COMPLEX HELE-SHAW MODEL. LOCAL SOLVABILITY FOR A … fileCOMPLEX HELE-SHAW MODEL. LOCAL SOLVABILITY FOR A DOUBLY

COMPLEX HELE-SHAW MODEL.LOCAL SOLVABILITY FOR A DOUBLY CONNECTED DOMAIN

SERGEI V. ROGOSIN AND TATYANA S. VAITEKHOVICH

Department of Mathematics and Mechanics, Belarusian State UniversityF. Skaryny av. 4, 220050 Minsk, Belarus

e-mail: [email protected]

Abstract. The paper is devoted to the study of the Hele-Shaw moving boundary valueproblem in doubly-connected domains. It is reformulated as a couple of problems, namely,the Schwarz boundary value problem for a doubly-connected domain and an abstractCauchy-Kovalevsky problem. Local existence and uniqueness of the classical solution tothe Hele-Shaw problem is shown by using a variant of the Schwarz alternation method forthe Schwarz problem and Nirenberg-Nishida theorem for the Cauchy-Kovalevsky problem.

1. Introduction

Hele-Shaw model as discussed in the classical paper by british physicist H.S.Hele-Shaw [15] consists in the description of the dynamics of the frontier between twoimmiscible fluids in a narrow channel. In [38], [31] a mathematical model was derivedfor describing Hele-Shaw flows with a free boundary produced by injection/suctionof fluid into/from a narrow channel supposing constant atmospheric pressure onthe moving boundary. This model can be represented in the following form, wheref = f(z, t) is the Riemann mapping function from the unit disk G1 onto the regionoccupied by fluid at the time t, and Q(t) is the rate of injection/suction:

Re

(1

z

∂f

∂t

∂f

∂z

)= Q(t), z ∈ ∂G1, t ∈ [0, T ), (1) model1

f(z, 0) = f0(z), z ∈ G1, (2) model2

f(0, t) = 0, t ∈ [0, T ). (3) model3

This model commonly called complex Hele-Shaw model has been generalized in anumber of direction (see e.g. [16], [36]). Local existence and uniqueness for (1)-(3) isshown in [38] under additional condition f ′z(0, t) > 0. In the case of rational initialmapping f0(z) an elementary proof of the local existence is proposed in [12]. In[30] the complex Hele-Shaw model is reformulated as an abstract Cauchy-Kovalevskyproblem. The solution of the later was found on the the base of the Nirenberg-Nishida theorem [20], [21] (c.f. [23]). This approach is also used in [24], [28], [29]for analytical and numerical treatment of certain variants of the complex Hele-Shaw

Date: September 19, 2004.1991 Mathematics Subject Classification. 76D27, 30E25, 35A10.Key words and phrases. Hele-Shaw flows, doubly connected domain, classical solutions, Schwarz

alternation method, abstract Cauchy-Kovalevsky problem.1

2 S. ROGOSIN AND T. VAITEKHOVICH

model. Among the result dealing with analytic solutions to the Hele-Shaw problemwe have to mention the classical paper [25] in which a first collection of the exactsolutions was constructed (c.f. also [17]).

The Hele-Shaw moving boundary value problem and its generalizations is studiedon the base of certain weak or variational formulations starting from the results in[7], [13], exploiting the celebrated Baiocchi transformation [3]. These formulations ofthe Hele-Shaw model may be viewed also as an enthalpy formulation of the Stefanproblem [8].

Several attempts was made to attack the Hele-Shaw problem in the case of whenthe fluid occupies a multiply-connected region. In [14] an ill-posed moving boundaryproblem for doubly connected domains in RN . It is proposed a weak formulation ofthis problem, the local in time solvability is shown. This problem is connected withHele-Shaw model when N = 2 and with electro-chemical machining model as N ≤ 3.Another model which also lead naturally to the moving boundary value problem ina multiply-connected domain is that of freezing/solidification model (see e.g. [2]).In [32], [33] the Hele-Shaw problem for different type of multiply-connected domainsis reduced by applying the Cauchy transform to the functional equation in complexdomains (for developing of this technique and other application of the method of thefunctional equations see [19]).

The idea of the constructing a conformal mapping from a canonical N -connectedregion to the fluid domain domain, especially automorphic functions invariant withrespect to corresponding Schottky groups, is employed in [34] for the study of thesingularity-driven Hele-Shaw flows of a multiply-connected fluid region.

In [5] the theory of algebraic curves and quadrature domains is used to constructexact solutions to the problem of the squeeze flow of multiply-connected fluid domainsin a Hele-Shaw cell.

In our paper we consider a generalization of the problem (1)-(3) to the case ofdoubly connected domains: given a function f0(z), holomorphic and univalent in aneighbourhood of the domain

G(0) ≡ {z ∈ C : |z − aj| > rj(0), j = 1, 2; |a1 − a2| > r1(0) + r2(0)} ,

functions Q1(t), Q2(t), continuous in a right-sided neighbourhood of t = 0, and anumber a ∈ [0, 2π), find f(z, t), holomorphic and univalent in z in a neighbourhoodof the domain

G(t) ≡ {z ∈ C : |z − aj| > rj(t), j = 1, 2; |a1 − a2| > r1(t) + r2(t)} ,

Γj(t) ≡ {z ∈ C : |z − aj| = rj(t)}, j = 1, 2,

and continuously differentiable in a right-sided neighbourhood of t = 0, satisfying thefollowing conditions:

