HIGH ORDER EXPANSIONS IN QUASINEUTRAL LIMIT OF THE EULER-POISSON SYSTEM FOR A POTENTIAL FLOW

HIGH ORDER EXPANSIONS IN QUASINEUTRAL LIMIT OF THE EULER-POISSON SYSTEM FOR A POTENTIAL FLOW

INGRID VIOLET

Laboratoire de Math´ematiques, CNRS UMR 6620

Universit´e Blaise Pascal (Clermont-Ferrand 2), 63177 Aubi`ere cedex, France [email protected]

Abstract : We are interested in steady-state Euler-Poisson system for a potential flow used in the mathematical modeling of plasmas and semiconductors. In the case of quasineutral limit, boundary layers can appear. We study this limit by using an asymptotic expansion. We show the existence and uniqueness of each profile and give the justification of the asymptotic expansion up to any order.

Keywords : Steady-state Euler-Poisson system, potential and subsonic flow, quasineutral limit, asymptotic expansion and justification, semiconductors and plasmas.

AMS Subject Classification (2000) : 35B25, 35B40, 35C20, 35D05, 35J25, 35Q35.

1. Introduction

The hydrodynamic model is widely used in mathematical modeling and numerical sim- ulation for plasmas [3] and semiconductors [15]. It consists in two non linear equations given by the conservation laws of momentum and density, called Euler equations, plus a Poisson equation for the electrostatic potential. Due to the hyperbolicity of the transient non linear Euler equations, the weak solution is only studied in one space dimension. In such a situation, the existence of global weak solution is shown in the set of bounded functions [14].

In this paper we only consider the unipolar steady-state case for a potential flow. Then the Euler-Poisson system reads as follows (see [5, 16, 17]) :

ε

2|∇ψ|²+h(n) =φ+εψ τ , (1.1)

−div(n∇ψ) = 0, (1.2)

−λ²∆φ =b(x)−n.

(1.3)

This system will be studied in an open and bounded domain Ω in R^d (d=2 or d=3 in practice) with sufficiently smooth boundary Γ. Here n = n(x), ψ = ψ(x), φ = φ(x) represent respectively the electron density, the velocity potential and the electrostatic potential. The function h=h(n) is the enthalpy for the system and is defined by :

h⁰(n) = p⁰(n)

n , n >0, and h(1) = 0,

1

where p = p(n) is the pressure function, supposed to be sufficiently smooth and strictly increasing for n > 0. The function b = b(x) represents the doping profile for a semicon- ductor and the ion density for a plasma. The parametersλ, , τ represent respectively the scaled Debye length, electron mass and relaxation time of the system. They are dimen- sionless and small compared to the caracteristic length of physical interest. Then it is important to study the limitsλ→0, →0, τ →0.In [17] we used asymptotic expansions to study the zero-electron-mass limit and the zero-relaxation-time limit and to show the convergence of the Euler-Poisson system to the incompressible Euler equations. Here we study the quasineutral limitλ →0. In all the following we take τ ≡1 and we keep ε >0 as a small parameter independent of λ in the equations.

By eliminating φ of (1.1) and (1.3) and using (1.2) we have :

∆h(n) − ε n

d

X

i,j=1

ψ_xiψ_xjn_xixj+ ε

n²(∇ψ.∇n)²+ ε

n(∇ψ.∇n)

− ε n

d

X

i,j=1

ψ_xiψ_xixjn_xj− 1

λ²(n−b) +εQ(ψ) = 0.

(1.4)

where Q is given by :

(1.5) Q(ψ) =

d

X

i,j=1

ψ²_xixj.

Forn >0, it is easy to see that (n, ψ, φ) is a smooth solution to the system (1.1)-(1.3) if and only if (n, ψ) is a smooth solution to (1.2) and (1.4). Moreover, forψ given, the equation (1.4) is elliptic if and only if the flow is subsonic, i.e., the condition |∇ψ| < p

p⁰(n)/ε holds.

We supplement the system (1.1)-(1.3) by Dirichlet boundary conditions :

(1.6) n=

m

X

j=0

λ^jn^j_D +n^m_D,λ, ψ =

m

X

j=0

λ^jψ^j_D+ψ_D,λ^m , on Γ =∂Ω, where n^m_D,λ and ψ_D,λ^m are smooth enough and defined on ¯Ω.

The Euler-Poisson system and its asymptotic limits have been studied by a lot of authors. In [5] it is shown the existence and uniqueness of solutions for a potential flow under an assumption on the smallness of data, which implies that the problem is in the subsonic region. In [16] the author shows that the smallness condition corresponds to the smallness of ε. Then the existence and uniqueness hold for large data provided that ε is small enough.

The quasineutral limit has been studied in several special cases. In one-dimensional steady-state Euler-Poisson system it was performed in [19] for well-prepared boundary data. The steady problem in several space variables for a potential flow without the forma- tion of boundary layers was investigated in [16]. In [4] the authors use pseudo-differential techniques to study this limit in transient Euler-Poisson model. The quasineutral limit has also been studied in the bipolar case in the drift-diffusion equations (see [7, 8, 12]).

See also [2] for the study of this limit in a semi-linear Poisson equation in which the elec- tron density is described by the Maxwell-Boltzmann relation. This relation is also used in [4].

