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Abstract and Applied Analysis
Volume 2011, Article ID 948236, 10 pages
http://dx.doi.org/10.1155/2011/948236
Research Article

A Special Class of Univalent Functions in Hele-Shaw Flow Problems

1Faculty of Economics and Business Administration, Babeş-Bolyai University, 400591 Cluj-Napoca, Romania
2Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania

Received 7 December 2010; Accepted 23 February 2011

Academic Editor: Yoshikazu Giga

Copyright © 2011 Paula Curt and Denisa Fericean. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection. Applying methods from the theory of univalent functions, we prove the invariance in time of Φ-likeness property (a geometric property which includes starlikeness and spiral-likeness) for two basic cases: the inner problem and the outer problem. We study both zero and nonzero surface tension models. Certain particular cases are also presented.

1. Introduction

The time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection was studied by many authors. By using methods of univalent functions theory, they proved that certain geometric properties (such as starlikeness and directional convexity) are preserved in time [16]. In this paper, we continue their study by proving the invariance in time of another geometric property: Φ-likeness.

In the first section of the paper, we review the main results that are needed later. We start by presenting the basic notions regarding the bounded case.

In this case, we study the flow of a viscous fluid in a planar Hele-Shaw cell under injection through a source (of constant strength 𝑄, 𝑄<0 in case of injection) which is situated at the origin. Suppose that at the initial moment the domain Ω(0) occupied by the fluid is simply connected and is bounded by an analytic and smooth curve Γ(0)=𝜕Ω(0). By using the well-known Riemann mapping theorem, the domain Ω(𝑡) (occupied by the fluid at the moment 𝑡) can be described by a univalent function 𝑓(𝜁,𝑡) of the unit disk 𝑈={𝜁|𝜁|<1} onto Ω(𝑡) normalized by 𝑓(0,𝑡)=0, 𝑓(0,𝑡)>0. We denote Γ(𝑡)=𝜕Ω(𝑡). The function 𝑓(𝜁,0)=𝑓0(𝜁) produces a parametrization of Γ0={𝑓0(𝑒𝑖𝜃),𝜃[0,2𝜋)}, and the moving boundary is parameterized by Γ(𝑡)={𝑓(𝑒𝑖𝜃,𝑡),𝜃[0,2𝜋)}.

The equation satisfied by the free boundary was first derived by Galin [7] and Polubarinova-Kochina [8, 9]:̇Re𝑓(𝜁,𝑡)𝜁𝑓𝑄(𝜁,𝑡)=2𝜋,𝜁=𝑒𝑖𝜃(1.1) (in the previous equality we have used the notations: 𝑓=𝜕𝑓/𝜕𝜁, ̇𝑓=𝜕𝑓/𝜕𝑡).

A classical solution in the interval [0,𝑇) is a function 𝑓(𝜁,𝑡), 𝑡[0,𝑇), that is univalent on 𝑈 and 𝐶1 with respect to 𝑡 in [0,𝑇). It is known that, starting with an analytic and smooth boundary Γ(0), the classical solution exists and is unique locally in time (see [10]; see e.g., [11, Chapter 1]). Note that 𝑇 is called the blow-up time.

In the case of the problem of injection (𝑄<0) of the fluid into a bounded simply connected with small surface tension 𝛾>0, the Polubarinova-Galin equation [6] is of the form:̇Re𝑓(𝜁,𝑡)𝜁𝑓𝑄(𝜁,𝑡)=𝑖2𝜋+𝛾𝐻𝜕𝑘𝑒𝜕𝜃𝑖𝜃,𝑡(𝜃),𝜁=𝑒𝑖𝜃,(1.2) where 𝑘(𝑒𝑖𝜃,𝑡)=(1/|𝑓(𝑒𝑖𝜃,𝑡)|)Re(1+(𝑒𝑖𝜃𝑓(𝑒𝑖𝜃,𝑡)/𝑓(𝑒𝑖𝜃,𝑡))), 𝜃[0,2𝜋), and the Hilbert transform in (1.2) is given by (1/𝜋)P.V.𝜃02𝜋Φ(𝑒𝑖𝜃)/(1𝑒𝑖(𝜃𝜃))𝑑𝜃=𝐻[Φ](𝜃).

