Nova S´erie
BOUNDARY LAYERS IN WEAK SOLUTIONS OF HYPERBOLIC CONSERVATION LAWS.
III. VANISHING RELAXATION LIMITS
K.T. Joseph and P.G. LeFloch Recommended by J.P. Dias
Abstract: This is the third part of a series concerned with boundary layers in solutions of nonlinear hyperbolic systems of conservation laws. We consider here self- similar solutions of the Riemann problem, following a pioneering idea by Dafermos.
The system under study is strictly hyperbolic but no assumption of genuine nonlinearity is made. The boundary is possibly characteristic, that the sign of the characteristic speed near the boundary is not known a priori. We investigate the effect of vanishing relaxation terms on the solutions of the Riemann problem. We show that the boundary Riemann problem with relaxation admits continuous solutions that remain uniformly bounded in the total variation norm. Following the second part of this series, we derive the necessary uniform estimates near the boundary which allow us to describe the structure of the boundary layer even when the boundary is characteristic. Our analysis provides still a new approach to the existence of Riemann solutions for systems of conservation laws.
1 – Introduction
We continue our investigation [6, 7, 8] of the boundary and initial value prob- lem for nonlinear hyperbolic systems of conservation laws
(1.1) ∂tu+∂xf(u) = 0, u=u(x, t)∈ B(u∗, δ0) ,
whereB(∗, δ0)⊂RN is the open ball with centeru∗ and (small) radiusδ0, and the
Received: February 4, 2002.
Mathematics Subject Classification: Primary35L65; Secondary76L05.
Keywords and Phrases: conservation law; shock wave; boundary layer; vanishing relaxation method; self-similar solution.
flux-functionf: B(u∗, δ0) →RN is a smooth mapping such that A(u) : =Df(u) admitsN real and distinct eigenvalues denoted by
λ1(u)< ... < λN(u) ,
and corresponding basis of left- and right-eigenvectorslj(u) andrj(u), 1≤j ≤N. It is well-known that weak solutions of (1.1) are not uniquely determined by their boundary and initial data. Parts I and II of this series were concerned with the selection of admissible solutions via the vanishing viscosity method. Here, we aim at constructing weak solutions by the zero-relaxation method. Mathe- matical studies of the effect of relaxation on discontinuous solutions of nonlinear hyperbolic equations go back to the works of Liu [14] and Jin and Xin [9], in particular. See also the review by Natalini [16]. For general properties of systems of conservation laws we refer to the monographs [10, 11, 20].
Given a constant a >0 such that
(1.2) −a < λ1(u)< ... < λN(u)< a , u∈ B(u∗, δ0) , we consider the relaxation approximation associated with (1.1) (1.3)
∂tuε+∂xvε= 0 ,
∂tvε+a2∂xuε= 1 ε
³f(uε)−vε´,
whereuε=uε(x, t) andvε=vε(x, t) are the unknowns andε >0 (the relaxation) is a parameter tending to zero. As in [8], we restrict attention to self-similar solutions, that is, solutions depending on the variableξ=x/tonly:
(1.4a) −ξ uε0+vε0= 0 ,
−ξ vε0+a2uε0 = 1 ε
³f(uε)−vε´ .
We search for a smooth solution (uε, vε) defined on a bounded interval [b, c] and satisfying the boundary conditions
(1.4b) uε(b) =uL, uε(c) =uR ,
whereuL and uR are given inB(u∗, δ0), andb andc are chosen such that
−a < b < c < a , sup
u∈B(u∗,δ0)
λN(u) < c .
The first condition is fundamental for the point of view of linear stability of the relaxation approximation. We stress that no inequality is imposed between b and the eigenvalues λj, so that the boundary ξ = b may be characteristic.
On the other hand, for simplicity in the presentation and without loss of gener- ality, we assume that the boundaryξ =cis not characteristic. Our purpose is to extend the analysis in [8] (concerned with the vanishing viscosity method) to the relaxation approximation, which introduces new technical difficulties.
First of all, we prove in this paper that the boundary-value problem (1.4) admits a smooth solution (uε, vε) which is of uniformly bounded total variation.