The zero-electron-mass limit (ε →0) and the zero-relaxation-time limit (τ → 0) have also been studied a lot. See [1, 11, 17] for different results on these two limits. In [9], the authors study the combined relaxation-time limit and the vanishing Debye length.

This article is based on the method of asymptotic expansions presented in [18]. In [18] the justification of these asymptotic expansions is only given up to first order in the one-dimensional case. Here we will give their justification up to any order in the multidimensional case by using the Schauder fixed point Theorem. The main difficulty is to verify the assumptions in this theorem which are achieved by the Leray Schauder fixed point Theorem.

The paper is organised as follow. In Section 2, we give the formal asymptotic expansions and the systems verified by each boundary layer profiles under our assumptions. Section 3 is devoted to the justification up to any order of the asymptotic expansions.

2. Formal asymptotic expansions

In this section we study the formal asymptotic expansions of a solution to (1.1)-(1.3).

We use for this the method of asymptotic expansions presented in [18]. We assume : (H1)b ∈C^∞( ¯Ω), 0< n≤b(x)≤n, x¯ ∈Ω, n,¯ ¯n∈R,

(H2)n^j_D ∈C^∞( ¯Ω) for 0≤j ≤m, (H3)ψ^j_D ∈C^2,δ( ¯Ω) for 0≤j ≤m, (H4)n⁰_D(x) = b(x), n¹_D(x) = 0, x∈Ω,¯

(H5) (λ^−m−1n^m_D,λ)_λ>0 is bounded inW^2,q(Ω), q > _1−δ^d , δ∈(0,1), (H6) (λ^−m+1ψ_D,λ^m )_λ>0 is bounded in C^2,δ( ¯Ω).

The assumption (H4) is a compatibility condition for the first and second order terms.

It assures that there will not appear any boundary layers in these two terms. The case without any compatibility conditions presents some difficulties which we didn’t succeed in this study (see Remark 3.2).

2.1. Internal expansion. Let : n(x) =X

k≥0

λ^knk(x); ψ(x) = X

k≥0

λ^kψk(x); φ(x) = X

k≥0

λ^kφk(x).

We inject this into (1.1)-(1.3) and obtain : ε

2|∇X

k≥0

λ^kψ_k(x)

|²+hX

k≥0

λ^kn_k(x)

=X

k≥0

λ^kφ_k(x) +εX

k≥0

λ^kψ_k(x), (2.1)

−divX

k≥0

λ^knk(x)∇X

k≥0

λ^kψk(x)

= 0, (2.2)

−λ²∆X

k≥0

λ^kφ_k(x)

=b(x)−X

k≥0

λ^kn_k(x), in Ω.

(2.3)

Formally,

divX

k≥0

λ^kn_k(x)∇X

k≥0

λ^kψ_k(x)

=X

k≥0

λ^k k

X

i=0

div(n_i∇ψk−i)

,

|∇X

k≥0

λ^kψ_k(x)

|² =X

k≥0

λ^{k k}

X

i=0

∇ψ_i.∇ψk−i

,

hX

k≥0

λ^kn_k(x)

=X

k≥0

λ^kh_k(n), where n= (n_i)i≥0 and :

h_k(n) = 1 k!

d^kh P

k≥0λ^kn_k

dλ^k

λ=0

, k ≥0.

As shown in [17] : h_k(n) = h⁰(n₀)n_k+ ¯h_k((n_i)0≤i≤k−1), k ≥ 1 where ¯h_k is smooth and

¯h₁ ≡0. Then :

h(n_λ) =h(n₀) +X

k≥1

λ^kh⁰(n₀)n_k+X

k≥2

λ^k¯h_k((n_i)_{0≤i≤k−1}).

Hence the system (2.1)-(2.3) becomes : ε

2 X

k≥0

λ^k k

X

i=0

∇ψ_i.∇ψk−i

+h(n₀) + X

k≥1

λ^kh⁰(n₀)n_k+X

k≥2

λ^kh¯_k((n_i)0≤i≤k−1)

= X

k≥0

λ^kφ_k(x) +εX

k≥0

λ^kψ_k(x), (2.4)

−X

k≥0

λ^{k k}

X

i=0

div(n_i∇ψ_k−i)

= 0, (2.5)

−X

k≥2

λ^k∆φk−2 =b(x)−X

k≥0

λ^kn_k. (2.6)

We identify the order in λ and obtain the system verified by (n_k, ψ_k, φ_k) for all k. For k = 0 we have :

φ₀ =−ε

2|∇ψ₀|²−h(n₀) +εψ₀, (2.7)

div(n0∇ψ0) = 0, (2.8)

n₀ =b(x).

(2.9)

Fork = 1 we have :

φ1 =−ε∇ψ0.∇ψ1−h⁰(n0)n1+εψ1, (2.10)

−div(n₀∇ψ₁) = div(n₁∇ψ₀), (2.11)

n₁ = 0.

