Nonlinear Parabolic Equations with Spatial Discontinuities

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Nonlinear Parabolic Equations with Spatial Discontinuities

Clément Cancès

To cite this version:

Clément Cancès. Nonlinear Parabolic Equations with Spatial Discontinuities. Nonlinear Differential

Equations and Applications, Springer Verlag, 2008, 15, pp.427-456. �hal-01713524�

Nonlinear Parabolic Equations with Spatial Discontinuities

Clément Cancès

Abstract

We consider a two phase flow involving no capillary barriers in a hetero- geneous porous media, composed by an apposition of several homogeneous porous media. We prove the existence of a weak solution for such a flow using the convergence of a finite volume approximation. Then under the as- sumption that the equations governing the flows in each homogeneous porous media degenerate in not too different ways, we prove the uniqueness of the weak solution, using a doubling variable method. We also prove that such a solution belongs toC([0, T], L^p(Ω))for anyp∈[1,+∞).

Keywords.finite volume methods, porous media flow, spatial discontinuity.

AMS subject classification.35B05, 35K65, 35R05, 65M12

1 Presentation of the problem and main results

1.1 Presentation of the problem

In this paper, we are interested by the parabolic equation obtained by modeling a two phase flow in a heterogeneous porous media. Let Ωbe an open polygonal subset ofR^d (d≤3), letT >0. One assumes that there exists a finite number of polygonal open subsetsΩi⊂Ω,i∈ {1, ..., N}such that:

[

i

Ωi= Ω and Ωi∩Ωj=∅ if i6=j. (1) For all(i, j)∈ {1, ..., N}², i6=j, one defines Γi,j the subset of Ωgiven byΓi,j = Ω_i∩Ω_j.

∗Univertsité de Provence, Marseille; cances@cmi.univ-mrs.fr

EachΩi will represent a porous media with its own physical characteristics, uwill represent the saturation of the oily phase. We aim to solve the problem:















∂tu−div(λi(u)∇πi(u)) = 0 in Ωi×(0, T), πi(u) = πj(u) onΓi,j×(0, T), λ_i(u)∇πi(u)·n_i+λ_j(u)∇πj(u)·n_j = 0 onΓ_i,j×(0, T), λ_i(u)∇πi(u)·n_i = 0 on∂Ω×(0, T),

u(x,0) = u₀(x) in Ω,

(2)

where πi is an increasing Lipschitz continuous function associated to Ωi, λi is a non negative continuous function with λ_i|]0,1[ > 0, the initial data u₀ ∈ L^∞(Ω) with0≤u₀≤1 a.e. inΩ.

Remark 1.1 We choose to consider only homogeneous Neumann boundary condi- tions on∂Ω, but this work can be easily generalized for non-homogeneous Dirichlet condition, the same way as in [5, 7].

1.2 Mathematical definition of the problem

In this part, one defines all the functions necessary to explicit the problem (2) and the notion of weak solution.

Particularly, one can associate to eachΩ_itwo functions, the capillary pressure π_i and the global mobilityλ_i, on which we do the following assumptions:

Assumption 1

1. For alli∈ {1, ..., N}, the function πi:R→Ris continuous and satisfies:

• ∀s≤0, πi(s) =πi(0)

• ∀s≥1, πi(s) =πi(1)

• π_i|[0,1] ∈C¹([0,1],R)is an increasing function

• ∀(i, j)∈ {1, ..., N}², πi(0) =πj(0)

• ∀(i, j)∈ {1, ..., N}², πi(1) =πj(1).

Let m0≥1. For alli∈ {1, ..., N}, for all s∈[0,1], we might chooseπi(s) = β_is^m0+ (1−β_i)s^mi for any β_i∈]0,1]and any m_i> m₀.

2. For alli∈ {1, ...N}, the functionλ_i:R→R⁺ is continuous and satisfies:

• ∀s≤0, λ_i(s) =λ_i(0)

• ∀s≥1, λi(s) =λi(1)

• ∀s∈]0,1[, λi(s)>0.

One can now define a functionλ : S

iΩi×R→R⁺ by(x, s)7→λi(s), for allx∈Ωi, for alls∈R. One denotes Cλ= maxi(sup_s∈R(|λi(s)|)).

A classical choice for λi is: ∀i ∈ {1, ..., N},∀s ∈ [0,1], λi(s) = αis(1−s) withαi>0.

We can now define the functionsϕ_i andΠ_i, i∈ {1, ..., N}.

Definition 1.1 Under Assumptions 1, for all i∈ {1, ..., N}, we define:

ϕ_i :

R → R⁺ s 7→ Rs

0 λ_i(a)π⁰_i(a)da (3)

• ∀i∈ {1, ..., N}, ϕ_i|[0,1] is a derivable increasing function

• ∀s≤0, ϕ_i(s) =ϕ_i(0) = 0

• ∀s≥1, ϕi(s) =ϕi(1).

We denote byLϕ a Lipschitz constant for allϕi, i∈ {1, ..., N}.

We also define the functionΠi:

Π_i :







R → R⁺ s 7→

Z π_i(s) π_i(0)

r min

j∈{1,...,N}(λj◦π⁽⁻¹⁾_j (a))da (4)

• ∀i∈ {1, ..., N}, Π_i|[0,1] is a derivable increasing function

• ∀s≤0, Π_i(s) = Π_i(0) = 0

• ∀s≥1, Π_i(s) = Π_i(1)

• ∀(i, j)∈ {1, ..., N}, ∀(s_i, s_j)∈R, π_i(s_i) =π_j(s_j)⇔Π_i(s_i) = Π_j(s_j).