Re

(1

z

∂f

∂t

∂f

∂z

)= Qj(t), z ∈ Γj(t), j = 1, 2, t ∈ [0, T ), (4) model_d1

f(z, 0) = f0(z), z ∈ G(0), (5) model_d2

COMPLEX HELE-SHAW MODEL 3

∫

Γ1(t)

∣∣∣∣∂f

∂z(τ, t)

∣∣∣∣−2

Q1(t)dτ

τ − a1

+

∫

Γ2(t)

∣∣∣∣∂f

∂z(τ, t)

∣∣∣∣−2

Q2(t)dτ

τ − a2

= 0, t ∈ [0, T ). (6) model_d3

We follow here the strategy of the paper [29] in which an ill-posed Hele-Shaw prob-lem for simply connected domain was treated by reduction to an abstract Cauchy-Kovalevsky problem and nonlinear Riemann-Hilbert-Poincare problem (see also [30],[35]). The case of doubly connected domains leads to certain additional difficultieswe have to overcome.

First, it is known (see [11]) that the conformal mapping of a doubly connecteddomain onto canonical domain (in particular onto the exterior of two discs) dependson so called conformal modulus (in our case - the fraction of the radii of the circlesΓ1(t), Γ2(t)). In our study we use an analytic dependence of the conformal moduluson small perturbation of Γ1(t), Γ2(t) (see [18]).

Second, to get the unique conformal mapping of a doubly connected domain ontothe canonical domain one have to add a (real) uniqueness condition which need tohave a physical sense. It is proposed in the form (6).

Third, the problem (4)-(6) can be reduced to a couple of problems, namely, theSchwarz problem for a doubly connected domain G(t) and evolution equation tobe solved successively. The first of these problems has a “global” and stationarycharacter. The later means that it should be solved for each fixed time instance, butthe former means that we have to construct solution in the whole domain. Moreoverwe need to show the possibility to continue the obtained solution analytically into aneighbourhood of the initial domain and regularity dependence of the solution on thesmall perturbation of the boundary curves. These properties and the solution itselfare obtained on the base of a variant of the Schwarz alternation method as proposed in[19] and by using the analyticity of the solution to the Schwarz problem with respectto the boundary curves shown in [26]. We have to remark that in the contrary to thesimply connected case there no suitable exact formula for the solution to the Schwarzproblem. The second problem, namely, evolution equation, in the contrary, has a“local” character, i.e. is studied in a neighbourhood of each boundary curve. Thisproblem is reduced to an abstract problem which is solved by using the followingtheorem which is a tool for the study abstract nonlinear operator equations in a scaleof Banach spaces

theoNirNish Theorem 1. ([21]) Let {Bs, ‖ · ‖s}0<s≤1 be a scale of Banach spaces, i.e. a family ofcontinuously embedded Banach spaces such that for all 0 < s′ ≤ s ≤ 1 the norm ofthe canonical embedding Is→s′ : Bs → Bs′ is not greater than 1 (‖Is→s′‖ ≤ 1).

Let us consider the abstract Cauchy-Kovalevsky problem

dw

dt= L(t, w), w(0) = 0, (7) CauKov

satisfying the following conditions in {Bs, ‖ · ‖s}0<s≤1 (where C, K, R, and T be po-sitive constants independent of s, s′, t):

(i) the nonlinear operator L(t, w) is continuous in t and maps

[0, T ]× {w ∈ Bs : ‖w‖s ≤ R} into Bs′ for all 0 < s′ ≤ s ≤ 1; (8) NirNish1


(ii) the continuous function L(t, 0) satisfies

‖L(t, 0)‖ ≤ K

1− s, for all 0 < s < 1; (9) NirNish2

(iii) for all 0 < s′ ≤ s ≤ 1, t ∈ [0, T ), and w1, w2 belonging to the open ball{w ∈ Bs : ‖w‖s < R} we have

‖L(t, w1)− L(t, w2)‖s′ ≤C

s− s′‖w1 − w2‖s. (10) NirNish3

Under these assumptions there exists one and only one solution

w ∈ C1 ([0, a0(1− s)),Bs)0<s<1 , ‖w‖s < R,

where a0 is a suitable positive constant.

Our paper is organized as follows. We start (Sec. 2) with reformulation of the initialproblem (4)-(6) in the equivalent form, namely, in the form of the linear Schwarzboundary value problem for a doubly connected domain and an abstract Cauchy-Kovalevsky problem to be solved successively. We also discuss the structure of theobtained couple of problems. In Sec. 3 a suitable scale of Banach spaces is constructedand its properties are studied. Solution of the Schwarz problem in a doubly connecteddomain is presented in Sec. 4. We conclude our investigation (Sec. 5) by theapplication of Theorem 1 to the Cauchy-Kovalevsky problem in the Banach scaledefined in Sec. 3.