(2.12)

And fork ≥2 :

φk=−ε 2

k

X

i=0

∇ψi.∇ψk−i−h⁰(n0)nk−¯hk((ni)0≤i≤k−1) +εψk, (2.13)

−div(n₀∇ψ_k) =

k

X

i=1

div(n_i∇ψ_k−i), (2.14)

n_k= ∆φk−2. (2.15)

All the profiles (n_k, ψ_k, φ_k) can be determined, uniquely and sufficiently smooth, by in- duction onk with boundary conditions given later. First, we obtainn₀ by (2.9), then we have ψ₀ by (2.8) and φ₀ by (2.7). We use the same way for determining the solutions of (2.10)-(2.12) and (2.13)-(2.15). Then the internal solution is constructed. For m ≥2 let us denote :

n^λ_I,m =

m

X

k=0

λ^kn_k; ψ^λ_I,m =

m

X

k=0

λ^kψ_k; φ^λ_I,m =

m

X

k=0

λ^kφ_k.

By construction, if (n_k, ψ_k, φ_k) are smooth enough, then the error equations are of order O(λ^m+1). Since n_k = ∆φk−2, for k ≥ 2, and is not necessarly equal to n^k_D on Γ, then a boundary layer can appear.

2.2. External expansion. We follow the notations in [6]. For x ∈ Ω, we note t(x) the distance from Γ to x and s(x) the point of Γ nearest from x. For θ > 0, let Ω_θ be the boundary layer of size θ :

Ωθ ={x∈Ω;|x−y|< θ, y∈Γ}.

If θ is small enough, s(x) is defined uniquely for all x ∈ Ω_θ. In Ω_θ, we define the fast variable byξ(x, λ) = t(x)/λ.Forx∈Ωθ,letν(x) = (ν1, ..., νd) the unit interior-directional normal vector of Γ passing from x. Then from :

t(x) =kx−s(x)k, x−s(x) =t(x)ν(x),

and due to the fact that for all i= 1, ..., d, ∂s(x)/∂x_i is orthogonal to ν(x), it is easy to see that ∇_xt = ν(x). Hence the partial derivative of a function w(s(x), ξ(x, λ)) may be decomposed as :

(2.16) ∂w(s(x), ξ(x, λ))

∂x_i =λ⁻¹νi

∂w

∂ξ +Diw,

whereDi is a first order differential operator ins defined by : Diw=∇sw._∂x^∂s

i.Similarly : (2.17) ∂²w(s(x), ξ(x, λ))

∂x_i∂x_j =λ⁻²ν_i∂²w

∂ξ² +λ⁻¹D_ji∂w

∂ξ +D_jD_iw+∇_sw. ∂²s

∂x_i∂x_j, where D_ji =ν_iD_j +ν_jD_i+∂ν_i/∂x_j. Note that for all i, j we have : D_ji =D_ij.

For every functionw(x) defined in Ω_θ the equivalent function of (s, t) is designated by

˜

w i.e. w(x) = ˜w(s(x), t(x)) = ˜w(s(x), λξ(x, λ)). We develop ˜w(s(x), λξ(x, λ)) formally to obtain :

˜

w(s(x), λξ(x, λ)) = ˜w(s(x),0) +O(λ).

Let ¯w(s) = ˜w(s,0). Then the ansatz of an approximate solution up to order m of (1.1)- (1.3) in Ω_θ are given by :

˜

n^λ_a,m(x) =n^λ_I,m(x) + ˜n^λ_B,m(s(x), ξ(x, λ)), ψ˜_a,m^λ (x) = ψ_I,m^λ (x) + ˜ψ_B,m^λ (s(x), ξ(x, λ)), φ˜^λ_a,m(x) = φ^λ_I,m(x) + ˜φ^λ_B,m(s(x), ξ(x, λ));

where the boundary layers (˜n^λ_B,m,ψ˜^λ_B,m,φ˜^λ_B,m) have the expansions :

˜ n^λ_B,m=

m

X

k=0

λ^kn^b_k, ψ˜_B,m^λ =

m+1

X

k=0

λ^kψ_k^b, φ˜^λ_B,m =

m

X

k=0

λ^kφ^b_k,

in which each term (n^b_k(s, ξ), ψ_k^b(s, ξ), φ^b_k(s, ξ)) will be chosen to decay exponentially when ξ tends to +∞. They are determined by setting (˜n^λ_a,m,ψ˜_a,m^λ ,φ˜^λ_a,m) in (1.1)-(1.3) and by identification of the order inλ. Let∂/∂ν =Pd

i=1νi∂/∂xi. After computation we obtain : ψ^b₀ ≡0,

the system for (n^b₀, ψ₁^b, φ^b₀) :

(S₀)



















(¯n₀+n^b₀)∂²ψ₁^b

∂ξ² + ∂ψ¯₀

∂ν +∂ψ^b₁

∂ξ ∂n^b₀

∂ξ = 0, ε

2 ∂ψ₁^b

∂ξ 2

+ε∂ψ₁^b

∂ξ

∂ψ¯₀

∂ν +h(¯n₀+n^b₀)−φ^b₀ = ¯φ₀+εψ¯₀,

∂²φ^b₀

∂ξ² =n^b₀,

and the system for (n^b_k, ψ^b_k+1, φ^b_k) withk ≥1 :

(S_k)























ε∂ψ^b_k+1

∂ξ

∂ψ₁^b

∂ξ +∂ψ¯0

∂ν

+h⁰(¯n₀+n^b₀)n^b_k−φ^b_k =F_1,k(n^b_l, ψ_l+1^b ,0≤l ≤k−1), (¯n₀+n^b₀)∂²ψ_k+1^b

∂ξ² + ∂ψ¯₀

∂ν +∂ψ₁^b

∂ξ

∂n^b_k

∂ξ +n^b_k∂²ψ₁^b

∂ξ² + ∂n^b₀

∂ξ

∂ψ^b_k+1

∂ξ =

F_2,k(n^b_l, ψ_l+1^b ,0≤l≤k−1),

−∂²φ^b_k

∂ξ² +n^b_k =F_3,k(φ^b_l, k−1≤l ≤k−2),

where F_i,k, i = 1,2,3, are given functions of (n^b_l, ψ^b_l+1)0≤l≤k−1 for F_1,k, F_2,k, and of (φ^b_l)k−1≤l≤k−2 for F_3,k.