Our analysis here generalizes previous works on self-similar, vanishing viscosity approximations by Dafermos [1], Dafermos and DiPerna [2], Fan [4], Fan and Slemrod [5], LeFloch and Rohde [12], LeFloch and Tzavaras [13], Slemrod [17, 18], Slemrod and Tzavaras [19], Tzavaras [21], and the authors in [8].
Next, the limiting behavior of uε, asεgoes to zero, is investigated by distin- guishing between three different regimes:
(i) There is no effect due to the boundary when
(1.5a) b <infλ1 .
(ii) There is some effect due to the boundary, and the boundary may be characteristic, when there exits an integerpsuch that
(1.5b) infλp< b <supλp .
(iii) There is some effect of the boundary but the boundary ξ = b is not characteristic when
(1.5c) supλp−1< b <infλp(u) .
We will see that, when b satisfies (1.5a) the limit of uε solves the standard Riemann problem associated with the data (1.4b). In the cases (1.5b) and (1.5c), the boundary conditionu(b) =uLis not satisfied(in general) by the limit-function usince aboundary layerarises nearξ=b. In fact, we show that the limit-function
u(ξ) = lim
ε→0uε(ξ)
satisfies theboundary Riemann problemin the interval [b, c]
(1.6a) −ξ u0+f(u)0 = 0,
(1.6b) u(c) =uR ,
(1.6c) u(b+)∈ E(uL) ,
where the boundary set E(uL) is determined from the boundary data uL. We recall that the initial and boundary value problem for the nonlinear hy- perbolic equation (1.6a) is usually not well-posed when the boundary data are required in the (strong) sense u(b+) =uL. This latter condition must be weak- ened, as was pointed out by Dubois and LeFloch [3]. We will also rely here on the technique developed in [7] for vanishing viscosity limits, to rigorously derive the boundary setE(uB) and to describe its local structure.
We conclude this introduction with the basic reduction which allows us to reduce the first-order system of 2N equations (1.4) to a second-order system of N equations. Taking derivatives with respect to ξ in both equations (1.4a) we find
−ξ u0u00−u0u0+v00= 0 ,
−ξ v00−v0+a2u00 = 1 ε
³Df(u)u0−v0´. Eliminatingv we obtain a single equation for u
−ξ(ξ u00+u0)−ξ u0+a2u00= 1 ε
³Df(u)u0−ξ u0´, which can be rewritten in the form
(1.7) ε(a2−ξ2)u00−³Df(u) + (2ε−1)ξ´u0 = 0.
We will search for a solutionuεof (1.7), defined on the interval [b, c] and satisfying the boundary conditions (1.4b). The functionvε is recovered from uε thanks to the relation
(1.8) vε= ε(a2−ξ2)uε0+f(uε),
which follows from (1.2a) by computing vε0= ξ uε0 from the first equation and substituting in the second one.
2 – Scalar Conservation Laws
In this section we consider the scalar case f: R1 →R1. The equations (1.7) becomes
(2.1) ε(a2−ξ2)uε00−³f0(uε) + (2ε−1)ξ´uε0 = 0 on [b, c] where−a < b < c < awith boundary conditions (2.2) uε(b) =uL, uε(c) =uR .
We assume thatuL and uR satisfy
(2.3) f0(uL)∈[b, c], f0(uR)∈[b, c].
To solve (2.1) with the boundary conditions (2.2) we reformulate the problem in an integral form. Precisely, we rewrite (2.1) as
ε uε00= f0(uε) + (2ε−1)ξ (a2−ξ2) uε0 . Setting
(2.4) gε(ξ) =
Z ξ α
(1−2ε)ξ−f0(uε) (a2−ξ2) ds
for some givenα in [b, c], we can integrate the above equation and get (2.5) uε(ξ)0 = (uR−uL) e−gε(ξ)ε
Z c
b e−gεε(s)ds .
In integrating (2.5) once and using the boundary conditions (2.2), we find
(2.6) uε(ξ) = uL+ (uR−uL) Z ξ
b
e−gkε(s) ds Z c
b e−gε(s)ε ds .