2. The structure of the problem

Let us consider the structure of problem (4)-(6). The equation (4) can be rewrittenin the form

Re

(1

z

∂f

∂t(z, t)

(∂f

∂z(z, t)

)−1)

=

∣∣∣∣∂f

∂z(z, t)

∣∣∣∣−2

Qj(t), z ∈ Γj(t), j = 1, 2, (11) Sch1

which have to be solved for each fixed t ∈ [0, T ). It is known [1] that under certainadditional condition (e.g. under condition (6) or the condition

Im

(1

z

∂f

∂t(z, t)

(∂f

∂z(z, t)

)−1)∣∣∣

z=z0

= 0,

where z0 is a point in G(t)) the problem (11) has a unique solution which is defined bymeans of the Schwarz operator for the circular doubly connected domain T. Only fewexact formulas are known for the Schwarz operator, namely the celebrated Villat’sformula for concentric annulus (see [1]) and a formula for multiply connected circulardomain (a form of this formula is presented in [19]). The first form of the Schwarzoperator is an integral operator with the Weierstrass zeta-function in the kernel, butin the last one the kernel is represented as a series in all elements of certain Schottkygroup. Both forms are not suitable for our aims. Therefore our goal is to find animplicit formula for the Schwarz operator T for our domain G(t) which will allow usto get the necessary properties of the solution.


Let us suppose that the operator T is already constructed. Then the equation (11)can be rewritten in the following equivalent form

∂f

∂t(z, t) = z

∂f

∂z(z, t)T

(∣∣∣∣∂f

∂z

∣∣∣∣−2

Qj(t)

), z ∈ G(t), j = 1, 2. (12) CauKov1

For each fixed domain G(t) we introduce the space H(G(t)) ∩ C1,α(G(t)), as well as

the spaces H(G(t)) ∩ Cα(G(t)), H(G(t)) ∩ C(G(t)). Then the following lemma is ananalog of Lemma 1 [30, p. 104] which can be proved by the change of the unknownfunction in the same manner as in [30].

CauKov2 Lemma 1. Let f(z, t) ∈ C1([0, a0),H(G(t)) ∩ C1,α(G(t))

)for each t ∈ [0, a0) be a

univalent function in G(t) solving in G(t)× (0, a0) the problem (12), (5) (or equiva-lently (4)-(6)).

Then the function φ(z, t) ≡ (∂f∂z

(z, t))−1 ∈ C1

([0, a0),H(G(t)) ∩ Cα(G(t))

)is a

solution to the problem

∂φ

∂t(z, t) = zTG(t)(φ)(z, t)φ(z, t)

∂

∂zφ(z, t)− φ(z, t)

∂

∂z

(zTG(t)(φ)(z, t)

), (13) CauKov3

(z, t) ∈ G(t)× (0, a0),

φ(z, 0) = φ0(z) ≡(

∂f0

∂z(z)

)−1

, z ∈ G(t), (14) CauKov4

where φ(z, t) 6= 0, and TG(t)(φ) denotes the nonlinear operator

TG(t)(φ) ≡ T(|φ|2 Q(t)

). (15) CauKov5

Conversely, if φ(z, t) ∈∈ C1([0, a0),H(G(t)) ∩ Cα(G(t))

)is a solution to (13)-

(14) such that φ(z, t) 6= 0 in G(t) × [0, a0). Then f(z, t) =z∫0

dζφ(ζ,t)

belongs to

C1([0, a0),H(G(t)) ∩ C1,α(G(t))

)and represents a locally univalent solution to the

problem (12), (5) (or equivalently (4)-(6)) in G(t)× (0, a0).

By this lemma we reduce the starting problem (4)-(6) to the problem (13)-(14)which is a scale-type problem. Thus it remains to find a way how to interpret theproblem (13)-(14) as a special case of (7). In Sec. 5 we will prove the existence ofthe solution φ(z, t) to the problem (13)-(14) which imply the following properties ofthe solution f(z, t):

a) the family f(z, t) belongs to the spaces H(G(t′)) ∩ C1,α(G(t′)) in the domains

G(t′) ≡ {z ∈ C : |z − aj| > rj(t′), j = 1, 2; |a1 − a2| > r1(t

′) + r2(t′)} ,

which are situated in a neighbourhood of G(t);b) f(z, t) is univalent for small t in the above said neighbourhood.


3. Scale of Banach spaces

Let us introduce analogously to [29] the following space of holomorphic functions(cf. [35]). In what follows we will assume for determines that Q1(t), Q2(t) are negativein a right-sided neighbourhood of t = 0:

Q1(t), Q2(t) < 0, t ∈ [0, T ). (16) qu

Such assumption corresponds to the shrinkage of both holes in the Hele-Shaw cell.All other situation can be considered analogously.

Let us fix constants ri,j i, j = 1, 2; rj(0) > r1,j > r2,j > 0, a positive numberb > 0 and introduce parameter s ∈ (0, 1). Denote by H (

G(s)

)the space of functions,

holomorphic in the following circular doubly connected domain:

G(s) ≡ {z ∈ C : |z − aj| > r1,j − s(r1,j − r2,j), j = 1, 2} .

Then we define the space

B :={

g = g(z, t) ∈⋃

0<s<1

C ([0, b(1− s)),H(G(s)) ∩ C1,α(G(s))

):

‖g‖B = max

{sup

s∈(0,1),h<b(1−s)

maxt∈[0,h]

‖g(·, t)‖Cα(G(s));

sups∈(0,1),t<b(1−s)

∥∥∥∥∂g

∂z(·, t)

∥∥∥∥Cα(G(s))

(1− t

b(1− s)

) 12

}< ∞

}.

The following lemmas are simply generalization of [29, Lemma 1, Lemma 2].

Banach Lemma 2. The function space B is a Banach space.