Hence the approximate solution is constructed in Ω_θ.To complete the definition of the approximate solution in ¯Ω, letσ ∈C^∞(0,∞) be a smooth function such thatσ(t) = 1 for 0≤t≤θ/2 and σ ≡0 fort≥θ and we define :

(n^λ_B,m(x), ψ^λ_B,m(x), φ^λ_B,m(x)) =

( (˜n^λ_B,m(s(x), t(x)/λ),ψ˜^λ_B,m(s(x)t(x)/λ),φ˜^λ_B,m(s(x)t(x)/λ))σ(t(x)), for x∈Ω_θ, 0, for x∈Ω−Ω_θ.

Then, (n^λ_B,m, ψ^λ_B,m, φ^λ_B,m) has the same regularity as (˜n^λ_B,m,ψ˜_B,m^λ ,φ˜^λ_B,m). If each (n^b_k(s, ξ), ψ^b_k(s, ξ), φ^b_k(s, ξ)) decays exponentially when ξ tends to +∞ it is easy to see

that the difference between (n^λ_B,m, ψ_B,m^λ , φ^λ_B,m) and (˜n^λ_B,m,ψ˜_B,m^λ ,φ˜^λ_B,m) is uniform of order of e^−µ/λ for a constant µ >0.

Finally, the boundary conditions (1.6) give for s∈Γ :

(2.18) n₀ =n⁰_D, n₁ =n¹_D, n^b₀(s,0) = n^b₁(s,0) = 0,n¯_k(s) +n^b_k(s,0) =n^k_D, k ≥2, (2.19) ψ₀ =ψ_D⁰, ψ₁ =ψ_D¹, ψ₂ =ψ_D², ψ₁^b(s,0) =ψ^b₂(s,0) = 0,ψ¯_k(s) +ψ_k^b(s,0) =ψ^k_D, k ≥3.

We refer to [18] for the scheme of determination of (n_k, ψ_k, φ_k, n^b_k, ψ_k+1^b , φ^b_k). We can show under the assumption (H4) :

n^b₀ =n^b₁ =ψ^b₁ =ψ₂^b = 0.

The approximate solution up to order m is constructed in the form : (2.20) (n^a_λ, ψ_λ^a, φ^a_λ) = (n^λ_I,m+n^λ_B,m, ψ^λ_I,m+ψ^λ_B,m, φ^λ_I,m+φ^λ_B,m), in ¯Ω.

By construction : n^a_λ =Pm

k=0λ^kn^k_D, ψ_λ^a=Pm

k=0λ^kψ_D^k on Γ and : n^a_λ =n₀+

m

X

j=2

λ^j(n_j +n^b_j), (2.21)

ψ_λ^a=ψ₀+λψ₁+λ²ψ₂+

m

X

j=3

λ^j(ψ_j+ψ_j^b) +λ^m+1ψ^b_m+1. (2.22)

The existence and uniqueness of boundary layers (n^b_k, ψ_k+1^b ) with exponential decay have been shown in [18] for each k ≥0.Thus we obtain,

Theorem 2.1. Under the assumptions (H1)-(H6), there exists a unique asymptotic ex- pansion (2.20) up to order m, sufficiently smooth, satisfying (2.21)-(2.22).

3. Justification of the quasineutral limit

We have seen in the introduction that (1.1)-(1.3), (1.6) is equivalent to (1.1)-(1.2), (1.4), (1.6). The main result of this paper is :

Theorem 3.1. Under the assumption (H1)-(H6), for λ small enough, there is an ε₀ >0 independent of λsuch that for allε ∈[0, ε₀], the problem (1.2), (1.4), (1.6) has a solution (n_λ, ψ_λ)∈W^2,q(Ω)×C^2,δ( ¯Ω) which satisfies :

(3.1) knλ−n^a_λk_W^2,q_(Ω) ≤Aλ^m−1, kψλ−ψ^a_λk_C^2,δ_{( ¯Ω)} ≤Aλ^m−1, where A is a constant independent of λ.

Remark 3.2. Using equation (1.1), the continuity of h and the estimates from Theorem 3.1, we can easily show, for λ small enough :

kφ_λ −φ^a_λk_C1,δ( ¯Ω) ≤Aλ^m−1, where A is a constant independent of λ.

Remark 3.3. Without the assumption (H4), some boundary layers of order 0 and 1 appear, and we are not able to estimate n_λ −n^a_λ and ψ_λ −ψ_λ^a in the same spaces as in (3.1).

