REGULARIZATION OF POINT VORTICES PAIRS FOR THE EULER
EQUATION IN DIMENSION TWO
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
Abstract. In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This
procedure is carried out by constructing solutions to the following elliptic problem

m
n
X
¡
¡
¢p
κ−
κ+
1 ¢p X
1

j
i
−ε2 ∆u =
χΩ+ u − q −
ln + −
χΩ− q −
ln − u + ,
x ∈ Ω,
i
j
2π
ε
2π
ε
i=1
j=1


u = 0,
x ∈ ∂Ω,
−
where p > 1, Ω ⊂ R2 is a bounded domain, Ω+
i and Ωj are mutually disjoint subdomains
−
of Ω and χΩ+ (resp. χΩ− ) are characteristic functions of Ω+
i (resp. Ωj ), q is a harmonic
i
j
function.
We show that if Ω is a simply-connected smooth domain, then for any given C 1 -stable
+
−
+
−
critical point of Kirchhoff-Routh function W(x+
1 , · · · , xm , x1 , · · · , xn ) with κi > 0 (i =
−
1, · · · , m) and κj > 0 (j = 1, · · · , n), there is a stationary classical solution approximating stationary
vortex solution of incompressible Euler equations with total
Pm m + nPpoints
n
−
vorticity i=1 κ+
−
κ
i
j=1 j . The case that n = 0 can be dealt with in the same way as
−
well by taking each Ωj as empty set and set χΩ− ≡ 0, κ−
j = 0.
j
1. Introduction and main results
The incompressible Euler equations
(
vt + (v · ∇)v = −∇P,
∇ · v = 0,
(1.1)
describe the evolution of the velocity v and the pressure P in an incompressible flow. In
R2 , the vorticity of the flow is defined by ω := ∇ × v = ∂1 v2 − ∂2 v1 , which satisfies the
equation
ωt + v · ∇ω = 0.
The velocity v of an incompressible fluid in two dimensions admits a stream function ψ
such that v = J∇ψ = ( ∂x∂ 2 ψ, − ∂x∂ 1 ψ), where J denotes the symplectic matrix
µ
¶
0 1
J=
.
−1 0
By the definitions, ψ is a solution of the Poisson equation −∆ψ = ω.
1
2
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
Suppose that ω is known, then the velocity v can be recovered by the following BiotSavart law
1 −Jx
v=ω ∗
.
2π |x|2
P
One special singular solution of Euler equations is given by ω = m
i=1 κi δxi (t) ( κi 6= 0 is
called the strength of the ith vortex xi ), which is related to
m
X
κi J(x − xi (t))
v=−
.
2π |x − xi (t)|2
i=1
The positions of the vortices xi : R → R2 satisfy the following Kirchhoff law
κi
dxi
= J∇xi W,
dt
where W is the so called Kirchhoff-Routh function defined by
m
1
1 X κi κj
log
.
W(x1 , · · · , xm ) =
2 i6=j 2π
|xi − xj |
For a simply connected bounded domain Ω ⊂ R2 , let vn be the normal component of
the velocity v on ∂Ω, that is vn (x) = v(x)
R · ν(x), where ν(x) is the unit outward normal
on ∂Ω at x ∈ ∂Ω. Then by ∇ · v = 0, ∂Ω vn = 0. It turns out that the Kirchhoff-Routh
function for bounded domain Ω is associated with Green function and vn . Suppose that
v0 is the unique harmonic field whose normal component on the boundary ∂Ω is vn . If Ω
is simply-connected, then v0 can be represented by v0 = J∇ψ0 , and ψ0 is determined up
to a constant by

−∆ψ0 = 0, in Ω,
(1.2)
∂ψ
− 0 = vn , on ∂Ω,
∂τ
0
where ∂ψ
denotes the tangential derivative on ∂Ω. The Kirchhoff-Routh function associ∂τ
ated to the vortex dynamics then is given by (see Lin [23])
m
m
m
X
1X
1X 2
W(x1 , · · · , xm ) =
κi κj G(xi , xj ) +
κi H(xi , xi ) +
κi ψ0 (xi ),
2 i6=j
2 i=1
i=1
(1.3)
where G is the Green function of −∆ on Ω with 0 Dirichlet boundary condition and H is
its regular part (the Robin function).
For m clockwise vortices motion (corresponding to κ+
i > 0) and n anti-clockwise vortices
−
motion (corresponding to −κj < 0), the Kirchhoff-Routh function associated to the vortex
REGULARIZATION OF POINT VORTICES
3
dynamics becomes
−
+
−
W(x+
1 , · · · , xm , x1 , · · · , xn )
m
n
1 X
1 X − −
+ +
+
+
−
κi κk G(xi , xk ) +
κj κl G(x−
=
j , xl )
2 i,k=1,i6=k
2 j,l=1,j6=l
m
n
1X + 2
1X − 2
+
+
−
+
(κ ) H(xi , xi ) +
(κ ) H(x−
j , xj )
2 i=1 i
2 j=1 j
m X
n
m
n
X
X
X
+ −
+
−
+
+
−
−
κi κj G(xi , xj ) +
κi ψ0 (xi ) −
κ−
j ψ0 (xj ).
i=1 j=1
i=1
(1.4)
j=1
It is known that critical points of the Kirchhoff-Routh function W give rise to stationary
vortex points solutions of the Euler equations(see the Kirchhoff law). As for the existence
and multiplicity of critical points of W given by (1.3), we refer to [5] and the references
therein, where the case ψ0 ≡ 0 was studied. We expect that it is possible to obtain, at least
by add assumptions on ψ0 , the multiplicity and even non-degeneracy of critical points for
W.
There exist huge literatures dealing with the stationary incompressible Euler equations,
see [1, 2, 4], [6]-[11], [16]-[18], [19, 21, 22], [28]-[31] and references therein. Roughly speaking, there are two methods to construct stationary solutions of the Euler equation, which
are the vorticity method and the stream-function method. The vorticity method was first
established by Arnold (see [3]). Benjamin [6] developed a new approach based on a variational principle for the vorticity to study the existence of vortex rings in three dimensions
which was adapted successfully by Burton [8] and Turkington [31].
The stream-function method consists in observing that if ω = λf (ψ), that is, if ψ satisfies
(
−∆ψ = λf (ψ),
x ∈ Ω,
u = ψ0 ,
x ∈ ∂Ω,
for some arbitrary function f ∈ C 1 (R), then v = J∇ψ and P = λF (ψ) − 21 |∇ψ|2 is a
Rt
stationary solution to the Euler equations, where F (t) = 0 f (s)ds. Moreover, the velocity
v is irrotational on the set where f (ψ) = 0. f is called vorticity function and λ the vortex
strength parameter.
Set q = −ψ0 and u = ψ − ψ0 , then u satisfies the following Dirichlet boundary value
problem
(
−∆u = λf (u − q),
x ∈ Ω,
(1.5)
u = 0,
x ∈ ∂Ω.
One of our motivation to study (1.5) is to justify the weak formulation for point vortex
solutions of the incompressible Euler equations by approximating these solutions with
classical solutions. A lot of work has been done in this respect, see [1, 2, 4, 7, 19, 25, 27,
28, 30, 31, 33, 34] and the references therein. Our work is also motivated by a recent paper
of Smets and Van Schaftingen [30], which will be described in more detail later.
4
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
In [18] Elcrat and Miller, by a rearrangements of functions, have studied steady, inviscid
flows in two dimensions which have concentrated regions of vorticity. In particular, they
studied such flows which ”desingularize” a configuration of point vortices in stable equilibrium with an irrotational flow, which generalized their earlier work for one vortex [16][17].
Saffman and Sheffield [29] have found an example of a steady flow in aerodynamics with
a single point vortex which is stable for a certain range of the parameters. This has been
generalized in [16], where some examples computationally of stable configurations of two
point vortices were briefly discussed. Further examples of multiple point vortex configurations are given in [26], where a theorem on the existence of such configurations is also
given.
It is worthwhile to point out that except [18] the above approximations can just give
explanation for the formulation to single point vortex solutions. Smets and Van Schaftingen
[30] investigated the following problem
(
¢p
¡
κ
ln 1ε + , in Ω,
−ε2 ∆u = u − q − 2π
(1.6)
u = 0,
on ∂Ω,
where p > 1. They gave the exact asymptotic behavior and expansion of the least energy
solution by estimating the upper bounds on the energy. The solutions for (1.6) in [30]
were obtained by finding a minimizer of the corresponding functional in a suitable function
space, which gives approximation to a single point non-vanishing vortex.
Concerning regularization of pairs of vortices, Smets and Van Schaftingen [30] also studied the following problem

³
´
−ε2 ∆u = u − q − κ+ ln 1 p − (q − κ− ln 1 − u)p , in Ω,
+
2π
ε
2π
ε
+
(1.7)
u = 0,
on ∂Ω,
and obtained solution with least energy among sign-change solutions of (1.7). They also
obtained the exact asymptotic behavior and expansion of such solutions by similar methods
for (1.6).
In this paper, we try to establish, by a different method, existence of stationary solu−
+
−
tions concentrating near C 1 -stable critical points of W(x+
1 , · · · , xm , x1 , · · · , xn ) with both
clockwise and anti-clockwise point vortex. To achieve our goal, we consider the following
semilinear elliptic problem

m
n
X
¡
¢p
¡
κ−

κ+ 1 ¢p X
1
j
 −ε2 ∆u =
χΩ+i u − q − i ln + −
ln − u + , in Ω,
χΩ−j q −
2π ε
2π ε
(1.8)
i=1
j=1


u = 0,
on ∂Ω,
where p > 1, q ∈ C 2 (Ω), Ω+i (i = 1, · · · , m) and Ω−j ( j = 1, · · · , n) are mutually disjoint
+
−
−
subdomains of Ω such that x+
i,∗ ∈ Ω i , and xj,∗ ∈ Ω j .
REGULARIZATION OF POINT VORTICES
5
To be more precise, we will consider an equivalent problem of (1.8) instead. Let w =
2π p−1
and δ = ε( | ln
) 2 , then (1.8) becomes
ε|

m
n
X
2π q p X
2π q

p
+
−δ 2 ∆w =
χΩ+i (w − κi −
)+ −
χΩ−j (
− κ−
j − w)+ , in Ω,
|
ln
ε|
|
ln
ε|
(1.9)
i=1
j=1


