Lobachevskii Journal of Mathematics Vol. 1, 1998, 3–12 \begin{matrix}c \\ ◯ \end{matrix}Yu.E. Hohlov, D.V. Prokhorov, A.Ju. Vasil’ev

Yu.E. Hohlov, D.V. Prokhorov, A.Ju. Vasil’ev

ON GEOMETRICAL PROPERTIES OF FREE BOUNDARIES IN THE HELE-SHAW FLOWS MOVING BOUNDARY PROBLEM

(submitted by Ph.G.Avhadiev)

§1. Introduction

The global behaviour of the solutions in dynamic models of mathematical physics with free boundaries is the most complicated foremost due to the nonregular behaviour of the boundary. So the properties of that free boundary which are being kept continually are of interest. One of such models [1-3] is the Hele-Shaw flows of viscous fluid with taking away (pumping in) this fluid by point sewer (source) (see [4]). The other typical example is the Stefan-type problem with undercooling.

In our article we discuss the geometrical properties of the moving boundary for two basic cases in the plain problem of the Hele-Shaw flows: for the inner problem for the flows in a bounded simply connected domain; and for the exterior problem for dynamics of an aerofoil connected with the flows in the exterior part of a bounded simply connected domain. We prove the invariance of the properties of starlikeness in case of the inner problem of pumping; of convexity in case of the exterior problem of tightening of an aerofoil. We also adduce some examples for the problem of tightening where the corresponding properties of starlikeness, convexity and close-to-convexity are not inherited by the moving boundary.

Earlier the investigation of topological properties and the classification of possible scenarii which can be realized for the Hele-Shaw flows were held in [5], [6]. We also mention the work [7] where the result on invariance of the properties of starlikeness was obtained with the generalized formulation of the inner problem; and the work [8] where the invariance of the properties of convexity was pointed for the problem on a tightening aerofoil.

§2. Mathematical model

We consider the two-dimensional potential flow of fluid that can be described by a velocity field V = (V_{1}, V_{2}). The proposal of flow potentiality means that the velocity is linearly proportional to the pressure gradient,

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This research is partially supported by Russian Foundation for Basic Research, Grant 96-01-00345 A

V = −∇p. Here we consider divergentless fields so the relation div V = 0 is valid that means the harmonicity of the pressure function p out of singularities of the velocity field; i.e., −Δp = Q(t)δ(x, y). We note that in one-phase Stefan problem [4] the state equation is parabolic and there exists a derivation c\frac{∂p}{∂t}. We consider the domain D_{0} and the power of sewer (source) Q as prescribed and model taking away (or pumping in) by a real function Q = Q(t); Q < 0 (Q > 0) correspondently. Further we consider only the source with the constant power Q(t) ≡ Q = const because the case of variable power can be reduced by the change of variable. In addition, we suppose that on the moving boundary ∂D(t) given by the equation ϕ(x, y, t) = 0 the cinematic and dynamic boundary condition \frac{∂ϕ}{∂t} − ∇ϕ · ∇p = 0; p = 0 are valid.

The inner problem can be reduced (see [9], [10]) to a non-linear boundary problem for analytic functions by introducing a complex potential and a subsidiary univalent mapping f of the unit disk U = {z ∈ ℂ : |z| < 1} onto the flow domain D(t). Denote z = f(ζ, t) a mapping normalized by the conditions f(0, t) = 0, f′(0, t) > 0, then the moving boundary in the model is parametrized by the curve f(∂U, t) where f is a solution of the non-linear boundary problem

(1)  ℜe[\frac{∂f(ζ, t)}{∂t}\overline{(ζ \frac{∂f(ζ, t)}{∂ζ})}] = Q, |ζ| = 1

which is satisfying the initial condition

(2)  f(ζ, 0) = f_{0}(ζ), ζ ∈ U,

generated by the source power Q and the initial domain D(0) = f_{0}(U).