Proof of Theorem 3.1 : We search a solution of the problem (1.2), (1.4), (1.6) of the form :

n_λ =n^a_λ+λ^m−1r_λ, ψ_λ =ψ^a_λ+λ^m−1p_λ,

where n^a_λ and ψ_λ^a are the expansions defined in (2.21)-(2.22). To this end we define the two following operators :

N(n, ψ) := L(n, ψ)− 1

λ²(n−b) +εQ(ψ), M(n, ψ) := −div(n∇ψ),

where :

L(n, ψ) := ∆h(n) − ε n

3

X

i,j=1

ψ_xiψ_xjn_xixj+ ε

n∇ψ.∇n+ ε

n²(∇ψ.∇n)²

− ε n

3

X

i,j=1

ψ_xiψ_xixjn_xj.

Then the system (1.2),(1.4),(1.6) can be written as : N(n_λ, ψ_λ)(x, λ) = 0,

(3.2)

M(n_λ, ψ_λ)(x, λ) = 0, in Ω, (3.3)

n_λ =

m

X

k=0

λ^kn^k_D+n^m_D,λ, ψ_λ =

m

X

k=0

λ^kψ_D^k +ψ^m_D,λ, on Γ.

(3.4)

By construction it is easy to check that (see [18]) :

(3.5) N(n^a_λ, ψ_λ^a) = O(λ^m−1), M(n^a_λ, ψ_λ^a) =O(λ^m), inL^∞(Ω), (3.6) N(n^a_λ, ψ_λ^a) = O(λ^m−2), M(n^a_λ, ψ_λ^a) = O(λ^m−1), in C¹( ¯Ω), uniformly with respect to λ.

We search now the system verified by (r_λ, p_λ) by replacing formallyn_λ byn^a_λ+λ^m−1r_λ and ψ_λ byψ_λ^a+λ^m−1p_λ in (3.2)-(3.4). From (3.3) we have :

(3.7) −div[(n^a_λ+λ^m−1r_λ)∇p_λ] =λ^−m+1[−M(n^a_λ, ψ^a_λ) +λ^m−1div(r_λ∇ψ_λ^a)].

Similarly, (3.2) gives :

∆H(r_λ, x, λ)− ε n^a_λ+λ^m−1r_λ

3

X

i,j=1

∂ψ_λ^a

∂x_i +λ^m−1∂p_λ

∂x_i

∂ψ_λ^a

∂x_j +λ^m−1∂p_λ

∂x_j

∂²r_λ

∂x_i∂x_j

+ ε

(n^a_λ+λ^m−1r_λ)²

λ^m−1(∇(ψ_λ^a+λ^m−1p_λ).∇r_λ)²+ 2∇(ψ_λ^a+λ^m−1p_λ).∇r_λ× (∇ψ_λ^a.∇n^a_λ+λ^m−1∇p_λ.∇n^a_λ)

+ ε

n^a_λ+λ^m−1r_λ∇(ψ_λ^a+λ^m−1p_λ).∇r_λ

− ε

n^a_λ +λ^m−1r_λ

3

X

i,j=1

∂ψ_λ^a

∂x_i +λ^m−1∂p_λ

∂x_i

∂²ψ_λ^a

∂x_i∂x_j +λ^m−1 ∂²p_λ

∂x_i∂x_j ∂r_λ

∂x_j (3.8)

− 1

λ²rλ(1−λ²β(rλ, x)) = −λ^−m+1n^a_λ

n^a_λ+λ^m−1r_λN(n^a_λ, ψ_λ^a)−g1(n^a_λ, ψ_λ^a, rλ, pλ)−g2(n^a_λ, ψ_λ^a, rλ)

with :

H(r_λ, x, λ) :=r_λ Z 1

0

h⁰(n^a_λ(x) +λ^m−1r_λt)dt,

g₁(n^a_λ, ψ^a_λ, r_λ, p_λ) := − ε n^a_λ+λ^m−1r_λ

3

X

i,j=1

∂ψ_λ^a

∂x_i

∂p_λ

∂x_j +∂p_λ

∂x_i ∂ψ^a_λ

∂x_j +λ^m−1∂p_λ

∂x_j

∂²n^a_λ

∂x_i∂x_j

+ ε

(n^a_λ+λ^m−1r_λ)²

λ^m−1(∇p_λ.∇n^a_λ)²+ 2(∇ψ^a_λ.∇n^a_λ)(∇p_λ.∇n^a_λ)

+ ε

n^a_λ+λ^m−1r_λ∇p_λ.∇n^a_λ − ε n^a_λ+λ^m−1r_λ

" ₃ X

i,j=1

∂ψ^a_λ

∂x_i

∂n^a_λ

∂x_j

∂²p_λ

∂x_i∂x_j

+

3

X

i,j=1

∂p_λ

∂x_i

∂n^a_λ

∂x_j

∂²ψ_λ^a

∂x_i∂x_j +λ^m−1 ∂²p_λ

∂x_i∂x_j #

+ ε

"

λ^m−1Q(p_λ) + 2

3

X

i,j=1

∂²ψ_λ^a

∂x_i∂x_j

∂²p_λ

∂x_i∂x_j

# ,

g₂(n^a_λ, ψ_λ^a, r_λ) :=− εr_λ

n^a_λ(n^a_λ+λ^m−1rλ)²(∇ψ_λ^a.∇n^a_λ)²+ εr_λ n^a_λ+λ^m−1rλ

Q(ψ_λ^a), and,

β(r_λ, x) = 1

n^a_λ(x) +λ^m−1rλ

∆h(n^a_λ(x))− 1

λ²(n^a_λ(x) +λ^m−1rλ)(n^a_λ(x)−b(x)).