w = 0,
on ∂Ω.
2π
u
| ln ε|
We will use a reduction argument to prove our main results. To this end, we need to
construct an approximate solution for (1.9). 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 [14, 15] to construct an approximate solution. Since there are two parameters δ, ε in (1.9) and two terms in nonlinearity, which causes some difficulty, we must take
this influence into careful consideration in order to perform the reduction argument. Let us
point out that, when studying pair of vortex solutions with both anti-clockwise and clockwise point vortex, except the difficulties mentioned above we have additional difficulties
due to the interactions between the positive part and the negative part of those solutions.
+
−
−
For example, one can easily see that in the expressions of s+
1,δ , · · · , sm,δ , s1,δ , · · · , sn,δ and
+
−
−
a+
1,δ , · · · , am,δ , a1,δ , · · · , an,δ in Lemma 2.1 the effect of interactions has to be taken into
careful consideration and the estimates are much more involved. The method used in
the present paper could be applied to study related problems, for example, shallow water
vortices, which was studied recently by De Valeriola and Van Schaftingen in [32].
Our first main results concerning (1.1) is the following:
Theorem 1.1. Suppose that Ω ⊂ R2 is a bounded simply-connected smooth
domain. Let
R
vn : ∂Ω → R be such that vn ∈ Ls (∂Ω) for some s > 1 satisfying ∂Ω vn = 0. Let
−
1
κ+
i > 0, κj > 0, i = 1, · · · , m, j = 1, · · · , n. Then, for any given C -stable critical point
−
+
−
(x+
1,∗ , · · · , xm,∗ , x1,∗ , · · · , xn,∗ ) of Kirchhoff-Routh function W defined by (1.4), there exists
ε0 > 0, such that for each ε ∈ (0, ε0 ), (1.1) has a stationary solution vε with outward
m
n
X
X
boundary flux given by vn , such that its vorticities ωε =
ωi,+ε +
ωj,−ε satisfies for
i=1
j=1
small ε,
supp(ωi,+ε ) ⊂ B(x+
i, ε , Cε),
for i = 1, · · · , m,
supp(ωj,−ε ) ⊂ B(x−
j, ε , Cε),
for j = 1, · · · , n,
+
−
−
where x+
i, ε ∈ Ωi (i = 1, · · · , m), xj, ε ∈ Ωj (j = 1, · · · , n) and C > 0 is a constant independent of ε.
Furthermore as ε → 0,
+
−
−
+
+
−
−
(x+
1, ε , · · · , xm, ε , x1, ε , · · · , xn, ε ) → (x1,∗ , · · · , xm,∗ , x1,∗ , · · · , xn,∗ ),
Z
ωi,+ε → κ+
i , i = 1, · · · , m,
B(x+
i, ε ,Cε)
6
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
Z
B(x−
j, ε ,Cε)
ωi,−ε → −κ−
j , j = 1, · · · , n,
Z
ωε →
Ω
m
X
i=1
κ+
i
−
n
X
κ−
j .
j=1
Remark 1.2. The case m = n = 1, corresponding to pairs of vortices, was studied by Smets
and Van Schaftingen [30] by minimizing the corresponding energy functional in the Nehari
−
−
W(x+
sup
manifold. In their paper W(x+
1 , x1 ) as ε → 0. Our result
1, ε , x1,ε ) →
−
+
−
x+
1 , x1 ∈Ω, x1 6=x1
extends theirs to more general critical points (with additional assumption that the critical
point is non-degenerate or stable in the sense of C 1 ). The method used here is constructive
and different from theirs.
−
Remark 1.3. In the case that m = n = 1, suppose that (x+
1,∗ , x1,∗ ) is a strict local max−
imum(or minimum) point of Kirchhoff-Routh function W(x+
1 , x1 ) defined by (1.4), then
1
the C stability is not needed and statement of Theorem 1.1 still holds which can be proved
similarly (see Remark 1.5). Thus we can re-obtain corresponding existence result in [30].
Theorem 1.1 is proved via the following result concerning problem (1.8).
−
Theorem 1.4. Suppose q ∈ C 2 (Ω). Then for any given κ+
i > 0, κj > 0, i = 1, · · · , m, j =
−
+
−
1, . . . , n and for any given C 1 -stable critical point (x+
1,∗ , · · · , xm,∗ , x1,∗ , · · · , xn,∗ ) of KirchhoffRouth function W defined by (1.4), there exists ε0 > 0, such that for each ε ∈ (0, ε0 ), (1.8)
κ+
+
1
i
has a solution uε , such that the set Ω+
ε,i = {x : uε (x) − 2π ln ε − q(x) > 0} ⊂⊂ Ωi , i =
κ−
−
j
1
1, · · · , m, Ω−
ε,j = {x : uε (x) − 2π ln ε − q(x) > 0} ⊂⊂ Ωj , j = 1, · · · , n and as ε → 0, each
−
+
+
Ω+
ε, i (resp. Ωε, j ) shrinks to xi,∗ ∈ Ω(resp. xj,∗ ∈ Ω).
−
Remark 1.5. For the case m = n = 1, suppose that (x+
1,∗ , x1,∗ ) is a strict local maximum(or
+
−
minimum) point of Kirchhoff-Routh function W(x1 , x1 ) defined by (1.4), statement of
Theorem 1.4 still holds, which can be proved by making corresponding modification of the
proof of Theorem 1.4 in obtaining critical point of K(Z) defined by (4.1)(see Propositions
2.3, 2.5 and 2.6 in [13] for detailed arguments).
Remark 1.6. (1.9) can be considered as a free boundary problem. Similar problems have
been studied extensively. The reader can refer to [12, 14, 15, 22] for more results on this
kind of problems.
By the same way for the proof of Theorem 1.1 but simpler, we can extend the result
of least energy solution obtained via constrained minimization problem in [30] to nonleast energy solutions and show that multi-point vortex solutions can be approximated
by stationary classical solutions. Indeed we have the following result concerning problem
(1.1):
Theorem 1.7. Suppose that Ω ⊂ R2 is a bounded simply-connectedR smooth domain. Let
vn : ∂Ω → R be such that vn ∈ Ls (∂Ω) for some s > 1 satisfying ∂Ω vn = 0. Let κi >
REGULARIZATION OF POINT VORTICES
7
0, i = 1, · · · , m. Then, for any given C 1 - stable critical point (x∗1 , · · · , x∗m ) of KirchhoffRouth function W(x1 , · · · , xm ) defined by (1.3), there exists ε0 > 0, such that for each
ε ∈ (0, ε0 ), (1.1) has a stationary solution vε with outward boundary flux given by vn , such
m
X
that its vorticities ωε =
ωi, ε satisfies for small ε,
i=1
supp(ωi, ε ) ⊂ B(xi, ε , Cε),
for i = 1, · · · , m,
where xi, ε ∈ Ωi (i = 1, · · · , m), C > 0 is a constant independent of ε.
Furthermore as ε → 0,
Z
ωi, ε → κi , i = 1, · · · , m,
B(xi, ε ,Cε)
Z
ωε →
Ω
m
X
κi ,
i=1
(x1, ε , · · · , xm, ε ) → (x∗1 , · · · , x∗m ).
Remark 1.8. The case m = 1, corresponding to a point vortex, was studied by Smets and
Van Schaftingen [30] by minimizing the corresponding energy functional. In their paper
W(x1, ε ) → sup W(x) as ε → 0.
x∈Ω
We end this section by outlining the organization of our paper. In section 2, we construct
the approximate solution for (1.9). We will carry out a reduction argument in section 3
and the main results will be proved in section 4. We put some basic estimates used in
sections 3 and 4 in the appendix.
2. Approximate solutions
In this section, as preliminary, we will construct approximate solutions for (1.9).
Let R > 0 be a large constant, such that for any x ∈ Ω, Ω ⊂⊂ BR (x). For any given
a > 0, it is well-known that the following problem
(
−δ 2 ∆w = (w − a)p+ , in BR (0),
(2.1)
w = 0,
on ∂BR (0),
has a unique (positive) solution Wδ,a , which can be represented by
(
−2/(p−1) ¡ |x| ¢
a + δ 2/(p−1) sδ
φ sδ , |x| ≤ sδ ,
Wδ,a (x) =
|x|
sδ
a ln R / ln R ,
sδ ≤ |x| ≤ R,
where and henceforth φ(x) = φ(|x|) is the unique solution of
¡
¢
−∆φ = φp , φ > 0, φ ∈ H01 B1 (0)
(2.2)
8
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
and sδ ∈ (0, R) is determined by
−2/(p−1) 0
δ 2/(p−1) sδ
φ (1) =
a
.
ln(sδ /R)
(2.3)
By (2.3) we can obtain
sδ
→
δ| ln δ|(p−1)/2
µ
|φ0 (1)|
a
¶(p−1)/2
> 0,
as δ → 0.
Moreover, by Pohozaev identity, we can get
Z
Z
π(p + 1) 0
p+1
2
φ
=
|φ (1)| ,
φp = 2π|φ0 (1)|.
2
B1 (0)
B1 (0)
For any z ∈ Ω, define Wδ,z,a (x) = Wδ,a (x − z). Because Wδ,z,a does not vanish on ∂Ω, we
need to make a projection as follows. Let P Wδ,z,a be the solution of
(
−δ 2 ∆w = (Wδ,z,a − a)p+ , in Ω,
w = 0,
on ∂Ω,
then
P Wδ,z,a = Wδ,z,a −
a
g(x, z),
ln sRδ
(2.4)
where g(x, z) satisfies
(
−∆g = 0,
in Ω,
R
g = ln |x−z| , on ∂Ω.
It is easy to see that
g(x, z) = ln R + 2πh(x, z),
where h(x, z) = −H(x, z).
+
+
+
Let Z = (Zm
, Zn− ), where Zm
= (z1+ , · · · , zm
), Zn+ = (z1− , · · · , zn− ). We will construct
solutions for (1.9) of the form
m
X
i=1
P Wδ,zi+ ,a+ −
δ,i
n
X
j=1
P Wδ,zj− ,a− + ωδ ,
δ,j
−
where zi+ , zj− ∈ Ω, a+
δ,i > 0, aδ,j > 0 for i = 1, · · · , m, j = 1, · · · , n, ωδ is a perturbation
−
term. To make ωδ as small as possible, we need to choose a+
δ,i , aδ,j properly. Indeed we will
+
−
−
+
−
choose aδ,i , aδ,j such that, together with (s1 (Z), · · · , s+
m (Z), s1 (Z), · · · , sn (Z)), the system
in Lemma 2.1 to be given in the following will be satisfied.
In this paper, we always assume that zi+ , zj− ∈ Ω satisfies
REGULARIZATION OF POINT VORTICES
d(zi+ , ∂Ω) ≥ %, d(zj− , ∂Ω) ≥ %,
|zj− − zl− | ≥ %L̄ ,
|zi+ − zk+ | ≥ %L̄ ,
|zi+ − zj− | ≥ %L̄ ,
9
i, k = 1, · · · , m, i 6= k
j, l = 1, · · · , n, j 6= l,
(2.5)
where % > 0 is a fixed small constant and L̄ > 0 is a fixed large constant.
+
−
−
Lemma 2.1. For δ > 0 small, there exist (s+
δ,1 (Z), · · · , sδ,m (Z), sδ,1 (Z), · · · , sδ,n (Z)) and
−
−
+
(a+
δ,1 (Z), · · · , aδ,m (Z), aδ,1 (Z), · · · , aδ,n (Z)) satisfying the following system
−2/(p−1) 0
δ 2/(p−1) (s+
φ (1) =
i )
−2/(p−1) 0
δ 2/(p−1) (s−
φ (1) =
j )
a+
i
ln
s+
i
R
a−
j
ln
s−
j
R
,
i = 1, · · · , m,
(2.6)
,
j = 1, · · · , n,
(2.7)
n
m
a+
i
=
κ+
i
2πq(zi+ ) g(zi+ , zi+ ) + X Ḡ(zi+ , zα+ ) + X Ḡ(zi+ , zl− ) −
+
ai −
aα +
al ,
+
| ln ε|
ln sR+
ln sR+
ln sR−
l=1
α6=i
α
i
−
a−
j = κj −
l
n
m
Ḡ(zβ− , zj− ) − X
Ḡ(zj− , zk+ ) +
2πq(zj− ) g(zj− , zj− ) − X
a
−
a
+
ak ,
+
j
β
R
R
| ln ε|
ln sR−
ln
ln
−
+
s
s
β6=j
k=1
j
where Ḡ(x, y) = ln
R
|x−y|
i = 1, · · · , m,
β
(2.8)
j = 1, · · · , n,
k
(2.9)
− g(x, y) for x 6= y.
Proof. We will prove that if δ > 0 is small enough then system (2.6)-(2.9) has a solution
−
−
+
− +
+
−
(s+
1 , · · · , sm , s1 , · · · , sn , a1 , · · · , am , a1 , · · · , an ) in D defined by
¸m+n Y
¸ Y
¸
·
m ·
n ·
1 + 3 +
1 − 3 −
δ
×
, δ| ln δ|
κ , κ ×
κ , κ .
D :=
| ln δ|
2 i 2 i
2 j 2 j
i=1
j=1
−
+
−
∗
It is not difficult to see, for any fixed (s+
1 , · · · , sm , s1 , · · · , sn ) and 0 < δ < δ small
+
−
+
−
enough, that the subsystem (2.8)-(2.9) has a solution (a1 , · · · , am , a1 , · · · , an ) depending
−
+
−
1 +
3 + 1 −
3 −
+
−
on (s+
1 , · · · , sm , s1 , · · · , sn ) such that 2 κi ≤ ai ≤ 2 κi , 2 κj ≤ aj ≤ 2 κj for i = 1, · · · , m
+
−
−
and j = 1, · · · , n. For such (a1 , · · · , a+
m , a1 , · · · , an ) define
2
θi+ (s+
1 ,···
−
, s+
m , s1 , · · ·
, s−
n)
(s+ ) p−1
φ0 (1) 2
= i R + + δ p−1 , i = 1, · · · , m,
ai
ln s+
i
2
θj− (s+
1 ,···
−
, s+
m , s1 , · · ·
, s−
n)
=
p−1
(s−
j )
ln sR−
j
+
2
φ0 (1) p−1
, j = 1, · · · , n,
− δ
aj
10
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
then it is easy to verify that for i, ` = 1, · · · , m, j, k = 1, · · · , n,
 + +
−
+
+
−
δ
−
θi (s1 , · · · , s+

m , s1 , · · · , sn ) > 0, si = δ| ln δ|, s` , sj ∈ [ | ln δ| , δ| ln δ| ] for ` 6= i,


 θ+ (s+ , · · · , s+ , s− , · · · , s− ) < 0, s+ = δ , s+ , s− ∈ [ δ , δ| ln δ| ] for ` 6= i,
1
m 1
n
i
i
j
`
| ln δ|
| ln δ|
− +
−
−
+
−
δ
+
−
,
s
∈
[
θ
(s
,
·
·
·
,
s
,
s
,
·
·
·
,
s
)
>
0,
s
=
δ|
ln
δ|,
s
, δ| ln δ| ] for k 6= j,

1
m 1
n
j
j
`
k

| ln δ|

 θ− (s+ , · · · , s+ , s− , · · · , s− ) < 0, s− = δ , s+ , s− ∈ [ δ , δ| ln δ| ] for k 6= j.
1
m 1
n
j
j
`
k
| ln δ|
| ln δ|
+
−
−
By the Poincaré-Miranda Theorem in [20, 24], we can get (s+
δ,1 , · · · , sδ,m , sδ,1 , · · · , sδ,n )
−
−
+
− +
−
−
+
such that θi+ (s+
δ,1 , · · · , sδ,m , sδ,1 , · · · , sδ,n ) = 0, θj (sδ,1 , · · · , sδ,m , sδ,1 , · · · , sδ,n ) = 0 for i =
1, · · · , m, j = 1, · · · , n. This completes our proof of Lemma 2.1.
¤
±
±
±
In the sequel, to simplify notation, we will use a±
δ,i , sδ,i to denote aδ,i (Z), sδ,i (Z) for
−
+
−
+
given Z = (Zm
, Zn− ). From now on we will always choose (a+
δ,1 , · · · , aδ,m , aδ,1 , · · · , aδ,n ) and
−
+
+
−
−
+
−
(s+
δ,1 , · · · , sδ,m , sδ,1 , · · · , sδ,n ) such that (2.6)–(2.9) hold. For (aδ,1 , · · · , aδ,m , aδ,1 , · · · , aδ,n )
−
+
−
and (s+
δ,1 , · · · , sδ,m , sδ,1 , · · · , sδ,n ) chosen in such a way define
+
−
Pδ,Z,i
= P Wδ,zi+ , a+ , Pδ,Z,j
= P Wδ,zj− , a− .
δ,i
δ,j
Remark 2.2. It is not difficult to obtain the following asymptotic expansions:
µ
¶
1
ln | ln ε|
1
= R +O
, i = 1, · · · , m,
| ln ε|2
ln sR+
ln ε
(2.10)
(2.11)
δ,i
a+
δ,i
=
m
+
+ +
X
+ 2πq(zi ) g(zi , zi )
κi +
+
−
| ln ε|
ln Rε
α6=i
n
³ ln | ln ε| ´
Ḡ(zi+ , zα+ ) X Ḡ(zi+ , zl− )
+
+O
, i = 1, · · · , m,
| ln ε|2
ln Rε
ln Rε
l=1
(2.12)

¶
¶
µ
µ
∂a+
∂ s+

1
ε
δ,i
δ,i


,
, i, k = 1, · · · , m, h = 1, 2,

+ = O
+ = O

| ln ε|
| ln ε|
∂zk,h
 ∂zk,h
µ
¶
µ
¶


∂ a+
∂s+

1
ε
δ,i
δ,i


,
, i = 1, · · · , m, j = 1, · · · , n, h = 1, 2.

