Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions

BLAISE PASCAL

Marie-Josée Jasor, Laurent Lévi

Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed Dirichlet-Neumann Boundary Conditions

Volume 10, n^o2 (2003), p. 269-296.

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Singular Perturbations for a Class of Degenerate Parabolic Equations with Mixed

Dirichlet-Neumann Boundary Conditions

Marie-Josée Jasor Laurent Lévi

Abstract

We establish a singular perturbation property for a class of quasi- linear parabolic degenerate equations associated with a mixed Dirichlet- Neumann boundary condition in a bounded domain of R^p, 1 ≤ p <

+∞. In order to prove the L¹-convergence of viscous solutions to- ward the entropy solution of the corresponding first-order hyperbolic problem, we refer to some properties of bounded sequences in L^∞ together with a weak formulation of boundary conditions for scalar conservation laws.

1 Introduction

This paper is devoted to the study of the singular limit, as goes to0⁺, for the class of second-order degenerate equations

∂_tu+Div_x(B(t, x)ϕ(u)) +ψ(t, x, u) =∆φ(u) in]0, T[×Ω, >0, (1.1) whereΩis a bounded domain ofR^p,1≤p <+∞, with a Lipschitz boundary Γ and T a positive real. Moreover, B is a vector field on ]0, T[×Ω, ϕ is a non-decreasing function andφis aC¹-class function with

φ⁰≥0,L₁({x∈R, φ⁰(x) = 0}) = 0, (1.2) whereL_p refers to the Lebesgue measure onR^p, p≥1.

Equations (1.1) are associated with the mixed boundary conditions:

u=u_Γe on]0, T[×Γ_e,∇φ(u).ν = 0on ]0, T[×(Γ\Γ_e), (1.3)

where ν is the outward unit vector defined a.e. on Γ and Γ_e is a part of Γ with a positive dΓ-measure.

In petroleum engineering within the context of a secondary recovery pro- cess, theDead Oil isothermal model leads us to consider a two-phase (oil- water) flow incompressible and immiscible through a porous medium (the reservoir rockΩ) and a partitioning of its boundaryΓin three open separated areasΓ_e (with a positive dΓ-measure),Γ_s (with a non-negativedΓ-measure) and Γ_L and corresponding respectively to the water injection wells, the oil production ones and the waterproof walls of the deposit with:

Γ = Γ_e∪Γ_s∪Γ_L∪∂Γ_L,Γ_e∩Γ_s=∅.

Then the oil-reduced saturation is the solution of an (1.1)-type equation reladocumentclassted to boundary conditions (1.3), where ϕ denotes the oil flood ratio, (−B) is a pressure gradient, for example the global pressure P introduced by G.Chavent & P.Jaffré [6], which is the solution of a linear parabolic equation with the mixed Neumann-Robin boundary conditions

d∇P.ν=f on Γ_e,∇P.ν= 0 onΓ_L, d∇P.ν =−λP on Γ_s, (1.4) whered is a positive constant,f a given smooth stationary filtering velocity at the injection wells and λ the permeability coefficient at the production wells satisfying:

f ≥0 onΓ_e, dΓ−meas{x∈Γ_e, f(x)>0}>0, λ≥0 onΓ_s, dΓ−meas{x∈Γ_s, λ(x)>0}>0.

A detailed presentation may be found in G.Gagneux & M.Madaune-Tort’s book [10].

As(1.1)is written, the capillary pressure between the oil and water phases has been taken into account through the functionφ. Usually, these capillary effects are negligible in favor of transport ones, meaning that is small; the case when is equal to zero signifying that the capillary effects are com- pletely neglected. Hence, our aim is to compare both models whether the viscous parameteris positive (and(1.1)is a quasilinear degenerate parabolic equation) or nul (and(1.1)is an hyperbolic first-order quasilinear equation).

We can summarize the properties obtained in this work through the next theorem whose proof will be developed in the following sections:

Theorem 1.1:

(i) For any positive, the degenerate parabolic-hyperbolic equation(1.1)asso- ciated with the mixed Dirichlet-Neumann boundary conditions(1.3) and the initial datau₀ has an unique solution u.

(ii) When goes to 0⁺ the sequence (u)_>0 gives an L¹-approximation of the weak entropy solution to the quasilinear first-order hyperbolic problem formally described by:

∂_tu+Div_x(B(t, x)ϕ(u)) +ψ(t, x, u) = 0 in ]0, T[×Ω, (1.5) u=u_Γe on an unknown part of ]0, T[×Γ but including ]0, T[×Γ_e, (1.6)

u(0, .) =u₀ on Ω. (1.7)

Numerous works have been achieved on the study of the behavior of the viscous sequence(u)_>0 as goes to 0⁺, especially when the diffusion term is linear [3, 18] or for an homogeneous Dirichlet boundary condition [12] and even for obstacle problems [15, 16, 17, 20]. However, few works have dealt with the particular framework considered here. In fact, this paper refers to the study in [13] whenBis stationary and sufficiently smooth to obtain local a priori estimates for the sequence (u)_>0 in the space of bounded func- tions with bounded variations on ]0, T[×Ω, by using weight-functions non- increasing along the characteristics of the linear operator v → B.∇v. But since these smoothness conditions are not generally satisfied by the pressure gradient∇P, we try to release in this paper the assumptions onB bearing in mind that hereB is time-and-space depending and considering that due to the concept of Young measures [7, 8, 18, 21] a uniform L^∞-estimate of approximate solutions is sufficient to characterize the-limit of(1.1,1.3).

Thence our first objective is to prove the existence and uniqueness of the solutionu to(1.1,1.3)associated with the initial datumu₀. With this view we introduce, for each value of the parameterδ in ]0,1], a regularized prob- lem obtained by turning the nonlinearityφintoφ_δ=φ+δId_R. We establish a priori estimates of u_,δ that are independent from δ so as to provide an

existence result for the degenerate problem. In fact, the mathematical con- text considered here only permits to obtain δ-uniform Hilbertian estimates for φ_δ(u_,δ) and in order to establish the uniqueness proof for (1.1,1.3) an assumption on the local behavior ofϕ◦φ⁻¹is necessary. Namely, we assume that

ϕ◦φ⁻¹ is Lipschitzian on [φ(−M(T)), φ(M(T))],

with a constant M_ϕ◦φ⁰ −1. (1.8) where M(T) is defined by (1.9).