From (1.6) the boundary conditions associated to (3.7)-(3.8) are : (3.9) p_λ =λ^−m+1ψ^m_D,λ, r_λ =λ^−m+1n^m_D,λ, on Γ.

Note that by definition, using the assumptions, the applicationH(., x, λ) : r_λ 7→H(r_λ, x, λ) is continuous and strictly increasing and then invertible.

To prove Theorem 3.1, it suffices to prove the following Lemma.

Lemma 3.4. Under the assumptions (H1)-(H6), for λ small enough, there is an ε₀ > 0 independent of λ such that for all ε ∈ [0, ε₀], the problem (3.7)-(3.9) has a solution (r_λ, p_λ)∈W^2,q(Ω)×C^2,δ( ¯Ω) which satisfies :

kr_λk_W^2,q_(Ω) ≤A₁, kp_λk_C2,δ( ¯Ω) ≤A₁, where A₁ is a constant independent of λ.

Proof : Let σ_λ ∈S with :

S ={ρ∈C^1,δ( ¯Ω); kρk_C^1,δ_{( ¯Ω)} ≤K},

where K is a constant independent of λ which will be fixed later. Here S is a closed and convex set. We define the mapping T :σ_λ 7→p_λ 7→r_λ by :

(A) the solution of :

−div[(n^a_λ+λ^m−1σ_λ)∇p_λ] =f₁(σ_λ, ψ^a_λ, n^a_λ), in Ω, p_λ =λ^−m+1ψ_D,λ^m , on Γ

with

f₁(σ_λ, ψ^a_λ, n^a_λ) := λ^−m+1

−M(n^a_λ, ψ_λ^a) +λ^m−1div(σ_λ∇ψ_λ^a)

;

(B) let r_λ =G(v_λ, x, λ) where G(., x, λ) = H⁻¹(., x, λ) with v_λ being solution of :

∆v_λ− εG⁰(H(σ_λ, x, λ), x, λ) n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψ_λ

∂x_i

∂ψ_λ

∂x_j

∂²v_λ

∂x_i∂x_j

− εG⁰⁰(H(σ_λ, x, λ), x, λ) (n^a_λ+λ^m−1σ_λ)G⁰(H(σ_λ, x, λ), x, λ)

3

X

i,j=1

∂ψ_λ

∂x_i

∂ψ_λ

∂x_j

∂σ_λ

∂x_i

∂v_λ

∂x_j

− ε

n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψ_λ

∂x_i

∂ψ_λ

∂x_j

∂G⁰(H(σ_λ, x, λ), x, λ)

∂x_j

∂v_λ

∂x_i +∂G⁰(H(σ_λ, x, λ), x, λ)

∂x_i

∂v_λ

∂x_j

−εG⁰⁰(H(σλ, x, λ), x, λ) n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψλ

∂x_i

∂ψλ

∂x_j

∂H(σλ, x, λ)

∂x_i

∂vλ

∂x_j

+εG⁰(H(σ_λ, x, λ), x, λ) (n^a_λ+λ^m−1σλ)²

λ^m−1(∇ψ_λ.∇σ_λ)∇ψ_λ+ 2(∇ψ^a_λ.∇n^a_λ+λ^m−1∇p_λ.∇n^a_λ)∇ψ_λ

.∇v_λ

+εG⁰(H(σ_λ, x, λ), x, λ)

n^a_λ+λ^m−1σ_λ ∇ψ_λ.∇v_λ−εG⁰(H(σ_λ, x, λ), x, λ) n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψ_λ

∂x_i

∂²ψ_λ

∂x_i∂x_j

∂v_λ

∂x_j

− 1

λ²G(v_λ, x, λ)(1−λ²β(σ_λ, x)) = f₂(σ_λ, x, λ), x∈Ω, vλ =H(λ^−m+1n^m_D,λ, x, λ), on Γ,

where we note :

G⁰(v_λ, x, λ) = ∂G

∂v_λ(v_λ, x, λ), G⁰⁰(v_λ, x, λ) = ∂²G

∂v²_λ(v_λ, x, λ), f₂(σ_λ, x, λ) = −λ^−m+1 n^a_λ

n^a_λ+λ^m−1σ_λN(n^a_λ, ψ^a_λ)−g₁(n^a_λ, ψ_λ^a, σ_λ, p_λ)

− g₂(n^a_λ, ψ^a_λ, σ_λ) + ε n^a_λ +λ^m−1σλ

3

X

i,j=1

∂ψ_λ

∂xi

∂ψ_λ

∂xj

∂²G

∂xi∂xj

(H(σ_λ, x, λ), x, λ)

− ε

(n^a_λ +λ^m−1σ_λ)²

λ^m−1(∇ψ_λ.∇σ_λ)∇ψ_λ+ 2(∇ψ^a_λ.∇n^a_λ+λ^m−1∇p_λ.∇n^a_λ)∇ψ_λ

.∇_xG(H(σ_λ, x, λ), x, λ)

− ε

n^a_λ+λ^m−1σ_λ∇ψ_λ.∇_xG(H(σ_λ, x, λ), x, λ)

+ ε

n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψ_λ

∂x_i

∂²ψ_λ

∂x_i∂x_j

∂G

∂x_j(H(σ_λ, x, λ), x, λ).