− = O
− = O
| ln ε|
| ln ε|
∂ zj,h
∂ zj,h
(2.13)
−
−
Moreover, aδ,j and sδ,j have similar expansions.
To simplify notations, set
+
Pδ,Z
=
m
X
α=1
+
−
Pδ,Z,α
, Pδ,Z
=
n
X
−
Pδ,Z,β
.
(2.14)
β=1
Then, for any fixed constant L > 0, we have that for x ∈ BLs+ (zi+ ),
δ,i
REGULARIZATION OF POINT VORTICES
+
Pδ,Z,i
(x)
−
κ+
i
11
a+
2πq(x)
2πq(x)
δ,i
g(x, zi+ ) − κ+
−
= Wδ,zi+ , a+ (x) −
i −
R
δ,i
| ln ε|
| ln ε|
ln s+
δ,i
a+
δ,i
ln sR+
δ,i
=Wδ,zi+ , a+ (x) − κ+
i −
δ,i
g(zi+ , zi+ ) −
a+
δ,i
ln sR+
δ,i
³­
´
®
Dg(zi+ , zi+ ), x − zi+ + O(|x − zi+ |2 )
®
¢
2πq(zi+ )
2π ¡­
−
Dq(zi+ ), x − zi+ + O(|x − zi+ |2 )
| ln ε|
| ln ε|
®
2π ­
2πq(zi+ )
+
=Wδ,zi+ , a+ (x) − κi −
−
Dq(zi+ ), x − zi+
δ,i
| ln ε|
| ln ε|
à + !
+
+
­
®
a
(sδ,i )2
aδ,i
δ,i
+ +
+ +
+
g(zi , zi ) −
Dg(zi , zi ), x − zi + O
,
−
| ln ε|
ln sR+
ln sR+
−
δ,i
δ,i
and for k 6= i and x ∈ BLs+ (zi+ ), by (2.2)
δ,i
+
Pδ,Z,k
(x)
a+
δ,k
= R
ln s+
δ,k
= Wδ,z+ ,a+ (x) −
k
Ḡ(zi+ , zk+ )
δ,k
+
a+
δ,k
ln sR
+
δ,k
­
a+
δ,k
ln
g(x, zk+ )
R
s+
δ,k
DḠ(zi+ , zk+ ), x
=
a+
δ,k
ln sR
+
Ḡ(x, zk+ )
δ,k
−
zi+
®
+O
³ (s+ )2 ´
δ,i
| ln ε|
,
and
−
Pδ,Z,j
(x) =
a−
δ,j
ln sR−
Ḡ(zi+ , zj− ) +
δ,j
a−
δ,j ­
ln sR−
³ (s+ )2 ´
®
δ,i
DḠ(zi+ , zj− ), x − zi+ + O
.
| ln ε|
δ,j
So, by using (2.8), we obtain
2πq(x)
| ln ε|
®
®
a+
2π ­
δ,i ­
+
+
+
+ +
+
=Wδ,zi+ ,a+ (x) − aδ,i −
,
z
),
x
−
z
Dq(zi ), x − zi −
Dg(z
i
i
i
δ,i
| ln ε|
ln sR+
+
−
Pδ,Z
(x) − Pδ,Z
(x) − κ+
i −
δ,i
+
m
X
a+
δ,k ­
®
DḠ(zi+ , zk+ ), x − zi+ −
ln sR
+
δ,k
à + !
(sδ,i )2
,
+O
| ln ε|
k6=i
n
X
a−
δ,l ­
l=1
x ∈ BLs+ (zi+ ).
δ,i
ln sR−
δ,l
DḠ(zi+ , zl− ), x − zi+
®
(2.15)
12
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
Similarly, we have
2πq(x)
| ln ε|
®
®
a−
2π ­
δ,j ­
−
−
− −
−
+
Dq(zj ), x − zj −
Dg(z
,
z
),
x
−
z
j
j
j
| ln ε|
ln sR−
−
+
Pδ,Z
(x) − Pδ,Z
(x) − κ−
j +
=Wδ,zj− ,a− (x) − a−
δ,j
δ,j
δ,j
+
n
X
a−
δ,l
ln sR−
l6=j
δ,l
Ã
+O
­
2
(s−
δ,j )
| ln ε|
®
DḠ(zj− , zl− ), x − zj− −
!
,
m
X
a+
δ,k
ln sR
+
k=1
δ,k
­
DḠ(zj− , zk+ ), x − zj−
(2.16)
®
x ∈ BLs− (zj− ).
δ,j
We end this section by giving the following expansion which can be obtained by direct
computation and will be used in the next two sections.
∂Wδ,zi± ,a± (x)
δ,i
∂z ±
 i,h
=
µ
¶
´(p+1)/2 ¡ |x − z ± | ¢ z ± − xh
a±
1³
1

δ,i
i,h
i
0


, x ∈ Bs± (zi± ),
φ

±
± +O
0 (1)|| ln R |
δ,i

δ
|
ln
ε|
s
|x
−
z
|
|φ

i
δ,i
s±

δ,i
µ
¶

±

a±

1
δ,i zi,h − xh


−
+O
,

 ln R± |x − zi± |2
| ln ε|
s
δ,i
x ∈ Ω \ Bs± (zi± ).
δ,i
(2.17)
3. the reduction
+
−
In this section, letting Pδ,Z
and Pδ,Z
be given as in (2.14), we are to look for solutions of
+
−
the form Pδ,Z −Pδ,Z +ωδ,Z , where ωδ,Z is a small perturbation when δ is small. We will show
+
−
the existence of ωδ,Z for given Z so that Pδ,Z
−Pδ,Z
+ωδ,Z solves (1.9) in W 2, p (Ω)∩H01 (Ω)\H∗ ,
where H∗ is a finite dimension subspace of W 2, p (Ω) ∩ H01 (Ω). In the next section, we will
show that if Z is a critical point of
Z
δ2
+
−
|D(Pδ,Z
− Pδ,Z
+ ωδ,Z )|2
K(Z) =
2 Ω
¶p+1
Z µ
m
X
1
2πq(x)
+
−
+
−
Pδ,Z − Pδ,Z + ωδ,Z − κi −
+
p
+
1
| ln ε| +
Ω
i
i=1
µ
¶p+1
Z
n
X 1
2πq(x)
−
+
−
− κj − Pδ,Z + Pδ,Z − ωδ,Z
−
,
p + 1 Ω−j
| ln ε|
+
j=1
REGULARIZATION OF POINT VORTICES
13
−
+
then Pδ,Z
− Pδ,Z
+ ωδ,Z is indeed a solution of (1.9). Finding critical points of K(Z) is
a problem of finite dimension, so in this sense, we are trying to reduce the problem to a
finite dimension one in this section.
+
−
To this end we need to study the kernel of linearized problem of (1.9) at Pδ,Z
− Pδ,Z
.
2
First let us study the kernel of the following problem in R :
p−1
−∆v − pw+
v = 0,
where
v ∈ L∞ (R2 ),
(3.1)
(
φ(|x|),
|x| ≤ 1,
w(x) =
0
φ (1) ln |x|, |x| > 1,
is the solution of
p
−∆w = w+
,
in R2 .
(3.2)
Since φ0 (1) < 0 and ln |x| is harmonic for |x| > 1, we see that w ∈ C 1 (R2 ). Moreover,
since w+ is Lip-continuous, by the Schauder estimate, w ∈ C 2,α for any α ∈ (0, 1).
∂w
It is easy to see that for i = 1, 2, ∂x
is a solution of (3.1). Moreover, from Dancer
i
and Yan [15], we know that w is also non-degenerate, in the© sense that
ª the kernel of the
p−1
∂w ∂w
1,2
2
operator Lv := −∆v − pw+ v, v ∈ D (R ) is spanned by ∂x1 , ∂x2 .
+
−
Let Pδ,Z,i
, Pδ,Z,j
be the functions defined in (2.10). Set
(
Fδ,Z =
Z
+
∂Pδ,Z,i
p
u : u ∈ L (Ω),
Ω
+
∂zi,h
Z
−
∂Pδ,Z,j
u = 0,
Ω
−
∂zj,h
u = 0,
)
i = 1, · · · , m, j = 1, · · · , n, h = 1, 2 ,
and
(
Eδ,Z =
u : u ∈ W 2,p (Ω) ∩ H01 (Ω),
Ã
Z
∆
Ω
+
∂Pδ,Z,i
+
∂zi,h
!
Ã
Z
u = 0,
∆
Ω
−
∂Pδ,Z,j
−
∂zj,h
!
u = 0,
)
i = 1, · · · , m, j = 1, · · · , n, h = 1, 2 .
For any u ∈ Lp (Ω), define Qδ u as follows:
Ã
Ã
!
!
m X
2
n X
2
³ ∂P − ´
³ ∂P + ´
X
X
δ,Z,j
δ,Z,i
Qδ u = u −
b−
−δ 2 ∆
b+
−δ 2 ∆
−
,
i,h
+
−
j,
h̄
∂z
∂z
i,h
j,h̄
i=1 h=1
j=1
h̄=1
−
where the constants b+
i,h , bj,h̄ satisfy
14
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
!
³ ∂P + ´ ∂P +
δ,Z,k
δ,Z,i
b+
−δ 2 ∆
i,h
+
+
∂zi,h
∂zk,
Ω
i=1 h=1
ĥ
! Z
Ã
Z
n
2
+
−
´ ∂P +
³ ∂P
XX
∂Pδ,Z,k
δ,Z,k
δ,Z,j
−
2
+
u
bj,h̄ −δ
∆
=
+
+ ,
∂zj,−h̄
∂zk,
∂zk,
Ω
Ω
j=1
ĥ
ĥ
Ã
m X
2
X
Z
(3.3)
h̄=1
and
Ã
!
³ ∂P + ´ ∂P −
δ,Z,i
δ,Z,l
−δ 2 ∆
b+
i,h
+
∂zi,h
∂zl,−h̃
Ω
i=1 h=1
Ã
! Z
Z
2
n X
−
³ ∂P − ´ ∂P −
X
∂Pδ,Z,l
δ,Z,j
δ,Z,l
2
+
b−
−δ
∆
=
u
−
−
− .
j,h̄
∂z
∂z
∂z
Ω
Ω
j,h̄
l,h̃
l,h̃
j=1
m X
2
X
Z
Since
h̄=1
+
∂Pδ,Z,k
+ Qδ u
Ω ∂zk,ĥ
p
Z
Z
(3.4)
−
∂Pδ,Z,l
= 0,
Qδ u = 0, the operator Qδ can be regarded as a
∂zl,−h̃
projection from L (Ω) to Fδ,Z . In order to show that we can solve (3.3) and (3.4) to obtain
−
b+
i,h and bj,h̄ , we just need the following estimate ( by (2.13) and (2.17))
Ω
Z
³ ∂P + ´ ∂P +
δ,Z,i
δ,Z,k
−δ
∆
+
+
∂z
∂z
Ω
i,h
k,ĥ
!
Ã
Z
+
+
+ +
∂W
¡
¢
∂Pδ,Z,k
∂a
δ,z
,a
p−1
δ,i
i
δ,i
=p
−
Wδ,zi+ ,a+ − a+
δ,i +
+
+
+
δ,i
∂zi,h
∂zi,h
∂zk,
Ω
ĥ
µ
¶
c
ε
+O
=δikhĥ
,
p+1
| ln ε|
| ln ε|p+1
2
(3.5)
where c > 0 is a constant, δikhĥ = 1, if i = k and h = ĥ; otherwise, δijhĥ = 0.
Similarly,
µ
¶
Z
³ ∂P − ´ ∂P −
ε
c
δ,Z,j
δ,Z,l
2
−δ
∆
+O
= δjlh̄h̃
,
| ln ε|p+1
| ln ε|p+1
∂zj,−h̄
∂zl,−h̃
Ω
where c > 0 is a constant, δjlh̄h̃ = 1, if j = l and h̄ = h̃; otherwise, δjlh̄h̃ = 0.
Set
2
Lδ u = −δ ∆u −
−
m
X
i=1
n
X
j=1
µ
pχΩ+i
+
Pδ,Z
−
−
Pδ,Z
−
κ+
i
µ
pχΩ−j
−
Pδ,Z
−
+
Pδ,Z
−
κ−
j
2πq(x)
−
| ln ε|
2πq(x)
+
| ln ε|
¶p−1
u
+
¶p−1
u,
+
(3.6)
REGULARIZATION OF POINT VORTICES
and
15
³
´ [³
´
+
−
n
+ (z )
− (z ) .
Bδ,Z = ∪m
B
∪
B
i=1 Ls
j=1 Ls
i
j
δ,i
δ,j
We have the following lemma.
Lemma 3.1. There are constants ρ0 > 0 and δ0 > 0, such that for any δ ∈ (0, δ0 ], Z
satisfying (2.5), u ∈ Eδ,Z with Qδ Lδ u = 0 in Ω \ Bδ,Z for some L > 0 large, then
2
ρ0 δ p
kQδ Lδ ukLp (Ω) ≥
| ln δ|
(p−1)2
p
kukL∞ (Ω) .
±
Proof. Set s±
N,j = sδN ,j . Henceforth, we will use k · kp and k · k∞ to denote k · kLp (Ω) and
k · kL∞ (Ω) respectively.
We argue by contradiction. Suppose that there are δN → 0, ZN satisfying (2.5) and
uN ∈ EδN ,ZN with QδN LδN uN = 0 in Ω \ BδN ,ZN and kuN k∞ = 1 such that
2
δNp
1
kQδN LδN uN kp ≤
2 .
N | ln δ | (p−1)
p
N
−
First, we estimate the b+
i,h,N , bj,h̄,N corresponding to LδN uN , which satisfy
QδN LδN uN = LδN uN −
m X
2
X
Ã
b+
i,h,N
i=1 h=1
−
n X
2
X
2
−δN
∆
Ã
b−
j,h̄,N
2
−δN
∆
∂Pδ+N ,ZN ,i
+
∂zi,h
∂Pδ−N ,ZN ,j
j=1 h̄=1
For each fixed k, multiplying (3.7) by
∂Pδ+
N ,ZN ,k
∂z +
!
!
∂zj,−h̄
, noting that
k,ĥ
Z
¡
Ω
QδN LδN uN
¢ ∂Pδ+N ,ZN ,k
+
∂zk,
ĥ
= 0,
we obtain
Ã
Z
u N L δN
∂Pδ+N ,ZN ,k
!
Z
=
¡
LδN u N
¢ ∂Pδ+N ,ZN ,k
+
+
∂zk,
∂zk,
Ω
ĥ
ĥ
!
Ã
Z
m
2
+
+
XX
∂PδN ,ZN ,i ∂PδN ,ZN ,k
2
=
b+
−δ
∆
N
i,h,N
+
+
∂zi,h
∂zk,
Ω
i=1 h=1
ĥ
Ã
!
Z
n
2
−
+
XX
∂PδN ,ZN ,j ∂PδN ,ZN ,k
2
b−
−δN
∆
+
.
+
j,h̄,N
∂zj,−h̄
∂zk,
Ω
j=1
ĥ
Ω
h̄=1
Using (2.15), (2.16) and Lemma A.1, we obtain
(3.7)
.
16
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
Ã
Z
Ω
u N L δN
∂Pδ+N ,ZN ,k
Z "
=
−
Ω
2
δN
∆
!
+
∂zk,
ĥ
à +
!
∂PδN ,ZN ,k
+
∂zk,
ĥ
−
m
X
i=1
µ
pχΩ+i
Pδ+N ,ZN
−
Pδ−N ,ZN
−
κ+
i
2πq(x)
−
| ln εN |
#
¶p−1
+
∂P
2πq(x)
δN ,ZN ,k
−
pχΩ−j Pδ−N ,ZN − Pδ+N ,ZN − κ−
uN
j +
+
|
ln
ε
|
∂z
N
+
j=1
k,ĥ
!
Ã
Z ³
+
´p−1 ∂WδN ,z+ ,a+
∂a
k,N
δ
,k
δ
,k
N
N
−
uN
=p
WδN ,z+ ,a+ − a+
δN ,k
+
+
k,N δN ,k
∂z
∂z
+
Ω
k,ĥ
k,ĥ
Ã
Ã
!!p−1
m Z
+
X
∂Pδ+N ,ZN ,k
s
N,α
+
−p
WδN ,z+ ,a+ − aδN ,α + O
uN
+
α,N δN ,α
+
| ln εN |
∂zk,
α=1 Ωα
ĥ
+
Ã
à − !!p−1
n Z
X
sN,β
∂Pδ+N ,ZN ,k
−
−
−
WδN ,z ,a
− aδN ,β + O
−p
uN
+
β,N δN ,β
−
| ln εN |
∂zk,
ĥ
β=1 Ωβ
+
µ
¶
ε2N
=O
.
| ln εN |p
n
X
µ
¶p−1
+
∂Pδ+N ,ZN ,k
+
∂zk,
ĥ
−
2
2
Using (3.5) and (3.6), we obtain that b+
i,h,N = O (εN | ln εN |). Similarly, bi,h,N = O (εN | ln εN |).
Therefore,
Ã
!
!
Ã
m X
2
n X
2
+
−
X
X
∂P
∂P
δN ,ZN ,i
δN ,ZN ,j
2
2
b+
−δN
∆
+
b−
−δN
∆
i,h,N
+
j,
h̄,N
∂zi,h
∂zj,−h̄
i=1 h=1
j=1 h̄=1
!
Ã
m X
2
+
´p−1 ∂WδN ,z+ ,a+
³
X
∂a
i,N
δ
,i
δ
,i
N
N
+
− a+
−
=p
b+
,a+
i,h,N WδN ,zi,N
δN ,i
+
+
δN ,i
∂z
∂z
+
i,h
i,h
i=1 h=1
Ã
!
n X
2
−
³
´p−1 ∂WδN ,z− ,a−
X
∂a
j,N δN ,j
δN ,j
+p
b−
WδN ,z− ,a− − a−
−
δN ,j
−
−
j,h̄,N
j,N δN ,j
∂z
∂z
+
j,h̄
j,h̄
j=1 h̄=1