Solving the integral equation (2.6) is equivalent to finding a fixed point of the map
(2.7) F(u) : = uL+ (uR−uL) Z ξ
b
e−gkε(s) ds Z c
b
e−gε(s)ε ds
withg given by (2.4). LetK be the set of all continuous functions on [b, c] which take values in the interval [min(uL, uR),max(uL, uR)]. This set is a bounded closed and convex subset of the Banach space C[b, c] of all continuous functions on [b, c] endowed with the uniform topology. It is clear that F maps K into K because the right-hand side of (2.7) is a convex combination ofuL and uR. Let us also show that the mapF: K →K is compact. Let un be a sequence in K.
Let
(2.8a) F(uεn)(ξ) : = uL+ (uR−uL) Z ξ
b e−gn−1(ε s) ds Z c
b e−gn−ε1(s)ds
where
(2.8b) gnε(ξ) : = Z ξ
α
(1−2ε)s−f0(un) (a2−s2) ds . Since
(2.9) F(uεn)(ξ)∈hmin(uL, uR),max(uL, uR)i from (2.8b) it follows that
|gnε(ξ)| ≤ (1−2ε)a+a
a2−a2∗ 2a ≤ 4a2 a2−a2∗ ,
where 0 < a∗ < a is a constant such that f0(uL), f0(uR) ∈[−a∗, a∗]. Using the above estimate together with
F(uεn)0(ξ) = (uR−uL) e−gn−1(ε ξ) Z c
b
e−gn−1(ε s)ds we have
(2.10) |F(uεn)0(ξ)| ≤ |uB−uL| (b−c) e
8a2 ε(a2−a2
∗) .
Hence, for each fixedε >0 (2.9) and (2.10) provide us with uniform estimates for uεnand its derivatives. By Ascoli Theorem, the sequenceF(un) is compact. Now, by Schauder’s fixed point theorem there must exits u ∈K such that F(u) =u.
This completes the existence of a solution to the equation (2.6). Furthermore, this solution is twice continuously differentiable iff(u) is anduεsatisfy the estimates (2.11a) uε(ξ)∈hmin(uL, uR),max(uL, uR)i,
Z c
b |uε(ξ)0|ds≤ |uR−uL|. Using (2.11a) invε0=ξ uε0 we get
(2.11b)
Z c
b |vε(ξ)0|ds ≤ b|uR−uL|. Additionally, by (1.8) we find
(2.12) vε−f(uε) = ε(a2−ξ2)uε0 .
This completes the proof of existence of the solution (uε, vε) to the problem for (1.2), together wih the uniform total variation estimates.
Now, to study the singular limitε→0 we proceed the following way. Because of the estimate (2.11a) the right-hand side of (2.12) tends to 0 inL1. So, it follows that (f(uε)−vε) → 0 as ε → 0 in L1 and, hence, almost everywhere in (b, c) along a subsequence. But, by the estimates (2.11),uε is compact and there is a subsequence which converges almost everywhere to a functionu. It follows that, along a subsequence,vε converges and it limits coincides with f(u). In fact, by (2.11a) and (2.12),
|vε−f(uε)|L1[b,c] ≤ C ε . Then, from the first equation in (1.4a) we get
−ξ u0+f(u)0 = 0
in the sense of distributions in (b, c). Furthermore, the limitusatisfies the entropy condition
−ξ p(u)ξ+q(u)ξ ≤ 0
for all entropy pairs (p(u), q(u)) with p(u) convex. This follows on passing to the limit in
(2ε−1)ξ p(uε)ξ+q(uε)ξ ≤ ε(a2−x2)p(uε)ξξ .
With regard to the boundary condition for u, we distinguish between sereval cases. When b < λm = min(f0(uL), f0(uR)), u satisfies the boundary conditions (2.2). In fact, we even have the property
(2.13) u(ξ) =
(uL, ξ < λm, uR, ξ > λM .