Proof. Let {gk} be a Cauchy sequence from B. Then it is a Cauchy sequence inthe spaces C (

[0, h],H(G(s)) ∩ Cλ(G(s)))

for each h < b(1 − s). It follows from thecompleteness of every of the latter spaces that there exists a uniquely determinedlimit element g = g(h,s) ∈ C (

[0, h],H(G(s)) ∩ Cα(G(s))). This element satisfies the

estimate maxt∈[0,h] ‖g(h,s)‖Cα(G(s))≤ ‖gk0‖B + ε for a suitable index k0 uniformly for

all h < b(1 − s) and s ∈ (0, 1). The limit element of our Cauchy sequence {gk} isthe function g = g(z, t) with the property that g coincides with g(h,s) for all h <b(1− s), s ∈ (0, 1). Consequently

sups∈(0,1),h<b(1−s)

maxt∈[0,h]

‖g‖Cα(G(s))< +∞.

If h < b(1 − s) for a certain s ∈ (0, 1), then there exists a larger number p ∈ (s, 1)such that h < b(1− p), too. Using Cauchy’s integral formula for g in G(p)

g(z, t) = − 1

2πi

∫

Γ1(p)

g(τ, t)

τ − zdτ − 1

2πi

∫

Γ2(p)

g(τ, t)

τ − zdτ + C,

where Γ1(p), Γ

2(p) are boundary circles of the domain G(p), C is a constant, and esti-

mating ∂g∂z

in G(s) gives

g ∈ C ([0, b(1− s)),H(G(s)) ∩ C1,α(G(s))

).


Finally,∥∥∥∥∂g

∂z

∥∥∥∥Cα(G(s))

(1− t

b(1− s)

) 12

≤∥∥∥∥∂gk0

∂z

∥∥∥∥Cα(G(s))

(1− t

b(1− s)

) 12

+ ε

≤ ‖gk0‖B + ε

gives the property

sups∈(0,1),t<b(1−s)

∥∥∥∥∂g

∂z(·, t)

∥∥∥∥Cα(G(s))

(1− t

b(1− s)

) 12

< ∞,

i.e. g ∈ B. ¤

algebra Lemma 3. The function space B is an algebra with

‖g · h‖B ≤ 2‖g‖B‖h‖B for all g, h ∈ B.

Proof. We know that g, h ∈ C ([0, b(1− s)),H(G(s)) ∩ C1,α(G(s))

).

The statement follows from inequalities

‖g · h‖Cα(G(s))≤ ‖g‖Cα(G(s))

· ‖h‖Cα(G(s)),

∥∥∥∥∂

∂z(g · h)

∥∥∥∥Cα(G(s))

≤∥∥∥∥

∂

∂zg

∥∥∥∥Cα(G(s))

‖h‖Cα(G(s))+ ‖g‖Cα(G(s))

∥∥∥∥∂

∂zh

∥∥∥∥Cα(G(s))

,

and the definition of ‖g‖B and ‖h‖B. ¤

In the case of simply connected domains the pre-image of the domains occupiedby one of the fluids in the Hele-Shaw can be fixed (e.g. the unit disc). It is not thecase when the domain is doubly connected. For each fixed t one can choose two discsas pre-images of the “holes” in the Hele-Shaw cell. With time changing the ratioof the radii of these discs is changing too (one can suppose that one of these radiusis constant but another one is changing; alternatively, both radii can be supposedchanging).

Later in the paper we need also the following result:

radius Lemma 4. Let

Z ≡{

ζ ∈ C1(T,C) : inf

{∣∣∣∣ζ(s)− ζ(t)

s− t

∣∣∣∣ : s, t ∈ T, s 6= t

}> 0

}

be the set of simple curves ζ of class C1(T,C).Then the operator which assigns to each pair of simple smooth curves (ζ1, ζ2) from

Z ≡ {ζ ≡ (ζ1, ζ2) ∈ Z2 : ζ1(T) ⊂ E[ζ2], ζ2(T) ⊂ E[ζ1]

}

the conformal modulus of the corresponding doubly connected domain is real analyticlocally near any fixed pair (ζ1

0 , ζ20 ) ∈ Z.

The proof in the case of annulus is given in [18, Theorem 6.1] on the base of theImplicit Function Theorem. The proof in the case of any other doubly connecteddomain follows from conformal equivalence under Mobius transformation.


4. Schwarz alternation method

In order to get the corresponding properties of the solution to the Schwarz prob-lem we introduce the following notations and results following mainly to [19]. Weformulate them in the case of an arbitrary circular n-connected domain.

Let D ≡ C\(

n⋃k=1

clDk

)be a circular plane multiply connected domain, Dk ≡ {z ∈

C : |z − ak| < rk}. Let the formula

z∗(km,km−1,...,k1) =(z∗(km−1,...,k1)

)∗(km)

(17) symm

defines successive symmetries with respect to the circles Tk1 = ∂ Dk1 , . . . ,Tkm =∂ Dkm . If in the sequence km, km−1, . . . , k1 no two neighbouring indexes are equalthen the number m is called the level of the mapping z 7→ z∗(km,km−1,...,k1). When m iseven, these mappings are Mobius transformations. When m is odd, these mappingsare anti-Mobius transformations, i.e. Mobius with respect to z. The above describedsuccessive symmetries forms for each fixed domain D so called Schottky group K ofsymmetries (see [9]). In the following we denote by G the subgroup of all but notidentity elements of K of even level, and by F the family of all elements of K of oddlevel.

