Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas

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Citation for the original published paper (version of record):

Bernhoff, N. (2012)

Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas.

Journal of statistical physics, 147(6): 1156-1181 https://doi.org/10.1007/s10955-012-0513-y

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Niclas Bernhoff

Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-Condensable Gas

Received: date / Accepted: date

Abstract We consider a non-linear half-space problem related to the con- densation problem for the discrete Boltzmann equation and extend some known results for a single-component gas to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian at in- finity, as the non-condensable gas tends to zero at infinity. We assume that the vapor is completely absorbed and that the non-condensable gas is diffusively reflected at the condensed phase and that the vapor molecules leaving the condensed phase are distributed according to a given distribution. The condi- tions, on the given distribution, needed for the existence of a unique solution of the problem are investigated. We also find exact solvability conditions and solutions for a simplified six+four-velocity model, as the given distribution is a Maxwellian at rest, and study a simplified twelve+six-velocity model.

Keywords Boltzmann equation · boundary layers · discrete velocity models · half-space problem · non-condensable gas

Mathematics Subject Classification (2000) 82C40 · 76P05 · 35Q20

1 Introduction

In this paper we consider the condensation problem for a single-component gas or vapor when a non-condensable gas is present [15]. Formulation and motivation of the problem can be found in [15]. The vapor is assumed to tend to an assigned Maxwellian M_∞^A, with a flow velocity towards the condensed phase, at infinity, while the non-condensable gas tends to zero at infinity.

N. Bernhoff

Department of Mathematics, Karlstad University, 65188 Karlstad, SWEDEN Tel.: +46-54-7002024

Fax: +46-54-7001851

E-mail: niclas.bernhoff@kau.se

Steady condensation of the vapor takes place at the condensed phase, which is held at a constant temperature. We assume that the vapor is completely absorbed and that the non-condensable gas is diffusively reflected at the condensed phase, i.e. there is no net flow across the condensed phase and the gas molecules leaving the condensed phase are distributed according to a non-drifting Maxwellian M₀^B at the condensed phase. The vapor molecules leaving the condensed phase are distributed according to a given distribution.

The conditions, on the given distribution at the condensed phase, needed for the existence of a unique solution of the problem are investigated. We assume that the given distribution is sufficiently close to the Maxwellian M_∞^A at the infinity and that the total mass of the non-condensable gas relatively this distance is sufficiently small. The explicit number of conditions on the given distribution is given in Theorem 2, under some assumptions on the discrete velocity models for the gases.The typical case is that the given distribution is the Maxwellian at the condensed phase [15]. However, we can’t be sure that there is any Maxwellian at rest close enough to the Maxwellian at infinity, but if there is, of course our results are valid also in this case.

Similar problems have been studied for the discrete Boltzmann equation for single species (a vapor in the absence of a non-condensable gas) [4],[3], and references therein, and binary mixtures of two vapors [5], as well as for the full Boltzmann equation for single species [19],[1],[20] and binary mixtures [18], and references therein. For the discrete Boltzmann equation, one obtain for binary mixtures of two vapors a similar structure as for single- component gases [5]. One can then extend results for half-space problems of single-component gases [2], [4] to yield also for binary mixtures of two vapors. However, though, both complete absorption and diffuse reflection conditions are considered (at least implicitly) for both single-component gases and binary mixtures of two vapors in [4] and [5], the situation will be different when one of the gases is non-condensable. The fact that the distribution function for the non-condensable gas tends to zero at infinity changes the situation. First of all we can not use the standard transformation used in [4] and [5], but we use instead a slight modification of it, which changes the structure of the obtained system. Secondly, the trivial case when the non- condensable gas is absent, i.e. the case of a single-component gas considered in [4] and [19], is a trivial solution of the system. Therefore, in difference to the case of a vapor, we need, in the case of a non-condensable gas, to have a free parameter, which will later be settled by fixing the amount of the non-condensable gas. Hence, even if our proof is influenced by the proof in [4] (and [19]) for single-component gases, we have to take these differences into account. To our knowledge, there is no corresponding results for the full Boltzmann equation up to now.

The paper is organized as follows. In Section 2 we present the discrete velocity model for binary mixtures and some of its properties. We make a transformation and obtain a transformed system, presented with some of its properties in Section 3. In Section 4 we present our assumptions and our main result in Theorem 2. The proof of our main result (Theorem 2) is presented in Section 5. In Section 6 we find an exact solvability condition and the solution for a simplified six+four-velocity model, for which the non-linear

problem becomes linear. Here the vapor molecules leaving the condensed phase are distributed according to the Maxwellian at the condensed phase.

