Multiple Vortices for the Shallow Water Equation

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by  Daomin Cao

Multiple Vortices for the Shallow Water Equation
Daomin Cao∗ and Zhongyuan Liu†
3
1
0
Abstract
2

n
In this paper, we construct stationary classical solutions of the shallow water equa-
a
J tion with vanishing Froude number Fr in the so-called lake model. To this end we

8 need to study solutions to the following semilinear elliptic problem
2
−ε2div( u)= b(u−qlog 1)p, in Ω,
] ∇b ε +
P
(u= 0, on ∂Ω,
A
.
th for small ε> 0, where p > 1, div ∇bq = 0 and Ω ⊂ R2 is a smooth bounded domain,.
a
m We showed that if qb2 has m(cid:16) str(cid:17)ictly local minimum(maximum) points z¯i, i =
1,··· ,m, then there is a stationary classical solution approximating stationary m
[
pointsvortexsolutionofshallowwaterequationswithvorticity m 2πq(z¯i). Moreover,
1 i=1 b(z¯i)
0v strictly local minimum points of qb2 on the boundary can also gPive vortex solutions for
the shallow water equation. As a further study we construct vortex pair solutions as
2
4 well.
6 Existence and asymptotic behavior of single point non-vanishing vortex solutions
.
1 were studied by S. De Valeriola and J. Van Schaftingen in [9].
0
3
Mathematics Subject Classification: 35J20, 35J40, 35J60, 35R45.
1
: Keywords: Shallow water equation; Free boundary; Stream function; Vortex.
v
i
X
r 1 Introduction and Main Results
a
We consider fluid contained in a basin by a uniform gravitational acceleration g and fixed
vertical lateral boundaries(i.e. no sloping beaches). Suppose that (x,y) is horizontal spatial
coordinate which is confined to a fixed bounded domain Ω with boundary ∂Ω.The vertical
coordinate is chosen so that the mean height of the fluid’s free upper surface is at z = 0. Let
z = −b(x,y) give the fixed bottom topography, so b is a strict positive function over Ω. Let
z = h(x,y) be the free upper surface. We assume that both b and ∂Ω vary over distances L
which are large compared to the mean depth B, that is, the ratio δ = B is small.
L
∗[email protected], Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing
100190,P.R. China
†[email protected],AcademyofMathematicsandSystemsScience,ChineseAcademyofSciences,Beijing
100190,P.R. China
1
Let uandw denote thehorizontal andvertical components respectively of the fluidveloc-
ity. We will consider only those motion for which u,w and h each vary in (x,y)over distances
L, in other words, we will make the long-wave approximation. The ”Froude number” is de-
noted as Fr = U , where U is the characteristic magnitude of u. We will consider the case
√gB
of small ”Froude number” Fr and h is small compared to B. In such cases, from [1, 3, 4, 15],
the leading-order evolution of u(x,y,t) and h(x,y,t) will be governed by equations that have
the non-dimensional form
∂ v+(v·∇)v = −∇h,
t
(1.1)
(div(bv) = 0,
where ∇ is the horizontal gradient. Since these equations apply to a domain which is shallow
compared to its width and whose free surface exhibits negligible surface motion, they are
called the ’lake’ equations(see [4], for instance).
The first equation in (1.1) can be rewritten in terms of the vorticity ω = ∇×v as
|v|2
∂ v+ω ×v = −∇ +h .
t
2
(cid:18) (cid:19)
This model is analogous to the two-dimensional Euler equation for an idea incompressible
fluid and has been recently studied by many authors. For instance, see [1, 3, 4, 15] and the
references cited therein.
