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

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 [1–6]. 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(𝑒^𝑖𝜃),𝜃∈[0,2𝜋)}, and the moving boundary is parameterized by Γ(𝑡)={𝑓(𝑒^𝑖𝜃,𝑡),𝜃∈[0,2𝜋)}.

The equation satisfied by the free boundary was first derived by Galin [7] and Polubarinova-Kochina [8, 9]:Re𝑓̇(𝜁,𝑡)𝜁𝑓^(𝜁,𝑡)=−𝑄2𝜋,𝜁=𝑒^𝑖𝜃(1.1) (in the previous equality we have used the notations: 𝑓^=𝜕𝑓/𝜕𝜁, 𝑓̇=𝜕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:Re𝑓̇(𝜁,𝑡)𝜁𝑓^(𝜁,𝑡)=−𝑄2𝜋+𝛾𝐻𝑖𝜕𝜕𝑘𝜃𝑒^𝑖𝜃,𝑡(𝜃),𝜁=𝑒^𝑖𝜃,(1.2) where 𝑘(𝑒^𝑖𝜃,𝑡)=(1/|𝑓^(𝑒^𝑖𝜃,𝑡)|)Re(1+(𝑒^𝑖𝜃𝑓^(𝑒^𝑖𝜃,𝑡)/𝑓^(𝑒^𝑖𝜃,𝑡))), 𝜃∈[0,2𝜋), and the Hilbert transform in (1.2) is given by (1/𝜋)P.V.𝜃∫^02𝜋Φ(𝑒^𝑖𝜃′)/(1−𝑒^{𝑖(𝜃−𝜃′)})𝑑𝜃^=𝐻[Φ](𝜃).