In Section 7 we prove all the necessary conditions for existence, except one, which still is most likely to be fulfilled, for a simplified twelve+six-velocity model.

2 Discrete velocity models (DVMs) for binary mixtures

We first remind some properties of the discrete Boltzmann equation, or the general discrete velocity model (DVM), for binary mixtures [5].

The planar stationary discrete Boltzmann equation for a binary mixture of the gases A and B reads











ξ_i^A,1dF_i^A

dx = Q^AA_i (F^A, F^A) + Q^BA_i (F^B, F^A), i = 1, ..., n_A, ξ^B,1_j dF_j^B

dx = Q^AB_j (F^A, F^B) + Q^BB_j (F^B, F^B), j = 1, ..., nB,

(1)

where Vα = {ξ₁^α, ..., ξ_n^αα} ⊂ R^d, α, β ∈ {A, B} are finite sets of velocities, F_i^α = F_i^α(x) = F^α(x, ξ_i^α) for i = 1, ..., nα, and F^α = F^α(x, ξ) represents the microscopic density of particles (of the gas α) with velocity ξ at position x ∈ R. We denote by mα the mass of a molecule of the gas α. Here and below, α, β ∈ {A, B}.

For a function g^α= g^α(ξ) (possibly depending on more variables than ξ), we will identify g^αwith its restrictions to the set V^α, but also when suitable consider it like a vector function

g^α= (g^α₁, ..., g^α_nα), with g^α_i = g^α(ξ_i^α) . The collision operators Q^βα_i (F^β, F^α) in (1) are given by

Q^βα_i (F^β, F^α) =

nα

X

k=1 n_β

X

j,l=1

Γ_ij^kl(β, α) (F_k^αF_l^β− F_i^αF_j^β) for i = 1, ..., nα,

where it is assumed that the collision coefficients Γ_ij^kl(β, α), with 1 ≤ i, k ≤ nαand 1 ≤ j, l ≤ nβ, satisfy the relations

Γ_ij^kl(α, α) = Γ_ji^kl(α, α) and Γ_ij^kl(β, α) = Γ_kl^ij(β, α) = Γ_ji^lk(α, β) ≥ 0, with equality unless the conservation laws

m_αξ_i^α+m_βξ_j^β= m_αξ_k^α+m_βξ_l^β and m_α|ξ_i^α|²+m_β  ^ξ

β j

2

= m_α|ξ_k^α|²+m_β  ^ξ

β l

2

are satisfied. We denote F = F^A, F^B

= F^A(ξ) , F^B(ξ)

and Q(F, F )

= Q^AA(F^A, F^A) + Q^BA(F^B, F^A), Q^AB(F^A, F^B) + Q^BB(F^B, F^B)
.

Then the system (1) can be rewritten as DdF

dx = Q (F, F ) , where

D =

DA 0 0 DB



, with Dα= diag(ξ^α,1₁ , ..., ξ^α,1_nα).

We consider the case of non-zero ξ_i^α,1, ξ_i^α,1 6= 0, and we can then (without loss of generality) assume that

D_α=

D⁺_α 0 0 −D_α⁻

 , where

D⁺_α = diag(ξ^α,1₁ , ..., ξ^α,1

n⁺α

) and D_α⁻= −diag(ξ^α,1

n⁺α+1, ..., ξ_n^α,1

α), with ξ₁^α,1, ..., ξ^α,1

n⁺_α > 0 and ξ^α,1

n⁺_α+1, ..., ξ_n^α,1_α < 0.

The collision operator Q(f, f ) can be obtained from the bilinear expres- sions

Qi(F, G) = 1 2

nA

X

j,k,l=1

Γ_ij^kl(A, A)(F_k^AG^A_l + G^A_kF_l^A− F_i^AG^A_j − G^A_i F_j^A)

+1 2

nA

X

k=1 nB

X

j,l=1

Γ_ij^kl(B, A)(F_k^AG^B_l + G^A_kF_l^B− F_i^AG^B_j − G^A_i F_j^B), i = 1, ..., nA,

and

QnA+i(F, G) = 1 2

nB

X

k=1 nA

X

j,l=1

Γ_ij^kl(A, B)(F_k^BG^A_l + G^B_kF_l^A− F_i^BG^A_j − G^B_i F_j^A)

+1 2

nB

X

j,k,l=1

Γ_ij^kl(B, B)(F_k^BG^B_l + G^B_kF_l^B− F_i^BG^B_j − G^B_i F_j^B), i = 1, ..., nB.