Our second objective is to pass to the limit whengoes to0⁺. The maxi- mum principle ensures that the sequence(u)_>0is bounded inL^∞(]0, T[×Ω).

Accordingly, the behavior of bounded sequences inL^∞ and its consequences that we owe to R.Eymard, T.Gallouët & R.Herbin [8] permit to pass to the L^∞-weak star limit. This provides the desired singular perturbations property: as goes to 0⁺ the sequence of viscous solutions (u)_>0 gives anL¹-approximation of the solution uto the hyperbolic first-order problem (1.5,1.6,1.7).

Remark:The main feature of this paper is to deal with some mixed Dirichlet- Neumann boundary conditions on ]0, T[×∂Ω. The nonhomogeneous condi- tions are given by the physical model but they are not essential to compre- hension.

Of course all the results exposed here still hold in the situation of Dirichlet data for the same kind of operator. Especially the singular perturbations property stated in theorem 3.1 applies with the same smoothness assumption on the boundary data and with the same mathematical tool. Concerning the existence and uniqueness of theorem 2.1, the hypothesis (1.8) falls given that that every weak solution in the sense of theorem 2.1 fulfills implicitly an entropy inequality which is chosen as the starting point for the uniqueness (see [10]). It is also possible to weaken (1.2) and consider a strong degenerate operator for which φ⁰ ≥ 0, L₁({x ∈ R, φ⁰(x) = 0}) ≥ 0, assuming thereby thatφ⁻¹ is not a function necessary. In this special context we need to refer to a weak entropy formulation for the second-order problem which is the state of the art at the moment (see [4],[5],[9],[19]...). At our knowledge there is no result concerning the existence and uniqueness of the weak entropy solution

to a strongly degenerate parabolic-hyperbolic operator associated with mixed Dirichlet-Neumann boundary conditions.

1.1 Main notations and hypotheses on data

For anyt of ]0, T], Q_t denotes the cylinder]0, t[×Ω knowing that Q_T = Q.

SimilarlyΣ_t=]0, t[×ΓandΣ_T = Σ. Thus, in the rest of this paper we assume the following hypotheses are fulfilled:

•Ba is vector field of(W^1,+∞(Q))^p, this regularity being actually given when one follows the Kruskov uniqueness method for first-order quasilinear equa- tion [14]. Furthermore we suppose thatDiv_xB= 0, which is not restrictive given that(1.1)involves a reaction term.

•Let us define:

Γ₋={σ∈Σ,B(σ).ν <0} withdΓ(Γ₋)>0

and thus, as a result of(1.4) with ∇P =−B, Γ_e is an open part ofΓ such that

Γ₋⊆Γ_e and dΓ(Γ_e)>0.

It is important to notice thatΓ₋, which is the part of Γcorresponding to the inward characteristics for the linear hyperbolic operatorv→B.∇v, may be considered independent from the time variable t. Indeed, the physical model considered shows that, due to(1.4),

B.ν =−f /don Γ_e.

•u₀ is a measurable and bounded functionΩ.

•u_Γeis smooth enough to reduce(1.3)to an homogeneous Dirichlet boundary condition on ]0, T[×Γ_e by means of a translation procedure. Namely, it is sufficient to assume that u_Γe is the trace on ]0, T[×Γ of a function u_Γe of H¹(Q)∩L^∞(Q)such that∂_tu_Γe and∂_tφ(u_Γe) belong toH¹(Q).

Let us observe that these assumptions ensure the existence of a time- depending functionM :t→M(t)defined for any tof[0, T] by

M(t) =max(ku_Γek_L^∞_(Σ),ku₀k_L^∞_(Ω))e^Mψ0t+ e^Mψ0t−1

M_ψ⁰ kψ(t, x,0)k_L^∞_(Q) (1.9)

•ϕis a non-decreasing Lipschitzian function on[−M(T), M(T)]with a con- stantM_ϕ⁰.

• φ is an increasing W^1,+∞(−M(T), M(T))-class function such that (1.2) holds. Let us note that these hypotheses only entail the continuity ofφ⁻¹.

We remind thatϕ◦φ⁻¹ is Lipschitzian on [φ(−M(T)), φ(M(T))]with a constant M_ϕ◦φ⁰ −1.

•ψ belongs toW^1,+∞(Q×R),M_ψ⁰ =ess sup

(t,x,u)∈Q×R

|∂_uψ(t, x, u)|

Eventually, we introduce "sgn_λ", the Lipschitzian and bounded approxi- mation of the function "sgn", given for any positive parameter λ and for all positive realxby:

sgn_λ(x) = minx λ,1

and sgn_λ(−x) =−sgn_λ(x).

2 The vanishing viscosity method

In order to take the-limit in(1.1), we seeka priori estimates for the sequence (u)_>0. This is the purpose of the next theorem which is the first main result of this paper:

Theorem 2.1: For any positive, the parabolic degenerate equation(1.1)has a unique solutionuassociated with initial datumu₀ and boundary condition u_Γe on Γ_e. Precisely, u belongs to L^∞(Q), φ(u) and ∂_tu are respectively elements ofL²(0, T;H¹(Ω)) andL²(0, T;V⁰), and

for a.e. t of]0, T[ u=u_Γe a.e. onΓ_e, ess lim

t→0⁺

Z

Ω

|u(t, x)−u₀(x)|dx= 0. (2.1)

In addition, u satisfies the variational equality for a.e. t of ]0, T[ and for anyv ofV:

< ∂_tu, v > − Z

Ω

ϕ(u)B(t, x).∇vdx+ Z

Γ

ϕ(u)vB(t, γ).νdγ

+

Z

Ω

∇φ(u).∇vdx+ Z

Ω

ψ(t, x, u)vdx= 0. (2.2)

where < ., . > stands for the duality brackets V⁰-V. What is more, for any positive ,

kuk_L^∞_(Q)≤M(T), (2.3)

whereM(T) is defined through (1.9).