Recently, De Valeriola and Van Schaftingen [9] studied the desingularization of vortices
for (1.1) with stream function method, which consists in observing that if ψ satisfies
∇ψ
−div = bf(ψ)
b
(cid:18) (cid:19)
for f ∈ C1(R), then v = curlψ and h = −F(ψ)−|v|2 with F(s) = sf(s)ds forma stationary
b 2 0
solution to the shallow water equation. Moreover, the velocity v is irrotational on the set
R
where f(ψ) = 0. It is easy to see that if ψ satisfies div ψ0 = 0, then v = curlψ0 is an
0 ∇b 0 b
irrotaional stationary solution of (1.1). In [9], they studied the asymptotics of solutions of
(cid:0) (cid:1)
−ε2div( ψ) = bψp, in Ω,
∇b + (1.2)
(ψ = ψ0ln 1ε, on ∂Ω,
where p > 1, Ω ⊂ R2 is smooth bounded domain.
To obtain their results, De Valeriola and Van Schaftingen investigated the following
problem
−ε2div( u) = b(u−q )p, in Ω,
∇b ε + (1.3)
(u = 0, on ∂Ω,
where p > 1, q = −ψ , q = qln 1, Ω ⊂ R2 is smooth bounded domain.
0 ε ε
More precisely, they first obtained the existence of solutions by using mountain pass
lemma and studied the asymptotic behavior of solutions by giving exact estimates to the
upperandlowerenergyboundsoftheleastenergysolutions. Asaconsequence, theyobtained
2
that the “vortex core” shrinks to a point x which is the minimum point of q2. However, it
0 b
is hard to apply their method to construct multiple vortices for (1.1).
Motivated by [9], our goal in this paper is to construct multiple stationary vortices for
shallow water equations. More specifically, we want to find some highenergy solutions whose
“vortex core” consists of multiple components which shrink to several distinct points in Ω¯
ψ2
as ε → 0 under some additional assumptions on 0.
b
Our main results in this paper can be stated as follows:
Theorem 1.1. Suppose that Ω ⊂ R2 is a smooth bounded domain. Let b ∈ C1(Ω¯),ψ ∈
0
C2(Ω¯) be such that div( ψ0) = 0 and let v = curlψ . If inf b > 0 and sup ψ < 0, then for
∇b 0 0 Ω Ω 0
ψ2
any given strictly local minimum(maximum) points z¯ ,··· ,z¯ of 0, there exists ε > 0, such
1 m b 0
that for each ε ∈ (0,ε ), there exists a family of solutions v ∈ C1(Ω,R2) and h ∈ C1(Ω) of
0 ε ε
div(bv ) = 0, in Ω,
ε
(v ·∇)v = −∇h , in Ω,
 ε ε ε
v ·n = v ·nln 1, on ∂Ω,
ε 0 ε
where n is the unit outward normal. Furthermore the corresponding vorticity ω := curlv
ε ε
satisfying
suppω ⊂ ∪m B(z ,Cε) forz ∈ Ω, i = 1,··· ,m
ε i=1 i,ε i,ε
and as ε → 0,
m
2πψ (z¯)
ω → − 0 i ,
ε
b(z¯)
ZΩ i=1 i
X
(z ,··· ,z ) → (z¯ ,··· ,z¯ ).
1,ε m,ε 1 m
ψ2
The next result shows that strictly local minimum points of 0 on the boundary ∂Ω can
b
also give vortex solutions for (1.1).
Theorem 1.2. Suppose that Ω ⊂ R2 is a smooth bounded domain. Let b ∈ C1(Ω¯),ψ ∈
0
C2(Ω¯) be such that div( ψ0) = 0 and let v = curlψ . If inf b > 0 and sup ψ < 0, then,
∇b 0 0 Ω Ω 0
ψ2
for any given strictly local minimum points zˆ ,··· ,zˆ of 0 on the boundary ∂Ω, there exists
1 n b
ε > 0, such that for each ε ∈ (0,ε ), there exists a family of solutions v ∈ C1(Ω,R2) and
0 0 ε
h ∈ C1(Ω) of
ε
div(bv ) = 0, in Ω,
ε
(v ·∇)v = −∇h , in Ω,
 ε ε ε
v ·n = v ·nln 1, on ∂Ω,
ε 0 ε
where n is the unit outward normal. The corresponding vorticity ωε := curlvε satisfying
suppω ⊂ ∪n B(z ,Cε) forz ∈ Ω, i = 1,··· ,n
ε i=1 i,ε i,ε
and as ε → 0,
n
2πψ (zˆ)
ω → − 0 i .