By construction n^a_λ and ψ^a_λ being sufficiently smooth, f₁ ∈ C^0,δ( ¯Ω) and is bounded in C^0,δ( ¯Ω) uniformly in λ. Moreover by assumption n₀ ≥ n > 0, hence for λ small enough

we have : n^a_λ+λ^m−1σ_λ ≥ ¹₂n > 0. Finally, using the Theorem 6.6 in [10] we obtain that the problem (A) has a unique solution p_λ ∈C^2,δ( ¯Ω) and :

kp_λk_C2,δ( ¯Ω) ≤C(K),

with C(K) a constant independent of λ.Therefore, there exists ε₁ >0, independent ofλ, such that for all ε∈[0, ε₁],(B) is an elliptic problem.

Here, we cannot use the classical result used in [5] due to the fact that the function G depends not only on v_λ, but also on the variable x.The idea here is to use the Leray- Schauder fixed point Theorem to show first the existence and boundedness of solutionsv_λ to (B). This implies the existence and boundedness of r_λ =G(v_λ, x, λ) which are needed to prove thatT is an application fromS toS,and then to apply the Schauder fixed point Theorem. More precisely, we use the Leray-Schauder fixed point Theorem to prove the following Lemma.

Lemma 3.5. Under the assumptions (H1)-(H6), for λ small enough, there is an ε0 >0, independent of λ such that for all ε ∈ [0, ε₀], (B) has a unique solution v_λ ∈ W^2,q(Ω) which satisfies :

kv_λk_W^2,q_(Ω) ≤A₂, with A₂ independent of λ, K.

Proof : As mention previously, to prove this lemma, we will use the Leray-Schauder fixed point theorem. Let τ ∈ [0,1] and w ∈ W^2,q(Ω). We define the application ˜T : W^2,q(Ω)×[0,1]→C^1,δ( ¯Ω) by (w, τ)7→v where v solves the problem :

∆v−εG⁰(H(σ_λ, x, λ), x, λ) n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψ_λ

∂x_i

∂ψ_λ

∂x_j

∂²v

∂x_i∂x_j

− τ εG⁰⁰(H(σ_λ, x, λ), x, λ) (n^a_λ+λ^m−1σ_λ)G⁰(H(σ_λ, x, λ), x, λ)

3

X

i,j=1

∂ψ_λ

∂x_i

∂ψ_λ

∂x_j

∂σ_λ

∂x_i

∂w

∂x_j

− τ ε n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψ_λ

∂x_i

∂ψ_λ

∂x_j

∂G⁰(H(σ_λ, x, λ), x, λ)

∂x_j

∂w

∂x_i + ∂G⁰(H(σ_λ, x, λ), x, λ)

∂x_i

∂w

∂x_j

−τ εG⁰⁰(H(σ_λ, x, λ), x, λ) n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψ_λ

∂x_i

∂ψ_λ

∂x_j

∂H(σ_λ, x, λ)

∂x_i

∂w

∂x_j

+τ εG⁰(H(σ_λ, x, λ), x, λ) (n^a_λ +λ^m−1σ_λ)²

λ^m−1(∇ψλ.∇σλ)∇ψλ+ 2(∇ψ_λ^a.∇n^a_λ+λ^m−1∇pλ.∇n^a_λ)∇ψλ

.∇w

+τ εG⁰(H(σ_λ, x, λ), x, λ)

n^a_λ+λ^m−1σ_λ ∇ψλ.∇w−τ εG⁰(H(σ_λ, x, λ), x, λ) n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψ_λ

∂x_i

∂²ψ_λ

∂x_i∂x_j

∂w

∂x_j

−τ

λ²G(w, x, λ)(1−λ²β(σ_λ, x)) =τ f₂(σ_λ, x, λ), x∈Ω, v =τ H(λ^−m+1n^m_D,λ, x, λ), on Γ.

This problem is elliptic and admits a unique solution for ε small enough, hence ˜T is well-defined. We have : ˜T(w,0) =v₁ where v₁ is a solution of the linear problem :

∆v₁− εG⁰(H(σ_λ, x, λ), x, λ) n^a_λ+λ^m−1σ_λ

3

X

i,j=1

∂ψ_λ

∂x_i

∂ψ_λ

∂x_j

∂²v₁

∂x_i∂x_j = 0 v₁ = 0, on Γ.

Then v1 = 0 and hence ˜T(w,0) = 0. Furthermore, it is not difficult to check that ˜T is continuous and also compact (due to the compact injection from W^2,q(Ω) to C^1,δ( ¯Ω)).

To achieve the proof of Lemma 3.5, we need to prove that if v₂ is a fixed point of ˜T then we have

kv₂k_W^2,q_(Ω) ≤A₃,

with A₃ being independent of λ, K. This is the statement of Lemma 3.6. Since its proof being technical, it is moved to Appendix.