2.1 Proof of theorem 2.1: uniqueness property

The uniqueness proof for the solution of parabolic degenerate equations have been achieved by many authors, principally for the Cauchy-Dirichlet problem.

Various techniques have been used to avoid the difficulties owing to the lack of regularity of first-order partial derivatives for a weak solution and to the nonlinear context considered. At the moment, the state-of-art consists in considering a strongly degenerate problems where φ is only non-decreasing.

Following the original ideas of J.Carrillo [5] many works dealt with entropy formulation for parabolic degenerate problems (see [4, 9, 11, 19] and the corresponding references). Then the uniqueness proof refers to the Kruskov method [14] classically used to study the uniqueness of the weak entropy solution to quasilinear conservation laws by splitting the time and the space variables in two.

In this work, as we take into account a mixed Dirichlet-Neumann bound- ary condition, we have at our disposal the works of M.J.Jasor [13] where B is stationary and those of G.Gagneux & M.Madaune-Tort [10] for varia- tional inequality. In each case, in order to deal with the convective term, the functionϕ◦φ⁻¹is supposed Hölder-continuous with an exponent of at least 1/2. However, given that here B is time depending, we have been forced to assume thatϕ◦φ⁻¹ is Lipschitzian on a bounded interval of R. In this weakly degenerate framework, the next statement holds:

Proposition 2.2: Suppose that (1.8) holds and let u and v be two weak solutions to degenerate equation(1.1)corresponding respectively to the couple of boundary data (u₀, u_Γe) and (v₀, u_Γe). Then, for a.e. tof ]0, T[:

Z

Ω

(u(t, x)−v(t, x))⁺dx≤ Z

Ω

(u₀− v₀)⁺dx e^Mψ0t.

Proof: The demonstration draws its inspiration from that presented in [10] by splitting only the time variable in two. Let us briefly describe the mathematical tools: letζ be an element of D₊(0, T) and for any positive µ, letρ_µbe the standard mollifier with support in[−µ,+µ]. For anytand ˜tof ]0, T[, we consider

ζ_µ(t,˜t) =ζ

t+ ˜t 2

ρ_µ

t−˜t 2

,

withµsufficiently small such thatζ_µ belongs toD₊(]0, T[×]0, T[).

In order to simplify the writing, we drop the index temporarily and we add a tilde superscript to any function with the variable ˜t. Thus, in variational formulation (2.2) for u written in the variables (t, x) we may choose the test-function sgn⁺_λ(φ(u)−φ(˜v))ζ_µ and in the one satisfied by v and written in the variables (˜t, x), the test-function−sgn⁺_λ(φ(u)−φ(˜v))ζ_µ. Then we integrate on the time variables over]0, T[×]0, T[. By adding up and taking into account the positiveness of the diffusive terms, it comes:

T

Z

0 T

Z

0

< ∂_tu−∂˜t˜v, sgn⁺_λ(φ(u)−φ(˜v))> ζ_µdtd˜t

−

T

Z

0

Z

Q

{ϕ(u)B−ϕ(˜v) ˜B}.∇sgn⁺_λ(φ(u)−φ(˜v))ζ_µdxdtd˜t

+

T

Z

0

Z

Σ

{ϕ(u)B−ϕ(˜v) ˜B}.νsgn⁺_λ(φ(u)−φ(˜v))ζ_µdσdtd˜t

≤ −

T

Z

0

Z

Q

{ψ(t, x, u)−ψ(˜t, x,v)}sgn˜ ⁺_λ(φ(u)−φ(˜v))ζ_µdxdtd˜t. (2.4)

In order to pass to the limit with respect toλ and then toµin (2.4), let us develop the following transformations in the right-hand side:

• For the duality brackets V⁰-V, thanks to an adaptation of the Mignot- Bamberger Lemma [2], one gets an integration-by-parts formula which pro- vides the next integral:

−

T

Z

0

Z

Q





u

Z

˜ v

sgn⁺_λ(φ(τ)−φ(˜v))dτ ∂_tζ_µ

+

u

Z

˜ v

sgn⁺_λ(φ(u)−φ(τ))dτ ∂˜tζ_µ



dxdtd˜t.

When the λ-limit is taken, it comes:

−

T

Z

0

Z

Q

(u−v)˜ ⁺(∂_tζ_µ+∂˜tζ_µ)dxdtd˜t.

•For the second line:

−

T

Z

0

Z

Q

{ϕ(u)−ϕ(˜v)}B.∇sgn⁺_λ(φ(u)−φ(˜v))ζ_µdxdtdt˜

−

T

Z

0

Z

Q

ϕ(˜v){B−B}.∇sgn˜ ⁺_λ(φ(u)−φ(˜v))ζ_µdxdtdt.˜

Thus, thanks to hypothesis(1.8)and to the Sacks Lemma, the first term goes to0 withλ. For the second one a Green formula is used since(1.8) ensures thatϕ(˜v)belongs toL²(0, T;H¹(Ω)).

Finally, when λ goes to0⁺, by expressing the time-partial derivative of ζ_µand taking the monotonicity of φinto account, it comes:

Z

]0,T[×Q

n

−(u−v)˜⁺ζ⁰ρ_µ+∇ϕ(˜v).{B−B}sgn˜ ⁺(u−˜v)ζρ_µ o

dxdtd˜t

+ Z

]0,T[×Σ

{ϕ(u)−ϕ(˜v)}B.νsgn⁺(u−˜v)ζρ_µdσd˜t

≤ M_ψ⁰ Z

]0,T[×Q

(u−˜v)⁺ζρ_µdxdtd˜t.

Let us note that due to the partitioning ofΓ and to the monotonicity of ϕthe boundary integral is non-negative. Then, by referring to the notion of a Lebesgue point for an integrable function, we may take the µ-limit in the previous inequality. As the second term in the left-hand side goes to 0, it follows:

− Z

Q

(u−v)⁺ζ⁰dxdt≤M_ψ⁰ Z

Q

(u−v)⁺ζdxdt,

for any ζ of D₊(0, T) and by density, for any ζ of W₊^1,1(0, T) with ζ(0) = ζ(T) = 0.

Now the conclusion is classical: it uses a piecewise linear approximation ofI]0,t[, withtgiven outside a set of measure zero. Thanks to the definition of initial condition(2.1)foru andv, and to the Gronwall Lemma we complete the proof of theorem.