Denoting

Q(F, G) = (Q1(F, G) , ..., Qn(F, G)), with n = nA+ nB, we see that, for arbitrary F and G

Q (F, G) = Q (G, F ) . A vector φ = φ^A, φ^B

is a collision invariant if and only if φ^α_i + φ^β_j = φ^α_k + φ^β_l,

for all indices 1 ≤ i, k ≤ n_α, 1 ≤ j, l ≤ n_β and α, β ∈ {A, B}, such that Γ_ij^kl(β, α) 6= 0.

We consider below only DVMs, such that the DVMs for the gases A and B are normal, i.e. the only collision invariants of the forms φ = φ^A, 0

and φ = 0, φ^B

, respectively, fulfills

φ^α= φ^α(ξ) = aα+ mαb · ξ + cmα|ξ|²,

for some constant aα, c ∈R^{and b ∈}R^d. It is also preferable that any general collision invariant of our DVMs is of the form

φ = φ^A, φ^B

, with φ^α= φ^α(ξ) = a_α+ m_αb · ξ + cm_α|ξ|², (2) for some constant aA, aB, c ∈R^{and b ∈}R^d. In this case the equation

hφ, Q (F, F )i = 0

has the general solution (2). Here and below, we denote by h·, ·i the Euclidean scalar product onRⁿ. Such DVMs, being normal both considering the gases together as a mixture as well as considering them separately as single species, is called supernormal [7]. This property is fulfilled for the continuous Boltz- mann equation. In the discrete case we can obtain so called spurious (unphys- ical) collision invariants. However, possible spurious collision invariants (for the mixture) don’t seem to affect the qualitative properties of our results.

We would also like our DVMs to fulfill that the equation φ, Q^AB F^A, F^B


= 0 (3)

has the general solution φ = a, where a is constant. We call a supernormal DVM fulfilling condition (3) for optinormal. This property is fulfilled for the continuous Boltzmann equation [14], but not necessarily for a DVM. However, we will below see that we can relax this assumption a little.

Example 1 The DVM, with

mA= 2mB,

where the vapor, gas A, is modeled by the twelve-velocity model with veloc- ities

(±1, ±1), (±1, ±3), and (±3, ±1),

and the non-condensable gas B is modeled by the six-velocity model with velocities

(±2, 0) and (±2, ±4), is optinormal.

A binary Maxwellian distribution (or just a bi-Maxwellian) is a function M = M^A, M^B

, such that

Q(M, M ) = 0 and M_i^α≥ 0 for all 1 ≤ i ≤ n_α.

All bi-Maxwellians are of the form M = e^φ, where φ is a collision invariant, i.e. for normal models we will have

M = M^A, M^B

, with M^α= e^φα= e^{aα+mαb·ξ+cmα|ξ|2}. (4) We will study distributions F , such that

F → M^A, 0

as x → ∞, where M^A= e^φA= e^{aA+mAb·ξ+cmA|ξ|2}. (5)

3 Transformed system For a bi-Maxwellian

M = M^A, ²M^B
,

where M^α= e^φα= e^{aα+mαb·ξ+cmα|ξ|2}and  is a so far undetermined positive constant less or equal to 1, 0 <  ≤ 1, we obtain, by denoting

F = M^A, 0
+√

M f , (6)

in Eq.(1), the system









 DA

df^A

dx + LAAf^A= −LBAf^B+ SAA(f^A, f^A) + SBA(f^B, f^A) D_Bdf^B

dx + L_ABf^B= S_BB(f^B, f^B) + S_AB(f^A, f^B)

,

where

LAAf^A

i= −2

nA

X

k=1 nA

X

j,l=1

q

M_j^AΓ_ij^kl(A, A)(

q

M_l^Af_k^A−q

M_j^Af_i^A),

LABf^B

i⁰ = −

nB

X

k=1 nA

X

j,l=1

q

M_j^AΓ_i^kl0j(A, B)(

q

M_l^Af_k^B−q

M_j^Af_i^B0), and

LBAf^B

i= −

nA

X

k=1 nB

X

j,l=1

q

M_j^BΓ_ij^kl(B, A)(

q

M_k^Af_l^B− q

M_i^Af_j^B), for i = 1, ..., n_Aand i⁰= 1, ..., n_B,

and the quadratic parts Sαβ are given by

Sαβ(f^α, f^β)

i=

nβ

X

k=1 n_α

X

j,l=1

q

M_j^αΓ_ij^kl(α, β)(f_k^αf_l^β− f_i^αf_j^β), i = 1, ..., nβ.