Note that it is convenient to parametrize the flow domain D(t) for the exterior problem by a subsidiary univalent mapping of the exterior part of the unit disk U onto the exterior part of a simply connected bounded domain - an aerofoil B(t). This mapping has an expansion in the neighbourhood of the infinity F (ζ, t) = F′(∞, t)ζ+α_{0}(t)+α_{1}(t)ζ^{−1}+…, where F (ζ, t) satisfies the same equation (1) as for the inner problem.

It is known [11] that in the Hele-Shaw flows problem the classic solution exists locally in time in both cases: pumping and taking. The proof by Vinogradov and Kufarev is based on the reduction of the problem to the system of non-linear integral-differential equations. The decision of the problem is ascertained by the method of successive approximations using a lot of difficult estimates of singular integrals. Recently [12]-[14] it has become clear that those models (and even more complicated) could also be interpreted as particular cases of the abstract Cauchy problem, so the classical solvability (locally in time) may be proved using the non-linear abstract Cauchy-Kowalewskaya theorem [15]. The Vinogradov-Kufarev reduction based on the Schwarz integral representation in the problem (1) can be written as follows:

(3)  \frac{∂f(ζ, t)}{∂t} = Qζ \frac{∂f(ζ, t)}{∂ζ} \frac{1}{2π} ∫_{0}^{2π}|f′(e^{iθ}, t)|^{−2} \frac{e^{iθ} + ζ}{e^{iθ} − ζ} dθ,

and will be used further.

§3. Starlike boundaries

We remind that a bounded domain D ∋ 0 is starlike with respect to the origin if every segment linking any bounary point of the boundary ∂D with the origin is in D. A function f mapping U onto the starlike domain is said to be starlike and satisfies the necessary and sufficient condition of starlikeness (see [16,17]):

(4)  ℜe\frac{ζf′(ζ)}{f(ζ)} ≥ 0, ζ ∈ U.

The analagous criterion can be adduced for functions which are mapping the exterior part of U onto the exterior part of a starlike domain. The solutions f(ζ, t) describing the dynamics of the Hele-Shaw flows with the source at the origin preserve the geometrical property of starlikeness with respect to the point ζ = 0. Namely, the following statement is valid.

Theorem 1. If a function f_{0}(ζ) is starlike, then the solutions f(ζ, t) of the problem of the Hele-Shaw flows with the source (Q > 0) at the center of starlikeness and the initial condition f(ζ, 0) = f_{0}(ζ) are starlike for all t > 0 from the interval of the solution existence.

Proof. The proof is based on the consideration of the critical mapping f that has the zero-order of starlikeness. We propose the opposite statement to Theorem 1. Then there exists t_{0} > 0 and ζ_{0} = e^{iφ_{0}} such that

(5)  ℜe\frac{ζ_{0}f′(ζ_{0}, t_{0})}{f(ζ_{0}, t_{0})} = 0, ℑm\frac{ζ_{0}f′(ζ_{0}, t_{0})}{f(ζ_{0}, t_{0})} ≠ 0.

Let t_{0} be the first such point from zero. Without loss of generality we assume

(6)  ℑm\frac{ζ_{0}f′(ζ_{0}, t_{0})}{f(ζ_{0}, t_{0})} > 0, Q = 1.

Since ζ_{0} is a critical point and the image of the unit disk U under the mapping \frac{ζf′(ζ,t_{0})}{f(ζ,t_{0})} touches the imagine axis at the point ζ_{0}, we state that

\frac{∂}{∂φ} arg \frac{e^{iφ}f′(e^{iφ}, t_{0})}{f(e^{iφ}, t_{0})}|_{φ=φ_{0}} = 0,

\frac{∂}{∂r} arg \frac{re^{iφ_{0}}f′(re^{iφ_{0}}, t_{0})}{f(re^{iφ_{0}}, t_{0})}|_{r=1} ≥ 0.