We mention the following technical results: (𝜕𝑘/𝜕𝜃)(𝑒𝑖𝜃,𝑡)=Im(𝑒2𝑖𝜃𝑆𝑓(𝑒𝑖𝜃,𝑡))/|𝑓(𝑒𝑖𝜃,𝑡)| where 𝑆𝑓 is the Schwarzian derivative given by 𝑆𝑓(𝜁)=(𝑓(𝜁,𝑡)/𝑓(𝜁,𝑡))(1/2)(𝑓(𝜁,𝑡)/𝑓(𝜁,𝑡))2 and𝜕𝑖𝜕𝜃𝜕𝑘𝑒𝜕𝜃𝑖𝜃𝜕,𝑡(𝜃)=𝑒𝜕𝜃𝑖Im2𝑖𝜃𝑆𝑓𝑒𝑖𝜃||𝑓𝑒𝑖𝜃||[]=𝐻𝑖𝐴(𝜃),(1.3) where 1𝐴(𝜁)=||𝑓||(𝜁)Re2𝜁2𝑆𝑓𝑓(𝜁)+𝜁(𝜁)𝑓(𝜁)𝑓(𝜁)𝑓𝑓(𝜁)(𝜁)𝑓(𝜁)+Im𝜁𝑓(𝜁)𝑓(𝜁)Im𝜁2𝑆𝑓.(𝜁)(1.4)

The case of unbounded domain with bounded complement can be viewed as the dynamics of a contracting bubble in a Hele-Shaw cell since the fluid occupies a neighbourhood of infinity and injection (of constant strength 𝑄<0) is supposed to take place at infinity. Again, we denote by Ω(𝑡) the domain occupied by the fluid at the moment 𝑡, Γ(𝑡)=𝜕Ω(𝑡). By using the Riemann mapping theorem, the domain Ω(𝑡) can be described by a univalent function 𝐹(𝜁,𝑡) from the exterior of the unit disk 𝑈={𝜁|𝜁|>1} onto Ω(𝑡), 𝐹(𝜁,𝑡)=𝑎𝜁+𝑎0+𝑎1/𝜁+, 𝑎>0.

The equation satisfied by the free boundary is [4, 6]̇Re𝐹(𝜁,𝑡)𝜁𝐹=𝑄(𝜁,𝑡)2𝜋,𝜁=𝑒𝑖𝜃(1.5) for the zero tension surface model anḋRe𝐹(𝜁,𝑡)𝜁𝐹=𝑄(𝜁,𝑡)𝑖2𝜋𝛾𝐻𝜕𝑘𝑒𝜕𝑡𝑖𝜃,𝑡(𝜃),𝜁=𝑒𝑖𝜃(1.6) for the small surface tension model.

2. The Inner Problem (Bounded Domains)

In this section, we obtain the invariance in time of Φ-likeness property for the inner problem. Starting with an initial bounded domain Ω(0) which is Φ-like, we prove that at each moment 𝑡[0,𝑇) the domain Ω(𝑡) is Φ-like (both for zero and nonzero surface tension models).

Definition 2.1. Let 𝑓 be a holomorphic function on 𝑈 such that 𝑓(0)=0 and 𝑓(0)0. Let Φ be a holomorphic function on 𝑓(𝑈) such that Φ(0)=0 and ReΦ(0)>0. We say that 𝑓 is Φ-like on 𝑈 (or Φ-like) if Re𝑧𝑓(𝑧)Φ(𝑓(𝑧))>0,foreach𝑧𝑈.(2.1)

We remark that a Φ-like function is univalent. In fact, any univalent function is Φ-like for some Φ.

Remark 2.2. (a) The concept of Φ-likeness was introduced and studied by Brickman in 1973 [12] and generalizes the notions of starlikeness and spiral-likeness. Applications of this notion in the study of univalence may be found in [13].
(b) If Φ(𝑤)=𝑤 in the above definition, then 𝑓 is starlike.
(c) If Φ(𝑤)=𝜆𝑤 and Re𝜆>0, then 𝑓 is spiral-like of type arg𝜆.

We restate that a holomorphic function 𝑓 on 𝑈 such that 𝑓(0)=0 and 𝑓(0)0 is said to be spiral-like of type 𝛼(𝜋/2,𝜋/2) if Re(𝑒𝑖𝛼𝑧𝑓(𝑧)/𝑓(𝑧))>0, 𝑧𝑈 [13, 14].

The following result is a generalization of [1,Theorem 1] to the case of Φ-like functions. The mentioned theorem may be obtained by taking Φ(𝑤)𝑤 in Theorem 2.3 below.