ε
b(zˆ)
ZΩ i=1 i
X
3
Moreover,
ln|lnε| 1
|z −zˆ| ≤ C , dist(z ,∂Ω) ≥ ,
i,ε i |lnε| i,ε |lnε|α
(cid:18) (cid:19)
where α is a positive constant.
ψ2
Remark 1.3. If 0 has strictly local minimum points in Ω and on the boundary ∂Ω, then
b
there is, from Theorem 1.1 and 1.2, a stationary solution of the shallow water equation such
that its vorticity set shrinks to corresponding strictly local minimum points .
It is worthwhile to pointing out that although the structure of shallow water equations is
very analogous to that of two dimensional Euler equations for an ideal incompressible fluid,
the position of vortex for (1.1) exhibits a striking difference with that of the Euler equations.
The position of vortex for Euler equation is closely related to Kirchhoff-Routh function. The
interested reader can refer to [5, 6, 16, 14] for more results on this problem.
Theorem 1.1 and Theorem 1.2 are proved via the following results concerning problem
(1.3):
Theorem 1.4. Suppose that Ω ⊂ R2 is a smooth bounded domain. Let b ∈ C1(Ω¯), q ∈
C2(Ω¯), inf b > 0 and inf q > 0. Then, for any given strictly local minimum(maximum)
Ω Ω
points z¯ ,··· ,z¯ of q2, there exists ε > 0, such that for each ε ∈ (0,ε ), (1.3) has a solution
1 m b 0 0
u , such that the set Ω = {x : u −qln 1 > 0} has exactly m components Ω , i = 1,··· ,m
ε ε ε ε ε,i
and as ε → 0, each Ω shrinks to the point z¯.
ε,i i
Theorem 1.5. Suppose that Ω ⊂ R2 is a smooth bounded domain. Let b ∈ C1(Ω¯), q ∈
C2(Ω¯), inf b > 0 and inf q > 0. Then, for any given strictly local minimum points
Ω Ω
zˆ ,··· ,zˆ of q2 on the boundary ∂Ω, there exists ε > 0, such that for each ε ∈ (0,ε ),
1 n b 0 0
(1.3) has a solution u , such that the set Ω = {x : u −qln 1 > 0} has exactly m components
ε ε ε ε
Ω , i = 1,··· ,m and as ε → 0, each Ω shrinks to the point zˆ.
ε,i ε,i i
Not as in [9] where (1.3) is investigated directly, we prove Theorem 1.4 and Theorem
1−p
1.5 by considering the following equivalent problem of (1.3) instead. Set δ = ε ln 1 2 ,
ε
w = u , then (1.3) becomes
lnε (cid:0) (cid:1)
| |
−δ2div( w) = b(w −q)p, in Ω,
∇b + (1.4)
(u = 0, on ∂Ω.
We will use a reduction argument to prove Theorem 1.4 and Theorem 1.5. To this end,
we need to construct an approximate solution for (1.4). For the problem studied in this
paper, the corresponding “limit” problem in R2 has no bounded nontrivial solution. So,
we will follow the method in [7, 8] to construct an approximate solution. Since there are
two parameters δ, ε and b in problem (1.4), which causes some difficulty, we must take
this influence into careful consideration and give delicate estimates in order to perform the
reduction argument.
We will also apply the above idea and techniques to construct vortex pairs to shallow
water equations in section 5, which has never been addressed before.
4
As a final remark, our results seem connected with the work of Wei, Ye and Zhou [17,
18, 19] on the anisotropic Emden-Fowler equation
div(a(x)∇u)+ε2a(x)eu = 0, in Ω,
(u = 0, on ∂Ω.
Theyconstructed(boundary)bubblingsolutionsshowingastrikingdifferencewiththeisotropic
case(a ≡ constant). Moreover, we point out that problem (1.4) can be considered as a free
boundary problem. Similar problems have been studied extensively. The reader can refer to
[5, 6, 7, 9, 8, 11, 13, 16] for more results on this kind of problems.
