), q(u)) be a entropy pair for (23). Then, we have in sense

• f distributions

∂tη(uε) + div(vεq(uε)) − ε ∆η(uε) = −ε η′′(u) |∇η(u)|2. Since η is a convex function, then ∂tη(uε) + div(vεq(uε)) − ε ∆η(uε) ≤ 0.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

For instance, we could take the entropy pair (η(u), q(u)) = F+(u, v) for all v ∈ R, given in (20). Then, we have in sense of distributions ∂t(u−v)++div(vε sgn(u−v)+ g(u)−g(v)

• )−ε ∆(u−v)+ = −mε

+,

(28) where mε

+ is a real nonnegative Radon measure given by

mε

+(t, x, v) = ε |∇uε(t, x)|2 δv=uε(t,x)([0, 1]).

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Now, if we differentiate in the distribution sense (28) with respect to v, we get as now a standard procedure in the Kinetic formulation, the following linear transport equation ∂tf ε + g′(v) vε · ∇f ε − ε ∆f ε = ∂vmε

+,

(29) where f ε(t, x, v) := sgn(uε(t, x) − v)+. Let us point out that, by definition 0 ≤ f ε(t, x, v) ≤ 1 in ΩT × R, and since 0 ≤ uε(t, x) ≤ 1, if v / ∈ [0, 1], then mε

+(t, x, v) ≡ 0 in

the distributional sense, that is, the support of mε is contained in ΩT × [0, 1]. Also, we have for each v ∈ R

• ΩT

mε

+(t, x, v) dxdt ≤

• ΩT

|√ε ∇uε|2 dxdt ≤ C, i.e. mε

+ is uniformly bounded with respect to ε > 0.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Passing to subsequences, if necessary, we obtain ∂tf + g′(v) v · ∇f = ∂vm+ in D′(ΩT × R). (30) Moreover, we proved the strong traces for the initial-boundary values of the kinetic function. Indeed, we have the following

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Proposition The function f = f (t, x, v) has the trace f 0 at the time t = 0, such that f 0 = f 0(x, v) ≡ lim

δ→0+

1 δ δ f (s, x, v) ds, f 0 =

• f 02 .

The function f = f (t, x, v) has the trace f b on ΓT × R, such that f b = f b(t, x, v) ≡ lim

δ→0+

1 δ δ f (t, x − s n(x), v) ds, f b =

• f b2

for a. a. (t, x, v) ∈ ΓT × R, where g′(v)bn(t, x) ≤ 0.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Similar result we obtain for η(u) = (u − v)−, in particular ∂t(1 − f ) + g′(v) v · ∇(1 − f ) = −∂vm− in D′(ΩT × R). (31) Now, for each θ > 0, let ρθ(x) be an even mollifier and denote by zθ(x) = (z ∗ ρδ)(x) :=

• Rd z(y) ρθ(x − y) dy

be the regularization of z = z(x) on the variable x ∈ Ω.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Then, mollifying (30) and (31), we obtain ∂tf θ + g′(v) v · ∇f θ = ∂vmθ

+ + Rθ +

in D′(ΩT × R), (32) and ∂t(1−f θ)+g′(v) v·∇(1−f θ) = −∂vmθ

−+Rθ +

in D′(ΩT × R), (33) with Rθ

± → 0 in L1 loc(ΩT × (0, 1)) as θ → 0+, in view of

DiPerna-Lions’ approach for Transport Equations (recall the regularity of the velocity field v).

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Now, we conveniently multiply (32), (33) respectively by (1 − f θ), f θ, and deduce that the function F := f (1 − f ), satisfies ∂tF + g′(v) v · ∇F ≤ 0 in D′(ΩT × R). (34) Therefore, applying the strong-trace results for the kinetic function, we obtain for almost all t0 ∈ [0, T]

• Ω×R

F dxdv ≤ 0. Since F ≥ 0, we deduce that F ≡ 0 a.e. on ΩT × R. Consequently, f (t, x, v) ∈ {0, 1} a.e. on ΩT × R. (35)

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

The limit transition on τ → 0+

For a given viscous parameter ν > 0, we consider the following initial-boundary value problem, denoted as IBVPτ=0: Find a pair (u, v) = (u(t, x), v(t, x)) : ΩT → R × Rd solution to the quasi-stationary Stokes-Buckley-Leverett system in the domain ΩT ∂tu + div

• v g(u)
• =

0, (36) −ν∆v + h(u)v = −∇p, div (v) = 0, (37) satisfying the boundary conditions (u, v) = (ub, b)

• n ΓT,

(38) and the initial condition u = u0 in Ω. (39)

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Theorem If the data g, h, ub, u0, b fulfills the regularity properties, then the IBVPτ=0 has a weak solution (u, v), satisfying

• u 1

a . e. in ΩT, v, ∂tv ∈ L2(0, T; V1(Ω)). Proposition There exists a pair (u, v) ∈ L∞(ΩT) × L2(0, T; V1(Ω)), with ∂tv ∈ L2(0, T; V1(Ω)), and a subsequence of {uτ, vτ}τ>0 , such that uτ ⇀ u ∗ -weakly in L∞(ΩT), (40) vτ → v strongly in L2(ΩT). (41)

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

The limit transition on ν → 0+???

This is the last limit process to solve the Muskat Problem! Recall the scalar conservation law, and using the incompressibility condition, we have ∂tu + v · ∇xg(u) = 0. In fact, we should consider first the more simple case of Transport Equations, that is ∂tu + v · ∇xu = 0, div(v) = 0, (42) to understand better the difficult involved.

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

Chen G.-Q., Frid H., Divergence measure fields and hyperbolic conservation laws. Arch. Rational Mech. Anal. 147 (1999) 89–118. DiPerna R.J., Lions P.L., Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, (1989) 511–547. Fursikov A., Simader M., Hou L., Trace theorems for the three-dimensional time-dependent solenoidal vector fields and their applications, Transactions of AMS, Vol. 354, N. 3, 1079-1116 (2001).

W Neves - UFRJ - Brazil Generalized Buckley-Leverett system

Introduction Non-linear porous-media theory Muskat problem Remarks on Muskat problem Generalized Buckley-Leverett system Statement of the Buckley-Leverett problem Existence of generalized solution

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