The matrices L_Aα are symmetric and semi-positive. Furthermore, LABf^B = 0 if f^B ∈ span(√

M^B), LAAf^A= 0 if and only if f^A=

√ M^Aφ^A, where φ = φ^A, 0

is a collision invariant, D

L_BAf^B,√ M^AE

=D

S_BA(f^B, f^A),√ M^AE

=D

S_αB(f^α, f^B),√ M^BE

= 0,

and D

SAA(f^A, f^A),

√

M^Aφ^AE

= 0.

In the continuous case ker(L_AB) = span(√

M^B) [14], so for an optimal model N (L_AB) = span(

√ M^B),

(cf. assumption (3)). We will, however, relax this assumption below. Here and below, we denote by N (Lαβ) the null-space of Lαβ.

By assumption (5)

f → 0 as x → ∞.

We denote by n^±_α, where n⁺_α+ n⁻_α = nα, and m^±_α, with m⁺_α+ m⁻_α = qα, the numbers of positive and negative eigenvalues (counted with multiplicity) of the matrices Dα and D⁻¹_α LAα respectively, and by m⁰_α the number of zero eigenvalues of D_α⁻¹LAα. Moreover, we denote by k⁺_α, k⁻_α, and lα, with k⁺_α + k⁻_α = kα, where kα+ lα = pα, the numbers of positive, negative, and zero eigenvalues of the pα× pα matrix Kα, with entries k_ij^α =

y^α_i, y^α_j

Dα = y^α_i, D_αy_j^α

, such that

y^α₁, ..., y^α_p

α

is a basis of the null-space of L_Aα, i.e.

in our case,

pA= d + 2, pB≥ 1, and span y^A₁, ..., y_d+2^A

= N (LAA)

= span(√ M^A,√

M^Aξ^A,1, ...,√

M^Aξ^A,d,√ M^A

ξ^A 

2).

We remind that we by h·, ·i denote the Euclidean scalar product onR^{n and} below we also denote

h·, ·i_D

α = h·, Dα·i .

We now remind a result by Bobylev and Bernhoff in [6] (see also [2]) and apply it in a specific case of interest for us.

Theorem 1 The numbers of positive, negative and zero eigenvalues of D_α⁻¹L_Aα are given by







m⁺_α = n⁺_α− k⁺_α− lα

m⁻_α = n⁻_α− k⁻_α − lα

m⁰_α= pα+ lα

.

In the proof of Theorem 1 bases

u^α₁, ..., u^α_qα, y₁^α, ..., y_k^α_α, z₁^α, ..., z_l^α_α, w^α₁, ..., w_l^α_α (7) ofR^nα, α ∈ {A, B}, such that

y_i^α, z_r^α∈ N (LAα), D_α⁻¹LAαw_r^α= z_r^αand D⁻¹_α LAαu^α_τ = λ^α_τu^α_τ, (8) and

hu^α_τ, u^α_νi_D

α = λ^α_τδ_{τ ν}, with λ^α₁, ..., λ^α

m⁺α > 0 and λ^α

m⁺α+1, ..., λ^α_q

α < 0, y^α_i, y^α_j

Dα= γ_i^αδij, with γ₁^α, ..., γ_k^α+

α > 0 and γ_k^α+

α+1, ..., γ^α_kα< 0, hu^α_τ, z_r^αi_D

α= hu^α_τ, w_r^αi_D

α= hu^α_τ, y^α_ii_D

α= hw_r^α, y_i^αi_D

α= hz_r^α, y_i^αi_D

α= 0, hw^α_r, w_s^αi_D

α= hz^α_r, z^α_si_D

α = 0 and hw^α_r, z^α_si_D

α = δrs, (9) are constructed.

If we assume that

n⁻_B= n⁺_B, or equivalently n_B = 2n⁺_B,

and that ξ_i+n^B +

B

= (−ξ_i^B,1, ξ_i^B,2, ..., ξ^B,d_i ), ξ_i^B,1> 0, for i = 1, ..., n⁺_B, (10) then

D⁻_B= D_B⁺.