Lemma 3.6. Assuming (H1)-(H6), let v₂ be a fixed point ofT .˜ Then for λ small enough there is an ε₀ >0, independent ofλ such that for all ε∈[0, ε₀] :

kv2k_W^2,q_(Ω) ≤A3, ∀τ ∈[0,1], where A₃ is a constant independent of λ and K.

Then by the Leray-Schauder fixed point theorem, for λ small enough, and ε ≤ ε₀, ε₀ independent ofλ, T˜₁ = ˜T(w,1) has a fixed point v_λ ∈W^2,q(Ω) such that :

kvλk_W^2,q_(Ω) ≤A2, with A2 independent ofλ, K.

By definition of ˜T , v_λ is a solution to the problem (B). For the uniqueness of solution, we assume that there exist two solutions v_λ,1, v_λ,2, we substract the two systems. Using Theorem 9.15 in [10], we show that the obtained system has the unique solution zero. This gives v_λ,1 =v_λ,2, and then we obtain the uniqueness of solutions for (B). This completes the proof of Lemma 3.5.

Now using Lemma 3.5 and the injection W^2,q(Ω) ,→ C^1,δ( ¯Ω), with K = cA2, where c is the injection constant, we obtain v_λ ∈S which implies r_λ ∈S and then T :S →S.To complete the verification of the assumptions in the Schauder fixed point Theorem for T we need to show thatT is continuous and compact. This is the statement of the following Lemma.

Lemma 3.7. The application T, as an application fromC^1,δ( ¯Ω)toC^1,δ( ¯Ω), is continuous and compact.

Proof :

We define T =I◦G(., x, λ)◦χ◦γ by :

I :W^2,q(Ω),→C^1,δ( ¯Ω), which is continuous and compact, G(., x, λ) :v_λ 7→r_λ, which is continuous by assumption, γ :C^1,δ( ¯Ω)−→C^2,δ( ¯Ω), σλ 7→pλ, where pλ is solution of (A), χ:C^2,δ( ¯Ω)−→W^2,q(Ω), p_λ 7→v_λ, where v_λ is solution of (B).

We have to show that χ and γ are continuous.

Let p_λ,1 and p_λ,2 be two solutions of (A). Then by substraction of the two systems we obtain the following one :

−div[(n^a_λ+λ^m−1σ_λ,1)∇(p_λ,1−p_λ,2)] =f, in Ω, p_λ,1−p_λ,2 = 0, on Γ,

with :

f =f₁(σ_λ,1, ψ^a_λ, n^a_λ)−f₁(σ_λ,2, ψ_λ^a, n^a_λ)−λ^m−1div[(σ_λ,1−σ_λ,2)∇p_λ,2].

It is clear that :

kλ^m−1div[(σ_λ,1−σ_λ,2)∇p_λ,2]k_C0,δ( ¯Ω) ≤Ckσ˜ _λ,1−σ_λ,2k_C1,δ( ¯Ω), since kp_λ,2k_C2,δ( ¯Ω)≤C.˜

Moreover, by definition we have :

f1(σλ,1, ψ^a_λ, n^a_λ)−f1(σλ,2, ψ_λ^a, n^a_λ) = div[(σλ,1 −σλ,2)∇ψ^a_λ], and by construction : kψ_λ^ak_C^2,δ_{( ¯Ω)} ≤C.˜ Then

kf₁(σ_λ,1, ψ^a_λ, n^a_λ)−f₁(σ_λ,2, ψ_λ^a, n^a_λ)k_C0,δ( ¯Ω) ≤Ckσ˜ _λ,1−σ_λ,2k_C1,δ( ¯Ω). Hence we obtain that :

kfk_C^0,δ_{( ¯Ω)} ≤Ckσ˜ λ,1−σλ,2k_C^1,δ_{( ¯Ω)}. Using Theorem 6.6 in [10]

kp_λ,1−p_λ,2k_C2,δ( ¯Ω) ≤ kfk_C0,δ( ¯Ω). Hence :

kp_λ,1−p_λ,2k_C2,δ( ¯Ω) ≤Ckσ˜ _λ,1−σ_λ,2k_C1,δ( ¯Ω), and the application γ is continuous from C^1,δ( ¯Ω) to C^2,δ( ¯Ω).

In a same way, with the pressure functionp_λ smooth enough we obtain thatχis a con- tinuous application fromC^2,δ( ¯Ω) to W^2,q(Ω).Therefore, T is continuous. The application I being compact, we have that T is also compact. This completes the proof of Lemma 3.7.

Finally all the assumptions in the Schauder fixed point Theorem are satisfied. As a consequence, for λ small enough and ε ≤ ε₀, ε₀ being independent of λ, T has a fixed point. This completes the proof of Lemma 3.4.

Theorem 3.8. Assume (H1)-(H6). If in addition,(λ^−m−1n^m_D,λ)_λ>0 is bounded inW^2,∞(Ω), and (λ^−m−1ψ^m_D,λ)_λ>0 is bounded in W^1,q(Ω). Then,

kn_λ−n^a_λk_L^∞_(Ω) ≤A₄λ^m+1, kψ_λ−ψ_λ^ak_W^1,q_(Ω) ≤A₄λ^m+1, where A₄ is a constant independent of λ.

Since the proof uses notations defined in the proof of Lemma 3.6, it is also moved in Appendix.