By these formulas we calculate

(7)  ℜe[1 + \frac{ζ_{0}f′′(ζ_{0}, t_{0})}{f′(ζ_{0}, t_{0})} − \frac{ζ_{0}f′(ζ_{0}, t_{0})}{f(ζ_{0}, t_{0})}] = 0

(8)  ℑm[1 + \frac{ζ_{0}f′′(ζ_{0}, t_{0})}{f′(ζ_{0}, t_{0})} − \frac{ζ_{0}f′(ζ_{0}, t_{0})}{f′(ζ_{0}, t_{0})}] ≥ 0

From the Hele-Shaw equation we obtain

(9)  \frac{∂ arg \frac{ζf′(ζ,t)}{f(ζ,t)}}{∂t} = ℑm [(1 + \frac{ζf′′}{f′} − \frac{ζf′}{f})p(ζ, t) + ζp′_{ζ}(ζ, t)] ,

where

p(ζ, t) = \frac{1}{2π} ∫_{0}^{2π} \frac{1}{|f′(e^{iθ}, t)|^{2}} \frac{e^{iθ} + ζ}{e^{iθ} − ζ} dθ.

Calculate by integrating by parts

ζp′ = \frac{ζ}{2π} ∫_{0}^{2π} \frac{1}{|f′(e^{iθ}, t)|^{2}} \frac{2e^{iθ}}{(e^{iθ} − ζ)^{2}} dθ =

(10)  −\frac{i}{2π} ∫_{0}^{2π} \frac{2}{|f′(e^{iθ}, t)|^{2}} ℑm (\frac{e^{iθ}f′′(e^{iθ}, t)}{f′(e^{iθ}, t)}) \frac{e^{iθ} + ζ}{e^{iθ} − ζ} dθ, |ζ| < 1.

This Schwarz representation for (iζp′) allows us to obtain (−ℑm(ζp′)) on the circle ∂U. So from (9), (10) taking into account (7) we get

\frac{∂ arg \frac{ζ_{0}f′(ζ_{0},t)}{f(ζ_{0},t)}}{∂t} |_{t=t_{0}} = −\frac{1}{|f′(ζ_{0}, t_{0})|^{2}} ℑm (\frac{ζ_{0}f′′(ζ_{0}, t_{0})}{f′(ζ_{0}, t_{0})} + \frac{ζ_{0}f′(ζ_{0}, t_{0})}{f(ζ_{0}, t_{0})}) .

The right-hand side of this equality is strictly negative because of (6), (8). Therefore it is negative in the neighbourhoods of ζ_{0} and other points possessing such critical properties. Thus, for t > t_{0} (close to t_{0}) in this neighbourhood the following inequality is valid

ℜe\frac{ζf′(ζ, t)}{f(ζ, t)} > 0.

For all ζ out of the mentioned neighbourhoods the same inequality is valid because

ℜe\frac{ζf′(ζ, t_{0})}{f′(ζ, t_{0})} ≥ δ > 0

for such ζ. This contradicts to the proposal and finishes the proof of Theorem

The solutions F (ζ^{∗}, t) describing the dynamics of the domains of the Hele-Shaw flows with neither source nor sewer at the infinity do not preserve the geometrical property of starlikeness with respect to the point ζ = ∞. Now we show this by examples. We pass to the inner problem by the change F (ζ^{∗}) → \frac{1}{F ( \frac{1}{ζ^{∗}})}, |ζ^{∗}| > 1. Since the condition ℜe\frac{ζ^{∗}F′(ζ^{∗})}{F (ζ^{∗})} > 0 is equivalent

to ℜe\frac{ζF′(ζ)}{F (ζ)} > 0, the exterior problem can be reduced to the inner problem where the density \frac{1}{|f′(e^{iθ},t)|^{2}} in the Schwarz integral can be changed by the

density (−\frac{|F (e^{iθ},t)|^{4}}{|F′(e^{iθ},t)|^{2}} ). We suppose again that there is a critical mapping F (ζ, t) like in Theorem 1. For this mapping the expressions (5)-(8) are valid. We deduce the formula (9) with

(11)  p(ζ, t) = −\frac{1}{2π} ∫_{0}^{2π} \frac{|F (e^{iθ}, t)|^{4}}{|F′(e^{iθ}, t)|^{2}} \frac{e^{iθ} + ζ}{e^{iθ} − ζ} dθ.