Theorem 2.3. Let 𝑄<0 and 𝑓0 be a function which is Φ-like on 𝑈 and univalent on 𝑈. Let 𝑓(𝜁,𝑡) be the classical solution of the Polubarinova-Galin equation (1.1) with the initial condition 𝑓(𝜁,0)=𝑓0(𝜁). Also let Ω=0𝑡<𝑇Ω(𝑡)=0𝑡<𝑇𝑓(𝑈,𝑡), where 𝑇 is the blow-up time. If Φ is holomorphic on Ω and satisfies the condition ReΦ(𝑤)>0,𝑤Ω,(2.2) then 𝑓(𝜁,𝑡) is Φ like for 𝑡[0,𝑇).

Proof. Taking into account that all the functions 𝑓(𝜁,𝑡) have analytic univalent extensions to 𝑈 for each 𝑡[0,𝑇) and in consequence their derivatives 𝑓(𝜁,𝑡) are continuous and do not vanish in 𝑈, we can replace with “≥” the inequality in the definition (2.1) of a Φ-like function. The equality can be attained only for |𝜁|=1.
We suppose by contrary that the conclusion of Theorem 2.3 is not true. Then there exist 𝑡00 and 𝜁0=𝑒𝑖𝜃0 such that arg(𝜁0𝑓(𝜁0,𝑡0)/Φ(𝑓(𝜁0,𝑡0)))=𝜋/2(or𝜋/2), which is equivalent to 𝜁Re0𝜁𝑓0,𝑡0Φ𝑓𝜁0,𝑡0𝜁=0,Im0𝜁𝑓0,𝑡0Φ𝑓𝜁0,𝑡00,(2.3) and for each 𝜀>0, there are 𝑡>𝑡0 and 𝜃(𝜃0𝜀,𝜃0+𝜀) such that 𝑒arg𝑖𝜃𝑓𝑒𝑖𝜃,𝑡Φ𝑓𝑒𝑖𝜃𝜋,𝑡2𝜋or2.(2.4)
Let 𝑡0 be the first such point, 𝑡0[0,𝑇). Without loss of generality, we assume that 𝜁Im0𝑓𝜁0,𝑡0Φ𝑓𝜁0,𝑡0>0.(2.5)
In fact, (1,𝜃0) is a maximum point for the function 𝑔(𝑟,𝜃)=arg(𝑟𝑒𝑖𝜃𝑓(𝑟𝑒𝑖𝜃,𝑡0)/Φ(𝑓(𝑟𝑒𝑖𝜃,𝑡0))), where 𝑟[0,1], 𝜃[0,2𝜋]. Hence, (𝜕/𝜕𝜃)𝑔(1,𝜃0)=0 and (𝜕/𝜕𝑟)𝑔(1,𝜃0)0 (the stationarity condition at an endpoint of an interval), and in consequence we obtain 𝜁Re1+0𝑓𝜁0,𝑡0𝑓𝜁0,𝑡0Φ𝑓𝜁0,𝑡0𝜁0𝑓𝜁0,𝑡0Φ𝑓𝜁0,𝑡0𝜁=0,(2.6)Im1+0𝑓𝜁0,𝑡0𝑓𝜁0,𝑡0Φ𝑓𝜁0,𝑡0𝜁0𝑓𝜁0,𝑡0Φ𝑓𝜁0,𝑡00.(2.7)
By straightforward calculations, we get 𝜕𝜕𝑡arg𝜁𝑓(𝜁,𝑡)Φ(𝑓(𝜁,𝑡))=Im(𝜕/𝜕𝑡)𝑓(𝜁,𝑡)𝑓Φ(𝜁,𝑡)(𝑓(𝜁,𝑡))(𝜕/𝜕𝑡)𝑓(𝜁,𝑡)Φ(𝑓(𝜁,𝑡)).(2.8)
By differentiating the Polubarinova-Galin equation with respect to 𝜃, we obtain Im((𝜕/𝜕𝑡)𝑓(𝜁,𝑡)𝑓(𝜁,𝑡)𝜁𝑓(𝜁,𝑡)(𝜕/𝜕𝑡)𝑓(𝜁,𝑡)𝜁2𝑓(𝜁,𝑡)(𝜕/𝜕𝑡)𝑓(𝜁,𝑡))=0 for 𝜁=𝑒𝑖𝜃. The previous equality yields the following relation: ||𝑓||(𝜁,𝑡)2Im(𝜕/𝜕𝑡)𝑓(𝜁,𝑡)Φ𝑓(𝜁,𝑡)(𝑓(𝜁,𝑡))(𝜕/𝜕𝑡)𝑓(𝜁,𝑡)Φ(𝑓(𝜁,𝑡))=Im𝜁𝑓̇(𝜁,𝑡)𝑓(𝜁,𝑡)1+𝜁𝑓(𝜁,𝑡)𝑓Φ(𝜁,𝑡)(𝑓(𝜁,𝑡))𝜁𝑓(𝜁,𝑡).Φ(𝑓(𝜁,𝑡))(2.9)
If we substitute (2.6), (2.7), (2.8), and (1.1) in the above expression and replace 𝜃 by 𝜃0 and 𝑡 by 𝑡0, we obtain 𝜕𝜕𝑡arg𝜁𝑓(𝜁,𝑡)||||Φ(𝑓(𝜁,𝑡))𝜁=𝜁0,𝑡=𝑡0=𝑄||𝑓2𝜋(𝜁0,𝑡0)||2𝜁Im0𝑓𝜁0,𝑡0𝑓𝜁0,𝑡0Φ𝑓𝜁0,𝑡0𝜁0𝑓𝜁0,𝑡0Φ𝑓𝜁0,𝑡0+2ReΦ𝑓𝜁0,𝑡0𝜁Im0𝑓𝜁0,𝑡0Φ𝑓𝜁0,𝑡0<0,(2.10) due to (2.2), (2.3) and (2.5). Finally, we get (𝜕/𝜕𝑡)arg(𝜁𝑓(𝜁,𝑡)/Φ(𝑓(𝜁,𝑡)))|𝜁=𝜁0,𝑡=𝑡0<0. Therefore, arg(𝑒𝑖𝜃𝑓(𝑒𝑖𝜃,𝑡)/Φ(𝑓(𝑒𝑖𝜃,𝑡)))<𝜋/2, for 𝑡>𝑡0 (close to 𝑡0) in some neighbourhood of 𝜃0. This contradicts the assumption (2.4) and completes the proof.