Let b be the first component of b in Eqs.(4). If we assume that b < 0, then k⁻_B ≥ 1,

since

D√ M^B,

√ M^BE

DB

=

n⁺_B

X

i=1

ξ_i^B,1(1 − e^−2bξB,1i )M_i^B< 0.

For an optimal model

k⁻_B = 1 (cf. condition (3)) and hence,

k_B⁺= l_B = 0 and m⁺_B= n⁺_B. (11) We will relax condition (3) by assuming

k_B⁻= pB≥ 1.

Then the conditions (11) are still satisfied.

4 Main result

We consider the non-linear system









 DA

df^A

dx + LAAf^A= −LBAf^B+ SAA(f^A, f^A) + SBA(f^B, f^A) DB

df^B

dx + LABf^B= SBB(f^B, f^B) + SAB(f^A, f^B)

, (12)

where the solution tends to zero at infinity, i.e.

f^A(x) → 0 and f^B(x) → 0 as x → ∞, (13) and

LBAf^B, SBA(f^B, f^A) ∈ span(

√

M^A)^⊥, SAA(f^A, f^A) ∈ N (LAA)^⊥, and SαB ∈ N (LAB)^⊥.

We define the projections R^α₊:R^{nα →}Rⁿ

+

α and R^α₋:R^nα→Rⁿ

−

α, n⁻_α = n_α− n⁺_α, by

R₊^αs = s^α₊= (s₁, ..., s_n+

α) and R^α₋s = s^α₋= (s_n+

α+1, ..., s_nα)

for s^α= (s1, ..., sn_α).

We will below assume that n⁻_B = n⁺_B, and that the symmetry relation (10) is fulfilled. Furthermore, we assume that

k⁻_B= p_B. Then

k_B⁺= lB= 0, m⁺_B = n⁺_B and D_B⁻= D_B⁺. At x = 0 we assume the boundary conditions

f₊^A(0) = h0and f₊^B(0) = Cf₋^B(0) (14) where C is the n⁺_B× n⁺_B matrix, with the elements

cij =

ξ^B,1_j q M^B

n⁺_B+jM_0i^B D⁻_BM₀₋^B, 1 p

M_i^B. and

h0= 1 q

M₊^A

(a0− M₊^A) ∈Rⁿ

+ A,

where M₀^B = K₀^Be^c0mB|^ξB|², with K₀^B > 0, and a0∈Rⁿ

+

A. This corresponds to the boundary conditions

(F₊^A(0) = a0

F₊^B(0) = C0F₋^B(0) where C0 is the n⁺_B× n⁺_B matrix, with the elements

c0ij = ξ_j^B,1M_0i^B D_B⁻M₀₋^B, 1

(the discrete version of the diffusive boundary conditions, cf. [10], [2], or [4]), before the expansion (6).

We consider the case of condensation, i.e. we assume that b < 0, where b is the first component of b in Eq.(4). For the Boltzmann equation there is a critical number b−< 0 (where −b− is the speed of sound) [9], such that







k_A⁺= 1 and lA= 0 if b−< b < 0 k_A⁺= 0 and lA= 1 if b = b₋ k_A⁺= lA= 0 if b < b−

. (15)

We assume that we have a DVM with a critical number b₋ < 0, such that Eq.(15) is fulfilled. In fact, this number can be explicitly calculated for a plane axially symmetric 12-velocity model (assuming that the solution is symmetric with respect to the x-axis) see Section 7 below.

Furthermore, for b₋< b < 0 we will assume that R^A₊

√

M^A ∈ R/ ^A₊U_A⁺,

with U_A⁺ = span(u : L_AAu = λD_Au, λ > 0) = span(u^A₁, ..., u^A

m⁺_A), (16) or, equivalently,

dim(R^A₊Ue_A⁺) = m⁺_A+ 1 = n⁺_A, withUe_A⁺= span(u^A₁, ..., u^A_n+ A

,

√ M^A).

In this case, we can assume that y^A_pA=√

M^Awithout loss of generality, since lA= 0.

Remark 1 In fact, we could instead of √

M^A take any vector y ∈ N (LAA), such that

R^A₊y /∈ R^A₊U_A⁺ and

L_BAf^B, y

=

S_BA(f^B, f^A), y

= 0, as y^A_pA.

We introduce the operator C :R^{nB →}Rⁿ

+

B, given by C = R^B₊− CR^B₋.