The following theorems represent the solution Ψ(z) ≡ T(D,q)(z) to the Schwarzproblem

Re Ψ(t) = q(t), t ∈ ∂ D,

Im Ψ(z0) = 0(18) sch_mc

for a multiply connected circular domain D in the above introduced notations, whereq ≡ (q1, . . . , qn), qk : Tk → C are given real-valued functions, z0 ∈ D is a given pointinside D. We denote by T(D,q)(·) : q 7→ Ψ the Schwarz operator of the domain Dwith density q.

Theorem 2. [19, Theorem 4.11] The Schwarz operator T(D,q)(·) of a multiply con-th sch

nected circular domain D ≡ C \(

n⋃k=1

clDk

)has the form

T(D,q)(z) ≡ Ψ (z) =1

2πi

n∑

k=1

∫

Tk

qk(ζ)

{∑j

′′[

1

ζ − γj (w)− 1

ζ − γj (z)

]

+

(rk

ζ − ak

)2 ∑j

′[

1

ζ − γj (z)− 1

ζ − γj (w)

]− 1

ζ − z

}dζ (19) 4.4sc

+1

2πi

n∑

k=1

∫

Tk

qk(ζ)∂A

∂ν(ζ)dσ +

n∑m=1

Am [log (z − am) + ψm(z)] + iς,

where

Am :=1

2πi

n∑

k=1

∫

Tk

qk(ζ)∂αm

∂ν(ζ)dσ, m = 1, 2, ..., n,


A(ζ) has the form

A(ζ) =n−1∑m=1

Re

[φm

((w∗

(n)

)∗(m)

)− φm

(w∗

(m)

)]+ log

∣∣ζ − w∗(n)

∣∣

−αn(ζ) log rn −n−1∑m=1

αm(ζ) log∣∣w∗

(n) − am

∣∣ , (20) 4bA

the harmonic measure corresponding to m-th circle αm(ζ) has the form

αl (z) =n∑

m=1,m6=l

Am [Re ψm (z) + log |z − am|] + A, (21) 4.4sch

ψm(z) are derived in

ψm (z) = log

[ ∞∏

j=1, j 6=m

ψjm (z)

], (22) c9b

where

ψjm (z) =

γj(z)−am

γj(w)−am, if γj ∈ G,

γj(w)−am

γj(z)−am, if γj ∈ F .

ς is an arbitrary real constant,∑′ contains γj of odd level ( γj ∈ F), and

∑′′ of evenlevel (γj ∈ G). The series converges uniformly in each compact subset of clD\ {∞} .

The single-valued part of the Schwarz operator appears in the modified Dirichletproblem:

Re Ψ (s) = qk(s) + ck, t ∈ Tk, k = 1, 2, ..., n, (23) 4bD

where a given function qk ∈ C(∂Dk), k = 1, . . . , n, ck are undetermined real constants.If one of the constants ck is fixed arbitrary, then the remaining ones are determineduniquely and Ψ (z) is determined up to an arbitrary additive purely imaginary con-stant. Thus we have

Theorem 3. [19, Theorem 4.12] The single-valued part of the Schwarz operatorth sch single

Tsingle(D,q)(·) of a multiply connected circular domain D ≡ C \(

n⋃k=1

clDk

)cor-

responding to the modified Dirichlet problem (23) has the form

Tsingle(D,q)(z) ≡ Ψ (z) =1

2πi

n∑

k=1

∫

Tk

(qk (ζ) + ck)

{∑j

′′[

1

ζ − γj (w)− 1

ζ − γj (z)

]

+

(rk

ζ − ak

)2 ∑j

′[

1

ζ − γj (z)− 1

ζ − γj (w)

]− 1

ζ − z

}dζ (24) 4.4sca

+1

2πi

n∑

k=1

∫

Tk

qk(ζ)∂A

∂ν(ζ)dσ + iς.


One of the real constants ck can be fixed arbitrarily; the remaining ones are determineduniquely from the linear algebraic system

n∑

k=1

∫

Tk

(qk(ζ) + ck)∂αm

∂ν(ζ)dσ = 0, m = 1, 2, ..., n− 1. (25) 4.4ba8

By applying the properties of the Cauchy type operators (see e.g. [10]) and the suc-cessive symmetries as above introduced (see e.g. [19]) one can establish the followingresult

sch_reg Theorem 4. The Schwarz operator T(D,q)(·) (as well as the single-valued Schwarz

operator Tsingle(D,q)(·)) of a multiply connected circular domain D ≡ C\(

n⋃k=1

clDk

)

is a bounded operator fromn∏

k=1

Cm,α (Tk,R) ton∏

k=1

H (extTk,C)⋂ Cm,α (Tk,C).

Proof. From the induction argument follows that it suffices to prove the statementin the case of a doubly connected domain (i.e. for n = 2) and for m = 1. The aboveTheorems 2, 3 give then the existence of the solution to corresponding problem. Theform of the Schwarz operators (see (19), (24)) does not allow to obtain boundednessof these operators as well as their regularity. Thus we use an another approach.