If in the previous theorem, we take Φ(𝑤)𝑒𝑖𝛼𝑤, 𝛼(𝜋/2,𝜋/2); then we obtain the following corollary.

Corollary 2.4. Let 𝑄<0, and let 𝑓0 be a function which is spiral-like of type 𝛼(𝜋/2,𝜋/2) on 𝑈 and univalent on 𝑈. Then the classical solution of the Polubarinova-Galin equation (1.1) with the initial condition 𝑓(𝜁,0)=𝑓0(𝜁) is spiral-like of type 𝛼 for 𝑡[0,𝑇), where 𝑇 is the blow-up time.

The following result is a generalization of [6,Theorem 1] to the case of Φ-like functions. The mentioned theorem may be obtained by taking Φ(𝑤)𝑤 in Theorem 2.5 below.

Theorem 2.5. Let 𝑄<0 and the surface tension 𝛾 be sufficiently small. If 𝑓0 is a function which is Φ-like on 𝑈 and univalent on 𝑈, then there exists 𝑡(𝛾)𝑇 such that the classical solution 𝑓(𝜁,𝑡) of (1.2) with the initial condition 𝑓(𝜁,0)=𝑓0(𝜁) is Φ-like for 𝑡[0,𝑡(𝛾)), where 𝑇 is the blow-up time, Ω=0𝑡<𝑡(𝛾)Ω(𝑡)=0𝑡<𝑡(𝛾)𝑓(𝑈,𝑡), and Φ is a holomorphic function on Ω which satisfies the condition (2.2).