To prove this, consider some small δ > 0. It is easy to see (see Theorem 3.1, estimate (3.16b) below) that, in the regionξ < λm−δ,
(2.14a) |uε(ξ)−uL| ≤ |uR−uL|C ε
Z ξ
b e−(x2εa−λm)22 dx
≤ |uR−uL|C
ε (c−b)e−δ
2 2εa2 . Similarly, forξ > λM+δ,λM= max(f0(uL), f0(uR)),
(2.14b)
|uε(ξ)−uR| ≤ |uR−uL|C ε
Z c
ξ
e−(x2εa−λM)22 dx
≤ |uR−uL|C
ε (c−b)e−δ
2 2εa2 .
From the estimate (2.14) it follows thatuεconverges uniformly outside the inter- val [λm−δ, λM+δ] to the function given by (2.13), for each δ >0. This proves (2.13). So, the limit functionu can be extended by continuity to the left of λm and coincides with uL and to the right of λM with uR. We arrive at a weak solution to the Riemann problem with left- and right-hand initial datauL and uR, respectively.
We now treat the case λm < b < λM. In this case, the boundary condition u(b) =uL is generally not satisfied and, in the passage to the limit, a boundary layer is formed and the admissible boundary value belongs to a boundary set defined from the boundary layer. The corresponding ODE will be rigorously derived later in section 4, for general systems. Here, we content ourselves with a leading-order perturbation argument. Introduce the new variabley = ξ−bε and setVε(y) =uε(b+ε y) for 0≤y≤ c−εb. From (2.1) we get
(2.15) ε³a2−(ε y+b)2´Vε00−³f0(Vε) + (2ε−1) (b+ε y)´Vε0 = 0 . Expanding in the formVε =V +o(1) and keeping higher-order terms only, we get fory >0
(2.16t) (a2−b2)V00= f(V)0−b V0 .
Sinceuε is of uniformly bounded variation, so isV. Thus, there existV0 andV∞ such thatV(∞) =V∞ and V(0+) =V0. It can be seen from (2.6) that V0 =uL. Integrating (2.16) fromy to∞we arrive at the equation for the boundary layer:
(2.17) (a2−b2)V0=f(V)−f(V∞)−b V +b V∞, y >0, V(0) =uL, V(∞) =V∞ .
Consider the special case when the flux f(u) is genuinely nonlinear, in other wordsf(u) is strictly convex. Letf∗(u) be the convex dual off. Letu∗ =f∗0(b).
GivenuL, letu∗Lbe the unique solution of
f(u)−b u = f(uL)−b uL ,
which is not equal touL itself. A straightforward application of Theorem 4.1 in [7] shows that the set of all statesV∞ for which (2.17) has a solution is the set (2.18) E˜(uL) =
((−∞, u∗L)∪ {uL}, uL> u∗, (−∞, u∗], uL≤u∗ .
We know from [7] and the references therein that, for convex conservation laws, the problem (1.6) together with the boundary setE(uL) = ˜E(uL)∪ {u∗L}, is well posed. A more careful derivation of the boundary layer (carried out in section 4) would show that the boundary value of the limit namelyu(b+) satisfies
f(V∞)−b V∞ = f³u(b+)´−b u(b+),
which shows that indeed the traceu(b+) belongs to the set E(uL).
3 – Wave Interaction Estimates
In this section, we study a linearized version of the system of equations (1.7).
Given datauLand uR∈ B(u∗, δ) for someδ < δ0, the unknown functionuεtakes its values in the ball B(u∗, C∗δ) with C∗δ < δ0. For δ0 sufficiently small the eigenvalues ofDf(u) are separated, in the sense that
(3.1) −a < λm1 < λ1(u)< λM1 < λm2 <· · ·< λMN−1< λmN < λN(u)< λMN < a , u∈ B(u∗, δ0) .
Since Df(u) depends smoothly upon u, one can ensure thatλMk −λmk =O(δ0).
GivenuL, uR∈ B(u∗, δ) for some δ < δ0, we are going to construct a solutionuε of (1.7) having uniformly bounded variation, i.e.,
(3.2) T V(uε) : =
Z c
b |uε0(ξ)|dξ ≤ C .
This is done in several steps by dealing, in this section, with a linearized version of (1.7) and, then in Section 4, with the fully nonlinear problem.