This paper is organized as follows. In Section 2, we construct the approximate solution
for (1.4). We will carry out a reduction argument in Section 3 and prove the main results in
Section 4. In Section 5, we give some further results on vortex pairs for the shallow water
equations. Some basic estimates that used in sections 4 and 5 will be given in Section 6.
2 Approximate Solutions
In the section, we will construct approximate solutions for (1.4).
Let R > 0 be a large constant such that for any x ∈ Ω, Ω ⊂⊂ B (x). Consider the
R
following Dirichlet problem:
−δ2∆w = (w−a)p, in B (0),
+ R (2.1)
(w = 0, on ∂BR(0),
where a > 0 is a constant. Then, (2.1) has a unique solution W , which can be written as
δ,a
W (x) = a+δp−21s−δ p−21φ |sxδ| , |x| ≤ sδ, (2.2)
δ,a allnn|sRxδ|, (cid:0) (cid:1) sδ ≤ |x| ≤ R,
R
where φ(x) = φ(|x|) is the unique solution of

−∆φ = φp, φ > 0, φ ∈ H1 B (0) (2.3)
0 1
(cid:0) (cid:1)
and s ∈ (0,R) satisfies
δ
2 2 a
δp−1s−δ p−1φ′(1) = ln sδ ,
R
which implies
p−1
sδ |φ′(1)| 2
→ > 0, as δ → 0.
δ|lnδ|p−1 a
2 (cid:18) (cid:19)
Moreover, by Pohozaev identity, we can get that
π(p+1)
φp+1 = |φ(1)|2 and φp = 2π|φ(1)|.
′ ′
2
ZB1(0) ZB1(0)
5
For given ˆb > 0 and qˆ> 0, let V (x) be the solution of the following Dirichlet problem
δ,ˆb,qˆ
−δ2∆v =ˆb2(v −qˆ)p, in B (0),
+ R (2.4)
(v = 0, on ∂BR(0).
By scaling, from (2.1) and (2.2), we obtain
2
Vδ,ˆb,qˆ(x) =ˆb−p−21Wδ,ˆbp−21qˆ(x) = qˆl+n|xˆb|−p−21 (cid:16)sδδ(cid:17)p−1 φ |sxδ| , |x| ≤ sδ, (2.5)
qˆlnsRδ , (cid:0) (cid:1) sδ ≤ |x| ≤ R.
R
For any z ∈ Ω, define V (x) =V (x−z). Because V does not vanish on ∂Ω, we
δ,ˆb,qˆ,z δ,ˆb,qˆ δ,ˆb,qˆ
need to make a projection. Let PV be the solution of
δ,ˆb,qˆ,z
−δ2∆v =ˆb2(V −qˆ)p, in Ω,
δ,ˆb,qˆ,z + (2.6)
(v = 0, on ∂Ω,
and h(x,z) be the solution of
−∆h = 0, in Ω,
h = 1 ln 1 , on ∂Ω.
(
2π x z
| − |
Then
qˆ
PV (x) = V (x)− g(x,z), (2.7)
δ,ˆb,qˆ,z δ,ˆb,qˆ,z ln R
s
δ
where g(x,z) = lnR+2πh(x,z).
We will construct solutions for (1.4) of the following form
m
PV +ω ,
δ,ˆbj,qˆδ,j,zj δ
j=1
X
where z ∈ Ω for j = 1,··· ,m, ω is a perturbation term. To obtain a good estimate for ω ,
j δ δ
we need to choose qˆ properly.
δ,j
Denote Z = (z ,··· ,z ) ∈ R2m. In this paper, we always assume that z ∈ Ω satisfies
1 m j
|z −z | ≥ ̺L¯,dist(z ,∂Ω) ≥ ̺ > 0, or
i j j
1 (2.8)
|z −zˆ | < η, dist(z ,∂Ω) ≥ , i,j = 1,··· ,m, i 6= j
j j j |lnε|α
where ̺,η > 0 is a fixed small constant and L¯,α > 0 is a fixed large constant.