We denote byΠ(s, x) = Πi(s)andϕ(s, x) =ϕi(s)if x∈Ωi.

Remark 1.2 The last point seen in the previous definition allows us to connect Πi andΠj instead ofπi andπj on Γi,j.

The definition ofϕi implies that∂tu−∆ϕi(u) = 0in Ωi, and we can rewrite the transmission conditions onΓi,j:πi(u) =πj(u)and∇ϕi(u)·ni+∇ϕj(u)·nj = 0, where ni represents the outward normal toΩi. We can now define the notion of weak solution for the problem (2):

Definition 1.2 (weak solution) Under assumptions 1, a function u is said to be a weak solution to problem (2) if it satisfies:

1. u∈L^∞(Ω×(0, T)), 0≤u≤1 a.e. inΩ×(0, T),

2. ∀i∈ {1, ..., N}, ϕi(u)∈L²(0, T;H¹(Ωi)), 3. Π(u,·)∈L²(0, T;H¹(Ω)),

4. for allψ∈ D(Ω×[0, T)), Z

Ω

Z T 0

u(x, t)∂_tψ(x, t)dxdt+ Z

Ω

u₀(x)ψ(x,0)dx

−

N

X

i=1

Z

Ω_i

Z T 0

∇ϕi(u(x, t))· ∇ψ(x, t)dxdt = 0. (5)

Remark 1.3 The transmission condition πi(u) = πj(u) on the interface Γi,j is now replaced by the point 3 in the previous definition. Because of the lack of reg- ularity on the solution, we cannot write ∇ϕi(u)·ni+∇ϕj(u)·nj = 0on Γi,j in a strong sense. But it is easy to check that, ifuis a regular enough weak solution of (2), this condition is imposed by point 4 of the previous definition.

1.3 Finite volume approximation and main convergence re- sult

Let us first define space and time discretization ofΩ×(0, T).

Definition 1.3 (Admissible mesh of Ω) An admissible mesh of Ωis given by a setT of open bounded convex subsets ofΩcalled control volumes, a familyE of subsets ofΩ contained in hyperplanes of R^d with strictly positive measure, and a family of points(xK)K∈T (the “centers” of control volumes) satisfying the following properties:

1. ∃i∈ {1, ..., N}, K⊂Ω_i. We denote byTi={K∈ T/K⊂Ω_i}.

2. S

K∈TiK= Ωi. Thus, S

K∈T K= Ω.

3. For anyK∈ T, there exists a subsetEK ofEsuch that∂K=K\=S

σ∈EKσ.

Furthermore,E=S

K∈T EK.

4. For any(K, L)∈ T²withK6=L, either the “length” (i.e. the(d−1)Lebesgue measure) ofK∩L is0orK∩L=σ for someσ∈ E. In the latter case, we shall writeσ=K|L, and

• Ei={σ∈ E, ∃(K, L)∈ T_i², σ=K|L}

• Eext={σ∈ E, σ⊂∂Ω}, Eext,i={σ∈ E, σ⊂∂Ωi∩∂Ω}

• EΓi,j ={σ∈ E, ∃(K, L)∈ Ti× Tj, σ=K|L}

• F=S

i,jEΓ_i,j.

For any i∈ {1, ..., N}, for anyK∈ Ti, we shall denote by:

• NK ={L∈ T, ∃σ∈ E, σ=K|L}

• NK,i={L∈ Ti, ∃σ∈ Ei, σ=K|L}

• FK={L∈ T, ∃j6=i, ∃σ∈ EΓ_i,j, σ=K|L}

• EK,i=EK∩ Ei

• EK,ext=EK∩ Eext.

5. The family of points (x_K)_K∈T is such thatx_K ∈K (for all K∈ T) and, if σ=K|L, it is assumed that the straight line(x_K, x_L)is orthogonal toσ.

For a control volumeK ∈ T, we will denote by m(K) its measure andEext,K = EK∩∂Ω. For all σ ∈ E,we denote by m(σ) the (d−1)-Lebesgue measure of σ.

If σ ∈ EK, we denote by τK,σ the transmissibility of K through σ, defined by τK,σ = _d(x^m(σ)

K,σ). We also defineτ_K|L = _d(x^m(σ)

K,x_L). The size of the mesh is defined by:

size(T) = max

K∈Tdiam(K),

and a geometrical factor, linked with the regularity of the mesh, is defined by

reg(T) = max

K∈T(card(EK),max

σ∈EK

diam(K) d(xK, σ)).

Remark 1.4 For all σ∈ E, 1

τ_K|L = 1 τK,σ

+ 1 τL,σ

.

Definition 1.4 (Uniform time discretization of (0, T))A uniform time dis- cretization of(0, T) is given by an integer value M and a sequence of real values (tⁿ)n=0,...,M+1. We define δt = _M+1^T and, ∀n∈ {0, ..., M}, tⁿ = nδt. Thus we havet⁰= 0 andt^M+1=T.

Remark 1.5 We can easily prove all the results of this paper for a general time discretization, but for the sake of simplicity, we choose to consider only uniform time discretization.

Definition 1.5 (Space-time discretization ofΩ×(0, T))A finite volume dis- cretizationDof Ω×(0, T)is the family

D= (T,E,(xK)_K∈T, N,(tⁿ)n∈{0,...,M}),

where(T,E,(xK)K∈T) is an admissible mesh of Ωin the sense of Definition 1.3 and(N,(tⁿ)n∈{0,...,M})is a discretization of (0, T) in the sense of Definition 1.4.

For a given meshD, one defines:

size(D) =max(size(T), δt), andreg(D) =reg(T).