Remark: The Lipschitz condition forϕ◦φ⁻¹ is only needed to transform the second line of inequality(2.4). WhenBis stationary an Hölder condition is sufficient to transform the convective term in (2.4)(see [10],[13],..)

2.2 Proof of theorem 2.1: existence property

The equation(1.1)being nonlinear and degenerated in the sense(1.2), we first introduce some auxiliary non-degenerate problems by turningφintoφ+δId_R, for each value of the parameterδin ]0,1]. Thus we look for estimates of u_,δ that are independent fromδ(the latter will fix the regularity ofu) and from as soon as possible (the latter will specify the behavior of the sequence (u)_>0, as goes to 0⁺). First and foremost, if we refer to G.Gagneux &

M.Madaune-Tort’s book [10], the next statement holds:

Lemma 2.3: For any positive andδthe non-degenerate parabolic problem:

findu_,δ inW(0, T;H¹(Ω);L²(Ω))∩L^∞(0, T;H¹(Ω))satisfying for any v in H¹(Ω), v|Γe= 0, the variational equality for a.e. t in ]0, T[,

Z

Ω

(∂_tu_,δ+Div_x{B(t, x)ϕ(u_,δ)}+ψ(t, x, u_,δ))vdx

= −

Z

Ω

∇φ_δ(u_,δ).∇vdx, (2.5)

and the boundary conditions,

u_,δ=u^δ_Γe dΣ-a.e. on Σ_e, (2.6) u_,δ(0, .) =u^δ₀ a.e. on Ω, (2.7) has a unique solution.

What is more, if u_,δ andv_,δ are two solutions associated with the bound- ary data (u^δ₀, u^δ_Γe)and (v₀^δ, v_Γ^δ_e), then for any t of[0, T]:

Z

Ω

(u_,δ(t, x)−v_,δ(t, x))⁺dx

≤ (kBk∞M_ϕ⁰ Z

(Σe)t

(u^δ_Γe−v^δ_Γe)⁺dσ+ Z

Ω

(u^δ₀(x)−v₀^δ(x))⁺dx)e^Mψ0t.(2.8)

In addition,φ_δ(u_,δ) belongs to L²(0, T;H^3/2−η(Ω)) for any positiveη.

Lastly for any positive and δ and for any tin [0, T]:

|u_,δ(t, x)| ≤M_δ(t)for a.e. xin Ω, whereM_δ(t)is given by

M_δ(t) =max(ku^δ_Γek_L^∞_(Σ),ku^δ₀k_L^∞_(Ω))e^Mψ0t+e^Mψ0t−1

M_ψ⁰ kψ(t, x,0)k_L^∞_(Q)

In formulations (2.6,2.7) of boundary conditions, the smooth data u^δ_Γe and u^δ₀ are defined by first introducing a.e. on Q the function u_0,Γe in the same spirit as in [18]:











u_0,Γe(t, r+sν) =u_Γe(t, r) fort∈]0, T[, r∈Γ,0≤s < min(t, δ), u_0,Γe(t, x) =u_Γe(t, x) formin(dist(x,Γ), δ)< t < T, x∈Ω, u_0,Γe(t, x) =u₀(x)for −δ < t < min(dist(x,Γ), δ), x∈Ω, u_0,Γe(t, x) = 0 elsewhere.

Then let us define forpin Q:

u^δ_0,Γe(p) = Z

R^p+1

u_0,Γe(˜p)ρ_δ(p−p)d˜ p,˜

whereρ_δis the usual mollifier. Let us now denote byu^δ_Γeandu^δ₀the restriction ofu^δ_0,Γe to Σand {0} ×Ω respectively. Observe that

(u^δ_Γe)_δ>0 and(u^δ₀)_δ>0 are uniformly bounded in the respective L^∞-norm, lim

δ→0⁺u^δ₀=u₀ inL¹(Ω).

Remark:

i) Owing to the smoothness ofu_,δ, one has a.e. on Q:

∂_tu_,δ+Div_x{B(t, x)ϕ(u_,δ)}+ψ(t, x, u_,δ) =∆φ_δ(u_,δ). (2.9) In this way, ∆φ_δ(u_,δ) belongs to L²(Q) and ∇φ_δ(u_,δ) is an element of L²(0, T;H(div,Ω)). So, ∇φ_δ(u_,δ).ν = 0 in L²(]0, T[×Γ\Γ_e) and a.e. on ]0, T[×Γ\Γ_e.

ii) It is essential to mention that by considering in the definition of M_δ(t) the respectiveL^∞-norm for the sequences(u^δ_Γe)_δ>0 and (u^δ₀)_δ>0, the and δ uniform estimate holds for everytof[0, T]:

|u_,δ(t, x)| ≤M(t)for a.e. x inΩ, (2.10) where M(t) is given by (1.9).

On the one hand, we assume in this work that the vector fieldBis unsta- tionary. Consequently it is well-known that in this situation, anL¹-estimate of∂_tu_,δ (in terms of differential ratio or continuity modulus) is linked to an L¹-estimate of ∇u_,δ, and conversely (see for example S.N.Kruskov [14] for the Cauchy problem or [3, 12] for the Dirichlet one). On the other hand, we release the smoothness assumptions onBand u₀ as much as possible; thus, even if B is independent fromt, we may not refer to F.Mignot & J.P.Puel’s weight-functions [20] associated with the operator v → B.∇v in order to obtain, as in [13], anL¹-local estimate uniformly with respect to and δfor

∇u_,δ. This explains why we solely focus on Hilbertian estimates for φ_δ(u_,δ) that are in fact based on an energy inequality. With this view, we introduce an arbitrary sequence(µ_n)_n∈N in]0, T] which converges to zero asn goes to +∞. Thus, for a fixedn,

Lemma 2.4: Then there exists a constant c₁, independent from n and δ in ]0, µ_n/2[, such that:

k∇Φ_δ(u_,δ)k²_L2(µn,T;L²(Ω)^p) ≤c₁, (2.11) k∂_tu_,δk_L²_(µn,T;V⁰₎ ≤c₁, (2.12) k√

t−µ_n∂_tΦ_δ(u_,δ)k²_L2(µn,T;L²(Ω))≤c₁, (2.13) whereΦ_δ(r) =

r

Z

0

(φ⁰_δ(τ))^1/2dτ and V denotes the Hilbert space

V ={v∈H¹(Ω), v|Γe= 0},

used with theH¹-equivalent norm: kvk_V =



 Z

Ω

[∇v]²dx





1/2

.