Calculate by integrating by parts

(12)

ζp′ = \frac{i}{2π} ∫_{0}^{2π} \frac{|F (e^{iθ}, t)|^{4}}{|F′(e^{iθ}, t)|^{2}} [2ℑm\frac{e^{iθ}F′′(e^{iθ}, t)}{F′(e^{iθ}, t)} − 4ℑm\frac{e^{iθ}F′(e^{iθ}, t)}{F (e^{iθ}, t)}] \frac{e^{iθ} + ζ}{e^{iθ} − ζ} dθ.

From (9), (12) taking into account (7) we obtain

(13)

\frac{∂ arg \frac{ζ_{0}F′(ζ_{0},t)}{F (ζ_{0},t)}}{∂t} |_{t=t_{0}} = \frac{|F (ζ_{0}, t_{0})|^{4}}{|F′(ζ_{0}, t_{0}|^{2}} ℑm (\frac{ζ_{0}F′′(ζ_{0}, t_{0})}{F′(ζ_{0}, t_{0})} − 3\frac{ζ_{0}F′(ζ_{0}, t_{0})}{F (ζ_{0}, t_{0})}) .

The proof will be complete if we show that the right-hand side of this equality can be either positive or negative. Let

F (ζ, t_{0}) = ζ exp ∫_{0}^{ζ} [(1 + aζ)^{\frac{π}{2 arcsin a}} − 1] \frac{dζ}{ζ} , 0 < a < 1.

Then \frac{ζF′(ζ,t_{0})}{F (ζ,t_{0})} = (1 + aζ)^{\frac{π}{2 arcsin a}} , |ζ| < 1; and there are two critical points ζ_{0}, \overline{ζ_{0}}; ζ_{0} = i exp(i arcsin a). Elementary calculations imply (14)

ℑm (3\frac{ζ_{0}F′(ζ_{0}, t_{0})}{F (ζ_{0}, t_{0})} − \frac{ζ_{0}F′′(ζ_{0}, t_{0})}{F′(ζ_{0}, t_{0})}) = 2(1 − a^{2})^{\frac{π}{4 arcsin a}} − \frac{πa}{2\sqrt{1 − a^{2}} arcsin a}.

If a → 0, then (14) is equal to 2 − \frac{π}{2} > 0; if a → 1, then (14) tends to (−∞). Thus (12) can be either positive or negative dependently on a. The proof is completed.

§4. Convex boundaries

The convexity is another important geometrical property of either the flow domain or its complement. It is known [5] that in case of the inner problem both for the sewer and source at the origin the solutions of the Hele-Show equation do not preserve the property of convexity of the initial mapping. So considering the exterior problem we obtain the following result.

Theorem 2. If the function F (ζ^{∗}) maps U^{−} = |ζ^{∗}| > 1 onto the initial flow domain with the convex complement, then the solutions F (ζ^{∗}, t) of the problem of the Hele-Shaw flows with the source (Q > 0) at the infinity and with the initial condition F (ζ^{∗}, 0) = F (ζ^{∗}) preserve this property for all t ≥ 0 from the interval of existence of solutions.

Proof. We suppose the contrary and reduce this problem to the inner problem by the change f(ζ^{∗}) → \frac{1}{F ( \frac{1}{ζ^{∗}})}, |ζ^{∗}| > 1. The condition ℜe (1 + \frac{ζ^{∗}f′′(ζ^{∗})}{f′(ζ^{∗})}) ≥ 0

is equivalent to the inequality ℜe [2\frac{ζF′(ζ)}{F (ζ)} − 1 − \frac{ζF′′(ζ)}{F′(ζ)}] > 0, ζ = \frac{1}{ζ^{∗}}. The

density in the Schwarz integral should be changed by −\frac{|F (e^{iθ},t)|^{4}}{|F′(e^{iθ},t)|^{2}} . We denote

h(ζ, t) = 2\frac{ζF′(ζ,t)}{F (ζ,t)} −1− \frac{ζF′′(ζ,t)}{F′(ζ,t)}. The contrary statement implies the existence of the critical mapping F and points t_{0} ≥ 0 and ζ_{0} = e^{iθ_{0}} such that