We may now define the finite volume discretization of problem (2). Let D be a finite volume discretization ofΩ×(0, T)in the sense of Definition 1.5. The initial condition is discretized by:

U_K⁰ = 1 m(K)

Z

K

u₀(x)dx, ∀K∈ T. (6)

Animplicit finite volume schemefor the discretization of problem (2) is given by the following set of nonlinear equations, whose discrete unknowns are U = (U_Kⁿ⁺¹)_K∈T,n∈{0,...,M}:∀K∈ T,∀n∈ {0, ..., M}

m(K)U_Kⁿ⁺¹−U_Kⁿ

δt + X

σ∈EK,i

τK,σ(ϕ(U_Kⁿ⁺¹, xK)−ϕ(U_K,σⁿ⁺¹, xK))

+ X

σ∈FK

τK,σ(ϕ(U_Kⁿ⁺¹, xK)−ϕ(U_K,σⁿ⁺¹, xK)) = 0.

(7)

where∀L∈ NK, U_K,K|Lⁿ⁺¹ , U_L,K|Lⁿ⁺¹ are the only values in[0,1]that satisfy the transmission conditions:







τK,σ(ϕ(U_Kⁿ⁺¹, xK)−ϕ(U_K,σⁿ⁺¹, xK))

+τL,σ(ϕ(U_Lⁿ⁺¹, xL)−ϕ(U_L,σⁿ⁺¹, xL)) = 0 Π(U_K,σⁿ⁺¹, x_K) = Π(U_L,σⁿ⁺¹, x_L).

(8)

Definition 1.6 Let D be an admissible discretization of Ω×(0, T) in the sense of Definition 1.5. The approximate solution of problem (2) associated to the dis- cretizationDis defined almost everywhere in Ω×(0, T)by:

∀x∈K, ∀t∈(tⁿ, tⁿ⁺¹), ∀K∈ T, ∀n∈ {0, ..., M},

u_D(x, t) =U_Kⁿ⁺¹, (9)

where(U_Kⁿ⁺¹)_K∈T,n∈{0,...,M} is the unique solution to (7).

We will now state an assumption which will be useful to prove the uniqueness of the weak-solution in section 3.

Assumption 2 For all i ∈ {1, ..., N}, (ϕ_i ◦Π⁽⁻¹⁾_i )⁰ is a Lipschitz continuous function on[0,1].

We will now state our main result:

Theorem 1.1 (Convergence to the weak solution) Let ξ ∈ R, consider a family of admissible discretizations of Ω×(0, T) in the sense of Definition 1.5 such that, for all D in the family, one has ξ ≥ reg(D). For a given admissible

discretization Dof this family, letu_D denote the associated approximate solution as defined in Definition 1.6. Then, under assumptions 1-2:

u_D →u∈L^p(Ω×(0, T))assize(D)→0, ∀p∈[1,+∞)

whereuis the unique weak solution to problem (2) in the sense of Definition 1.2.

Furthermore, we have the following regularity result:

u∈C⁰([0, T], L^p(Ω)), ∀p∈[1,+∞)

All this paper will be devoted to the proof of Theorem 1.1. First, we will use the work of [3] to prove the existence of a weak-solution. Then, in subsection 2.5, we will prove the existence of a time continuous solution, applying Ascoli theorem to a family of approximate solutions for a regular enough initial data. The uniqueness of the weak solution will be proven in the section 3 by using a doubling variable method inspired from [2, 6, 8].

2 Existence of a weak solution

The main work of this section has already been done in [3]. We only need to get enough results to prove the time-continuity in section 2.5. This proof will need estimates obtained by working on the scheme, so we prefer to give the whole proof of convergence. In this whole part, any sequence (Dm)m∈N of admissible discretizations ofΩ×(0, T)in the sense of Definition 1.5 will be supposed to have a bounded regularity.

∃ζ∈R, ∀m∈N, ζ≥reg(Dm). (10)

2.1 Existence, uniqueness of the approximate solution

We state here the properties and estimates which are satisfied by the scheme (7) which we introduced in the previous section and prove existence and uniqueness of the solution to this scheme. First, we will take some notations for the convenience of the reader:

Notations 1 for allK∈ T, for allσ∈ EK∩ FK, for alln∈ {0, ..., M+ 1},



















ϕⁿ_K = ϕ(U_Kⁿ, xK), ϕⁿ_K,σ = ϕ(U_K,σⁿ , xK),

Πⁿ_K = Π(U_Kⁿ, xK), Πⁿ_K,σ = Π(U_K,σⁿ , xK),

πⁿ_K = π(U_Kⁿ, x_K), πⁿ_K,σ = π(U_K,σⁿ , x_K).

We will need the following lemma:

Lemma 2.1 For all K ∈ T, for all L ∈ NK, for all n ∈ {0, ..., M}, for all (U_Kⁿ⁺¹, U_Lⁿ⁺¹)∈R², there exists an unique(U_K,σⁿ⁺¹, U_L,σⁿ⁺¹)∈[0,1]² solution of (8).

Proof

Suppose that there existsi∈ {1, ..., N}such that(xK, xL)∈Ω²_i. Then (8) can be written:

( τ_K,σ

ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ

+τ_L,σ

ϕⁿ⁺¹_L −ϕⁿ⁺¹_L,σ

= 0 U_K,σⁿ⁺¹=U_L,σⁿ⁺¹

which leads to

( U_K,σⁿ⁺¹=U_L,σⁿ⁺¹ ϕⁿ⁺¹_K,σ =_τ ¹

K,σ+τ_L,σ

τK,σϕⁿ⁺¹_K +τL,σϕⁿ⁺¹_L and _τ ¹

K,σ+τ_L,σ

τK,σϕⁿ⁺¹_K +τL,σϕⁿ⁺¹_L

admits an unique antecedent throughϕi in [0,1].