2
2
−1 −
−1 +
p
n X
2
m X
2
p
X
X
ε
|b
|
εN |bi,h,N |
N
j,h̄,N 
+O
= O
p
p
|
ln
ε
|
|
ln
ε
N
N|
j=1 h̄=1
i=1 h=1


2
+1
p
εN
 in Lp (Ω).
= O
| ln εN |p−1
Thus, we obtain
REGULARIZATION OF POINT VORTICES

LδN uN = QδN LδN uN + O 

2
+1
p

17
2
p

εN
δN
.
 = O 1
2
p−1
| ln εN |
N | ln δ | (p−1)
p
N
For any fixed i, j, define
+
+
ũ+
i,N (y) = uN (sN,i y + zi,N ),
Let
L̃±
Nu =
−
−
ũ−
j,N (y) = uN (sN,j y + zj,N ).
¶p−1
2πq
−
−
+
−∆u − p
−
+
u
χ Ω+
2
k
δ
|
ln
ε
N| +
N
k=1
µ
¶p−1
n
2
X
(s±
2πq
N,i )
±
±
−
+
−
±
±
−p
χΩ− PδN ,ZN (sN,i y + zi,N ) − PδN ,ZN (sN,i y + zi,N ) − κl +
u.
2
l
δN
| ln εN | +
l=1
m
2
X
(s±
N,i )
µ
Pδ−N ,ZN (s±
N,i y
±
)
zi,N
Pδ+N ,ZN (s±
N,i y
±
)
zi,N
κ+
k
Then
2
p
(s±
N,i ) ×
Noting that
we find that
Ã
2
δN
± ±
± 2 kL̃N ũi,N kp = kLδN uN kp .
(sN,i )
δN
s±
N,i
!2
µ
=O