Proof. If we consider 𝑓 in the closure of 𝑈, then the inequality sign in (2.1) can be replaced by “≥” where equality can be attained for |𝜁|=1.
Suppose by contrary that the conclusion of Theorem 2.5 is not true. Then there exist 𝑡00 and 𝜁0=𝑒𝑖𝜃0 such that (2.3), (2.4), (2.5), (2.6), and (2.7) are true. At the same time the equality (2.8) is fulfilled. By differentiating the Polubarinova-Galin equation (1.2) with respect to 𝜃, we obtain 𝜕Im𝑓𝜕𝑡(𝜁,𝑡)𝑓(𝜁,𝑡)𝜁𝑓𝜕(𝜁,𝑡)𝜕𝑡𝑓(𝜁,𝑡)𝜁2𝑓𝜕(𝜁,𝑡)[]𝜕𝑡𝑓(𝜁,𝑡)=𝛾𝐻𝑖𝐴(𝜃),(2.11) for 𝜁=𝑒𝑖𝜃. The previous equality is equivalent to the following: ||𝑓||(𝜁,𝑡)2Im(𝜕/𝜕𝑡)𝑓(𝜁,𝑡)𝑓(𝜁,𝑡)Φ(𝑓(𝜁,𝑡))(𝜕/𝜕𝑡)𝑓(𝜁,𝑡)||||Φ(𝑓(𝜁,𝑡))𝜁=𝜁0=𝑒0𝑖𝜃,𝑡=𝑡0=𝑄𝑖2𝜋𝛾𝐻𝜕𝑘𝑒𝜕𝜃𝑖𝜃×,𝑡Im𝜁𝑓𝑓Φ(𝑓)Φ(𝑓)𝜁𝑓+2ReΦ(𝑓)Im𝜁𝑓||||Φ(𝑓)𝜁=𝜁0,𝑡=𝑡0,𝜃=𝜃0[]𝛾𝐻𝑖𝐴(𝜃)𝜁=𝜁0,𝑡=𝑡0,𝜃=𝜃0.(2.12)
The right-hand side of this equality is strictly negative for small 𝛾 because of (2.5), (2.7), and the fact that ReΦ(𝑓)>0.
By using (2.8), we obtain that arg(𝑒𝑖𝜃𝑓(𝑒𝑖𝜃,𝑡)/Φ(𝑓(𝑒𝑖𝜃,𝑡)))<𝜋/2 for 𝑡>𝑡0 (close to 𝑡0) in a neighbourhood of 𝜃0. This is in contradiction with our assumption and ends the proof.

Remark 2.6. Let 𝑄<0, and let 𝑓0 be a function that is Φ-like on 𝑈 and univalent on 𝑈. If 𝑓0 satisfies the condition (2.1) for each 𝜁𝑈, then there exist a surface tension 𝛾 (which depends on 𝑓0) sufficiently small and 𝑡(𝛾)𝑇 such that the classical solution 𝑓(𝜁,𝑡) of (1.2) with the initial condition 𝑓(𝜁,0)=𝑓0(𝜁) are Φ-like for 𝑡[0,𝑡(𝛾)), where 𝑇 is the blow-up time.

Proof. The conclusion is immediate from the smoothness of the classical solution of (1.1).

If Φ(𝑤)𝑒𝑖𝛼𝑤 in Theorem 2.5, where 𝛼(𝜋/2,𝜋/2), then we obtain the following corollary:

Corollary 2.7. Let 𝑄<0 and the surface tension 𝛾 be sufficiently small. If 𝑓0 is a function which is spiral-like of type 𝛼(𝜋/2,𝜋/2) on 𝑈 and univalent on 𝑈, then there exists 𝑡(𝛾)𝑇 such that the classical solution 𝑓(𝜁,𝑡) of (1.2) with the initial condition 𝑓(𝜁,0)=𝑓0(𝜁) is spiral-like of type 𝛼 for 𝑡[0,𝑡(𝛾)), where 𝑇 is the blow-up time.

3. The Outer Problem (Unbounded Domain with Bounded Complement)

In this section, we obtain the invariance in time of the same geometric property (denoted by Φ) for the outer problem.

Definition 3.1. Let 𝐹 be a holomorphic function on 𝑈={𝜁|𝜁|>1} such that 𝐹(𝜁)=𝑎𝜁+𝑎0+𝑎1/𝜁+, where 𝑎0. Let Φ be a holomorphic function on 𝐹(𝑈) such that lim𝜁Φ(𝜁)= and lim𝜁Φ(𝜁)>0. We say that 𝐹 is Φ-like on 𝑈 if Re𝜁𝐹(𝜁)Φ(𝐹(𝜁))>0,𝜁𝑈.(3.1)

Remark 3.2. (a) If 𝐹 is a Φ-like function on 𝑈, then the function 𝑓𝑈𝐂 given by 𝑓(𝑧)=1/𝐹(1/𝑧), 𝑧0, and 𝑓(0)=0, is Φ-like on 𝑈, where Φ𝑓(𝑈)𝐂, Φ(𝑤)=𝑤2Φ(1/𝑤), for all𝑤𝑓(𝑈){0} and Φ(0)=0.
(b) If 𝑓 is a Φ-like function on 𝑈, then the function 𝐹𝑈𝐂, 𝐹(𝜁)=1/𝑓(1/𝜁) is Φ-like on 𝑈, where Φ𝐹(𝑈)𝐂, Φ(𝜔)=𝜔2Φ(1/𝜔), for all 𝜔𝐹(𝑈).
(c) Any Φ-like holomorphic function 𝐹 on 𝑈 is univalent on 𝑈.