Let us now suppose that(x_K, x_L)∈Ω_i×Ω_j with j6=i (8)⇔

( U_K,σⁿ⁺¹=π⁻¹_i (πⁿ⁺¹_L,σ) τK,σ

ϕⁿ⁺¹_K −ϕi(π⁻¹_i (πⁿ⁺¹_L,σ)) +τL,σ

ϕⁿ⁺¹_L −ϕⁿ⁺¹_L,σ

= 0 then

U_K,σⁿ⁺¹=π⁻¹_i (πⁿ⁺¹_L,σ)

τK,σϕi(π⁻¹_i (πⁿ⁺¹_L,σ)) +τL,σϕⁿ⁺¹_L,σ =τK,σϕⁿ⁺¹_K +τL,σϕⁿ⁺¹_L . Sinceπi(0) =πj(0)andπi(1) =πj(1)(Assumption 1), then:

∀U_L,σⁿ⁺¹∈[0,1], U_K,σⁿ⁺¹∈[0,1].

The application θ : z 7→ τK,σϕi(π⁻¹_i (πj(z))) +τL,σϕj(z) is increasing on [0,1], ensuring this way the uniqueness of the solution of (8). Furthermore, it satisfies:

θ(0) = 0

θ(1) =τ_K,σϕ_i(1) +τ_L,σϕ_j(1).

We conclude the proof by obtaining the existence of the solution by using the

intermediate value theorem.

L^∞-stability of the scheme

One assumesu0 ∈L^∞(Ω), 0≤u0≤1 a.e in Ω, then for allK ∈ T,U_K⁰ ∈[0,1]

whereU_K⁰ is given by (6).

Proposition 2.2 Let Dan admissible discretization ofΩ×(0, T)in the sense of Definition 1.5, let(U_Kⁿ⁺¹)_K∈T,n∈{0,...,M} be a solution of the scheme (7), then for allK∈ T, for alln∈ {0, ..., M}:

0≤U_Kⁿ⁺¹≤1. (11) Proof

Let us first remark that the scheme can be written:

U_Kⁿ⁺¹−U_Kⁿ

δt m(K) + X

L∈N_K,i

τ_K|L(ϕⁿ⁺¹_K −ϕⁿ⁺¹_L )

+ X

σ∈F_K

τK,σ(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ) = 0,

withτ_K|L=_d(x^m(K|L)

K,x_L). Let us now rewrite it once again.∀K∈ Ti, ∀n∈ {0, ..., M}:

U_Kⁿ⁺¹=HK

U_Kⁿ,(U_Lⁿ⁺¹)L∈T

with:

HK

a,(aL)_L∈T

=

a+λ_Ka_K+_m(K)^δt P

L∈NK,iτ_K|L(ϕi(aL)−ϕi(aK)) +P

σ∈FKτK,σ(ϕi(aK,σ)−ϕi(aK))

1 +λK

whereλK is given by:

λK = δtLϕ

m(K) X

L∈NK,i

τ_K|L+ X

σ∈FK

τK,σ

andL_ϕ is a Lipschitz constant for allϕ_i. However,a_K,σ=g(a_K, a_L)where g is a non-decreasing function. We deduce from it thatH_K is a nondecreasing function of all its arguments.

Letn∈ {0, ..., M}, let us assume0≤U_Kⁿ ≤1, ∀K∈ T. Let us assume that there existsKmax∈ T such that:

U_Kⁿ⁺¹

max = max

K∈T(U_Kⁿ⁺¹)>1, then:

1< U_Kⁿ⁺¹

max ≤H_K

1,(U_Kⁿ⁺¹

max)_L∈T

= 1 +λKU_Kⁿ⁺¹

max

1 +λ_K < U_Kⁿ⁺¹

max, a contradiction. Therefore:

U_Kⁿ⁺¹

max≤1.

We can prove exactly in the same way that U_Kⁿ⁺¹

min= min

K∈T(U_Kⁿ⁺¹)≥0.

Proposition 2.3 Let Dbe an admissible discretization ofΩ×(0, T)in the sense of Definition 1.5. There exists a unique solution to the scheme (7).

Proof

Existence of the discrete solution

Letn∈ {0, ..., M}Since(U_K,σⁿ⁺¹)_σ∈E in a function of(U_Lⁿ⁺¹)_L∈T, we shall see the scheme as a non linear system of equations only depending on(U_Lⁿ⁺¹)_L∈T.

Let us consider the application Ψ :

( R^]T ×[0,1] → R^]T (U_Kⁿ⁺¹)_K∈T, λ

7→ (V_K)_K∈T where:

VK = U_Kⁿ⁺¹−λU_Kⁿ

δt m(K) +λ X

L∈NK,i

τ_K|L(ϕⁿ⁺¹_K −ϕⁿ⁺¹_L )

+λ X

σ∈FK

τK,σ(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)

The linear system of equationΨ

(U_Kⁿ⁺¹)_K∈T,0

= (0)_K∈T admits a unique trivial solution. The continuity ofλ7→Ψ(·, λ)and theL^∞-estimate (11) allows us to use a topological degree argument, insuring the existence of a solution forλ= 1.