Let us represent the solutions to the problem

Re Ψ (s) = qk(s), t ∈ Tk, k = 1, 2, (26) sch

in the following formΨ(z) = Ψ1(z) + Ψ2(z), z ∈ clD, (27) sum

where the functions Ψk, k = 1, 2, are holomorphic in the domains extTk ≡ {z ∈ C :|z − ak| > rk}, continuous in cl extTk ≡ {z ∈ C : |z − ak| ≥ rk}. Then the firstequation of (26) can be rewritten as

Re Ψ1 (s) + Re Ψ2

(s∗(1)

)= q1(s), |s− a1| = r1, (28) sch_1

where s∗(1) is the point symmetric to s with respect to the first circle T1 (thus s∗(1) = s).

By applying to the boundary condition (28) the Schwarz operator of the exterior ofthe first circle, which is determined uniquely up to the real constant ς1 by the followingformula

T1q1(z) ≡ T(extT1, q1)(z) ≡ − 1

2πi

∫

|τ−a1|=r1

τ + z − 2a1

τ − z

q1(τ)dτ

τ − a1

+ iς1, (29) sch_op1

we arrive at the functional equation in the domain extT1

Ψ1(z) = −Ψ2(z∗(1)) + T1q1(z), |z − a1| ≥ r1. (30) sch_fe1

Analogously we obtain the functional equation in the domain extT2

Ψ2(z) = −Ψ1(z∗(2)) + T2q2(z), |z − a2| ≥ r2, (31) sch_fe2

where operator T2 is defined by the formula

T2q2(z) ≡ T(extT2, q2)(z) ≡ − 1

2πi

∫

|τ−a2|=r2

τ + z − 2a2

τ − z

q2(τ)dτ

τ − a2

+ iς2, (32) sch_op2


The relations (30), (31) constitute a system of functional equations in a class ofholomorphic functions. The system (30), (31) can be solved by successive approxi-mations. Let us consider now the method of successive approximations for (30), (31)from another point of view. We take zero-order approximation for (30), (31) in theform

Ψ01(z) ≡ T1q1(z), |z − a1| ≥ r1.

The function Ψ01(z) satisfies the first boundary condition of (26), but possibly does

not satisfy the second one. This function is defined and holomorphic in the exteriorof the first circle. Thus we can take it as a zero-order approximation to the solutionin the domain D ≡ {z ∈ C : |z − ak| > rk, k = 1, 2}, namely

Ψ0(z) ≡ Ψ01(z)|D , z ∈ D.

Substituting Ψ01(z) into the functional equation (31) we obtain the first order

Ψ12(z) ≡ −Ψ0

1(z∗(2)) + T2q2(z), |z − a2| ≥ r2,

and respectivelyΨ1(z) ≡ Ψ0(z) + Ψ1

2(z)|D , z ∈ D.

The last function does satisfy the second boundary condition of (26), but possiblydoes not the first one. Further

Ψ21(z) ≡ −Ψ1

2(z∗(1)), |z − a1| ≥ r1,

and consequently

Ψ2(z) ≡ Ψ0(z) + Ψ1(z) + Ψ21(z)|D , z ∈ D.

Finally, an N -th approximation of the solution can be given by the following formula:

ΨN(z) ≡ Ψ0(z) + Ψ1(z) + . . . + ΨN−1(z) + ΨN1 (z)|D , z ∈ D, N is even number, (33) N_even

ΨN(z) ≡ Ψ0(z) + Ψ1(z) + . . . + ΨN−1(z) + ΨN2 (z)|D , z ∈ D, N is odd number. (34) N_odd

The proof of the convergence of the sequence ΨN(z) in the space H(D)⋂ C(clD) for

each fixed pair of functions q ≡ (q1, q2) ∈ C0,α(T1)×C0,α(T1) is presented in [19, Sec.4.9]. The similar proof can be given in the case of the space H(D)

⋂ C1,α(clD) for afixed pair of data q ≡ (q1, q2) ∈ C1,α(T1)× C1,α(T1).

In order to show it we have to mention that the solution of the Schwarz problemfor a doubly connected circular domain D can be rewritten in the form of sum of twoseries

Ψ(z) ≡ Ψ1(z) + Ψ2(z),

where

Ψ1(z) ≡ T1q1(z) + T1q1(α(z)) + T1q1(α2(z)) + . . . + T1q1(α

n(z)) + . . . ,

Ψ2(z) ≡ T2q2(z∗(1)) + T2q2(β(z)) + T2q2(β

2(z)) + . . . + T2q2(βn(z)) + . . . ,

α(z) ≡ z∗(12), β(z) ≡ z∗(21), αn(z) ≡ α(αn−1(z)), βn(z) ≡ β(βn−1(z)). (35) alpha_beta

Both operators α : z 7→ α(z), β : z 7→ β(z) are compact in the spaceH(D)⋂ C1,α(clD)

(see [19, Subsec. 4.9.3]). Hence it remains to prove that operators T1,T2 are boundedas operators acting from C1,α(T1), C1,α(T2) to H(D)

⋂ C1,α(clD) respectively. This


result is basically a restatement of two technical lemmas from [18, Lemmas 5.1, 5.2].It completes the proof. ¤

5. Application of the abstract Cauchy-Kovalevsky problem

In this Section we apply Nirenberg-Nishida theorem (Theorem 1) to the abstractCauchy-Kovalevsky problem (13)-(14), where the operator TG(t) is defined by theformula (15). We follow here the ideas and machinery of the papers [30], [29]. Wecan formulate the main result of our paper in the following manner.