Let ˆb = b(z ) and qˆ (Z), j = 1,··· ,m be the solution of the following problem:
j j δ,j
qˆ qˆ
qˆ = q(z )+ i g(z ,z )− j G¯(z ,z ), (2.9)
i i ln R i i ln R i j
ε j=i ε
X6
6
where G¯(x,z ) = ln R −g(x,z ). It is not difficult to see that since ln R → ∞ as ε → 0 ,
j x zj j ε
| − |
(2.9) is a linear system with coefficient matrix, which is a small perturbation of a positively
definite diagonal matrix for small ε. Thus we can obtain the solution (qˆ (Z),··· ,qˆ (Z))
δ,1 δ,m
to (2.9). Moreover, we have
q(z )− qˆδ,j(Z)G¯(z ,z )
i j=i lnR i j
qˆ (Z) = 6 ε .
δ,i P1− g(zi,zi)
lnR
ε
For simplicity, for given Z = (z ,··· ,z ), in this paper, we will use qˆ instead of qˆ (Z).
1 m δ,i δ,i
Define
m
V = PV , V = V . (2.10)
δ,Z,j δ,Z δ,Z,j
δ,ˆbj,qˆδ,j,zj
j=1
X
Let s be the solution of
δ,i
2
ˆbp−1qˆ
δp−21s−p−21φ′(1) = iln sδ,i,
R
then we have
1 1 ln|lnε|
= +O .
ln R ln R |lnε|2
s ε (cid:18) (cid:19)
δ,i
Thus, we find that for x ∈ B (z ), where L > 0 is any fixed constant,
Ls i
δ,i
qˆ
δ,i
V (x)−q(x) = V (x)− g(x,z )−q(x)
δ,Z,i δ,ˆbi,qˆδ,i,zi ln R i
s
δ,i
qˆ s |Dg(z ,z )|
δ,i δ,i i i
=V (x)−q(z )− g(z ,z )+O(s )+O
δ,ˆbi,qˆδ,i,zi i ln R i i δ,i ln R !
s s
δ,i δ,i
qˆ ln|lnε|
δ,i
=V (x)−q(z )− g(z ,z )+O g(z ,z )
δ,ˆbi,qˆδ,i,zi i ln R i i |lnε|2 i i
ε (cid:18) (cid:19)
and for j 6= i and x ∈ B (z ),
Ls i
δ,i
qˆ qˆ
V (x) = V (x)− δ,j g(x,z ) = δ,j G¯(x,z )
δ,Z,j δ,ˆbj,qˆδ,j,zj ln R j ln R j
s s
δ,j δ,j
qˆ qˆ
= δ,j G¯(z ,z )+ δ,j G¯(x,z )−G¯(z ,z )
ln R i j ln R j i j
s s
δ,j δ,j
(cid:0) (cid:1)
qˆ ln|lnε|
= δ,j G¯(z ,z )+O .
ln R i j |lnε|2
ε (cid:18) (cid:19)
So, by using (2.9), we obtain
7
ln|lnε|
V (x)−q(x) = V (x)−qˆ +O g(z ,z ) , x ∈ B (z ). (2.11)
δ,Z δ,ˆbi,qˆδ,i,zi δ,i |lnε|2 i i Lsδ,i i
(cid:18) (cid:19)
We end this section by giving the following formula which can be obtained by direct
computation and will be used in the next two sections.