The second-order equation for u=uε: [b, c]→RN is (3.3a) ε u00 = Df(u) + (2ε−1)ξ
a2−ξ2 u0 and the boundary conditions read
(3.3b) u(b) =uL, u(c) =uR .
On the other hand, recall thatvε(appearing in (1.4)) is recovered from (1.8) and that a uniform bound onT V(vε) will be a direct consequence of (3.2) and
v0 = ξ u0.
We aim at proving the existence of the solution uε of (3.3), taking values in B(u∗, C∗δ) (withC∗δ < δ0) and satisfying the estimate (3.2). Following Tzavaras [21] we set
(3.4) uε0(ξ) =
N
X
k=1
aεk(ξ)rk(uε(ξ)),
where the “wave strengths”aεk are determined by aεk(ξ) =lk(uε(ξ))·uε0(ξ).
From (3.3a) and (3.4) we deduce that
N
X
k=1
³λk(uε)−(1−2ε)ξ´
a2−ξ2 aεkrk(uε) = ε Ã N
X
k=1
aεkrk(uε)
!0
= ε
N
X
k=1
aεk0rk(uε) +ε
N
X
j,k=1
aεjaεkDrk(uε)·rj(uε) . (3.5)
Multiplying (3.5) bylk(uε) (k= 1, ..., N) successively and setting (3.6) βijk(uε) : = lk(uε)·Dri(uε)·rj(uε) , we find
(3.7) aεk0+(1−2ε)ξ−λk(uε) ε(a2−ξ2) aεk =
N
X
i,j=1
βijk(uε)aεi aεj, k= 1, ..., N . The boundary conditions (3.3b) yield
N
X
k=1
Z c b
aεkrk(uε)dξ = uR−uL .
The uniform BV bound (3.2) onuε is equivalent to the uniformL1 bound
N
X
k=1
Z c
b |aεk|dξ ≤ C .
The relations (3.6)–(3.7) form a first-order system of coupled, ordinary differen- tial equations. The functionuε arising in the coefficients λk(uε) and βijk(uε) is determined implicitly by (3.4) and (3.3b), namely
(3.8) uε(ξ) = uL+
N
X
k=1
Z ξ
b aεk(x)rk(uε(x))dx .
We start by studying a set of decoupled, linearized homogeneous equations.
Consider the equation
(3.9) ϕεk0+(1−2ε)ξ−λk(w)
ε(a2−ξ2) ϕεk = 0
for k= 1, ..., N, where w: [0,∞)→ B(u∗, δ0) is a given, continuous function.
It admits a unique (positive) solution with “unit mass”, i.e., (3.10)
Z c
b ϕεk(x)dx = 1 , namely
(3.11) ϕεk(ξ) = e−hkε(ξ) Z c
b
e−hkε(x)dx
, hk(ξ) = Z ξ
ρk
(1−2ε)x−λk(w(x)) a2−x2 dx withρk∈[b, c] still to be determined. Now, hk can be written in a more conve- nient form:
(3.12)
hk(ξ) = Z ξ
ρk
Ã(1−2ε)x−λk(w(x)) a2−ξ2
! dx
= Z ξ
ρk
−2ε x a2−x2 dx +
Z ξ ρk
x−λk(w(x)) a2−x2 dx
= εloga2−ξ2 a2−ρ2k +
Z ξ ρk
x−λk(w(x)) a2−x2 dx . Using (3.12) in (3.11) we get
(3.13)
ϕεk(ξ) = (a2−ξ2)−1e−gkε(ξ)
Ikε ,
Ikε= Z c
b
(a2−x2)−1e−gkε(x)dx , gk(ξ) = Z ξ
ρk
Ã(x−λk(w(x)) a2−x2
! dx . When emphasis will be needed, we write explicitlyϕεk=ϕεk(ξ;w) andgk=gk(ξ;w).
Observe thatϕεk does not depend on the scalarρk. It will be convenient to choose ρk∈[b, c] to be any point achieving aglobal minimumof gk, i.e.,
gk(ρk) = min
[b,c]gk . Sincew is continuous, whenρk∈(b, c] we have
(3.14) gk(ξ)≥0 for all ξ , gk(ρk) = 0, gk0(ρk) = 0.