Proof. This remark can be obtained by straightforward computations and its proof is, therefore, left to the reader.

The following result is a generalization of [5,Theorem 3]. The mentioned theorem may be obtained by taking Φ(𝑤)𝑤 in Theorem 3.3 below.

Theorem 3.3. Let 𝐹0 be a function which is Φ-like on 𝑈 and univalent on 𝑈. Then the solution 𝐹(𝜁,𝑡) of the Polubarinova-Galin equation (1.5) with the initial condition 𝐹(𝜁,0)=𝐹0(𝜁) is Φ-like for 𝑡[0,𝑇), where 𝑇 is the blow-up time, Ω=0𝑡<𝑇Ω(𝑡)=0𝑡<𝑇𝐹(𝑈,𝑡), and Φ is a holomorphic function on Ω which satisfies the following conditions: ReΦ(𝜔)𝜔Φ>0,Re(𝜔)<2ReΦ(𝜔)𝜔,𝜔Ω.(3.2)

Proof. By considering the function 𝑓(𝜁,𝑡)=1/𝐹(1/𝜁,𝑡), the Polubarinova-Galin equation (1.5) can be rewritten in terms of 𝑓 as follows: ̇Re𝑓(𝜁,𝑡)𝜁𝑓𝑄||||(𝜁,𝑡)=𝑓(𝜁,𝑡)4,||𝜁||2𝜋=1.(3.3) Due to the previous remark, the function 𝐹(𝜁,𝑡), 𝜁𝑈, is Φ-like if and only if 𝑓(𝜁,𝑡), 𝜁𝑈, is Φ-like, where the relationship between Φ and Φ is Φ(𝜔)=𝜔2Φ(1/𝜔), for all 𝜔𝐹(𝑈)(orΦ(𝑤)=𝑤2Φ(1/𝑤), for all 𝑤𝑓(𝑈)). Thus, it suffices to prove that the functions 𝑓(𝜁,𝑡) are Φ-like for 𝑡[0,𝑇).
Suppose by contrary that the previous statement is not true. Then there exist 𝑡00 and 𝜁0=𝑒𝑖𝜃0 such that (2.3), (2.4), (2.5), (2.6), and (2.7) are true. At the same time, the equality (2.8) is fulfilled. We have to determine the sign of the function (𝜕/𝜕𝑡)arg(𝜁𝑓(𝜁,𝑡)/Φ(𝑓(𝜁,𝑡))) at the point (𝜁0,𝑡0).
By differentiating (3.3) and using (2.6), (2.8), and (3.3), we get 𝜕𝜕𝑡arg𝜁𝑓(𝜁,𝑡)||||Φ(𝑓(𝜁,𝑡))𝜁=𝜁0,𝑡=𝑡0=𝑄||𝑓||4||𝑓2𝜋||2Im𝜁𝑓𝑓Φ(𝑓)𝜁𝑓Φ(𝑓)+2Im𝜁𝑓Φ(𝑓)ReΦ(𝑓)+2ReΦ(𝑓)𝑓|||||𝜁=𝜁0,𝑡=𝑡0.(3.4)
The right-hand side of the previous equality is strictly negative in view of (2.5), (2.7), ReΦ(𝑓)>0, and Re(Φ(𝑓)/𝑓)>0 (the previous inequalities are easy consequences of (3.2)). Therefore, arg(𝑒𝑖𝜃𝑓(𝑒𝑖𝜃,𝑡)/Φ(𝑓(𝑒𝑖𝜃,𝑡)))<𝜋/2, for 𝑡>𝑡0 (close to 𝑡0) in some neighbourhood of 𝜃0. This contradiction completes the proof.

The following result is a generalization of [6,Theorem 3.1]. The mentioned theorem may be obtained by taking Φ(𝑤)𝑤 in Theorem 3.4 below.