L δN u N = o 
1
| ln δN |p−1
,

2
p
δN
| ln δN |
¶
(p−1)2
p
.
As a result,
±
L̃±
in Lp (Ω±
N ũi,N = o(1),
N ),
©
ª
±
±
where Ω±
N = y : sN,i y + zi,N ∈ Ω .
Since kũ±
i,N k∞ = 1, by the regularity theory of elliptic equations, we may assume that
±
ũ±
i,N → ui ,
1
in Cloc
(R2 ).
It is easy to see that
µ
¶p−1
m
2
X
(s+
2πq
N,i )
+
+
+
−
+
+
+
χΩ+ PδN ,ZN (sN,i y + zi,N ) − PδN ,ZN (sN,i y + zi,N ) − κk −
2
k
δ
| ln εN | +
N
k=1
Ã
!!p−1
Ã
+
2
(s+
s
N,i )
N,i
=
WδN ,z+ ,a+ − a+
+ o(1)
δN ,i + O
2
i,N δN ,i
δN
| ln εN |
+
→
p−1
w+
.
18
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
Similarly,
n
2
X
(s−
N,j )
l=1
2
δN
→
p−1
w+
.
µ
χ Ω−
l
Pδ−N ,ZN (s−
N,j y
+
−
zj,N
)
−
Pδ+N ,ZN (s−
N,j y
+
−
zj,N
)
−
κ−
l
2πq
+
| ln εN |
¶p−1
+
Then, by Lemma A.1, u±
i satisfies
p−1
u = 0.
−∆u − pw+
Now from Proposition 3.1 in [15], we have
±
u±
i = c1
∂w
∂w
+ c±
.
2
∂x1
∂x2
(3.8)
Taking limit of
Z
¡ ∂Pδ±N ,ZN ,i ¢
∆
uN = 0,
±
∂zi,h
Ω
we get
Z
R2
which, together with (3.8), gives
u±
i
φp−1
+
∂φ ±
u = 0,
∂zh i
≡ 0. Thus for any L > 0,
ũ±
i,N → 0,
in C 1 (BL (0)),
±
which implies that uN = o(1) on ∂BLs± (zi,N
).
N,i
On one hand, by our assumption, QδN LδN uN = 0 in Ω \ BδN ,ZN . On the other hand, by
Lemma A.1, for i = 1, · · · , m, j = 1, · · · , n, we have
¶
µ
2πq(x)
+
−
+
+
PδN ,ZN − PδN ,ZN − κi −
= 0, x ∈ Ω+
(zi,N
),
i \ BLs+
N,i
| ln εN | +
µ
¶
2πq(x)
−
−
+
−
PδN ,ZN − PδN ,ZN − κj +
= 0, x ∈ Ω−
(zj,N
).
j \ BLs−
N,j
| ln εN | +
Thus,
−∆uN = 0,
in Ω \ BδN ,ZN .
However, uN = 0 on ∂Ω and uN = o(1) on ∂BδN ,ZN . So by maximum principle uN = o(1).
This is a contradiction since kuN k∞ = 1.
¤
Proposition 3.2. Qδ Lδ is one to one and onto from Eδ,Z to Fδ,Z .
REGULARIZATION OF POINT VORTICES
19
Proof. Suppose that Qδ Lδ u = 0. Then, by Lemma 3.1, u ≡ 0. Thus, Qδ Lδ is one to one.
Next, we prove that Qδ Lδ is an onto map from Eδ,Z to Fδ,Z . Denote
(
u : u ∈ H01 (Ω),
Ẽ =
Ã
Z
D
Ω
+
∂Pδ,Z,i
!
+
∂zi,h
Ã
Z
Du = 0,
D
Ω
−
∂Pδ,Z,j
!
−
∂zj,h
Du = 0,
)
i = 1, · · · , m, j = 1, · · · , n, h = 1, 2 .
Note that Eδ,Z = Ẽ∩W 2,p (Ω). On one hand, for any h̃ ∈ Fδ,Z , by the Riesz representation
theorem there is a unique u ∈ H01 (Ω) such that
Z
Z
2
δ
DuDϕ =
h̃ϕ, ∀ ϕ ∈ H01 (Ω).
(3.9)
Ω
Ω
On the other hand, from h̃ ∈ Fδ,Z , we find that u ∈ Ẽ. Moreover, by the Lp -estimate
u ∈ W 2,p (Ω). So, u ∈ Eδ,Z . Thus, Qδ (−δ 2 ∆) = −δ 2 ∆ is a one to one and onto map from
Eδ,Z to Fδ,Z . Meanwhile, Qδ Lδ u = h is equivalent to
−2
−1
u = pδ (−Qδ ∆)
· µ
³
´p−1
Pm
+
−
+
2π
+
P
χ
−
P
Qδ
−
κ
−
q(x)
u
i
δ,Z
δ,Z
i=1 Ωi
| ln ε|
+
³
´p−1 ¶¸
Pm
−
−
+
2π
− j=1 χΩ−j | ln ε| q(x) − κj + Pδ,Z − Pδ,Z
u
(3.10)
+
+δ −2 (−Qδ ∆)−1 h,
It is easy to check that
−2
−1
δ (−Qδ ∆)
u ∈ Eδ,Z .
· µ
´p−1
³
Pm
+
−
+
2π
+
χ
q(x)
u
Qδ
P
−
P
−
κ
−
i
δ,Z
δ,Z
i=1 Ωi
| ln ε|
+ ¶¸
³
´p−1
P
−
−
+
2π
− nj=1 χΩ−j | ln
q(x)
−
κ
+
P
−
P
u
j
δ,Z
δ,Z
ε|
+
is a compact operator in Eδ,Z . By the Fredholm alternative, (3.10) is solvable if and only
if
−2
−1
u = pδ (−Qδ ∆)
· µ
³
´p−1
Pm
+
−
+
2π
+
χ
q(x)
u
Qδ
P
−
P
−
κ
−
i
δ,Z
δ,Z
i=1 Ωi
| ln ε|
+
³
´p−1 ¶¸
Pn
−
−
+
2π
− j=1 χΩ−j | ln ε| q(x) − κj + Pδ,Z − Pδ,Z
u
+
has only trivial solutions, which is true since Qδ Lδ is one to one. Thus the result follows.
¤
Now consider the equation
Qδ Lδ ω = Qδ lδ+ − Qδ lδ− + Qδ Rδ+ (ω) − Qδ Rδ− (ω),
(3.11)
20
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
where
lδ+
=
m
X
i=1
lδ−
=
n
X
j=1
µ
+
Pδ,Z
χΩ+i
−
−
Pδ,Z
−
κ+
i
µ
−
Pδ,Z
χΩ−j
−
+
Pδ,Z
−
κ−
j
2πq(x)
−
| ln ε|
2πq(x)
+
| ln ε|
¶p
−
+
m ³
X
¶p
−
+
n ³
X
j=1
Wδ,zi+ ,a+ −
δ,i
i=1
a+
δ,i
Wδ,zj− ,a− − a−
δ,j
δ,j
´p
+
´p
+
,
(3.12)
,
(3.13)
and
Rδ+ (ω)
=
m
X
i=1
"µ
χΩ+i
µ
+
− p Pδ,Z
Rδ− (ω)
=
n
X
j=1
2πq(x)
−
+ω−
−
| ln ε|
¶p−1 #
2πq(x)
−
− Pδ,Z
− κ+
ω ,
i −
| ln ε| +
+
Pδ,Z
−
Pδ,Z
"µ
χΩ−j
µ
−
+ p Pδ,Z
2πq(x)
+
−
−ω−
| ln ε|
¶p−1 #
2πq(x)
+
− Pδ,Z
− κ−
ω .
j +
| ln ε| +
−
Pδ,Z
+
Pδ,Z
¶p
κ+
i
µ
−
+
Pδ,Z
−
−
Pδ,Z
+
¶p
µ
−
+
− Pδ,Z
− Pδ,Z
κ−
j
+
−
κ+
i
2πq(x)
−
| ln ε|
¶p
+
(3.14)
¶p
2πq(x)
−
− κj +
| ln ε| +
(3.15)
Using Proposition 3.2, we can rewrite (3.11) as
¡
¢
ω = Gδ ω =: (Qδ Lδ )−1 Qδ lδ+ − lδ− + Rδ+ (ω) − Rδ− (ω) .
(3.16)
The next proposition will enable us to reduce the problem of finding a solution for (1.9)
to a finite dimensional problem.
Proposition 3.3. There is an δ0 > 0, such that for any δ ∈ (0, δ0 ] and Z satisfying (2.5),
(3.11) has a unique solution ωδ ∈ Eδ,Z , with
³
´
p−1
kωδ k∞ = O δ| ln δ| 2 .
Proof. It follows from Lemma A.1 that if L is large enough and δ is small then
µ
¶
2πq(x)
+
−
+
Pδ,Z − Pδ,Z − κi −
= 0, x ∈ Ω+
(zi+ ), i = 1, · · · , m,
i \ BLs+
δ,i
| ln ε| +
¶
µ
2πq(x)
−
+
−
= 0, x ∈ Ω−
(zj− ), j = 1, · · · , n.
Pδ,Z − Pδ,Z − κj +
j \ BLs−
δ,j
| ln ε| +
Let
REGULARIZATION OF POINT VORTICES
21
n
o
p−1
M = Eδ,Z ∩ kωk∞ ≤ δ| ln δ| 2 .
Then M is complete under L∞ norm and Gδ is a map from Eδ,Z to Eδ,Z . Next we show
that Gδ is indeed a contraction map from M to M by the following two steps.
Step 1. Gδ is a map from M to M .
For any ω ∈ M , similar to Lemma A.1, it is not difficult to show that for large L > 0
and δ small
µ
¶
2πq(x)
+
−
+
Pδ,Z − Pδ,Z + ω − κi −
= 0, x ∈ Ω+
(zi+ ),
i \ BLs+
δ,i
| ln ε| +
¶
µ
(3.17)
2πq(x)
+
−
−
−
−
Pδ,Z − Pδ,Z − ω − κj +
= 0, x ∈ Ωj \ BLs− (zj ).
δ,j
| ln ε| +
Note also that for any u ∈ L∞ (Ω),
Qδ u = u in Ω \ Bδ,Z .
Therefore, using Lemma A.1, (3.12)–(3.15), we have for any ω ∈ M ,
Qδ (lδ+ − lδ− ) + Qδ (Rδ+ (ω) − Rδ− (ω)) = lδ+ − lδ− + Rδ+ (ω) − Rδ− (ω) = 0,
in Ω \ BZ,δ .
So, we can apply Lemma 3.1 to obtain
¡
¢
k(Qδ Lδ )−1 Qδ (lδ+ − lδ− ) + Qδ (Rδ+ (ω) − Rδ− (ω)) k∞
≤
C| ln δ|
δ
(p−1)2
p
2
p
kQδ (lδ+ − lδ− ) + Qδ (Rδ+ (ω) − Rδ− (ω))kp .
Thus, for any ω ∈ M ,
¡
¢
kGδ (ω)k∞ = k(Qδ Lδ )−1 Qδ lδ+ − lδ− + Rδ+ (ω) − Rδ− (ω) k∞
(p−1)2
¡
¢
C| ln δ| p
≤
kQδ lδ+ − lδ− + Rδ+ (ω) − Rδ− (ω) kp .
2
(3.18)
δp
, corresponding to u ∈ L∞ (Ω), satisfies
It follows from (3.3)–(3.6) that the constant b±
k,ĥ
Ã
|b±
| ≤ C| ln δ|p+1
k,ĥ
!
+ ¯
− ¯
X Z ¯¯ ∂Pδ,Z,i
X Z ¯¯ ∂Pδ,Z,j
¯
¯
¯
¯
+ ¯|u| +
− ¯|u| .
∂z
∂z
Ω
Ω
i,h
j,
h̄
i, h
j, h̄
Since
lδ+ − lδ− + Rδ+ (ω) − Rδ− (ω) = 0,
in Ω \ Bδ,Z ,
we deduce that the constant b±
, corresponding to lδ+ − lδ− + Rδ+ (ω) − Rδ− (ω), satisfies
k,ĥ
22
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI

|b±
| ≤C| ln δ|p+1
k,ĥ
X

α=1
i, h
+ C| ln δ|p+1
m Z
X

X

≤Cε
p
| ln ε|
klδ+
δ,α
n Z
X
β=1
j, h̄
1− p2
BLs+
−

¯ ∂P + ¯
¯ δ,Z,i ¯ +
−
+
−

¯
+ ¯|lδ − lδ + Rδ (ω) − Rδ (ω)|
+
∂z
(zα )
i,h
lδ−
+
BLs−
δ,β

¯ ∂P − ¯
¯ δ,Z,j ¯ +
−
+
−

¯
− ¯|lδ − lδ + Rδ (ω) − Rδ (ω)|
−
∂z
(zβ )
j,h̄
Rδ+ (ω)
− Rδ− (ω)kp .
As a consequence,
kQδ (lδ+ − lδ− + Rδ+ (ω) − Rδ− (ω))kp
≤klδ+ − lδ− + Rδ+ (ω) − Rδ− (ω)kp + C
X
i, h
°
°
°
³ ∂P + ´°
°
°
δ,Z,i
2
|b+
°
i,h | °−δ ∆
+
°
∂zi,h °
°
°
°
³ ∂P − ´°
X
°
δ,Z,j °
+C
|b−
| −δ 2 ∆
°
j,h̄ °
°
∂zj,−h̄ °
j, h̄
p
¡ +
¢
−
+
≤C klδ kp + klδ kp + kRδ (ω)kp + kRδ− (ω)kp .
p
(3.19)
To estimate kGδ (ω)k∞ , by (3.18) it suffices to estimate each term in the right hand side
of (3.19). From Lemma A.1 and (2.15), we have
°
°
¶p X
µ
m ³
m
°X
´p °
2πq(x)
°
°
+
−
−
Wδ,zi+ ,a+ − a+
klδ+ kp = °
χΩ+i Pδ,Z
− Pδ,Z
− κ+
°
i −
δ,i
δ,i
°
| ln ε| + i=1
+°
i=1
p
m
°
X
¢p−1 °
Cs+
°
δ,i °¡
≤
° Wδ,zi+ ,a+δ,i − a+
δ,i + °
| ln ε|
p
i=1
!
Ã
2
δ 1+ p
.
=O
p−1
1
| ln δ| 2 + p
REGULARIZATION OF POINT VORTICES
23
For the estimate of kRδ+ (ω)kp , we have
kRδ+ (ω)kp
° m
·³
°X
2πq(x) ´p
+
−
+
°
=°
χΩ+i Pδ,Z − Pδ,Z + ω − κi −
| ln ε| +
i=1
³
2πq(x) ´p
+
−
+
− Pδ,Z − Pδ,Z − κi −
| ln ε| +
¸°
³
2πq(x) ´p−1 °
+
−
+
− p Pδ,Z − Pδ,Z − κi −
ω °
°
| ln ε| +
p
°
µ
¶p−2 °
m
°X
°
2πq(x)
°
+
−
+
2 °
≤Ckωk∞ °
χΩ+i Pδ,Z − Pδ,Z − κi −
°
°
| ln ε| + °
i=1
p
!
à 2
2
δ p kωk∞
.
=O
1
| ln δ|p−3+ p
Similarly,
Ã
klδ− kp = O
2
δ 1+ p
| ln δ|
p−1
+ p1
2
!
Ã
,
kRδ− (ω)kp = O
(3.20)
!
2
δ p kωk2∞
1
| ln δ|p−3+ p
.
Thus,
kGδ (ω)k∞ ≤
C| ln δ|
(p−1)2
p
2
δp
≤C| ln δ|
≤δ| ln δ|
(p−1)2
p
p−1
2
³
klδ+ kp + klδ− kp + kRδ+ (ω)kp + kRδ− (ω)kp
Ã
!
δ
kωk2∞
p−1
1
1 +
| ln δ|p−3+ p
| ln δ| 2 + p
´
.
Step 2. Gδ is a contraction map.
In fact, for any ωi ∈ M , i = 1, 2, we have
£
¤
Gδ ω1 − Gδ ω2 = (Qδ Lδ )−1 Qδ Rδ+ (ω1 ) − Rδ+ (ω2 ) − (Rδ− (ω1 ) − Rδ− (ω2 )) .
Noting that
Rδ+ (ω1 ) = Rδ+ (ω2 ) = 0,
+
in Ω \ ∪m
i=1 BLs+ (zi ),
Rδ− (ω1 ) = Rδ− (ω2 ) = 0,
in Ω \ ∪nj=1 BLs− (zj− ),
δ,i
and
δ,j
(3.21)
24
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
we can deduce as in Step 1 that
C| ln δ|
(p−1)2
p
(kRδ+ (ω1 ) − Rδ+ (ω2 )kp + kRδ− (ω1 ) − Rδ− (ω2 )kp )
δ
¶
µ
kω1 k∞
kω2 k∞
p−1
≤C| ln δ|
+
kω1 − ω2 k∞
| ln δ|p−2 | ln δ|p−2
p+1
1
≤Cδ| ln δ| 2 kω1 − ω2 k∞ ≤ kω1 − ω2 k∞ .
2
kGδ ω1 − Gδ ω2 k∞ ≤
2
p
By Step 1 and Step 2, Gδ is a contraction map from M to M and thus there is a unique
p−1
ωδ ∈ M such that ωδ = Gδ ωδ . Moreover, from (3.21), kωδ k∞ ≤ δ| ln δ| 2 .
¤
4. Proof of The main results
In this section, we will give proofs for our main results Theorem 1.1 and Theorem
1.7. As the first step, we need to show Theorem 1.4. First we will choose Z, such that
+
−
Pδ,Z
− Pδ,Z
+ ωδ is a solution of (1.9), where ωδ is the map obtained in Proposition 3.3.
Define
δ2
I(u) =
2
Z
2
|Du| −
Ω
−
m
X
i=1
n
X
j=1
1
p+1
1
p+1
µ
Z
Ω
χΩ+i u −
µ
Z
Ω
χΩ−j
κ+
i
2πq(x)
−
| ln ε|
¶p+1
2πq(x)
− κ−
j −u
| ln ε|
+
¶p+1
,
+
and set
¡ +
¢
−
K(Z) = I Pδ,Z
− Pδ,Z
+ ωδ .
(4.1)
+
−
It is well-known that if Z is a critical point of K(Z), then Pδ,Z
− Pδ,Z
+ ωδ is a solution of
(1.9). We will prove that K(Z) has a critical point. To do this we need two preliminary
lemmas, which together with estimates in the Appendix, give expansions of K(Z), ∂K(Z)
∂z +
i,h
and
∂K(Z)
−
∂zj,
h̄
respectively.
Lemma 4.1. We have
K(Z) = I
¡
+
Pδ,Z
−
−
Pδ,Z
¢
µ
+O
ε3
| ln ε|p
¶
.
REGULARIZATION OF POINT VORTICES
+
Proof. Recalling that Pδ,Z
=
K(Z) = I
¡
+
Pδ,Z
−
Pm
i=1
−
Pδ,Z
¢
Z
+δ
2
Z
m
X
Pn
+
−
Pδ,Z,i
and Pδ,Z
=
D
Ω
"µ
¡
+
Pδ,Z
−
j=1
−
Pδ,Z
¢
25
−
Pδ,Z,j
, we have
δ2
Dωδ +
2
Z
|Dωδ |2
Ω
1
+
−
−
χΩ+i
Pδ,Z
− Pδ,Z
+ ωδ − κ+
i −
p+1 Ω
i=1
¶p+1 #
µ
2πq(x)
−
+
− κ+
− Pδ,Z
− Pδ,Z
i −
| ln ε| +
"
µ
Z
n
X
1
−
+
−
χΩ−j
Pδ,Z
− Pδ,Z
− ωδ − κ−
j +
p
+
1
Ω
j=1
µ
¶p+1 #
2πq(x)
−
+
− Pδ,Z
− κ−
.
− Pδ,Z
j +
| ln ε| +
2πq(x)
| ln ε|
2πq(x)
| ln ε|
¶p+1
+
¶p+1
+
Using Proposition 3.3 and (3.17), we find
"µ
Z
Ω+
i
+
−
Pδ,Z
− Pδ,Z
"µ
Z
=
BLs+ (zi+ )
δ,i
Ã
=O
2πq(x)
+ ωδ − κ+
i −
| ln ε|
+
−
Pδ,Z
− Pδ,Z
2
(s+
δ,i ) kωδ k∞
| ln ε|p
!
=O
¶p+1
µ
+
−
− Pδ,Z
− Pδ,Z
+
2πq(x)
+ ωδ − κ+
i −
| ln ε|
¶p+1
2πq(x)
− κ+
i −
| ln ε|
µ
+
−
− Pδ,Z
− Pδ,Z
+
On the other hand,
Z
δ
Ω
=
+
DPδ,Z
Dωδ
i=1
m Z
X
i=1
=
m Z ³
X
Bs+ (zk+ )
δ,k
Ω
Wδ,zi+ ,a+ − a+
δ,i
δ,i
p
(Wδ,zi+ ,a+ − a+
δ,i )+ ωδ = O
δ,i
´p
+
+
2πq(x)
− κ+
i −
| ln ε|
³ ε3 ´
.
| ln ε|p
2
¶p+1 #
ωδ
³ ε3 ´
.
| ln ε|p
¶p+1 #
+
26
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
Next, we estimate δ 2
2
−δ ∆ωδ =
m
X
i=1
R
Ω
|Dωδ |2 . Note that
µ
χΩ+i
+
Pδ,Z
−
−
Pδ,Z
+ ωδ −
κ+
i
2πq(x)
−
| ln ε|
¶p
−
+
¶p
µ
m ³
X
i=1
Wδ,zi+ ,a+ −
δ,i
a+
δ,i
´p
+
n ³
´p
X
2πq(x)
+
Wδ,zj− ,a− − a−
δ,j
δ,j
| ln ε| + j=1
+
j=1
Ã
!
Ã
!
2
2
m X
n X
+
−
X
X
∂Pδ,Z,i
∂Pδ,Z,j
+
−
2
2
bi,h −δ ∆
+
+
bj,h̄ −δ ∆
.
+
−
∂z
∂z
i,h
j,
h̄
i=1 h=1
j=1
−
n
X
−
+
χΩ−j Pδ,Z
− Pδ,Z
− ωδ − κ−
j +
h̄=1
Hence, by (2.15)–(2.16), we have
#
"µ
¶p ³
Z
m Z
´p
X
2πq(x)
+
−
δ 2 |Dωδ |2 =
Pδ,Z
− Pδ,Z
+ ωδ − κ+
− Wδ,zi+ ,a+ − a+
ωδ
i −
δ,i
δ,i
+
|
ln
ε|
+
Ω
+
i=1 Ωi
"µ
#
¶p ³
n Z
´p
X
2πq(x)
−
+
−
− Pδ,Z
− ωδ − κ−
− Wδ,zj− ,a− − a−
ωδ
Pδ,Z
j +
δ,j
δ,j
−
|
ln
ε|
+
Ω
+
j
j=1
!
!
Z Ã
Z Ã
n X
2
m X
2
−
+
X
X
∂Pδ,Z,j
∂Pδ,Z,i
+
−
2
2
bi,h
ωδ +
bj,h̄
−δ ∆
ωδ
+
−δ ∆
+
∂zi,h
∂zj,−h̄
Ω
Ω
j=1 h̄=1
i=1 h=1
!
à m 2
à +
!
m Z
+
³
´p−1
X
X
X
ε|b
|kω
k
s
δ
∞
i,h
δ,i
=p
Wδ,zi+ ,a+ − a+
+ ωδ ωδ + O
δ,i
δ,i
+
|
ln
ε|
| ln ε|p
+
i=1 Ωi
i=1 h=1
à −
!
n Z
³
´p−1
X
s
δ,j
Wδ,zj− ,a− − a−
−p
+ ωδ ωδ
δ,j
δ,j
−
|
ln
ε|
+
Ω
j
j=1
à n 2
!
X X ε|b−
|kω
k
δ
∞
j,h̄
+O
| ln ε|p
j=1 h̄=1
µ
¶
ε4
=O
.
| ln ε|p−1
Other terms can be estimated as above. So our assertion follows.
¤
Lemma 4.2. We have
³ ε3 ´
¡ +
¢
∂K(Z)
∂
−
=
I
P
−
P
+
O
, i = 1, · · · , m,
δ,Z
δ,Z
+
+
| ln ε|p−1
∂zi,h
∂zi,h
³ ε3 ´
¡ +
¢
∂K(Z)
∂
−
, j = 1, · · · , n.
=
I
P
−
P
+
O
δ,Z
δ,Z
| ln ε|p−1
∂zj,−h̄
∂zj,−h̄
REGULARIZATION OF POINT VORTICES
27
Proof. We give her the proof of the first one only. The second one can be proved similarly.
By the definition, we have
+
−
´ ∂P +
∂Pδ,Z
∂ωδ
δ,Z
I
−
+ ωδ ,
+ −
+ +
+
∂zi,h
∂zi,h
∂zi,h
à +
*
+
!
Z
−
´
¢
¡
∂P
∂P
∂ω
∂ ³ +
δ
δ,Z
δ,Z
−
−
+
+ ωδ , + + δ 2 Dωδ D
− Pδ,Z
+ I 0 Pδ,Z
= + I Pδ,Z − Pδ,Z
+ −
+
∂zi,h
∂zi,h
∂zi,h
∂zi,h
Ω
"µ
¶p µ
¶p #
m Z
X
2πq(x)
2πq(x)
−
+
−
+
+ ωδ − κ+
− Pδ,Z
− κ+
−
Pδ,Z
− Pδ,Z
− Pδ,Z
k −
k −
+
|
ln
ε|
| ln ε| +
+
k=1 Ωk
à +
!
−
∂Pδ,Z
∂Pδ,Z
×
+ −
+
∂zi,h
∂zi,h
"µ
¶p µ
¶p #
n Z
X
2πq(x)
2πq(x)
+
+
−
−
− κ−
− ωδ − κ−
− Pδ,Z
− Pδ,Z
−
− Pδ,Z
Pδ,Z
l +
l +
−
|
ln
ε|
| ln ε| +
Ω
+
l
l=1
à −
!
+
∂Pδ,Z
∂Pδ,Z
×
−
.
+
+
∂zi,h
∂zi,h
∂K(Z)
=
+
∂zi,h
*
0
³
+
Pδ,Z
−
Pδ,Z
Since ωδ ∈ Eδ,Z , we have
Ã
!
Z ³
±
´p−1 ∂Wδ,z± ,a±
∂a
δ,k
k
δ,k
Wδ,z± ,a± − a±
− ± ωδ = 0.
δ,k
±
k
δ,k
+
∂z
∂z
Ω
k,h
k,h
+
Differentiating the above relation with respect to zi,h
, we deduce
*
+
¢
∂ω
δ
+
−
I Pδ,Z
− Pδ,Z
+ ωδ , +
∂zi,h
Ã
!
!
Z Ã
Z
m X
2
n X
2
−
+
X
X
∂P
∂P
∂ω
∂ωδ
δ
δ,Z,β
δ,Z,α
−
2
2
=
b+
−δ
∆
+
b
−δ
∆
+
+
−
+
β,
h̃
α,ĥ
∂zα,ĥ
∂zi,h β=1
∂zβ,h̃
∂zi,h
Ω
Ω
α=1 ĥ=1
h̃=1
Ã
!
Z ³
m X
2
+
´p−1 ∂Wδ,zα+ ,a+
X
∂a
∂ωδ
δ,α
δ,α
=
pb+
Wδ,zα+ ,a+ − a+
− +
δ,α
+
+
α,ĥ
δ,α
+
∂zα,ĥ
∂zα,ĥ ∂zi,h
Ω
α=1 ĥ=1
!
Ã
Z ³
n X
2
´p−1 ∂Wδ,z− ,a−
X
∂a−
∂ωδ
δ,β
β
δ,β
−
−
− −
+
pbβ,h̃
Wδ,z− ,a− − aδ,β
−
+
β
δ,β
+
∂zβ,h̃
∂zβ,h̃ ∂zi,h
Ω
β=1 h̃=1