∂V (x)
δ,ˆbi,qˆδ,i,zi
∂z
i,h
2
ˆb−p−21 1 δ p−1 φ |x−zi| zi,h −xh +O(1), x ∈ B (z ),
i s s ′ s |x−z | sδ,i i (2.12)
 δ,i (cid:18) δ,i(cid:19) δ,i i
=  (cid:0) (cid:1)

 − qˆδ,i zi,h −xh +O(1), x ∈ Ω\B (z ).
ln R |x−z |2 sδ,i i
s i
 δ,i





3 The Reduction
Let V be given as in (2.10), we are to find solutions of the form V +ω , where ω is
δ,Z δ,Z δ,Z δ,Z
a small perturbation(obtained in Proposition 3.3). We will show that for any given Z, there
exists ω such that w = V +ω satisfies
δ,Z δ,Z δ,Z δ,Z
δ2
∇w ∇v −b(w −q)pv = 0, for anyv ∈ H1(Ω)∩W2,p(Ω)\H , (3.1)
b δ,Z δ,Z + 0 ∗
ZΩ(cid:20) (cid:21)
where H is a finite dimensional subspace of H1(Ω)∩W2,p(Ω). In the next section, we will
∗ 0
choose Z properly so that V +ω is a solution of (1.4).
δ,Z δ,Z
To show (3.1), we need to study the kernel of Lw := −δ2div(∇bw)−pb(Vδ,Z −q)p+−1w. To
do this first we need to understand the kernel of the linearized equation of
−∆w = wp, in R2. (3.2)
+
Let
φ(|x|), |x| ≤ 1,
w(x) =
(φ′(1)ln|x|, |x| > 1,
where φ is the solution of (2.3), then w ∈ C1(R2) is the unique solution of (3.2). Since
φ(1) < 0 and ln|x| is harmonic for |x| > 1. Moreover, since w is Lip-continuous, by the
′ +
Schauder estimate, w ∈ C2,α for any α ∈ (0,1).
The linearized equation of (3.2) at w is as follows
−∆v −pw+p−1v = 0, v ∈ L∞(R2). (3.3)
It is easy to see that ∂w, i = 1,2, is a solution of (3.3). Moreover, from Dancer and Yan
∂xi
[8], we know that w is also non-degenerate, in the sense that the kernel of the operator
Lv := −∆v −pw+p−1v, v ∈ D1,2(R2) is spanned by ∂∂xw1, ∂∂xw2 .
(cid:8) (cid:9)
8
Let V be the function defined in (2.10). Set
δ,Z,j
∂V
F = u : u ∈ Lp(Ω), δ,Z,ju = 0,j = 1,··· ,m, h = 1,2 ,
δ,Z
∂z
(cid:26) ZΩ j,h (cid:27)
and
∂V
E = u : u ∈ W2,p(Ω)∩H1(Ω), ∆ δ,Z,j u = 0,j = 1,··· ,m, h = 1,2 .
δ,Z 0 ∂z
(cid:26) ZΩ (cid:18) j,h (cid:19) (cid:27)
Define, for any u ∈ Lp(Ω), Q u as follows:
δ
m 2
∂V
Q u := u− c −δ2∆ δ,Z,j , (3.4)
δ j,h
∂z
Xj=1 Xh=1 (cid:18) (cid:16) j,h (cid:17)(cid:19)
where the constants c (j = 1,··· ,m, h = 1,2) are chosen to satisfy
j,h
m 2
∂V ∂V ∂V
c −δ2 ∆ δ,Z,j δ,Z,i = u δ,Z,i. (3.5)
j,h
∂z ∂z ∂z
Xj=1 Xh=1 (cid:18) ZΩ (cid:16) j,h (cid:17) i,h¯ (cid:19) ZΩ i,h¯
Since ∂Vδ,Z,jQ u = 0, the operator Q can be regarded as a projection from Lp(Ω)
Ω ∂zj,h δ δ
to F . In order to show the existence of c satisfying (3.5), we just need the following
δ,Z R j,h
estimate ( by (2.12)):
∂V ∂V
−δ2 ∆ δ,Z,j δ,Z,i
∂z ∂z
ZΩ (cid:16) j,h (cid:17) i,h¯
=pˆb2j ZΩ Vδ,ˆbj,qˆδ,j,zj −qˆδ,j p+−1 ∂Vδ∂,ˆbzjj,qˆ,hδ,j,zj − ∂∂zqˆjδ,,hj! ∂∂Vzδi,,Zh¯,i +O(cid:18)|lnεε|p+1(cid:19) (3.6)
(cid:0) (cid:1)
c ε
′
=δ +O ,
ijhh¯|lnε|p+1 |lnε|p
(cid:18) (cid:19)
where c > 0 is a constant, δ = 1, if i = j and h = h¯; otherwise, δ = 0.