Theorem 5. [Main Theorem] There exists an in general small interval of time [0, b)mainsuch that the problem (13)-(14) has a uniquely determined solution φ = φ(z, t) whichhas no zeros on G(p)× [0, b) and belongs to the space C1

([0, b),H(G(p)) ∩ C1,α(G(p))

).

The constant p determined the doubly connected domain G(p) is taken as in the defi-nition of B.

col_main Corollary 1. There exists an in general small interval of time [0, b) such that theproblem (4)-(6) has a uniquely determined solution f . The function f = f(z, t)is univalent with respect to z on G(p) for each t ∈ [0, b) and belongs to the space

C1([0, b),H(G(p)) ∩ C2,α(G(p))

). The constant p determined the doubly connected do-

main G(p) is taken as in the definition of B.

This corollary is a direct consequence of the theorem. Univalence of the function ffollows from the properties of initial function f0, the choice of ri,j i, j = 1, 2; rj(0) >r1,j > r2,j > 0 (see definition of B) and the chosen norm of B which is stronger thanthe sup-norm.

To prove the main theorem we need a number of auxiliary results which we presentbelow. We apply Theorem 1 to the problem (13)-(14). Hence the operator L(t, φ)coincides with the operator in the right-hand side of (13). Thus the operator L(t, φ)contains mainly the combination of two operators: differentiation operator ∂z andthe Schwarz operator T for a doubly connected circular domain applied to |φ|2.

To study the unbounded operator ∂z in the scale B we use the following resultwhich is based on the idea formulated in [37]:

partial_z Lemma 5. Let φ ∈ B. Then the operator

A : φ 7→t∫

0

∂zφ(·, τ)dτ, t ∈ [0, b(1− s)), s ∈ (0, 1),

is a continuous operator mapping B into itself, and satisfying the estimate

‖Aφ‖B ≤ Cb‖φ‖B, (36) est_dz

where C = C(ri,j), i, j = 1, 2.

This lemma is a restatement of [29, Lemma 3].Let us now begin to establish the necessary properties of the operator TG(t)(φ) ≡

T(|φ|2 Qj(t)

). In the previous section we have shown the boundedness of this opera-

tor in the corresponding spaces for each fixed domain G(t). But we need in Theorem1 the fulfillment of stronger conditions.


First of all, the function T(|φ|2 Qj(t)

)(z) have to be continued analytically in a

neighbourhood of the initial doubly connected domain G(0) (thus to preserve theanalogous property of the “initial” function φ0). To see it we mention first that theSchwarz operator can be rewritten in the form

T(|φ|2 Qj(t)

)(z) = T

(φ(τ, t)φ

(1

τ, t

)Qj(t)

)(z). (37) sch_conj

If the function φ(z, t) can be continued in the domain bigger than G(t) then the con-

jugate function φ(z, t) ≡ φ(

1τ, t

)possesses the “continuation” in the smaller domain.

Therefore we introduce another function space different from B.We fix as before the constants ri,j i, j = 1, 2; rj(0) > r1,j > r2,j > 0, a positive

number b > 0 and introduce parameter s ∈ (0, 1). Denote by H (A(s)

)the space of

functions, holomorphic in the union of two annulus:

A(s) ≡ A1(s)

⋃A2

(s),

Aj(s) ≡

{z ∈ C : r1,j − s(r1,j − r2,j) < |z − aj| < 1

r1,j − s(r1,j − r2,j)

}, j = 1, 2.

Then we define the space

Ba :={

g = g(z, t) ∈⋃

0<s<1

C ([0, b(1− s)),H(A(s)) ∩ C1,α(A(s))

):

‖g‖B = max

{sup

s∈(0,1),h<b(1−s)

maxt∈[0,h]

‖g(·, t)‖Cα(A(s));

sups∈(0,1),t<b(1−s)

∥∥∥∥∂g

∂z(·, t)

∥∥∥∥Cα(A(s))

(1− t

b(1− s)

) 12

}< ∞

}.

It is not hard to see that the space Ba is a Banach space and Banach algebra withrespect to usual multiplication of complex-valued functions. The following lemma isof the straightforward verification.

conj_in_B_a Lemma 6. If φ belongs to B, then the function φ(z, t) ≡ φ(

1τ, t

)as well as the

product φφ belongs to Ba. Besides,

‖φ‖Ba ≤ C‖φ‖B.Now all is prepared for the study of TG(t)(φ) on B. Let

Gη1,η2 ≡ {z ∈ C : |z − a1| > η1r1, |z − a2| > η2r2}be a neighbourhood of a doubly connected domain G(0). Let the function v be inC (

[0, T ),H(Gη1,η2) ∩ C(Gη1,η2))

for certain η1, η2 < 1. Then the function TG(t)(v) ≡T(G(t),q|v|2) as defined in (15) belongs to H(G(t)) for each t ∈ [0, T ). Moreover,the following lemma is valid.

an_cont Lemma 7. For any v ∈ C ([0, T ),H(Gη1,η2) ∩ C(Gη1,η2)

)the image TG(t)(v) of the

nonlinear operator TG(t) possesses an analytic continuation into the bigger domainGη0

1 ,η02

for certain η01 ≤ η1 < 1, η0

2 ≤ η2 < 1.


Proof. First we have to show that both integrals T1, T2 (see (29), (32)) meet thisproperty. It suffices to prove only for T1. Let us rewrite first T1(extT1, Q1|v|2) in theform analogous to (37):

T1(extT1, Q1|v|2)(z) = T1(extT1, Q1v(τ, t)v

(1

τ, t

))(z).