(15)  ℜeh(ζ_{0}, t_{0}) = 0

(16)  ℑmh(ζ_{0}, t_{0}) ≥ 0

(17)  ℜe\frac{ζ_{0}h′(ζ_{0}, t_{0})}{h(ζ_{0}, t_{0})} = 0

(18)  ℑm\frac{ζ_{0}h′(ζ_{0}, t_{0})}{h(ζ_{0}, t_{0})} ≥ 0

From the Hele-Shaw equation (3) by the standard calculation we obtain

(19)  \frac{∂ arg h(ζ, t)}{∂t} = ℑm [−\frac{ζh′}{h} p + (\frac{1}{h} − 1)ζp′ + \frac{ζ^{2}p′′}{h} ] ,

with p(ζ, t) from (11).

We also have the same formula (12) for ζp′ as in §3. Then differentiating (12) by ζ and further integrating by parts we receive the following equality:

(20)

ζp′ + ζ^{2}p′′ = \frac{−1}{2π} ∫_{0}^{2π} \frac{|F (e^{iθ}, t)|^{4}}{|F′(e^{iθ}, t)|^{2}} [2ℜeζh′(e^{iθ}, t) − 4(ℑmh(e^{iθ}, t))^{2}] \frac{e^{iθ} + ζ}{e^{iθ} − θ} dθ.

From (19), (20) taking into account (12) and (15), (17) we obtain

\frac{∂ arg h(ζ_{0}, t)}{∂t}|_{t=t_{0}} = −\frac{|F (ζ_{0}, t_{0})|^{4}}{|F′(ζ_{0}, t_{0})|^{2}} ℑm [2h(ζ_{0}, t_{0}) + \frac{ζ_{0}h′(ζ_{0})}{h(ζ_{0})}] .

Foremost, we suppose ℑmh(ζ_{0}, t_{0}) ≠ 0. Then from (16), (18) the righthand side of the latter equality is negative. Then, analogously to §3, we have ℜeh(ζ, t) > 0 for all |ζ| < 1 and t > t_{0} close to t_{0}.

Now we consider the case h(ζ_{0}, t_{0}) = 0. Then the conditions (15)-(18) should be changed by h(ζ_{0}, t_{0}) = 0. Besides,

\frac{∂}{∂φ}ℜeh(e^{iφ}, t_{0})|_{φ=φ_{0}} = ℑmζ_{0}h′(ζ_{0}, t_{0}) = 0

and

(21)  \frac{∂}{∂r} ℜeh(re^{iφ}, t_{0})|_{r=1} = ℜeζ_{0}h′(ζ_{0}, t_{0}) ≤ 0

The assumption h′(ζ_{0}, t_{0}) = 0 implies that h(ζ, t_{0}) has its values in the whole neighbourhood of the origin. This contradicts to the conditions that ℜeh(ζ, t_{0}) ≥ 0.

From the Hele-Shaw equation we deduce

\frac{∂ℜeh(ζ, t)}{∂t} = ℜe[h′ζp − ζp′ − ζ^{2}p′′]

Taking into account (20), (21) we calculate

\frac{∂ℜeh(ζ, t)}{∂t}|_{t=t_{0}} = −\frac{|F (ζ_{0}, t_{0})|^{4}}{|F′(ζ_{0}, t_{0})|^{2}} ℜeζ_{0}h′(ζ_{0}, t_{0}) > 0.

Hence ℜeh(ζ, t) > 0 for all |ζ| < 1 and t > t_{0} close to t_{0}. This completes the proof of Theorem 2.

§5. Close-to-convex boundaries

Here we consider the inner problem of inheriting the close-to-convexity property of the initial domain of flows by the solutions of the problem of the Hele-Shaw flows with moving boundaries.