Uniqueness of the discrete solution

Assume that, for a given value ofn, there exist two solutions to the scheme (7), (U_Kⁿ⁺¹)_K∈T and (V_Kⁿ⁺¹)_K∈T. Then, for allK∈ T, using the monotony ofHK:

max(U_Kⁿ⁺¹, V_Kⁿ⁺¹)≤HK(U_Kⁿ,(max(U_Lⁿ⁺¹, V_Lⁿ⁺¹))L∈T), (12) min(U_Kⁿ⁺¹, V_Kⁿ⁺¹)≤H_K(U_Kⁿ,(min(U_Lⁿ⁺¹, V_Lⁿ⁺¹))_L∈T). (13) One multiplies (12) and (13) by(1 +λK)m(K), substracts (13) to (12) and sum on K∈ T. Remarking that all the exchange terms between neighboring control volume disappear, we get:

X

K∈T

|U_Kⁿ⁺¹−V_Kⁿ⁺¹|m(K)≤0.

2.2 Discrete L

(0, T ; H

(Ω)) estimates

In this section, we will prove some estimates on the approximate solution. We first have to define the spaceX(D)the solution belongs to.

Definition 2.1 Let Dbe an admissible discretization of Ω×(0, T)in the sense of the Definition 1.5. We denote byX(D)the functional space:

X(D) =

v∈L^∞(Ω×(0, T)),∀K∈ T, ∀n∈ {0, ..., M},

∃V_Kⁿ⁺¹, v(x, t) =V_Kⁿ⁺¹ a.e. in K×(tⁿ, tⁿ⁺¹)

.

Definition 2.2 (DiscreteL²(0, T;H¹(Ωi))semi-norm)We define the discrete L²(0, T;H¹(Ωi))semi norm onX(D)by:

|v|²_1,D,i =

M

X

n=0

δt X

K|L∈Ei

τK,σ(V_Kⁿ⁺¹−V_Lⁿ⁺¹)².

We will need the following lemma:

Lemma 2.4 Let D be an admissible discretization of Ω×(0, T) in the sense of the Definition 1.5. Let u_D be the discrete solution to (7). Let σ ∈ Γi,j for some i, j∈ {1, ..., N}²,σ=K|L, K∈ Ti, L∈ Tj. Then:

0≤(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_K,σ)≤(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_L ), (14) 0≤(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(Πⁿ⁺¹_K −Πⁿ⁺¹_K,σ)≤(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(Πⁿ⁺¹_K −Πⁿ⁺¹_L ). (15) Proof

∀K∈ T, ϕ(·, xK), andπ(·, xK)are increasing functions on[0,1], thus (ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_K,σ)≥0.

Furthermore,(πⁿ⁺¹_K −πⁿ⁺¹_L ) = (πⁿ⁺¹_K −πⁿ⁺¹_K,σ +πⁿ⁺¹_L,σ −πⁿ⁺¹_L )then:

(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_L ) = (ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_K,σ) +(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_L −πⁿ⁺¹_L,σ)

= (ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_K,σ) +τL,σ

τK,σ

(ϕⁿ⁺¹_L −ϕⁿ⁺¹_L,σ)(πⁿ⁺¹_L −πⁿ⁺¹_L,σ)

≥ (ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_K,σ).

This proof is also true withΠ instead ofπ.

One introduces the function ηi:s7→

Z s 0

pλi(a)π⁰_i(a)da,

which fulfills thanks to Cauchy-Schwarz inequality, for all (a, b) ∈ [0,1]², for all i∈[[1,N]],

(ηi(a)−ηi(b))²≤(ϕi(a)−ϕi(b))(πi(a)−πi(b)). (16) Proposition 2.5 (Discrete L²(0, T;H¹(Ωi)) estimate) Under Assumption 1, letD be an admissible discretization of Ω×(0, T) in the sense of Definition 1.5, letuD be the solution of the scheme (7). Then there exists C only depending on π_i, Ω_i, i∈ {1, ..., N} such that:

N

X

i=1

|ηi(uD)|²_1,D,i≤C. (17)

0≤

M

X

n=0

δt X

σ∈ F σ=K|L

τK,σ(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_L )≤C (18)

Proof

Accumulation term:

Let us multiply the equations (7) by δtπⁿ⁺¹_K and sum on K∈ T, n∈ {0, ..., M}.

We get:

M

X

n=0

X

K∈T





m(K)(U_Kⁿ⁺¹−U_Kⁿ) +δtP

L∈NK,iτ_K|L(ϕⁿ⁺¹_K −ϕⁿ⁺¹_L )+

P

s∈FKτK,σ(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ) πⁿ⁺¹_K



= 0.

Let i belong to {1, ..., N}, let K∈ Ti. Since πi is an increasing function, gi:s7→Rs

0πi(a)dais a convex function. Then:

(U_Kⁿ⁺¹−U_Kⁿ)πⁿ⁺¹_K ≥gi(U_Kⁿ⁺¹)−gi(U_Kⁿ).

Thus:

M

X

n=0

X

K∈T

m(K)(U_Kⁿ⁺¹−U_Kⁿ)πⁿ⁺¹_K ≥ X

K∈T

m(K)(g_i(U_K^M+1)−g_i(U_K⁰))

≥ −m(Ω) Z 1

0

max

i∈{1,...,N}|πi(a)|da.

Diffusion term:

One gets, thanks to (16)

M

X

n=0

δt

N

X

i=1

X

K|L∈Ei

τ_K|L(ϕⁿ⁺¹_K −ϕⁿ⁺¹_L )(πⁿ⁺¹_K −πⁿ⁺¹_L )

≥

M

X

n=0

δt

N

X

i=1

X

K|L∈Ei

τ_K|L(ηⁿ⁺¹_K −η_Lⁿ⁺¹)²,

where η_Kⁿ⁺¹ denotes ηi(U_Kⁿ⁺¹) if K ⊂ Ωi. Furthermore, for all σ = K|L ∈ F, Lemma 2.4 implies:

(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_L )≥0,

then, we have the following estimates:

N

X

i=1

|ηi(u_D)|²_1,D,i≤m(Ω) Z 1

0

max

i∈{1,...,N}|πi(a)|da,

0≤

M

X

n=0

δt X

σ=K|L∈F

(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_L )≤m(Ω) Z 1

0

max

i∈{1,...,N}|π_i(a)|da.