Some commentaries - These uniform a priori estimates are established by taking advantage of the regularity ofu_Γe.

i) As for(2.11), we choose the test-function v=u_,δ−u^δ_0,Γe in(2.5)and we integrate over ]µ_n, T[. We just mention that the convective term is written by developing the partial derivatives. Then if we denoteF(r) =

r

Z

0

ϕ⁰(τ)τ dτ,

the Green formula, the partitioning ofΓand the monotonicity of ϕ ensure Z

]µn,T[×Ω

B.∇F(u_,δ)dxdt≤ Z

]µn,T[×Γe

B.ν∇F(u^δ_Γe)dxdt

ii) As for(2.13), we consider theL²(]µ_n, T[×Ω)-scalar product between(1.1) and(t−µ_n)∂_t{φ_δ(u_,δ)−φ_δ(u^δ_0,Γe)} that is an element ofW(µ_n, T, V, L²(Ω)).

To transform the diffusion term, we establish a Green formula thanks to the density of D([µ_n, T];X) into W(µ_n, T;X;L²(Ω)), where X is the Hilbert space

X ={v∈V,∆v∈L²(Ω), Z

Ω

∆vwdx=− Z

Ω

∇v.∇wdx,∀w ∈V},

used with the scalar product((u, v))_X = Z

Ω

∇u.∇vdx+ Z

Ω

∆u∆vdx and the associated norm.

The other terms are bounded thanks to a Cauchy inequality and estimates (2.10,2.11).

iii) As for(2.12)we refer to the definition of theL²(µ_n, T;V⁰)-norm and we use(2.10,2.11).

Remark:WhenBis stationary then anL¹-uniform estimate of time-differentials ratio holds. Indeed, this particular context ensures - tanks to(2.8)- the ex- istence of a constantc(δ)such that for any hof]0, T[and anytof[0, T−h],

ku_,δ(t+h, .)−u_,δ(h, .)k_L1(Ω) ≤c(δ)h,

the dependence with respect to δ been released as soon as∆u₀ and ∆φ(u₀) are bounded Radon measures onΩ.

The estimates of lemma 2.4 ensure that the family(√

t−µ_nφ_δ(w_,δ))_δ∈]0,µn 2 [

remains at least in a bounded set ofW^1,1(]µ_n, T[×Ω). The compactness em- bedding of the latter space into L¹(Q) and the continuity of φ⁻¹ provide the existence of a measurable functionu_n,and a subsequence - still denoted (u_,δ)_δ∈]0,µn

2[ - such that whenδgoes to0⁺,(u_,δ)_δ∈]0,µn

2 [goes tou_n,for every (t, x)in (]µ_n, T[×Ω)\N_n withL^p+1(N_n) = 0

Let us denote N = S

n∈N

N_n and let u be the L^∞(Q) weak-∗ limit of (u_,δ)_δ>0 up to a subsequence. Then, as δ goes to zero, (u_,δ)_δ>0 goes to u for every (t, x) in Q\N, and so in L^q(Q), 1 ≤ q < +∞. What is more due to (2.11) and (2.12) and to the fact that c₁ does not depends on n, (∇φ_δ(u_,δ))_δ>0 and (∂_tu_,δ)_δ>0 weakly converge up to a subsequence respec- tively toward∇φ(u)in L²(0, T, L²(Ω)^p) and∂_tu inL²(0, T, V⁰). Hence:

Proposition 2.5: The degenerate problem (1.1,1.3) has at least a weak so- lutionu associated with the initial data u₀ in the sense(2.1). This solution belongs to L^∞(Q), with φ(u) in L²(0, T;H¹(Ω)) and ∂_tu in L²(0, T;V⁰), and is characterized by the variational equality (2.2).

Proof: There is no difficulty in establishing variational equality (2.2).

We use the continuity of φ⁻¹ and for the boundary term the continuity of the trace operator from H^s(Ω) into L²(Γ) for 1/2 < s < 1. So let us focus on weak formulation (2.1) for the initial condition. The demonstration is inspired by that presented by F.Otto in [18] for the case of weak entropy solutions to quasilinear scalar conservation laws. Here, we take in(2.5)the test-function v= sgn_λ(u_,δ−k)αζ where α and ζ are respectively elements ofD₊(]− ∞, T[)andD₊(Ω)andkbelongs toR. By integrating with respect totfrom0to T, it comes:

− Z

Q





u,δ

Z

k

sgn_λ(τ−k)dτ



ζ∂_tαdxdt− Z

Q

F_λ(u_,δ, k)B.∇ζαdxdt

+ Z

Q

ψ(t, x, u_,δ)sgn_λ(u_,δ−k)αζdxdt

− Z

Q





u,δ

Z

k

φ⁰_δ(τ)sgn_λ(τ−k)dτ



∆ζαdxdt

≤ Z

Ω





u₀

Z

k

sgn_λ(τ−k)dτ



ζα(0)dx

where,

F_λ(v, w) =

v

Z

w

sgn_λ(τ−w)ϕ⁰(τ)dτ.

Thus, whenδand thenλtend to0⁺, an integration by parts with respect totprovides :

−

T

Z

0



 Z

Ω

|u−k|ζdx+θ(t)



α⁰(t)dt≤ Z

Ω

|u₀−k|ζα(0)dx, (2.14)

with:

θ(t) = Z

Ω

(

t

Z

0

[−|ϕ(u(τ, x))−ϕ(k)|B(τ, x).∇ζ

+ ψ(τ, x, u(τ, x))sgn(u(τ, x)−k)−|φ(u(τ, x))−φ(k)|∆ζ]dτ)dx.