µ
¶
m X
2 ε|b+ |
n X
2 ε|b− |
3
X
X
ε
α,
ĥ
β,
h̃
=O
.
=O 
+
p
p
p−1
|
ln
ε|
|
ln
ε|
|
ln
ε|
α=1
β=1
¡
0
ĥ=1
h̃=1
28
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
On the other hand, using (3.20) (for the definition of Rδ+ (ω), see (3.14)), we obtain
m Z
X
"µ
+
k=1 Ωk
"µ
m Z
X
+
−
Pδ,Z
− Pδ,Z
2πq(x)
+ ωδ − κ+
k −
| ln ε|
¶p
µ
+
−
− Pδ,Z
− Pδ,Z
+
2πq(x)
− κ+
k −
| ln ε|
¶p #
+
+
∂Pδ,Z,i
+
∂zi,h
¶p µ
¶p
2πq(x)
2πq(x)
+
−
+
−
+ ωδ −
=
−
− Pδ,Z − Pδ,Z − κk −
+
| ln ε| +
| ln ε| +
k=1 Ωk
#
µ
¶p−1
+
∂Pδ,Z,i
2πq(x)
+
−
+
− p Pδ,Z − Pδ,Z − κk −
ωδ
+
| ln ε| +
∂zi,h
# +
¶p−1
Z "µ
m
X
¡
¢
∂Pδ,Z,i
2πq(x)
p−1
+
−
+
+ + − a
+
p
− Pδ,Z
− κ+
−
Pδ,Z
−
W
δ,zk ,aδ,k
k
δ,k +
+ ωδ
| ln ε| +
∂zi,h
Ω+
k
k=1
à +
!
(sδ,k )2 kωδ k∞
+O
| ln ε|p
¶p−1
Z "µ
Z
m
+
X
∂P
2πq(x)
δ,Z,i
+
−
+
+
p
Pδ,Z − Pδ,Z − κk −
= Rδ (ωδ )
+ +
+
| ln ε| +
∂zi,h
Ω
Ω
k
k=1
# +
µ 3 ¶
¡
¢
∂Pδ,Z,i
ε
+ p−1
− Wδ,z+ ,a+ − aδ,k +
ωδ + O
+
k
δ,k
| ln ε|p
∂zi,h
¶
µ
ε3
.
=O
| ln ε|p−1
+
Pδ,Z
κ+
k
−
Pδ,Z
In addition, we have
¶p µ
¶p # −
∂Pδ,Z,i
2πq(x)
2πq(x)
+
−
+
−
+ ωδ −
− Pδ,Z − Pδ,Z − κl −
−
−
| ln ε| +
| ln ε| + ∂zi,h
Ω+
l
¶p−1 −
µ
¶
Z µ
∂Pδ,Z,i
2πq(x)
ε4
+
−
+
=p
Pδ,Z − Pδ,Z − κl −
− ωδ + O
|
ln
ε|
| ln ε|p−1
∂z
Ω+
i,h
+
l
µ 3 ¶
ε
=O
.
| ln ε|p
Z
"µ
+
Pδ,Z
−
Pδ,Z
κ+
l
Other teams can be estimated as above. Thus, the estimate follows.
¤
REGULARIZATION OF POINT VORTICES
29
+
Proof of Theorem 1.4. Recall that Z = (Zm
, Zn− ). Set
+
Φ(Zm
, Zn− )
=
m
X
+
4π 2 κ+
i q(zi )
i=1
+
−
4π 2 κ−
j q(zj )
+
m
X
j=1
n
X
− −
2
π(κ−
j ) g(zj , zj ) −
j=1
+
−
n
X
+ +
2
π(κ+
i ) g(zi , zi )
i=1
X
+
+ +
πκ+
i κk Ḡ(zi , zk ) −
i6=k
n
m X
X
X
−
− −
πκ−
j κl Ḡ(zl , zj )
j6=l
+ −
−
2πκ+
i κj Ḡ(zi , zj ).
i=1 j=1
Note that the Kirchhoff–Routh function associated to the vortex dynamics now is
+
, Zn− )
W(Zm
m
n
1 X − −
1 X
+ +
+ +
=
κ κ G(zi , zk ) +
κ κ G(zj− , zl− )
2 i,k=1,i6=k i k
2 j,l=1,j6=l j l
m
n
1X − 2
1X + 2
+
(κi ) H(zi+ , zi+ ) +
(κ ) H(zj− , zj− )
2 i=1
2 j=1 j
−
m X
n
X
−
+ −
κ+
i κj G(zi , zj )
i=1 j=1
+
m
X
+
κ+
i ψ0 (zi )
−
i=1
Recalling that h(zi , zj ) = −H(zi , zj ), we get
+
+
Φ(Zm
, Zn− ) = −4π 2 W(Zm
, Zn− ) + π ln R
n
X
−
κ−
j ψ0 (zj ).
j=1
à m
X
2
(κ+
i ) +
i=1
n
X
!
2
(κ−
j )
.
j=1
+
+
Hence, Φ(Zm
, Zn− ) and W(Zm
, Zn− ) possess the same critical points.
By Lemma 4.1, 4.2 and Proposition A.2, A.3, we have
à m
!
µ 2
¶
n
Cδ 2 π(p − 1)δ 2 X + 2 X − 2
δ ln | ln ε|
δ2
K(Z) = R +
Φ(Z)+O
(κi ) +
(κj ) +
, (4.2)
| ln ε|2
| ln ε|3
ln ε
4(ln Rε )2
i=1
j=1
and
4π 2 δ 2 ∂W(Z)
∂K(Z)
=
−
+O
±
±
| ln ε|2 ∂zi,h
∂zi,h
µ
δ 2 ln | ln ε|
| ln ε|3
¶
.
(4.3)
−
+
−
1
Thus, suppose that (x+
1, ∗ , · · · , xm, ∗ , x1, ∗ , · · · , xn, ∗ ) is a C -stable critical point of Kirchhoff−
+
−
Routh function W(Z), then K(Z) has a critical point (x+
1, ε , · · · , xm, ε , x1, ε , · · · , xn, ε ) =
−
+
−
(x+
1, ∗ , · · · , xm, ∗ , x1, ∗ , · · · , xn, ∗ ) + o(1).
³
´ 1−p
2
| ln ε|
| ln ε|
, it is not difficult
Thus we get a solution wδ for (1.9). Let uε = 2π wδ , δ = ε 2π
to check that uε has all the properties listed in Theorem 1.4 and thus the proof of Theorem
1.4 is complete.
¤
Now we are in the position to prove Theorem 1.1.
30
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
Proof of Theorem 1.1. By Theorem 1.4, we obtain that uε is a solution to (1.8).
Set
vε = J∇(uε − q),
ωε = ∇ × vε ,
µ
¶p+1
m
X 1
κ+
i | ln ε|
Pε =
χΩ+i uε − q −
p
+
1
2π
+
i=1
Ã
!p+1
n
X
κ−
1
1
j | ln ε|
+
χΩ−j q −
− uε
− |∇(uε − q)|2 .
p+1
2π
2
j=1
+
Then (vε , Pε ) forms a stationary solution for problem (1.1).
From our proof of Theorem 1.4, we see that as ε → 0
+
−
−
+
+
−
−
(x+
1, ε , · · · , xm, ε , x1, ε , · · · , xn, ε ) → (x1,∗ , · · · , xm,∗ , x1,∗ , · · · , xn,∗ ).
So we can find a positive constant C independent of ε so small that for small ε,
+
−
−
B(x+
i, ε , Cε) ⊆ Ωi , B(xj, ε , Cε) ⊆ Ωj for i = 1, · · · , m and j = 1, · · · , n.
We now verify that as ε → 0
Z
ωi,+ε → κ+
(4.4)
i , i = 1, · · · , m,
Z
B(x+
i, ε ,Cε)
B(x−
j, ε ,Cε)
and
ωi,−ε → −κ−
j , j = 1, · · · , n,
Z
ωε →
Ω
m
X
κ+
j
j=1
−
n
X
(4.5)
κ−
j .
(4.6)
j=1
By direct calculations, we have
µ
¶p
Z
Z
1
κ+
i | ln ε|
+
ωi, ε = 2
χ + uε − q −
ε Ω Ωi
2π
B(x+
+
i, ε ,Cε)
µ
¶p
Z
p
| ln ε|
2πq
=
wδ − κ+
i −
p
2
(2π) ε Ω+i
| ln ε| +
Ã
!
Z
³ s+ ´ p
| ln ε|p
δ,i
=
Wδ,zi+ ,a+ − a+
δ,i + O
δ,i
(2π)p ε2 B + +
| ln ε|
Ls
2
p
(s+
δ,i ) | ln ε|
=
(2π)p ε2
=
a+
δ,i | ln ε|
ln sR+
δ,i
→ κ+
i
δ,i
(z
Ã
i
δ
s+
δ,i
+ o(1)
as ε → 0.
+
)
2p
! p−1
Z
φp
B1 (0)
REGULARIZATION OF POINT VORTICES
31
Therefore, (4.4) follows. (4.5) can be proved exactly the same way. (4.6) follows directly
from (4.4) and (4.5). Therefore we complete our proof.
¤
Proof of Theorem 1.7. Theorem 1.7 can be proved simply by exactly the same arguments
−
above and taking κ−
≡ 0 for j = 1, · · · , n.
j = 0, Ωj = Ø and χΩ−
j
¤
Acknowledgements: Z. Liu was supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS. D.Cao was partially supported by Science Fund
for Creative Research Groups of NSFC(No.10721101)and NSFC grants(No.11271354 and
No.11331010). Both D. Cao and J. Wei were also supported by CAS Croucher Joint Laboratories Funding Scheme.
Appendix A. Energy expansion
¢
¡ +
−
and
− Pδ,Z
In this appendix, we give precise expansions of I Pδ,Z
which have been used in section 3 and section 4.
We always assume that zi+ , zj− ∈ Ω satisfy
d(zi+ , ∂Ω) ≥ %, d(zj− , ∂Ω) ≥ %,
|zj−
−
zl− |
L̄
≥% ,
|zi+
−
zj− |
|zi+ − zk+ | ≥ %L̄ ,
L̄
≥% ,
∂
± I
∂zi,h
¡
¢
+
−
− Pδ,Z
,
Pδ,Z
i, k = 1, · · · , m, i 6= k
j, l = 1, · · · , n, j 6= l,
where % > 0 is a fixed small constant and L̄ > 0 is a fixed large constant.
−
Lemma A.1. For x ∈ Ω+
i (i = 1, · · · , m) and x ∈ Ωj (j = 1, · · · , n), we have when ε > 0
small
2πq(x)
+
−
Pδ,Z
(x) − Pδ,Z
, x ∈ Bs+ (1−T s+ ) (zi+ ),
(x) > κ+
i +
δ,i
δ,i
| ln ε|
2πq(x)
−
+
Pδ,Z
, x ∈ Bs− (1−T s− ) (zj− ),
(x) − Pδ,Z
(x) > κ−
j −
δ,j
δ,j
| ln ε|
where T > 0 is a large constant; while
2πq(x)
,
| ln ε|
2πq(x)
−
+
Pδ,Z
(x) − Pδ,Z
(x) < κ−
,
j −
| ln ε|
where σ > 0 is a small constant.
+
−
Pδ,Z
(x) − Pδ,Z
(x) < κ+
i +
+
x ∈ Ω+
i \ Bs+ (1+(s+ )σ ) (zi ),
δ,i
x ∈ Ω−
j \ Bs−
δ,i
− σ
δ,j (1+(sδ,j ) )
(zj− ),
Proof. Suppose that x ∈ Bs+ (1−T s+ ) (zi+ ). It follows from (2.15) and φ01 (s) < 0 that
δ,i
δ,i
à + !
sδ,i
2πq(x)
+
−
+
+ + (x) − a
Pδ,Z
(x) − Pδ,Z
(x) − κ+
−
=
W
+
O
i
δ,zi ,aδ,i
δ,i
| ln ε|
| ln ε|
³ |x − z + | ´
³ ε ´
a+
δ,i
i
φ
> 0,
+
O
= 0
| ln ε|
s+
|φ (1)|| ln sR+ |
δ,i
δ,i
32
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
+
if T > 0 is large. On the other hand, if x ∈ Ω+
σ̃ (zi ), where σ̃ > σ > 0 is a fixed
i \ B(s+
δ,i )
small constant, then
+
−
Pδ,Z
(x) − Pδ,Z
(x) − κ+
i −
≤
m
X
a+
δ,i ln
i=1
2πq(x)
| ln ε|
2πq(x)
R
R
+
+ o(1)
+ / ln + − κi −
| ln ε|
|x − zi |
sδ,i
≤C σ̃ − κ+
i + o(1) < 0.
Finally, if x ∈ B(s+ )σ̃ (zi+ ) \ Bs+ (1+T (s+ )σ̃ ) (zi+ ) for some i and if T > 0 is large then
δ,i
δ,i
δ,i
+
−
(x) − Pδ,Z
(x) − κ+
Pδ,Z
i −
2π
q(x)
| ln ε| Ã
= Wδ,zi+ ,a+ (x) − a+
δ,i + O
δ,i
Ã
R
= a+
δ,i
≤
ln
|x−z + |
i
ln R
s+
δ,i
ln
ln
Ã
s
< 0.
+O
R
s+
δ,i
!
σ̃
(s+
δ,i )
− a+
δ,i + O
σ̃
ln(1+T (s+
δ,i ) )
−a+
R
δ,i
ln +
!
σ̃
(s+
δ,i )
R
s+
δ,i
!
σ̃
(s+
δ,i )
ln
δ,i
R
s+
δ,i
Note that by the choice of σ̃, Bs+ (1+(s+ )σ ) (zi+ ) ⊃ Bs+ (1+T (s+ )σ̃ ) (zi+ ) for small δ. We
δ,i
δ,i
δ,i
δ,i
therefore derive our conclusion.
¤
Proposition A.2. We have when ε > 0 small
I
¡
+
Pδ,Z
−
−
Pδ,Z
¢
Cδ 2 π(p − 1)δ 2
= R +
ln ε
4(ln Rε )2
−
−
+
à m
X
2
(κ+
i )
+
n
X
!
2
(κ−
j )
+
m
X
4π 2 δ 2 κ+ q(z + )
i
i=1
j=1
i=1
m
n
−
2 2 −
+
+
+
2
2
4π δ κj q(zj ) X πδ (κi ) g(zi , zi ) X
+
+
| ln ε|| ln Rε |
(ln Rε )2
i=1
j=1
j=1
m
n
−
−
−
−
2
+
+
+
+
2
X πδ κ κ Ḡ(z , z ) X πδ κj κl Ḡ(zl , zj )
i k
i
k
−
R 2
(ln ε )
(ln Rε )2
k6=i
l6=j
n
X
m X
n
−
+ −
X
2πδ 2 κ+
i κj Ḡ(zi , zj )
i=1 j=1
(ln Rε )2
where C is a positive constant.
Proof. Taking advantage of (2.4), we have
µ
+O
δ 2 ln | ln ε|
| ln ε|3
i
| ln ε|| ln Rε |
¶
,
− −
2
πδ 2 (κ−
j ) g(zj , zj )
(ln Rε )2
REGULARIZATION OF POINT VORTICES
Z
δ
2
Ω
+
m X
m
X
¯
¯
¯D(P + − P − )¯2 =
δ,Z
δ,Z
¡
Ω
Wδ,z− ,a− −
l
δ,l
¡
Wδ,z+ ,a+ − a+
δ,k
k
Ω
k=1 i=1
n Z
n X
X
l=1 j=1
Z
¢p −
a−
δ,l + Pδ,Z,j
−2
¢p
δ,k
n X
m Z
X
¡
+
Pδ,Z,i
Wδ,zi+ ,a+ − a+
δ,i
¢p
δ,i
Ω
j=1 i=1
+
33
+
−
Pδ,Z,j
.
First, we estimate
Z
³
δ,i
Bs+ (zi+ )
δ,i
Z
=
¡
−
R
s+
δ,i
+


Wδ,z+ ,a+ −
i
¢p+1
a+
δ,i
δ,i
Z
+
a+
δ,i
a+
δ,i
ln sR+
δ,i
¡
Bs+ (zi+ )
g(x, zi+ )
Wδ,zi+ ,a+ − a+
δ,i
¢p
δ,i
δ,i
Z
a+
δ,i
ln
Wδ,zi+ ,a+ −
δ,i
Bs+ (zi+ )
δ,i
Wδ,zi+ ,a+ − a+
δ,i
´p
¡
¢p
+
Wδ,zi+ ,a+ − a+
δ,i g(x, zi )
δ,i
Bs+ (zi )
δ,i
Z
Z
2p
³ δ ´ 2(p+1)
³ δ ´ p−1
p−1
+ 2
+ 2
+
p+1
(sδ,i )
(sδ,i )
= +
φ
+ aδ,i +
φp
sδ,i
s
B1 (0)
B1 (0)
δ,i
!
à +
Z
+ ³
2p
3
´
aδ,i
(s
)
δ p−1
δ,i
2
−
g(zi+ , zi+ )(s+
φp + O
δ,i )
+
R
| ln ε|p+1
ln s+ sδ,i
B1 (0)
δ,i
µ
¶
+
+
+
ε3
π(p + 1) δ 2 (aδ,i )2 2πδ 2 (aδ,i )2 2πδ 2 (aδ,i )2
+ +
=
+
−
g(zi , zi ) + O
.
2
| ln ε|p+1
(ln sR+ )2
ln sR+
(ln sR+ )2
δ,i
δ,i
δ,i
Next, for k 6= i,