However,ρkmay also be the boundary pointρk=b, butρk < cas can be checked from our non-characteristic assumption supλN < c.
Observe that the behavior at ξ =bdepends on the position of bwith respect to the eigenvaluesλj. For instance, ifb < λm1 then we haveρk> b. In general, we can define
(3.15) p(b) = minnk / b < λMk o.
Ifp(b)≥1,ρk=bfor allk < p(b) butρkis bounded away frombfor allk > p(b).
The characteristic case k=p(c) with λmp(c)≤b < λMp(c), for which we may have ρk=b orρk > b, will require careful estimates in the forthcoming analysis.
Givenb, cit is convenient to choosea∗such that−a <−a∗< b < λMN < c < a∗< a.
This choice ofa∗ is useful in the proof of the main properties on the functionsϕεk and their interactions stated in the following theorem.
Theorem 3.1. Forδ0 small enough, there exists a constant C >0 indepen- dent ofεfor which the following estimates hold. Letdk=λMk −λmk >0, then for allk < p(b)
(3.16a) 0< ϕεk(ξ)≤ C
ε e−(ξ2εa−b)2(ξ+b−2λMk ), b < ξ < c , while fork=p(b)
(3.16b) 0 < ϕεp(ξ) ≤
C
ε, b < ξ < λMp , C
ε e−
(ξ−λMp )2
2εa2 , λMp < ξ < c , and for allk > p(b)
(3.16c) 0 < ϕεk(ξ) ≤
C
ε e−
(ξ−λm k)2
2εa2 , b < ξ < λmk, C
ε, λmk < ξ < λMk , C
ε e−
(ξ−λM k )2
2εa2 , λMk < ξ < c . Suppose that λk is a constant. Then, ifk≤p(b), we have
(3.17a) ϕεk(ξ) = C
√εe−(ξ2εa−b)22 (ξ+b−2λk)
and ifp(b)< k,
(3.17b) ϕεk(ξ) = C
√εe−(ξ−λk
)2 2εa2 . Set
ck=
(λMk , k ≥p(b), 0, k < p(b),
and consider thewave interactioncoefficients (k, m, n= 1,2, ..., N) (3.18) Fkmnε (ξ) : = (a2−ξ2)−1e−gkε(ξ)
Z ξ ck
(a2−x2)egkε ϕεmϕεndx . Then the following uniform estimates hold
(3.19) |Fkmnε | ≤C
N
X
j=1
ϕεj .
The termsFkmnε will arise in estimating the coupling terms in the right-hand side of (3.7). Theorem 3.1 implies that, roughly speaking, the limiting measure
¯
ϕk: = limε→0ϕεk is supported in the interval spanned by the k-wave speed:
supp ¯ϕk⊂ {0} for all k < p(b) , (3.20a)
supp ¯ϕp(c)⊂[0, λMk ] for k=p(b) , (3.20b)
supp ¯ϕk⊂[λmk, λMk ] for all k > p(b). (3.20c)
In particular, for k < p(b), ¯ϕk either is a Dirac measure supported at ξ =b, or else vanishes identically.
Proof of Theorem 3.1: For simplicity we omit the explicit dependence in εthroughout this proof. We will first derive (3.16) in the casek < p(c). First we get a lower bound for the integral
(3.21)
Ik : = Z c
b
(a2−x2)−1e−gkε(x) dx
= √ ε
Z c−ρk√ ε b−ρk√ ε
³a2−(ρk+η√
ε)2´−1e−gk(ρk+η
√ε)
ε dη .