We will assume that the set

U_B⁺= span(Cu : LABu = λDBu, λ > 0) = span(Cu^B₁, ..., Cu^B_n+ B

) has non-zero codimension, i.e.

dim U_B⁺< n⁺_B, (17)

but, also that the set

Ue_B⁺= span(Cu^B₁, ..., Cu^B

n⁺_B, C

√ M^B) has codimension 0, i.e.

dimUe_B⁺= n⁺_B. (18)

Therefore, the set U_B⁺ has codimension 1, i.e.

dim U_B⁺= n⁺_B− 1.

We can without loss of generality assume that Cu^B_n+

B

∈ span(Cu^B₁, ..., Cu^B_n+ B−1).

If the set U_B⁺ would have had codimension 0, i.e. if dim U_B⁺ = n⁺_B, then the only possibility would have been f^B(x) = 0.

We fix  to be

 = min {|h₀| , 1} ,

and the total mass of the gas B to be m^tot_B , i.e.

mB nB

X

i=1

∞

Z

0

q

M_i^Bf_i^B(x) dx = m^tot_B , (19)

for a given positive constant m^tot_B . Clearly, the case m^tot_B = 0, corresponds to the case of single species considered in [4].

We now state our main result.

Theorem 2 Assume that we have a DVM with a critical number b₋ < 0, such that Eq. (15) is fulfilled, let conditions (17) and (18), for b₋ < b < 0 also condition (16), be fulfilled, and suppose that hh₀, h₀i_D⁺

A

is sufficiently small and that m^tot_B is sufficiently small relatively |h0|. Then with

k⁺_A+ l_A=

1 if b−≤ b < 0 0 if b < b−

conditions on h₀, the system (12) with the boundary conditions (13),(14) under the condition (19), has a locally unique solution.

Theorem 2 is proved in Section 5.

Remark 2 If

M₀^B

√

M^B ∈ U_B⁺ then condition (17) is fulfilled, since

C M₀^B

√

M^B = 0.

Half-space problems for the Boltzmann equation are of great importance in the study of the asymptotic behavior of the solutions of boundary value problems of the Boltzmann equation for small Knudsen numbers [12],[13].

Half-space problems provide the boundary conditions for the fluid-dynamic- type equations and Knudsen-layer corrections to the solution of the fluid- dynamic-type equations in a neighborhood of the boundary. Theorem 2 tells us that the number of parameters to be specified in the boundary conditions depends on whether the condensing vapor flow is subsonic or supersonic.

This behavior has earlier been found numerically in [16] and [17] as the vapor molecules leaving the condensed phase are distributed according to the Maxwellian at the condensed phase. We can’t be sure that there is any Maxwellian at rest close enough to the Maxwellian at infinity, but if this is the case, our results are still valid. To our knowledge, this is the first rigorous analytical result of this kind and no corresponding results exist for the full Boltzmann equation.

5 Proof of the main result

We add (cf. Refs. [19] and [4]) a damping term

−γ(ΨA, ΨB) = −γ(DAP_A⁺f^A, DBP_B⁺f^B), to the right-hand side of the system (12) and obtain









 D_Adf^A

dx + L_AAf^A= −L_BAf^B+ S_AA(f^A, f^A) + S_BA(f^B, f^A) − γΨ_A DB

df^B

dx + LABf^B= SBB(f^B, f^B) + SAB(f^A, f^B) − γΨB

,

(20) where γ > 0 and Ψα= DαP_α⁺f^α, with

P_A⁺f^A=















f^A(x), y^A_pA

D_A

y_p^A_A, y^A_pA

D_A

y_p^A_A if b−< b < 0 f^A(x), z^A₁

D_Aw₁^Aif b = b−

0 if b < b₋

, and

P_B⁺f^B = D

f^B(x),√ M^BE

DB

D√ M^B,√

M^BE

DB

√ M^B.

We can, without loss of generality, assume that y_p^B_B=

√ M^B, since l_B = 0.

First we consider the corresponding linearized inhomogeneous system









 D_Adf^A

dx + L_AAf^A= g_A− γD_AP_A⁺f^A D_Bdf^B

dx + L_ABf^B = g_B− γD_BP_B⁺f^B

, (21)

where gα= gα(x) :R+→R^nα are given functions such that D

g_α(x) ,√ M^αE

= 0. (22)

Below, we will consider the case

b₋ < b < 0.