The close-to-convexity property means [17] that the complement of f(U) can be covered by a family of non-intersecting rays having with ∂f(U) a common point or a common segment of a ray or a whole ray. It is known [18] that the close-to-convexity property is equivalent to the following analytic statement: for any θ_{1}, θ_{2} such that 0 < θ_{2} − θ_{1} < 2π the following inequality is valid

∫_{θ_{1}}^{θ_{2}}[1 + ℜe (\frac{re^{iθ}f′′(re^{iθ})}{f′(re^{iθ})})] dθ ≥ −π, 0 < r ≤ 1,

(the equality sign is possible if r = 1). We denote

H(θ_{1}, θ_{2}, f(ζ, t)) = ∫_{θ_{1}}^{θ_{2}}[1 + ℜe (ζ \frac{∂}{∂ζ} log (\frac{∂f(ζ, t)}{∂ζ}))] dθ; ζ = re^{iθ}.

Let f(ζ) = f(ζ, t_{0}) be a critical close-to-convex mapping; i.e., there are such θ_{1} and θ_{2} that 0 < θ_{2} − θ_{1} < 2π and H(θ_{1}, θ_{2}, f(e^{iθ})) = −π. Without loss of generality we assume t_{0} = 0. Therefore, J_{1}(θ) = H(θ, θ_{2}, f(e^{iθ})) and J_{2}(θ) = H(θ_{1}, θ, f(e^{iθ})) as differentiable functions of the first and the second argument H correspondently have the local minima J_{1}(θ) at the point θ_{1}; J_{2}(θ)– at the point θ_{2}; i.e., J′_{1}(θ_{1}) = 0, J′_{2}(θ_{2}) = 0. It means

(22)  ℜe (e^{iθ_{1}} \frac{f′′(e^{iθ_{1}})}{f′(e^{iθ_{1}})}) = ℜe (e^{iθ_{2}} \frac{f′′(e^{iθ_{2}})}{f′(e^{iθ_{2}})}) = −1

The function J_{3}(r) = H(θ_{1}, θ_{2}, f(re^{iθ})) locally decreases for r ∈ (0, 1] in the neighbourhood of 1 and J_{3}(r) is differentiable there. Hence

J′_{3}(1) = ℑm ∫_{θ_{1}}^{θ_{2}}d (\frac{e^{iθ}f′′(e^{iθ})}{f′(e^{iθ})}) =

(23)  ℑm (\frac{e^{iθ_{2}}f′′(e^{iθ_{2}})}{f′(e^{iθ_{2}})} − \frac{e^{iθ_{1}}f′′(e^{iθ_{1}})}{f′(e^{iθ_{1}})}) ≤ 0.

It is obvious that to check inheriting the close-to-convexity property it is equivalent to check the sign of

\frac{∂H(θ_{1}, θ_{2}, f(e^{iθ}, t))}{∂t}

at the point t = t_{0}. If it is negative, then there is a neighbourhood (t_{0}, t_{0} +ε) where H(θ_{1}, θ_{2}, f(ζ, t)) < −π which contradicts the condition of close-to-convexity.

Calculate

\frac{∂ \frac{e^{iθ}f′′(e^{iθ},t)}{f′(e^{iθ},t)}}{∂t} = −i\frac{∂}{∂θ} \{(1 + \frac{e^{iθ}f′′(e^{iθ}, t)}{f′(e^{iθ}, t)}) p + e^{iθ}p′_{ζ}\}

where we take the function p from (9).

Due to (22) we obtain from the Hele-Shaw equation (3)

\frac{∂}{∂t}H(e^{iθ_{1}}, e^{iθ_{2}}, f(e^{iθ}, t))|_{t=t_{0}} = [\frac{Q}{|f′(e^{iθ})|^{2}} ℑm\frac{e^{iθ}f′′(e^{iθ})}{f′(e^{iθ})}]_{θ_{1}}^{θ_{2}} + ℑm(e^{iθ}p′_{ζ})|_{θ_{1}}^{θ_{2}}.