Definition 2.3 (DiscreteL²(0, T;H¹(Ω)) semi-norm)LetDbe an admissible discretization of Ω×(0, T) in the sense of the Definition 1.5. One defines the discreteL²(0, T;H¹(Ω)) semi norm ofv∈ X(D)by:

|v|²_1,D =

N

X

i=1

|v|²_1,D,i+ X

σ=K|L∈F

h

τ_K|L(v(xK, tⁿ⁺¹)−v(xL, tⁿ⁺¹))²i .

We will need the following lemma:

Lemma 2.6 Under assumptions 1, for alli in{1, ..., N}, the function Πi◦η_i⁽⁻¹⁾ admits1 as Lipschitz constant.

Proof

Leti∈ {1, ..., N}, leta∈]0, ϕi(1)[, letb∈]0, ηi(1)[withb6=a.

We setA=η_i⁽⁻¹⁾(a)andB=η_i⁽⁻¹⁾(b). One has:

πi◦η_i⁽⁻¹⁾(b)−πi◦η_i⁽⁻¹⁾(a)

b−a = πi(B)−πi(A) ηi(B)−ηi(A).

One denote by I(A, B) the interval [A, B] if B ≥ A, and [B, A] if A ≥ B. The definition of the functionη_i implies:

min

C∈I(A,B)

pλ_i(C)(π_i(B)−π_i(A))≤η_i(B)−η_i(A)≤ max

C∈I(A,B)

pλ_i(C)(π_i(B)−π_i(A)).

Then, there existsC∈I(A, B)such that:

η_i(B)−η_i(A) =p

λ_i(C)(π_i(B)−π_i(A)).

So one gets:

πi◦η⁽⁻¹⁾_i (b)−πi◦η⁽⁻¹⁾_i (a)

b−a = 1

pλi(C).

Lettingb tend toa, we get, using the continuity ofη⁽⁻¹⁾_i : (π_i◦η⁽⁻¹⁾_i )⁰(a) = 1

q

λi◦η_i⁽⁻¹⁾(a) .

Remarking thatΠ_i = Ψ◦π_i with

Ψ :







[πi(0), πi(1)] → R⁺ p 7→ Rp

π_i(0) minj∈{1,...,N}

q

λj◦π⁽⁻¹⁾_j (a)

da we may obtain:

(Πi◦η⁽⁻¹⁾_i )⁰(a) = Ψ⁰(πi◦η_i⁽⁻¹⁾(a))(πi◦η_i⁽⁻¹⁾)⁰(a) =Ψ⁰(πi◦η_i⁽⁻¹⁾(a)) q

λi◦η⁽⁻¹⁾_i (a) .

Remarking that the definition ofΨimpliesΨ⁰(πi(y))≤p

λi(y), we get that (Πi◦η⁽⁻¹⁾_i )⁰(a)≤1.

Proposition 2.7 (Discrete L²(0, T;H¹(Ω)) estimate) Let D be an admissible discretization of Ω×(0, T) in the sense of Definition 1.5, let uD ∈ X(D) be the approximate solution given by the scheme (7). There exists a constant C only depending onΩ, πi,∀i∈ {1, ..., N}such that:

|Π(u_D,·)|²_1,D≤C.

Proof

Using inequality (17) proven in Proposition 2.5 and Lemma 2.6, we immediately get that:

N

X

i=1

|Πi(u_D)|²_1,D,i≤C.

Let us now consider the case σ =K|L ∈ F. Using πⁿ⁺¹_K,σ = πⁿ⁺¹_L,σ, inequal- ity (18) together with (8) leads to:

M

X

n=0

δt X

σ=K|L∈F

τK,σ(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_K,σ)+

τL,σ(ϕⁿ⁺¹_L −ϕⁿ⁺¹_L,σ)(πⁿ⁺¹_L −πⁿ⁺¹_L,σ)

≤C. (19)

For allσ∈ F, for allKsuch thatσ∈ EK, we have thanks to (16):

(ηⁿ⁺¹_K −η_K,σⁿ⁺¹)²≤(ϕⁿ⁺¹_K −ϕⁿ⁺¹_K,σ)(πⁿ⁺¹_K −πⁿ⁺¹_K,σ).

Lemma 2.6 implies:

(Πⁿ⁺¹_K −Πⁿ⁺¹_K,σ)²≤(ηⁿ⁺¹_K −η_K,σⁿ⁺¹)² and

(Πⁿ⁺¹_L −Πⁿ⁺¹_L,σ)²≤(ηⁿ⁺¹_L −η_L,σⁿ⁺¹)², thus (19) leads to

M

X

n=0

δt X

σ=K|L∈F

h

τK,σ(Πⁿ⁺¹_K −Πⁿ⁺¹_K,σ)²+τL,σ(Πⁿ⁺¹_L −Πⁿ⁺¹_L,σ)²i

≤C. (20)

The convexity of the functionx7→x² together with the relation

1 τ_K|L = _τ¹

K,σ +_τ¹

L,σ leads to:

M

X

n=0

δt X

σ=K|L∈F

τ_K|L(Πⁿ⁺¹_K −Πⁿ⁺¹_L )²≤C.

2.3 Some compactness results

We aim in this section to get enough compactness results to be able to letsize(D) tend to0. Proposition 2.2 insures that, up to a subsequence,∃u∈L^∞(Ω×(0, T)) such thatu_D * uin the L^∞(Ω×(0, T))-weak?topology.