Sincek < p(c) we have ρk=b, using the change of variablex=ρk+√
ε τ we get gk(ρk+η√ε)
ε = 1
ε
Z b+η√ε
b
µx−λk(w(x)) a2−x2
¶ dx
= Z η
0
Ãτ+ √1ε³b−λk(w(b+√ ε τ))´ a2−(b+√
ε τ)2
! dτ . Sinceb > λMk and we are interested in η≥0
gk(ρk+η√ ε) ≤
Z η 0
τ +√1ε(b−λmk) a2−a2∗ dτ
≤ η2
2(a2−a2∗) + η
√ε a2−a2∗ (b−λmk) . Using this in (3.21) we get,
(3.22)
Ik ≥
√ε (a2−a2∗)
Z c√−εb
0
e−
η2 2(a2−a2
∗)−√ε(ab−2λm−ka2
∗)η
dη
= ε
a2−a2∗ Z c−εb
0 e−
εη2 2(a2−a2
∗)−(ab−2λmk
−a2
∗)η
dη
≥ ε Z c−b
0
e−
η2 2(a2−a2
∗)−ab−2λmk
−a2
∗η
dη
= C ε , asεis small. Sinceξ > b, (3.23)
gk(ξ) = Z ξ
b
µx−λk(w(x))) a2−x2
¶ dx≥
Z ξ b
µx−λMk a2
¶
dx = (ξ−b)
2a2 (ξ+b−2λMk ). The estimate (3.16a) now follows from(3.21) and (3.22).
Consider next the casek≥p(c), for which eitherρk>0 ifk > por elseρk≥0 ifk=p. When ρk = 0, the same proof as above yieldsIk ≥Cε. When ρk >0, we havegk0(ρk) = 0 and thus ρk−λk(w(ρk)) = 0. So we obtain
gk(ρk+η√ ε)
ε = 1
ε
Z ρk+η√ ε ρk
µx−ρk+ρk−λk(v(x)) a2−x2
¶ dx . Now ifη≥0,
gk(ρk+η√ε)
ε ≤ 1
ε(a2−a2∗)
µ Z η√ε
0 x dx+ 1
ε(ρk−λmk)η√ ε
¶
≤ η2
2(a2−a2∗)+ η
√ε a2−a2∗ dk .
Similarly, ifη≤0,
gk(ρk+η√ ε)
ε ≤ η2
2(a2−a2∗) − η
√ε a2−a2∗ dk . These lead us to the lower bound:
Ik ≥ √ ε
ÃZ 0
b−ρk√ ε
e−
η2 2(a2−a2
∗)−√ε(aη2−a2
∗)dk
dη + Z c−ρk√ε
0
e−
η2 2(a2−a2
∗)+√ε(aη2−a2
∗)dk
dη
!
= ε ÃZ 0
b−ρk ε
e−
εη2 2(a2−a2
∗)−(a2η
−a2
∗)dk
dη + Z c−ρk
ε
0
e−
εη2 2(a2−a2
∗)+ η
(a2−a2
∗)dk
dη
!
(3.24)
≥ ε ÃZ 0
b−ρk
e−
η2 2(a2−a2
∗)−(a2η
−a2
∗)dk
dη + Z c−ρk
0
e−
η2 2(a2−a2
∗)+ η
(a2−a2
∗)dk
dη
!
= C ε .
Since 0< e−gkε(ξ) ≤1, the estimate ϕk≤C/ε in (3.16b)–(3.16c) is established.
On the other hand, for ξ≥λMk we have (3.25a) gk(ξ) =gk(λMk ) +
Z ξ λMk
(x−λk) a2−x2 dx≥
Z ξ λMk
³x−λMk
a2 )dx= (ξ−λMk )2 2a2 . Combining (3.24) with (3.25a), the estimates (3.16b)–(3.16c) in the regionξ ≥λMk are proven. Finally forρk> b andb < ξ ≤λmk a similar argument shows that
(3.25b) gk(ξ) ≥
Z ξ
λmk
(x−λmk)
a2 dx = (ξ−λmk)2 2a2 .
This leads us to the estimates (3.16b)–(3.16c) in the region b < ξ ≤ λmk. The proof of (3.16) is completed.
When λk is a constant a direct calculation gives the estimate (3.17). In fact we have a better lower estimate forIkε, namely
Ikε ≥ C√ ε .
In the rest of this proof we will often use the lower bound Ik≥C ε. We will also need the upper bound forIk. An easy direct calculation shows that
(3.26) Ik ≤ 1
2alog
·a+c a+b.a−b
a−c
¸ .