In this way, the time-depending function t → R

Ω

|u −k|ζdx +θ(t) is identified a.e. with a non-increasing and bounded function, so it has a limit whentgoes to0⁺,t∈]0, T[\O,L₁(O) = 0. As θ goes to0witht, it comes

ess lim

t→0⁺

Z

Ω

|u−k|ζdx≤ Z

Ω

|u₀−k|ζdx,

for any function ζ of D₊(Ω) and any real k. As a consequence, owing to F.Otto’s reasoning in [18] we may announce:

ess lim

t→0⁺

Z

Ω

|u(t, x)−K(x)|dx≤ Z

Ω

|u₀−K(x)|dx

for anyK ofL^∞(Ω). Condition (2.1)follows, which completes the proof.

Remark:

i) Since φ(u) belongs to L²(0, T;H¹(Ω)) and ∂_tφ(u)to L²(α, T;L²(Ω)) for any positiveα, thenφ(u)is a function ofC([α, T];L²(Ω))∩C_s([α, T];H¹(Ω)).

Thus we may define a trace foru(t, .)onΓ, for everyt∈]0, T], through:

trace(u)(t, .) =φ⁻¹(trace(φ(u))(t, .)).

In this sense,u=u_Γe onΣ_e.

ii) WhenBis independent from the time variable then the estimate of time- differential ratio pointed at the previous remark ensures, through the Ascoli Lemma, that the sequence (u_,δ)_δ>0 converges toward u in C([0, T];L¹(Ω)).

In this situation, the initial condition forucan be strongly formulated under the form: u(0, .) =u₀a.e. inΩ.

iii) The previous existence result for u doest not require any smoothness assumption forϕ◦φ⁻¹.

To complete this paragraph, let us remark that the uniqueness property foruensures that the whole sequence(u_,δ)_δ>0gives an approximation ofu. Therefore, given the-uniform estimate(2.10)satisfied byu_,δ one has (2.3) foru.

Remark: If the vector field B is independent fromt and if data are smooth enough (namely as soon as ∆φ(u₀) and ∆u₀ are bounded Radon measures onΩ) then theL¹-estimate of time-differential ratio pointed previously holds for u uniformly with respect to δ, which gives an a priori estimate of the sequence(u)_>0 inBV(0, T;L¹(Ω)).

3 A singular perturbations property

We now focus on the behavior of the sequence(u)_>0 as goes to 0⁺ that is precisely to prove that the sequence of solutions to parabolic degenerate equations(1.1)_>0associated with the couple of boundary data(u₀, u_Γe)pro- vides an L¹-approximation of the solution to the corresponding first-order problem(1.5,1.6,1.7). This property of singular perturbations extends that of M.J.Jasor [13] obtained in the special situation when Bis a smooth sta- tionary vector field so as to reason within the framework of bounded functions with locally bounded variations onQ.

We first recall that a mathematical formulation for (1.5,1.6,1.7) is pro- vided by bearing in mind that for a general first-order quasilinear equation, it is classical to refer to an entropy criterium that warrants uniqueness. With this view we refer to the works of F.Otto in [18], chapter 2, which introduce

a new formulation of boundary conditions for quasilinear hyperbolic equa- tions, which generalize to L^∞(Q)-solutions those of C.Bardos, A.Y.LeRoux

& J.C.Nedelec [3], only aviable for solutions with bounded variation on Q.

Thus, we denote

F(u, v) =|ϕ(u)−ϕ(v)|,

L(u, v, w) =|u−v|∂_tw+F(u, v)B.∇w−sgn(u−v)ψ(t, x, u)w.

We thus say:

Definition: A measurable functionuis the entropy solution to(1.5,1.6,1.7) if it satisfies:

i) the inner entropy condition, for allξ inD₊(]0, T[×Ω)and for any realk, Z

Q

L(u, k, ξ)dxdt≥0, (3.1)

ii)the initial condition, ess lim

t→0⁺

Z

Ω

|u(t, x)−u₀(x)|dx= 0, (3.2)

iii) the boundary condition in the weak sense ess lim

τ→0⁻

Z

Σ

F(u(σ+τ ν), u_Γe, k)B(σ).νζdσ≥0, (3.3)

for any real k and any functionζ ofL¹₊(Σ)where, for any reala, b, c, 2F(a, b, c) =F(a, b)−F(c, b) +F(a, c)

Remark: If it can be proved that for a.e. tof]0, T[, u(t, .)has bounded vari- ations onΩ, then(3.3)is equivalent to the classical formulation of boundary conditions owing to C.Bardos, A.Y.LeRoux & J.C.Nedelec [3], which is re- duced in our particular context for a.e. tof]0, T[ to:

∀k∈ I(u_Γe, γ_u),|ϕ(γ_u)−ϕ(k)|B.ν ≥0dΓ-a.e.,

where γ_u is the trace of u on Γ in the sense of bounded functions with bounded variations onΩ. Thus, by considering the definition ofΓ₋, we may note that ϕ(γ_u) = ϕ(u_Γe) a.e. on Γ− and since ϕ is non-decreasing, the Dirichlet condition is fulfilled as soon asϕ is not constant.

Here the next statement holds:

Theorem 3.1: When goes to 0⁺, the family (u)_>0 of solutions to de- generate equations(1.1)_>0associated with the couple of boundary conditions (u₀, u_Γe) strongly converges in L^q(Q), 1 ≤q < +∞ and a.e. on Q toward the weak entropy solutionu of first-order problem (1.5,1.6,1.7).

Before we must remind some properties of bounded sequences inL^∞.

3.1 Bounded sequences in L

LetObe an open bounded subset ofR^q(q≥1) and let(u_n)_n>0be a bounded sequence inL^∞(O). Clearly, for any continuous function hthere exists ¯hin L^∞(Q) such that for a subsequence,

h(u_n)*¯h weakly in L^∞(O).

Since the works of L.Tartar [21] and J.M.Ball [1] one has been able to de- scribe the composite limit¯h. Actually, thanks to the properties of the weak-∗

topology on the space of Radon measures, and to the properties of the gen- eralized inverse of the distribution function linked to a probability measure, the next compacity result holds (see [8]):

Proposition 3.2: Let (u_n)_n>0 be a sequence of measurable functions on O such that:

∃M >0,∀n >0,ku_nk_L∞(O) ≤M.