Since v ∈ C ([0, T ),H(Gη1,η2) ∩ C(Gη1,η2)

)then by Lemma 6 the functions v = v

(1τ, t

))

and vv possess an analytic continuation into an annulus A1(s) for certain 0 < s < 1.

By uniqueness of the analytic continuation we can conclude that the operator T1 canbe rewritten in the following form:

T1(extT1, Q1|v|2)(z) = − 1

2πi

∫

|τ−a1|=η1r1

τ + z − 2a1

τ − z

Q1(t)v(τ, t)v(

1τ, t

))dτ

τ − a1

+ iς1,

(38) T_1_an_contfor each t ∈ [0, T1) with certain T1 > 0. The kernel of this integral is holomorphic inthe exterior of the circle |τ − a1| = η1r1. It gives the desired result for the operatorT1. Similar consideration can be applied to the operator T2.

The functions α(z) and β(z) as defined in (35) are in fact Mobius transformationsand are holomorphic outside at most one point. Therefore each term in the series

Ψ1(z), Ψ2(z) has the analytic continuation properties as above formulated. The finalresult follows from the uniformity of the convergence of these series. ¤

bound Lemma 8. The operator

TG(t)(φ) ≡ T(|φ|2 Q(t)

)

satisfies in B the following inequalities∥∥∥TGη01 ,η0

2

(φ)∥∥∥B≤ C1‖φ‖2

B, (39) op_bound

∥∥∥TGη01 ,η0

2

(φ1)−TGη01 ,η0

2

(φ2)∥∥∥B≤ C1‖φ1 − φ2‖2

B. (40) op_Lipsch

Moreover the family of operators {TG(t)(φ)}t∈[0,T ) depends continuously on t ∈[0, t).

Proof. We have to start again with the operator T1. By rewriting it in the form (38)we can use then [29, Lemma 5]. Final result follows then from the properties of series

Ψ1(z), Ψ2(z). ¤

At last we need to show that the right-hand side of (13) does satisfy the conditionsof Theorem 1. To do it we rewrite equations (13) in suitable form

∂ω

∂t(z, t) ≡ L0(t, ω) = zTG

η01 ,η0

2

(ω + φ0)(z, t)∂

∂z(ω(z, t) + φ0(z))

− (ω(z, t) + φ0(z))∂

∂z

(zTG

η01 ,η0

2

((ω + φ0)(z, t))

,

ω(z, 0) ≡ φ(z, 0)− φ0(z) = 0.


ineq_NirNish Lemma 9. The operator L0 satisfies in the scale {Bs, ‖ · ‖s}0<s≤1 of Banach spaces,

{Bs}0<s≤1 :={

g = g(z, t) ∈ C ([0, b(1− s)),H(G(s)) ∩ C1,α(G(s))

):

‖g‖B = max

{sup

s∈(0,1),h<b(1−s)

maxt∈[0,h]

‖g(·, t)‖Cα(G(s));

sups∈(0,1),t<b(1−s)

∥∥∥∥∂g

∂z(·, t)

∥∥∥∥Cα(G(s))

(1− t

b(1− s)

) 12

}< ∞,

}.

the conditions (8)-(10) of Theorem 1.

Proof. The property (8) follows from the second statement of Lemma 8. Further,Lemma 5 gives that the operator ∂

∂zis bounded operator from Bs to Bs′ and

∥∥∥∥∂

∂z

∥∥∥∥s→s′

≤ 1

min{r1,1 − r2,1; r1,2 − r2,2}(s− s′). (41) partial

In order to get the properties (9)-(10) we rewrite the difference L0(t, ω1)−L0(t, ω2)in the following form:

L0(t, ω1) − L0(t, ω2) ≡ −(ω1 − ω2)∂

∂z

(zTG

η01 ,η0

2

(ω1 + φ0)(z, t))−

−(ω2 + φ0)∂

∂z

(z(TG

η01 ,η0

2

(ω1 + φ0)(z, t)−TGη01 ,η0

2

(ω2 + φ0)(z, t)))

+

+z(TGη01 ,η0

2

(ω1 + φ0)(z, t)−TGη01 ,η0

2

(ω2 + φ0)(z, t))∂

∂z(ω1 + φ0) +

+zTGη01 ,η0

2

(ω2 + φ0)(z, t)∂

∂z(ω1 − ω2).

Then inequality (9) follows from Lemma 8 and inequality (41).In the same manner we can prove that

‖L0(t, 0)‖s =

∥∥∥∥φ0∂

∂z

(zTG

η01 ,η0

2

(φ0)(z, t))− zTG

η01 ,η0

2

(φ0)(z, t)∂

∂zφ0(z)

∥∥∥∥s

≤ K

1− s.

¤

By applying Lemma 9 we obtain the existence and uniqueness of the local (in time)analytic solution to the Hele-Shaw for a doubly connected domain.

Acknowledgement. The work is partially supported by the Belarusian Fund forFundamental Scientific Research.

S.R. is grateful to Professor Michael Reissig for several valuable discussions of themathematical theory of Hele-Shaw flows. Special thanks to Professor Okay Celebi forthe brilliant organization of the Workshop “Recent Trends in the Applied ComplexAnalysis” (June 1-6, 2004, Ankara, Turkey).


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