Integrating by parts the derivation of the Schwarz integral p as in (10) we receive that the right-hand side of (24) is equal to

\frac{−Q}{|f′(e^{iθ})|^{2}} ℑm\frac{e^{iθ}f′′(e^{iθ})}{f′(e^{iθ})}|_{θ_{1}}^{θ_{2}}

Foremost, we assume Q < 0 and consider the example of the critical mapping

f(ζ) = ∫_{0}^{ζ} exp [−\frac{1}{2π} ∫_{0}^{2π}(γ(θ) − θ − \frac{π}{2} )S(θ, ζ)dθ] dζ,

where S(θ, ζ) is the Schwarz kernel, γ(θ) = \frac{3}{2}π(1 + sin(\frac{α}{π}(θ − π))), α = π − arcsin \frac{2}{3}; θ_{2} = −θ_{1} = π − \frac{π^{2}}{2α}. This function satisfies the condition f(ζ̄) = \overline{f(ζ)}, hence |f′(e^{iθ_{1}})| = |f′(e^{iθ_{2}})| and it is sufficient to prove that

ℑm\frac{e^{iθ}f′′(e^{iθ})}{f′(e^{iθ})} |_{θ_{1}}^{θ_{2}} ≠ 0. Then due to (23) we obtain that the both expressions

in (24) are negative.

Calculate integrating by parts

(25) ℑm\frac{e^{iφ}f′′(e^{iφ})}{f′(e^{iφ})}|_{θ_{1}}^{θ_{2}} = \frac{−3α^{2}}{2π^{2}} ∫_{0}^{2π}sin(\frac{α}{π}(θ − π)) log[1 + cos(θ + \frac{π^{2}}{2α})]dθ.

Due to the obvious inequality \frac{π}{2} < \frac{π^{2}}{2α} < π it is easy to obtain that the right-hand side of (25) is strictly negative and therefore is not equal to zero.

To complete the proof we show that f(z) is close-to-convex checking the condition ℜef′(z) ≥ 0. This condition is equivalent to the following inequality

−\frac{π}{2} ≤ ℜe\frac{1}{2π} ∫_{0}^{2π}(γ(θ) − θ − \frac{π}{2} )\frac{e^{iθ} + z}{e^{iθ} − z} dθ ≤ \frac{π}{2} .

The right-hand side of this inequality is equal to the obvious one: ∫_{0}^{2π}(γ(θ)−

θ − π)P (z, θ)dθ ≤ 0, where P (z, θ) is the Poisson kernel. The left-hand side inequality can be proved analogously. In case of the inner problem with the source at the origin (Q > 0) the example of the critical close-to-convex function

g(ζ) = \frac{f(\frac{ζ+a}{1+ζā}) − f(a)}{f′(a)(1 − |a|^{2})},

where a = (1 − ϵ)e^{iθ_{1}} and ϵ is sufficiently small, shows that the solutions of the Hele-Shaw equation (3) with the initial mapping g(ζ) and t ≥ 0 close to zero do not preserve the property of close-to-convexity.

The solutions f(ζ^{∗}, t) describing the dynamics of the domains of the Hele-Shaw flows with neither source nor sewer at the infinity do not preserve the geometrical property of close-to-convexity. By the change ζ^{∗} → \frac{1}{F ( \frac{1}{ζ^{∗}})}, |ζ^{∗}| > 1 rewriting the condition of close-to-convexity for the map-

ping F (ζ), ζ = \frac{1}{ζ^{∗}} we obtain the inequality

H̃(θ_{1}, θ_{2}, F (ζ, t)) =

= ∫_{θ_{1}}^{θ_{2}}[−1 − ℜe e^{iθ} \frac{F′′(re^{iθ})}{F′(re^{iθ})} + 2ℜe\frac{e^{iθ}F′(re^{iθ})}{F (re^{iθ})}] dθ ≥ −π, o < r < 1,

(the equality sign is possible if r = 1).