Then, there exists a subsequence (u_f(n))_n>0 and a measurable function π in L^∞(]0,1[×O)such that for all continuous and bounded functionshonO×]− M, M[,

∀ξ∈L¹(O), lim

n→+∞

Z

O

h(w, u_f(n))ξdw= Z

]0,1[×O

h(w, π(α, w))dαξdw.

Such a result - or the equivalent within the context of Young measure - has found its first application in the approximation through the artificial viscosity method of the Cauchy problem in R^p for scalar conservation laws, when only a uniformL^∞-control of approximate solutions holds (see [7, 21]).

It has also been applied to the numerical analysis of transport equations since

”Finite-Volume” schemes only give an L^∞-estimate uniformly with respect to the mesh length of the numerical solution (see e.g. [8, 9]...). Here, we refer to this concept when the approximating sequence is the sequence of solutions to viscous equations(1.1)_>0 associated with data(u₀, u_Γe).

3.2 Proof of theorem 3.1

Owing to (2.3) there exists a subsequence extracted from (u)_>0 - labelled (u)_>0- and a measurable and bounded functionπ on]0,1[×Qsuch that for any continuous and bounded functionf onQ×]−M, M[

lim

→0⁺

Z

Q

f(t, x, u)ξdxdt= Z

]0,1[×Q

f(t, x, π(α, t, x))ξdαdxdt,

for anyξ ofL¹(Q).

In fact, in order to establish the proof of theorem 3.2 we are going to demonstrate that the functionπis anentropy process solutionto(1.5,1.6,1.7), namely thatπ fulfills relations(3.1,3.2,3.3), where the integrations with re- spect to the Lebesgue measure on Q, Ω and Σ are respectively turned into integrations with respect to the Lebesgue measure on]0,1[×Q,]0,1[×Ωand ]0,1[×Σ.

Indeed by following the F.Otto’s works [18], we may prove that ifπ and ωare two entropy process solutions of(1.5,1.6,1.7)in the sense of definition 3.1, then for a.e. tof]0, T[:

t

Z

0

Z

Q×]0,1[²

|π(α, t, x)−ω(β, t, x)|dαdβdxdt= 0.

Thus, as mentioned in [8] the two processesπand ωare equal, for a.e. (t, x) onQ, to a common value u(t, x), which does not depend onαor β in]0,1[.

In addition, u is a measurable and bounded function on Q and is namely

the weak entropy solution of (1.5,1.6,1.7). Lastly, as a consequence of the uniqueness property, the whole sequence of approximate solutions strongly converges touinL^q(Q),1≤q <+∞and a.e. onQ(see e.g. [8] of [7] within the context of measure-valued solutions).

Remark: The uniqueness proof of an entropy process to (1.5,1.6,1.7) refers to the classical Kruskov method [14]: it split the time and the space variables in two. As pointed out in the introduction, we then need some smoothness assumptions onBin order to pass to the limit with mollifiers at once in the convective term and in the boundary ones (by using Otto’s techniques).

(i) Entropy inequality(3.1)and initial condition (3.2)for π

We take advantage of the approximation properties ofuthrough(u_,δ)_δ>0 to provide an entropy inequality foru, in which we can pass to the limit with respect to. So, let us come back to the regularized problem (2.5,2.6,2.7).

By multiplying a.e. on Q equation (2.9) with the function sgn_λ(u_,δ−k), k∈R, and by taking into account that a.e. onQ

sgn_λ(u_,δ−k)∆φ_δ(u_,δ)≤∆





u,δ

Z

k

sgn_λ(τ−k)φ⁰_δ(τ)dτ





and,

sgn_λ(u_,δ−k)Div_x(ϕ(u_,δ)B) =Div_x(F_λ(u_,δ, k)B) where,

F_λ(v, w) =

v

Z

w

sgn_λ(τ−w)ϕ⁰(τ)dτ.

Then, by lettingI_λ(u) =

u

Z

k

sgn_λ(τ−k)dτ, one has a.e. onQ,

∂_tI_λ(u_,δ) +Div_x(F_λ(u_,δ, k)B) +ψ(t, x, u_,δ)sgn_λ(u_,δ−k)

≤ ∆





u,δ

Z

k

sgn_λ(τ−k)φ⁰_δ(τ)dτ



.

In this way, for any functionζ ofD₊(]− ∞, T]×Ω),

− Z

Q

{I_λ(u_,δ)∂_tζ+F_λ(u_,δ, k)B.∇ζ−ψ(t, x, u_,δ)sgn_λ(u_,δ−k)ζ}dxdt

≤ Z

Q





u,δ

Z

k

sgn_λ(τ−k)φ⁰_δ(τ)dτ



∆ζdxdt+ Z

Ω

I_λ(u₀)ζ(0, x)dx.

Thence, we pass to the limit with respect toδand then with respect to- the first term in the right hand side of the above inequality being bounded by C^st thanks to(2.3). Lastly theλ-limit through the dominated convergence theorem gives

− Z

]0,1[×Q

L(π, k, ζ)dαdxdt≤ Z

Ω

|u₀−k|ζ(0, x)dx, (3.4)

for anyζ ofD₊(]− ∞, T]×Ω). So, one gets (3.1)forπ. Moreover F.Otto’s ideas [18] ensure that if(3.4)holds, then for any function K ofL^∞(Ω),

ess lim

t→0⁺sup Z

]0,1[×Ω

|π(α, t, x)−K(x)|dαdx≤ Z

Ω

|u₀(x)−K(x)|dx.