We now estimate the interaction coefficients Fkmn given by (3.18). First, suppose that at least one ofmorncoincide withk, for instance n=k. Then we find
(3.27)
Fkmk(ξ) = (a2−ξ2)−1e−gkε(ξ) Z ξ
ck
(a2−x2)egkε ϕmϕkdx
= ϕk Z ξ
ck
ϕmdx ≤ ϕk(ξ) .
To estimate Fkmn when bothm and nare not equal tok, we observe that
(3.28) |Fkmn| ≤ 1
2(Fkm+Fkn) , where for allk
(3.29) Fkj(ξ) =
(a2−ξ2)−1e−gkε(ξ) Z ξ
ck
(a2−x2)egkε ϕj(x)2dx, ξ≥ck, (a2−ξ2)−1e−gkε(ξ)
Z ck
ξ (a2−x2)egkε ϕj(x)2dx, ξ≤ck , forj =m, n. So, it is sufficient to estimate now the coefficients Fkmε fork < m andk > m.
Case k < m: In the region b≤ck≤ξ we have Fkm(ξ) = (a2−ξ2)−1e
−1 ε
Rξ ρk(y−λk
a2−x2)dyZ ξ ck
(a2−x2)e
1 ε
Rx ρk(y−λk
a2−x2)dy
ϕm(x)2 dx
= 1
Im2 (a2−ξ2)−1e−ε1 Rξ
ρm(y−λm
a2−x2)dy
· Z ξ
ck
(a2−x2)−1e−1ε Rξ
x(λm−λk
a2−x2 )dy
e−1ε Rx
ρm(a2y−λm
−x2)dy
dx
≤ O(1) ε ϕm(ξ)
Z ξ ck
e−1ε Rξ
x(λm−λk
a2−x2 )dy
dx
≤ O(1) ε ϕm(ξ)
Z ξ ck
e−
1
ε(λma2m−λMk
−a2
∗
)(ξ−x)
dx
= O(1)
(λmm−λMk )ϕm(ξ) µ
1−e−
1 ε
³λmm−λM
k a2−a2
∗
´
(ξ−ck)¶ ,
where we usedck ≤x≤ξ,Ikε≥Cε and that due to the choice of ρm
Z x
ρm
(y−λm)dy ≥ 0 .
Sinceλmm−λMk >0, it follows that
(3.30) Fkm(ξ)≤O(1)ϕm(ξ) for all ξ≥ck .
Next, consider the region b < ξ < ck. If ck=b, there is nothing to prove.
Ifk≥p(b) and thusck> b, we proceed as follows. An easy calculation based on the expression (3.29) ofFkm gives
Fkm(ξ) = Ikε Imε2 ϕk(ξ)
· Z ck
ξ
(a2−x2)−1e−1ε Rρk
ρm z−λm a2−z2dz
.e−1ε Rx
ρm z−λm a2−z2 dz
.e−1ε Rx
ρk λm−λk
a2−z2 dz
dx (3.31)
≤ O(1)
ε2 ϕk(ξ)e−ε1 Rρk
ρm z−λm a2−z2dz
. Z ck
ξ
e−1ε Rx
ρk λm−λk
a2−z2 dz
dx .
Here again we used Ikε≥Cε. Now since b < ck≤λMk < λmm ≤ρm and ρk≤x≤ck, we have,
Z ρm
ρk
(y−λm)
a2−y2 dy ≤ −(λmm−λMk )2 2a2 ,
− Z x
ρk
(λm−λk)
a2−y2 dy ≤ (λMm−λmk)(λMk −λmk) a2−a2∗ . Observe finally that
βkm : = −(λmm−λMk )2
2a2 + (λMk −λmk)(λMk −λmk)
a2−a2∗ = −(λmm−λMk )2
2a2 +O(δ0) < 0 . Using this and the fact thatck−ξ≤c−bin (3.31) it follows that
(3.32) Fkm(ξ) ≤ C O(1)
ε2 ϕk(ξ)eβkmε ≤ o(1)ϕk(ξ) for all ξ ≤ck . Combining (3.30) and (3.32) we get
(3.33) Fkm(ξ)≤O(1)hϕk(ξ) +ϕm(ξ)i for all b≤ξ ≤c .