We suppose F (ζ, t) to be a critical mapping; i,e., there are such θ_{1} and θ_{2} that 0 < θ_{2} − θ_{1} < 2π, t_{0} ≥ 0 and H(θ_{1}, θ_{2}, F (ζ, t_{0})) = −π. Without loss of generality we assume t_{0} = 0. Analogously (22), (23) we obtain

(26)  2ℜe\frac{e^{iθ_{j}}F′(e^{iθ_{j}})}{F (e^{iθ_{j}})} − ℜe\frac{e^{iθ_{j}}F′′(e^{iθ_{j}})}{F′(e^{iθ_{j}})} = 1, j = 1, 2,

(27)  2ℑm\frac{e^{iθ}F′(e^{iθ})}{F (e^{iθ})} − ℑm\frac{e^{iθ}F′′(e^{iθ})}{F′(e^{iθ})}|_{θ_{1}}^{θ_{2}} ≤ 0

Integrating by parts the derivation of the Schwarz integral p with the density \frac{|F (e^{iθ},t)|^{4}}{|F′(e^{iθ},t)|^{2}} in the Hele-Shaw equation and taking into account (26) we receive

\frac{∂}{∂t}H̃(e^{iθ_{1}}, e^{iθ_{2}}, F (e^{iθ}, t))|_{t=t_{0}} =

(28) Q [\frac{|F (e^{iθ}, t_{0})|^{4}}{|F′(e^{iθ}, t_{0})|^{2}} ℑm (\frac{e^{iθ}F′′(e^{iθ}, t_{0})}{F′(e^{iθ}, t_{0})} − 2\frac{e^{iθ}F′(e^{iθ}, t_{0})}{F (e^{iθ}, t_{0})})]_{θ_{1}}^{θ_{2}} .

Obviously we adduce the same examples in both cases Q > 0 and Q < 0 as in the inner problem f(ζ) and g(ζ) to make the right-hand side of (28) negative according to (27). This finishes the proof.

References

1. P.Ya. Polubarinova-Kochina, Dokl. Acad. Nauk SSSR 47 (1945), 254-257.

2. L.A. Galin, Dokl. Akad. Nauk SSSR 47 (1945), 246-249.

3. P.P. Kufarev, Dokl. Akad. Nauk SSSR 60 (1948), 1333-1334.

4. C.M. Eliot, J.R. Ockendon, Weak and variational methods for moving boundary problem, London: Pitman, 1992.

5. S.D. Howison, Yu.E. Hohlov, Dokl. Akad. Nauk Rossii 325 (1992), 1161-1166.

6. Yu.E. Hohlov, S.D. Howison, On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows, Quart. Appl. Math. 54 (1994), 777-789.

7. E. DiBenedetto, A. Friedman, Trans. AMS 282 (1984), 183-204.

8. V.M. Entov, P.L. Etingof, Quart. J. Mech. Appl. Math. 44, 507-535.

9. S.D. Howison, Complex variable methods in Hele-Shaw moving boundary problems, Europ. J. Appl. Math. 3 (1992), 209-224.

10. Yu.E. Hohlov, S.D. Howison, C.Huntingford, J.R. Ockendon, A.A. Lacey, Quart. J. Mech. Appl. Math. 47 (1994), 107-128.

11. Yu.P. Vinogradov, P.P. Kufarev, Prikl. Matem. Mech. 12 (1948), 181-198.

12. M.Reissing, L. von Wolfersdorf, Arkiv för Matematik 31 (1993), 101-116.

13. M.Reissing, Nonlinear Analysis, TMA 23 (1994), 565-576.

14. Yu.E. Hohlov, M. Reissig, Europ. J. Appl. Math. (to appear) 6 (1995).

15. T.Nishida, J. Diff. Geom. 12 (1977), 629-633.

16. I.A.Aleksandrov, Parametric Continuations in the Theory of Univalent Functions, Nauka, Moscow (in Russian), 1976.

17. Chr. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 1973.

18. W. Kaplan, Mich. Math. J. 1 (1952), 169-185.

Departamento de Matematicas Universidad de Los Andes Santafe de Bogota, Colombia

E-mail address: avassill@uniandes.edu.co

Department of Mathematics and Mechanics Saratov State University Saratov, Russia

E-mail address: prokhor@scnit.saratov.su

Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, Russia

E-mail address: hohlov@math.ru

Received November 4, 1997