Theorem 3.4. Let 𝑄<0, and let the surface tension 𝛾 be sufficiently small. If 𝐹0 is a function which is Φ-like on 𝑈 and univalent on 𝑈, then there exists 𝑡(𝛾)𝑇 such that the solution 𝐹(𝜁,𝑡) of (1.6) with the initial condition 𝐹(𝜁,0)=𝐹0(𝜁) is Φ-like for 𝑡[0,𝑡(𝛾)), where 𝑇 is the blow-up time, Ω=0𝑡<𝑡(𝛾)Ω(𝑡)=0𝑡<𝑡(𝛾)𝐹(𝑈,𝑡), and Φ is a holomorphic function on Ω which satisfies the conditions (3.2).

Proof. We introduce (as in the proof of Theorem 3.3) the function 𝑓(𝜁,𝑡)=1/𝐹(1/𝜁,𝑡),𝜁𝑈, which is Φ-like (Φ(𝑤)=𝑤2Φ(1/𝑤)) if and only if 𝐹(𝜁,𝑡) is Φ-like.
The Polubarinova-Galin equation can be written in terms of 𝑓 as ̇Re𝑓(𝜁,𝑡)𝜁𝑓||𝑓||(𝜁,𝑡)=4𝑄𝑖2𝜋𝛾𝐻𝜕𝑘,||𝜁||𝜕𝜃(𝜃)=1.(3.5) Suppose by contrary that the conclusion of Theorem 3.4 is not true. Then there exist 𝑡00 and 𝜁0=𝑒𝑖𝜃0 such that (2.3), (2.4), (2.5), (2.6), and (2.7) are true. At the same time, the equality (2.8) is fulfilled. We differentiate (3.5) with respect to 𝜃. Since the left side is differentiable with respect to 𝜃 and the solution of (3.5) exists and is unique, then the right-hand side is differentiable and its derivative is bounded on [0,2𝜋].
If we denote 𝐴(𝜃,𝑡)=(𝜕/𝜕𝜃)𝐻[𝑖𝜕𝑘/𝜕𝜃](𝜃); then by using (2.6), (2.8) and (3.5), we obtain 𝜕𝜕𝑡arg𝜁𝑓(𝜁,𝑡)||||Φ(𝑓(𝜁,𝑡))𝜁=𝜁0=𝑒0𝑖𝜃,𝑡=𝑡0=||𝑓||4||𝑓||2𝑄𝑖2𝜋𝛾𝐻𝜕𝑘×𝜕𝜃(𝜃)Im𝜁𝑓𝑓Φ(𝑓)𝜁𝑓Φ(𝑓)+Im𝜁𝑓Φ(𝑓)2ReΦ(𝑓)+4ReΦ(𝑓)𝑓||||𝜁=𝜁0,𝑡=𝑡0+||𝑓||4||𝑓||2|||||𝛾𝐴(𝜃,𝑡)𝜁=𝜁0,𝑡=𝑡0.(3.6)
The right-hand side of the previous equality is strictly negative for small 𝛾 because of (2.5), (2.7), ReΦ(𝑓)>0, and Re(Φ(𝑓)/𝑓)>0. Therefore, arg(𝑒𝑖𝜃𝑓(𝑒𝑖𝜃,𝑡)/Φ(𝑓(𝑒𝑖𝜃,𝑡)))<𝜋/2, for 𝑡>𝑡0 (close to 𝑡0) in some neighbourhood of 𝜃0. This contradiction completes the proof.

Remark 3.5. Let 𝑄<0, and let 𝐹0 be a function that is Φ-like on 𝑈and univalent on 𝑈. If 𝐹0 satisfies the condition (3.1) for each 𝜁𝑈, then there exist a surface tension 𝛾 (which depends on 𝐹0) sufficiently small and 𝑡(𝛾)𝑇 such that the classical solution 𝐹(𝜁,𝑡) of (1.6) with the initial condition 𝐹(𝜁,0)=𝐹0(𝜁) is Φ-like for 𝑡[0,𝑡(𝛾)), where T is the blow-up time.

Proof. The conclusion is immediate from the smoothness of the classical solution of (1.6)

Acknowledgments

The authors are indebted to Gabriela Kohr for suggesting this subject and for valuable discussions during the preparation of this paper. Denisa Fericean is supported by Contract no. POSDRU/88/1.5/S/60185-“Innovative Doctoral Studies in a Knowledge-Based Society”. The authors thank the referee for useful suggestions that improved the aspects of the paper.

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