′ ijhh¯ ijhh¯
Define
∇u
Lδu = −δ2div b −pb(Vδ,Z −q)p+−1u.
(cid:18) (cid:19)
For the operator Q L we have the following lemma.
δ δ
Lemma 3.1. There are constants ρ > 0 and δ > 0, such that for any δ ∈ (0,δ ], Z
0 0 0
satisfying (2.8), u ∈ E with Q L u = o(1) in Lp(Ω\∪m B (z )) for some L > 0 large,
δ,Z δ δ j=1 Lsδ,j j
then
2
kQ L uk ≥ ρ0εp kuk .
δ δ Lp(Ω) |lnε|p 1 L∞(Ω)
−
9
Proof. We will use k · k ,k · k to denote k · k and k · k respectively. We argue
p Lp(Ω) L∞(Ω)
∞
by contradiction. Suppose that there are δ → 0, Z = (z ,··· ,z ) satisfying (2.8) and
n n 1,n m,n
u ∈ E with Q L u = o(1) in Lp(Ω\∪m B (z )), such that
n δn,Zn δn δn n j=1 Lsn,j j,n
2
1 εp
n
kQ L u k ≤ ,
δn δn n p n|lnε |p 1
n −
and ku k = 1, where and in the sequel we set s = s to simplify notation.
n n,j δn,j
∞
Firstly, we estimate c corresponding to u in (3.4). By definition c satisfies:
j,h,n n j,h,n
m 2
∂V
Q L u = L u − c −δ2∆ δn,Zn,j . (3.7)
δn δn n δn n j,h,n n ∂z
j=1 h=1 (cid:18) j,h (cid:19)
XX
∂V
For each fixed i, multiplying (3.7) by δn,Zn,i, noting that
∂zi,h¯
∂V
Q L u δn,Zn,i = 0,
δn δn n ∂z
ZΩ i,h¯
(cid:0) (cid:1)
we obtain
∂V ∂V
u L δn,Zn,i = L u δn,Zn,i
n δn ∂z δn n ∂z
ZΩ (cid:18) i,h¯ (cid:19) ZΩ i,h¯
m 2 (cid:0) (cid:1)
∂V ∂V
= c −δ2∆ δn,Zn,j δn,Zn,i.
j,h,n n ∂z ∂z
j=1 h=1 ZΩ(cid:18) j,h (cid:19) i,h¯
XX
Using (2.11) and Lemma 6.1, we obtain
∂V
u L δn,Zn,i
n δn ∂z
ZΩ (cid:18) i,h¯ (cid:19)
1 ∂V ∂V
=ZΩun(cid:20)−δn2div(cid:18)ˆbi∇ ∂δzni,,Zh¯n,i(cid:19)−pˆbi(Vδn,Zn −q)p+−1 ∂δzni,,Zh¯n,i(cid:21)
1 1 ∂V
+ u δ2div − ∇ δn,Zn,i
ZΩ n(cid:20) n (cid:18)(cid:18)ˆbi b(cid:19) ∂zi,h¯ (cid:19)(cid:21)
∂V
+p un (ˆbi −b)(Vδn,Zn −q)p+−1 ∂δzn,Zn,i
ZΩ (cid:20) i,h¯ (cid:21)
ε ln2|lnε | ε2 ε2
=O n n +O n +O n
|lnε |p+1 |lnε |p 1 |lnε |p
(cid:18) n (cid:19) (cid:18) n − (cid:19) (cid:18) n (cid:19)
ε ln2|lnε |
n n
=O .
|lnε |p+1
(cid:18) n (cid:19)
Using (3.6), we find that
c = O ε ln2|lnε | .
i,h,n n n
(cid:0) (cid:1)
10