Let us now turn to the Kolmogorov compactness criterion (see e.g. [1]) which will allow us to pass to the limit in the nonlinear second order terms.

Theorem 2.8 (Kolmogorov) LetQbe an open bounded subset ofR^k, and(vn)n

be a bounded sequence inL²(R^k)such that:

δ→0lim[sup

n∈N

kvn(·+δ)−vn(·)] = 0, then there existsv∈L²(Q)such that, up to a subsequence,

v_n →v inL²(Q)asn→ ∞.

Let us now show that we are in position to apply the Kolmogorov compactness cri- terion to(ηi(u_Dm))_m∈Nwhere(Dm)_m∈Nis a sequence of admissible discretization ofΩ×(0, T), withlim_m→∞size(Dm) = 0.

Space translates estimates

We will now state a proposition proven in [3], which is a consequence of Proposition 2.5.

Proposition 2.9 LetDbe an admissible discretization ofΩ×(0, T)in the sense of Definition 1.5, letuD ∈ X(D)be the approximate solution given by the scheme (7), leti∈ {1, ..., N}, letξ∈R^d, andΩi,ξ the open subset of Ωi defined by:

Ωi,ξ ={x∈Ωi / [x, x+ξ]⊂Ωi}.

Then there existsC only depending onΩ_i such that:

Z T 0

Z

Ω_i,ξ

|ηi(uD(x+ξ, t)−ηi(uD(x, t)|²dxdt≤ |ξ|(|ξ|+Csize(D))|ηi(uD)|1,D,i. (21) Time translates estimates

We state here a first result on the time translates of ϕi(u), already proven in [3].

Proposition 2.10 Let Dbe an admissible discretization of Ω×(0, T). LetuD ∈ X(D)be the approximate solution obtained with the scheme (7). Letω_i⊂ω_i ⊂Ω_i be an open subset ofΩ_i. We assume that size(T) is small enough to ensure that:

ω_i⊂Ω_i,size(T)={x∈Ω_i / B(x, size(T))⊂Ω_i}.

We set:

η_D,ωi =

η_i(u_D) on ω_i×(0, T)

0 on R^d+1\(ω_i×(0, T)) then, for allτ∈R, we get the following inequality:

kη_D,ωi(·,·+τ)−η_D,ωik²≤Cτ, (22) whereC is a constant which only depends onT,Ω, d, ϕi, πi,Θi.

Thus we can apply the Kolmogorov compactness criterion, and claim that there exists a function f such that, up to a subsequence, η(u_D,·) converges to f in L²(Ω×(0, T)) as size(D) tends to 0. Furthermore, letting size(D) tend to 0 in estimates (21) ensures thatf ∈L²(0, T;H¹(Ωi))for all i∈ {1, ..., N}. The same way, we can prove that Π(u_D,·) converges to some g ∈ L²(0, T;H¹(Ω)) in the L²(Ω×(0, T))-topology.

Remark 2.1 Since the functions ηi◦ϕi(−1) are Lipschitz continuous, estimates (21) and (22) still hold with ϕ_i instead of η_i. Then ϕ(u_D,·) also converges to a functionhin L²(Ω×(0, T)).

2.4 Convergence to a weak solution

In this section, we aim to prove that, for an admissible sequence(Dm)of discetiza- tion ofΩ×(0, T)in the sense of the Definition 1.5, with lim_m→+∞size(D_m) = 0 the solution of the schemeu_D tends to a weak solution of the problem (2).

We have proven in the previous section thatϕ(u_Dm,·)(resp.Π(u_Dm,·)) con- verges in L²(Ω×(0, T)) to a function f (resp. g). Proposition 2.2 allows us to assume thatu_Dm converges touin L^∞(Ω×(0, T)) for the weak? topology. Us- ing Minty Lemma, stated below, let us show that f = η(u,·), g = Π(u,·) and h=ϕ(u,·).

Lemma 2.11 (Minty lemma)LetΩbe an open subset ofR^k, let Ψ :R→Rbe a monotonous continuous function. Let(un)n∈Nsuch that:

u_n* u in L^∞(Ω) weak−? Ψ(u_n)→f in L¹(Ω)

then

Ψ(u) =f.

From the Lemma 2.11, we can deduce that, for alli∈ {1, ..., N}:

η_i(u_D)→η_i(u) in L¹(Ω_i×(0, T)), thus:

η(u_D,·)→η(u,·) in L¹(Ω×(0, T)).

The same way, ϕ(u_D,·) and Π(u_D,·) converge in L²(Ω×(0, T)) to ϕ(u,·) and Π(u,·), respectively. Estimate (21) insures that, for all i ∈ {1, ..., N}, ηi(u) be- longs to L²(0, T;H¹(Ωi)), and thus ϕi(u) too. A straightforward adaptation of Lemma 2.9 withΠ(uD,·)instead ofηi(uD)allows us to claim thatΠ(u,·)belongs toL²(0, T;H¹(Ω)).

Proposition 2.12 (Convergence to a weak solution) Under assumption 1, let(Dm) be a sequence of admissible discretizations in the sense of Definition 1.5 which fulfill the assumption (10) and such thatsize(D_m)→0. Let (u_Dm) be the sequence of approximate solutions given by the scheme (7). Then there exists a subsequence of approximate solutions still denoted (u_Dm) which converges to u in L^q(Ω×(0, T)) for all q ∈ [1,+∞). Furthermore, u is a weak solution to the problem (2) in the sense of Definition 1.2.

The proof of this proposition is a straightforward adaptation of the proof stated in [3].

2.5 Time continuity of the approximation limit

The aim of this section is to prove the existence of a time continuous solution.