Z
¡
Bs+ (zk+ )
δ,k
Wδ,z+ ,a+ − a+
δ,k
k
δ,k
Z
2p
³ δ ´ p−1
a+
δ,i
= +
sδ,k
ln sR+ B +
δ,i
³ δ ´
= +
sδ,k
=
2p
p−1
δ,k
+
φp
Wδ,z+ ,a+ −
i
δ,i
³ |x − z + | ´
s+
δ,k
k
Z
+ 2
a+
δ,i (sδ,k )
+ +
Ḡ(zk , zi )
ln sR+
B1 (0)
δ,i
+
2πδ 2 a+
δ,i aδ,k
| ln sR+ || ln sR
+ |
δ,i
s
(zk+ )
¢p

δ,k
µ
Ḡ(zi+ , zk+ )
+O
a+
δ,i
ln sR+
δ,i
g(x, zi+ )
Ḡ(x, zi+ )
Ã
φp + O
ε3
| ln ε|p+1
¶
.
3
(s+
δ,k )
| ln ε|p+1
!
34
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
Moreover, we have

Z
¡
¢
+ p
Wδ,zi+ ,a+ − aδ,i
+
δ,i
Bs+ (zi+ )
δ,i

Wδ,z− ,a− −
j
δ,j
a−
δ,j
ln
R
s−
δ,j
g(x, zj− )
Z
2p
³ δ ´ p−1
³
+ ´
a−
δ,j
p |x − zi |
= +
Ḡ(x, zj− )
φ
sδ,i
s+
ln sR− Bs+ (zi+ )
δ,i
δ,j
δ,i
à +
!
Z
+ 2
2p
3
³ δ ´ p−1
)
(s
a−
)
(s
δ,j δ,i
δ,i
= +
Ḡ(zj− , zi+ )
φp + O
R
|
ln
ε|p+1
sδ,i
ln s−
B1 (0)
δ,j
µ
¶
−
2πδ 2 a+
ε3
δ,i aδ,j
+ −
=
.
Ḡ(zi , zj ) + O
| ln ε|p+1
| ln sR+ || ln sR− |
δ,i
δ,j
By Lemma A.1 and (2.15),
µ
m Z
X
=
k=1 Ω
m Z
X
k=1
=
m
X
=
m
X
k=1
m
X
k=1
−
−
Pδ,Z
−
κ+
k
2πq(x)
−
| ln ε|
µ
BLs+ (zk+ )
δ,k
+
Pδ,Z
−
−
Pδ,Z
Ã
BLs+ (zk+ )
−
κ+
k
¶p+1
+
2πq(x)
−
| ln ε|
µ
Wδ,z+ ,a+
k
δ,k
¶p+1
+
s+
δ,k
− a+
+
O
δ,k
| ln ε|
¶!p+1
δ,k
Ã
k=1
=
k
m Z
X
k=1
=
χ Ω+
+
Pδ,Z
Ã
δ
s+
δ,k
δ
s+
δ,k
!
2(p+1)
p−1
Z
p+1
Bs+ (zk+ )
φ
s+
δ,k
δ,k
! 2(p+1)
p−1
π(p +
2
³ |x − z + | ´
k
Ã
Z
2
(s+
δ,k )
+
1) δ 2 (aδ,k )2
2
(ln sR
+ )
δ,k
φp+1 + O
B1 (0)
µ
+O
ε3
| ln ε|p+1
+
Ã
+O
3
(s+
δ,k )
| ln ε|p+1
¶
.
Other terms can be estimated as above. So, we have proved
3
(s+
δ,k )
| ln ε|p+1
!
!
REGULARIZATION OF POINT VORTICES
¡ +
I Pδ,Z
35


m
+ + 2 + 2
2 + 2
2 + 2
X
¢
δ (aδ,i )
πδ (aδ,i )
πg(zi , zi )δ (aδ,i )
−
 π(p + 1)

− Pδ,Z
=
+
−
R 2
R
R 2
4
|
ln
|
ln
|
ln
+ |
+ |
+ |
i=1
sδ,i
sδ,i
sδ,i


n
− − 2 − 2
2 − 2
2 − 2
X
δ (aδ,j )
πδ (aδ,j )
πg(zj , zj )δ (aδ,j )
 π(p + 1)

+
+
−
R 2
R
R 2
4
|
ln
|
ln
|
ln
|
|
|
j=1
s−
s−
s−
δ,j
+
−
+
π Ḡ(zk+ , zi+ )δ 2 a+
δ,i aδ,k
| ln sR+ || ln sR
+ |
k6=i
δ,i
δ,k
+
−
π Ḡ(zl− , zj− )δ 2 a−
δ,l aδ,j
| ln sR− || ln sR− |
l6=j
δ,l
δ,j
−
2π Ḡ(zi+ , zj− )δ 2 a+
δ,i aδ,j
| ln sR+ || ln sR− |
i=1 j=1
δ,i
δ,j

n
−
πδ 2 X (aδ,j )2 
+O
R 2
2
|
ln
− |
j=1
s
δ,j
n
X
m X
n
X

−
δ,j
m
X
µ


m
+
2
πδ X (aδ,i ) 
−
2
| ln sR+ |2
i=1
2
δ,i
ε3
| ln ε|p+1
¶
.
δ,j
Thus, the result follows from Remark 2.2.
¤
Proposition A.3. We have
+ +
2
¡ +
¢
∂q(zi+ ) 2πδ 2 (κ+
∂
4π 2 δ 2 κ+
i ) ∂g(zi , zi )
i
−
I
P
−
P
+
=
δ,Z
δ,Z
+
+
+
∂zi,h
∂zi,h
| ln ε|| ln Rε | ∂zi,h
(ln Rε )2
µ 2
¶
m
n
+
+ +
−
+ −
X
X
2πδ 2 κ+
2πδ 2 κ+
δ ln | ln ε|
i κk ∂ Ḡ(zk , zi )
i κl ∂ Ḡ(zi , zl )
−
+
+O
,
+
+
R 2
R 2
3
|
ln
ε|
∂z
∂z
(ln
)
(ln
)
i,h
i,h
ε
ε
k6=i
l=1
− −
2
¡ +
¢
4π 2 δ 2 κ−
∂q(zj− ) 2πδ 2 (κ−
∂
j
j ) ∂g(zj , zj )
−
I
P
−
P
=
−
+
δ,Z
δ,Z
∂zj,−h̄
∂zj,−h̄
| ln ε|| ln Rε | ∂zj,−h̄
(ln Rε )2
−
n
−
− −
X
2πδ 2 κ−
j κl ∂ Ḡ(zl , zj )
l6=j
(ln Rε )2
∂zj,−h̄
Proof. Direct computation yields that
+
m
+
− +
X
2πδ 2 κ−
j κk ∂ Ḡ(zj , zk )
k=1
(ln Rε )2
∂zj,−h̄
+O
µ
δ 2 ln | ln ε|
| ln ε|3
¶
.
36
DAOMIN CAO, ZHONGYUAN LIU, AND JUNCHENG WEI
¡ +
¢
∂
−
+ I Pδ,Z − Pδ,Z
∂zi,h
·³
¶p ¸
m Z
+
´p µ
X
∂Pδ,Z
2πq(x)
+
−
+
+
=
Wδ,z+ ,a+ − aδ,k − Pδ,Z − Pδ,Z − κk −
+
k
δ,k
+
| ln ε| + ∂zi,h
+
k=1 B + (zk )
Ls
+
n Z
X
Wδ,z− ,a− −
l
δ,l
m Z
X
k=1
−
·³
BLs− (zl− )
l=1
−
δ,k
·³
BLs+ (zk+ )
δ,k
n Z
X
Wδ,z+ ,a+ −
k
δ,k
·³
Wδ,z− ,a− −
l
BLs− (zl− )
l=1
δ,l
a−
δ,l
δ,l
+
+
−
+
Pδ,Z
−
µ
´p
´p
−
Pδ,Z
−
+
a+
δ,k
a−
δ,l
µ
´p
−
+
− Pδ,Z
− Pδ,Z
κ−
l
−
−
+
Pδ,Z
¶p ¸
−
∂Pδ,Z
+
∂zi,h
¶p ¸
−
∂Pδ,Z
2πq(x)
+
− κk −
+
| ln ε| + ∂zi,h
µ
−
Pδ,Z
2πq(x)
+
| ln ε|
−
κ+
l
2πq(x)
+
| ln ε|
δ,l
+
¶p ¸
+
+
∂Pδ,Z
+
∂zi,h
For k 6= i, from (2.15), we have
·
¡
Z
BLs+ (zk+ )
δ,k
Z
=
k
"
=O
δ,k
3
δ,k
³
BLs+ (zk+ )
µ
Wδ,z+ ,a+ −
ε
| ln ε|p+1
Wδ,z+ ,a+
k
δ,k
¢p
a+
δ,k +
µ
−
+
Pδ,Z
−
−
Pδ,Z
−
#
´p−1 s+
C
δ,k
− a+
δ,k
| ln ε| ln sR+
δ,i
¶
.
Using (2.15), Lemma A.1 and Remark 2.2, we find that
κ+
k
2πq(x)
−
| ln ε|
¶p ¸
+
+
∂Pδ,Z,i
+
∂zi,h
.
REGULARIZATION OF POINT VORTICES
·³
Z
Wδ,zi+ ,a+ −
δ,i
BLs+ (zi+ )
δ,i
Z
·
¡
=
Bs+
Z
=p
δ,i
(1+(s+ )σ )
δ,i
¡
Bs+ (zi+ )
m
X
a+
δ,k
k6=i
ln sR
+
δ,k
µ
´p
+
Wδ,zi+ ,a+ −
δ,i
−
+
Pδ,Z
¢p
a+
δ,i +
−
−
Pδ,Z
−
κ+
i
µ
+
−
− Pδ,Z
− Pδ,Z
2πq(x)
−
| ln ε|
¶p ¸
+
∂Pδ,Z,i
+
∂zi,h
¶p ¸
+
∂Pδ,Z,i
2πq(x)
+
− κi −
+
| ln ε| + ∂zi,h
+
·
®
®
a+
2π ­
δ,i ­
Dq(zi+ ), x − zi+ +
Dg(zi+ , zi+ ), x − zi+
R
δ,i
| ln ε|
ln s+
δ,i
¸ +
n
−
® X aδ,l ­
® ∂Pδ,Z,i
­
+
+
+ +
+ −
DḠ(zi , zk ), x − zi +
DḠ(zi , zl ), x − zi
+
R
∂zi,h
ln
−
s
l=1
Wδ,zi+ ,a+ −
δ,i
−
(zi )
a+
δ,i
37
¢p−1
a+
δ,i +
δ,l
³ ε2+σ ´
+O
| ln ε|p+1
µ
m
+ +
+ +
X
pδ 2 a+
a+
a+
2π ∂q(zi+ )
δ,i
δ,i ∂g(zi , zi )
δ,k ∂ Ḡ(zi , zk )
=− 0
+
−
+
+
+
∂zi,h
∂zi,h
|φ (1)|| ln sR+ | | ln ε| ∂zi,h
ln sR+
ln sR
+
k6=i
δ,i
δ,i
δ,k
n
³ ε2+σ ´
+ − ¶Z
X
a−
x2h
δ,l ∂ Ḡ(zi , zl )
p−1
0
+
φ
(|x|)φ
(|x|)
+
O
+
|x|
| ln ε|p+1
∂zi,h
ln sR−
B1 (0)
l=1
δ,l
+
4π 2 δ 2 a+
δ,i ∂q(zi )
=
+
| ln ε|| ln sR+ | ∂zi,h
δ,i
+
+
2
+ +
2πδ 2 (a+
δ,i ) ∂g(zi , zi )
(ln sR+ )2
δ,i
n
X
−
+ −
2πδ 2 a+
δ,i aδ,l ∂ Ḡ(zi , zl )
+
∂zi,h
| ln sR− || ln sR+ |
l=1
δ,l
δ,i
+
∂zi,h
m
+
+ +
X
2πδ 2 a+
δ,i aδ,k ∂ Ḡ(zi , zk )
−
+
R
∂zi,h
| ln sR
+ || ln + |
s
k6=i
δ,k
δ,i
³ ε2+σ ´
,
+O
| ln ε|p+1
since
Z
x2h
2π
= − |φ0 (1)|.
φ (|x|)φ (|x|)
|x|
p
B1 (0)
Other terms can be estimated similarly. Thus, the result follows.
p−1
0
¤
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Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190,
P.R. China
E-mail address: dmcao@amt.ac.cn
Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190,
P.R. China
E-mail address: liuzy@amss.ac.cn
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong
Kong
E-mail address: wei@math.cuhk.edu.hk