Initial condition(3.2)forπ follows withK =u₀. (ii) Dirichlet boundary condition(3.3) forπ

The demonstration is based on F.Otto’s original proof [18]. With this view, we introduce for any l of N the family of boundary entropy-entropy flux pairs(H_l, Q_l) defined by:

H_l(a, b, c) =H_l^b(a, c) = (dist(a,I[b, c]))²+ 1

l

2!1/2

−1 l. and

Q_l(a, b, c) =Q^b_l(a, c) =

a

Z

c

∂₁H_l^φδ(b)(φ_δ(τ), φ_δ(c))ϕ⁰(τ)dτ,

hence(Q_l)_l∈Nconverges uniformly aslgoes to+∞, toF(a, b, c)for any non- negative δ. Again, we take advantage of the approximation properties of

u through (u_,δ)_δ>0 to come back to regularized problem (2.5,2.6,2.7). By multiplying equation(2.9)with∂₁H_l^φδ(w)(φ_δ(u_,δ), φ_δ(k))ζξ, whereζ belongs toH_0,+¹ (Ω), ξ to D₊(]0, T[×R^p), w and k to R, and by considering that a.e.

onQ(to simplify the writing we drop the indexe temporarily)

∂₁H_l^φ(w)(φ(u), φ(k))∆φ(u)≤Div_x[∂₁H_l^φ(w)(φ(u), φ(k))∇φ(u)]

and,

∂₁H_l^φ(w)(φ(u), φ(k))Div_x(ϕ(u)B) =Div_x(Q_lB) it comes:

− Z

Q





 (

u,δ

Z

k

∂₁H_l^φδ(w)(φ_δ(τ), φ_δ(k))dτ)∂_tξζ+Q^w_l (u_,δ, k)B.∇ξζ

−∂₁H_l^φδ(w)(φ_δ(u_,δ), φ_δ(k))ψ(t, x, u_,δ)ζξo dxdt

≤ Z

Q

Q^w_l (u_,δ, k)B.∇ζξdxdt− Z

Q

∇[H_l^φδ(w)(φ_δ(u_,δ), φ_δ(k))ξ].∇ζdxdt

+ 2 Z

Q

H_l^φδ(w)(φ_δ(u_,δ), φ_δ(k))∇ξ.∇ζdxdt

+

Z

Q

H_l^φδ(w)(φ_δ(u_,δ), φ_δ(k))ζ∆ξdxdt. (3.5)

Owing to the converge properties of the sequence (u_,δ)_δ>0 towardu, we may pass to the δ-limit in (3.5) and in order to take the-limit - that is to control the right-hand side of(3.5)- we consider the particular choice for the functionζ: for any positive,

ζ(x) = 1−exp

−M_ϕ◦φ⁰ −1+L

s(x)

(3.6) where for any positive parameterµsmall enough,

s(x) =

min(dist(x,Γ), µ)for x∈Ω

−min(dist(x,Γ), µ) forx∈R^p\Ω,

withL= sup

0<s(x)<µ

|∆s(x)|. That way (see [18]), for any ϕofW₊^1,1(R^p)

M_ϕ◦φ⁰ −1

Z

Ω

|∇ζ|ϕdx≤ Z

Ω

∇ζ.∇ϕdx+ (M_ϕ◦φ⁰ −1+L) Z

Γ

ϕdσ. (3.7)

Therefore, considering that a.e. onQ

|Q^w_l (u, k)| = |

φ(u)

Z

φ(k)

∂₁H_l^φ(w)(τ, φ(k))(ϕ◦φ⁻¹)⁰(τ)dτ|

≤ M_ϕ◦φ⁰ −1H_l^φ(w)(φ(u), φ(k)),

and using weak differential inequality(3.7)withϕ=H_l^φ(w)(φ(u), φ(k))ξ, we obtain a majoration of the right-hand side of(3.5)through:

2 Z

Q

H_l^φ(w)(φ(u), φ(k))∇ξ.∇ζdxdt+ Z

Q

H_l^φ(w)(φ(u), φ(k))ζ∆ξdxdt

+(M_ϕ◦φ⁰ −1+L) Z

Σ

H_l^φ(w)(φ(u_Γe), φ(k))ξdσ.

Thanks to (2.3) and as ζ goes to 1 in L¹(Ω) and ∇ζ goes to 0 in (L¹(Ω))^pwe may pass to the limit when tends to0⁺. For any functionξ of D₊(]0, T[×R^p), it comes:

Z

]0,1[×Q







π

Z

k

∂₁H_l^φ(w)(φ(τ), φ(k))dτ ∂_tξ+Q^w_l (π, k)B.∇ξ

−∂₁H_l^φ(w)(φ(π), φ(k))ψ(t, x, π)ξ o

dαdxdt

≥ −M_ϕ◦φ⁰ −1

Z

Σ

H_l^φ(w)(φ(u_Γe), φ(k))ξdσ.

When one refers to F.Otto’s works [18], p. 115, lemma 7.34, and uses the smoothness of vector fieldB, this inequality implies that for anyζ ofL^∞(Σ)

andξ ofL¹₊(Σ),

ess lim

τ→0⁻

Z

]0,1[×Σ

Q_l(π(α, σ+τ ν), ζ, k)B(σ).νξdαdσ

≥ −M_ϕ◦φ⁰ −1

Z

Σ

H_l(φ(u_Γe), φ(ζ), φ(k))ξdσ.

Boundary condition (3.3) for π follows with ζ = u_Γe, which completes the proof of theorem 3.1.

Remark: In the special situation of an homogeneous boundary condition on Γ_e, the demonstration of boundary condition(3.3)forπ can easily be estab- lished without referring to assumption(1.8). Indeed, in this particular con- text, for anyζ of D₊(]0, T[×R^p)and any realk, the function ∂₁H_l⁰(u_,δ, k)ζ belongs to W(0, T, V, L²(Ω)); so it can be taken as a test-function in varia- tional formulation (2.5) for u_,δ. We integrate with respect tot over [0, T], we use the convexity ofz →∂₁H_l⁰(z, k)to have a majoration of the diffusive term and to transform the convective one, we note that:

Z

Q

Div_x{ϕ(u_,δ)B}∂₁H_l⁰(u_,δ, k)ζ = − Z

Q

Q^∗_l(u_,δ, k)B.∇ζdxdt

+ Z

Σ

Q^∗_l(u_,δ, k)B.νζdσ, (3.8)

where

Q^∗_l(a, b) =

a

Z

0

ϕ⁰(τ)H_l⁰(τ, b)dτ.

Due to the partitioning of Γ and to the monotonicity of ϕ, we observe that the boundary integral in the right-hand side of (3.8) is non-negative.

That way, passing to the limit with respect toδ and then with provides:

Z

]0,T[×Q

H_l⁰(π, k)∂_tζ+Q^∗_l(π, k)B.∇ζ−ψ(t, x, π)∂₁H_l⁰(π, k)ζ

dαdxdt≥0.