This result will be fundamental to prove the uniqueness of the weak solution to the problem (2).

In order to prove the continuity of a time continuous solution to the problem (2), we will apply the Ascoli theorem on a family of approximate solutions obtained through the scheme (7). We need a classical CFL assumption on the family of space-time discretizations.

Assumption 3 Let (Dm)_m∈N be an admissible space-time discretization of Ω× (0, T)in the sense of Definition 1.5. In all this subsection, we furthermore assume that there existsS1>0 which does not depend onm such that:

maxm

δt_m

(size(Tm))² ≤S1.

Remark 2.2 Assumption 3 and (10) ensure us that the quantity

max_K∈T(max_L∈NK,i(^δtτ_m(K)^K|L))stays bounded asmtends to+∞. The assumption 3 is not hard to fulfill in the theorical framework. One just has to choose a conve- nient time step. Nevertheless, this assumption is very demanding in the numerical framework, but we will be able to relax it in the sequel of this work.

We also need to make an assumption on the rgularity of the initial data, but once again this assumption will be relaxed in the sequel, thanks to a density argument.

Assumption 4 The initial data u₀ belongs to L^∞(Ω),0 ≤u₀ ≤1, and further- more fulfills:ϕ(u0) is a piecewise Lipschitz function.

We will first need the following lemmas:

Lemma 2.13 (DiscreteH¹(0, T;L²(ωi))estimate)Let(Dm)_m∈Nbe a sequence of admissible discretizations of Ω×(0, T) fulfilling assumption 3. Let (u_Dm) the sequence of approximate solutions given by the scheme (7). Let Oi be an open subset of Ωi such that ϕi(u0)_|Oi is a Lipschitz continuous function. Let ωi be an open subset of Oi, with ωi ⊂ Oi. Then there exists C only depending on Oi, ωi, λi, πi, u0, S1, ζ such that:

M

X

n=0

X K∈ T K⊂ω_i

m(K)(ϕ_i(u_Dm(x_K, tⁿ⁺¹))−ϕ_i(u_Dm(x_K, tⁿ)))²≤Cδt.

Proof

We use the following notations:

EO_i ={σ∈ Ei, σ=K|L/ K ⊂Oi, L⊂Oi}, Eωi ={σ∈ Ei, σ=K|L/ K⊂ω_i, L⊂ω_i}.

LetΘ_i∈C_c^∞(Ω)such thatsupp(Θ_i)⊂O_i, Θ_i|ωi= 1, 1≥Θ_i≥0. For allK∈ T, we denoteΘ_i,K = Θ_i(x_K). We multiply the scheme (7) by

(ϕⁿ⁺¹_K −ϕⁿ_K)Θ_i,K²δt:

U_Kⁿ⁺¹−U_Kⁿ

δt m(K)(ϕⁿ⁺¹_K −ϕⁿ_K)Θ_i,K²+ X

L∈NK,i

τ_K|L(ϕⁿ⁺¹_K −ϕⁿ⁺¹_L ) (ϕⁿ⁺¹_K −ϕⁿ_K)Θi,K2

= 0,

thus:

m(K)(ϕⁿ⁺¹_K −ϕⁿ_K)²Θ_i,K²≤L_ϕiδt X

L∈NK,i

τ_K|L(ϕⁿ⁺¹_L −ϕⁿ⁺¹_K )(ϕⁿ⁺¹_K −ϕⁿ_K)Θ_i,K².

LetM₁∈ {0, ..., M}. We sum onK∈ T and onn∈ {0, M₁}:

M₁

X

n=0

X

K∈T

m(K)(ϕⁿ⁺¹_K −ϕⁿ_K)²Θi,K2

≤Lϕ_i M₁

X

n=0

δt X

K|L∈Ei

τ_K|L(ϕⁿ⁺¹_L −ϕⁿ⁺¹_K )

((ϕⁿ⁺¹_K −ϕⁿ_K)Θ_i,K²−(ϕⁿ⁺¹_L −ϕⁿ_L)Θ_i,L²)

≤













 Lϕ_i

M1

X

n=0

δt X

K|L∈Ei

"

τ_K|L(ϕⁿ⁺¹_L −ϕⁿ⁺¹_K )_Θ

i,K2+Θi,L2

2

((ϕⁿ_L−ϕⁿ_K)−(ϕⁿ⁺¹_L −ϕⁿ⁺¹_K ))

#

+Lϕ_i

2

M₁

X

n=0

δt X

K|L∈Ei

τ_K|L(ϕⁿ⁺¹_L −ϕⁿ⁺¹_K )(Θi,L2

−Θi,K2

) ((ϕⁿ⁺¹_K −ϕⁿ_K) + (ϕⁿ⁺¹_L −ϕⁿ_L))















M1

X

n=0

X

K∈T

m(K)(ϕⁿ⁺¹_K −ϕⁿ_K)²Θ_i,K²≤A₁+A₂, (23) with















A₁=L_ϕi

M₁

X

n=0

δt X

K|L∈Ei

"

τ_K|L(ϕⁿ⁺¹_L −ϕⁿ⁺¹_K )_Θ

i,K2+Θ_i,L² 2

((ϕⁿ_L−ϕⁿ_K)−(ϕⁿ⁺¹_L −ϕⁿ⁺¹_K ))

#

A₂= L_ϕi 2

M1

X

n=0

δt X

K|L∈E_i

τ_K|L(ϕⁿ⁺¹_L −ϕⁿ⁺¹_K )(Θ_i,L²−Θ_i,K²) ((ϕⁿ⁺¹_K −ϕⁿ_K) + (ϕⁿ⁺¹_L −ϕⁿ_L))

.