We mention the following technical results: (𝜕𝑘/𝜕𝜃)(𝑒^𝑖𝜃,𝑡)=−Im(𝑒^2𝑖𝜃𝑆_𝑓(𝑒^𝑖𝜃,𝑡))/|𝑓^(𝑒^𝑖𝜃,𝑡)| where 𝑆_𝑓 is the Schwarzian derivative given by 𝑆_𝑓(𝜁)=(𝑓^(𝜁,𝑡)/𝑓^(𝜁,𝑡))^−(1/2)(𝑓^(𝜁,𝑡)/𝑓^(𝜁,𝑡))² and𝜕𝜕𝜃𝑖𝜕𝜕𝑘𝜃𝑒^𝑖𝜃,𝑡(𝜃)=−𝜕𝜕𝜃𝑖Im𝑒^2𝑖𝜃𝑆_𝑓𝑒^𝑖𝜃||𝑓^𝑒^𝑖𝜃||=𝐻[𝑖𝐴](𝜃),(1.3) where 𝐴(𝜁)=1||𝑓^(𝜁)||Re2𝜁²𝑆_𝑓(𝜁)+𝜁𝑓^(𝜁)𝑓^(𝜁)^−𝑓^(𝜁)𝑓^(𝜁)𝑓^(𝜁)+Im𝜁𝑓_𝑓^(𝜁)^(𝜁)𝑓^(𝜁)Im𝜁²𝑆_𝑓(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]Re𝐹̇(𝜁,𝑡)𝜁𝐹^(𝜁,𝑡)=𝑄2𝜋,𝜁=𝑒^𝑖𝜃(1.5) for the zero tension surface model andRe𝐹̇(𝜁,𝑡)𝜁𝐹^(𝜁,𝑡)=𝑄2𝜋−𝛾𝐻𝑖𝜕𝜕𝑘𝑡𝑒^𝑖𝜃,𝑡(𝜃),𝜁=𝑒^𝑖𝜃(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 Re𝑧𝑓′(𝑧)Φ(𝑓(𝑧))>0,foreach𝑧∈𝑈.(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(𝑒^𝑖𝛼𝑧𝑓′(𝑧)/𝑓(𝑧))>0, 𝑧∈𝑈 [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=⋃_{0≤𝑡<𝑇}Ω(𝑡)=⋃_{0≤𝑡<𝑇}𝑓(𝑈,𝑡), where 𝑇 is the blow-up time. If Φ is holomorphic on Ω and satisfies the condition ReΦ′(𝑤)>0,∀𝑤∈Ω,(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 𝑡₀≥0 and 𝜁₀=𝑒^𝑖𝜃0 such that arg(𝜁₀𝑓^(𝜁₀,𝑡₀)/Φ(𝑓(𝜁₀,𝑡₀)))=𝜋/2(or−𝜋/2), which is equivalent to Re𝜁⁰𝑓′𝜁₀,𝑡₀Φ𝑓𝜁₀,𝑡₀=0,Im𝜁⁰𝑓′𝜁₀,𝑡₀Φ𝑓𝜁₀,𝑡₀≠0,(2.3) and for each 𝜀>0, there are 𝑡>𝑡₀ and 𝜃∈(𝜃₀−𝜀,𝜃₀+𝜀) such that arg𝑒^𝑖𝜃𝑓^𝑒^𝑖𝜃,𝑡Φ𝑓𝑒^𝑖𝜃,𝑡≥𝜋2or≤−𝜋2.(2.4)
Let 𝑡₀ be the first such point, 𝑡₀∈[0,𝑇). Without loss of generality, we assume that Im𝜁⁰𝑓^𝜁₀,𝑡₀Φ𝑓𝜁₀,𝑡₀>0.(2.5)
In fact, (1,𝜃₀) is a maximum point for the function 𝑔(𝑟,𝜃)=arg(𝑟𝑒^𝑖𝜃𝑓^(𝑟𝑒^𝑖𝜃,𝑡₀)/Φ(𝑓(𝑟𝑒^𝑖𝜃,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 Re1+𝜁⁰𝑓^𝜁₀,𝑡₀𝑓^𝜁₀,𝑡₀−Φ^𝑓𝜁₀,𝑡₀𝜁₀𝑓^𝜁₀,𝑡₀Im1+𝜁₀Φ𝑓𝜁₀,𝑡₀=0,(2.6)𝑓^𝜁₀,𝑡₀𝑓^𝜁₀,𝑡₀−Φ^𝑓𝜁₀,𝑡₀𝜁₀𝑓^𝜁₀,𝑡₀Φ𝑓𝜁₀,𝑡₀≥0.(2.7)
By straightforward calculations, we get 𝜕𝜕𝑡arg𝜁𝑓^(𝜁,𝑡)Φ(𝑓(𝜁,𝑡))=Im(𝜕/𝜕𝑡)𝑓^(𝜁,𝑡)𝑓^(𝜁,𝑡)−Φ^(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=𝑒^𝑖𝜃. The previous equality yields the following relation: ||𝑓^(𝜁,𝑡)||²Im(𝜕/𝜕𝑡)𝑓′(𝜁,𝑡)𝑓′(𝜁,𝑡)−Φ^(𝑓(𝜁,𝑡))⋅(𝜕/𝜕𝑡)𝑓(𝜁,𝑡)=ImΦ(𝑓(𝜁,𝑡))𝜁𝑓^(𝜁,𝑡)𝑓̇(𝜁,𝑡)1+𝜁𝑓^(𝜁,𝑡)𝑓^(𝜁,𝑡)−Φ^(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}=𝑄2𝜋||𝑓^(𝜁₀,𝑡₀)||²Im𝜁⁰𝑓^𝜁₀,𝑡₀𝑓^𝜁₀,𝑡₀−Φ^𝑓𝜁₀,𝑡₀𝜁₀𝑓^𝜁₀,𝑡₀+2ReΦ_Φ𝑓𝜁₀,𝑡₀𝑓𝜁₀,𝑡₀Im𝜁⁰𝑓^𝜁₀,𝑡₀Φ𝑓𝜁₀,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(𝑒^𝑖𝜃𝑓^(𝑒^𝑖𝜃,𝑡)/Φ(𝑓(𝑒^𝑖𝜃,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)≡𝑒^−𝑖𝛼𝑤, 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=⋃_{0≤𝑡<𝑡(𝛾)}Ω(𝑡)=⋃_{0≤𝑡<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 𝑡₀≥0 and 𝜁₀=𝑒^𝑖𝜃0 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 Im𝜕𝜕𝑡𝑓^(𝜁,𝑡)𝑓^(𝜁,𝑡)−𝜁𝑓^(𝜁,𝑡)𝜕𝜕𝑡𝑓(𝜁,𝑡)−𝜁²𝑓^(𝜁,𝑡)𝜕𝜕𝑡𝑓(𝜁,𝑡)=𝛾𝐻[𝑖𝐴](𝜃),(2.11) for 𝜁=𝑒^𝑖𝜃. The previous equality is equivalent to the following: ||𝑓^(𝜁,𝑡)||²Im(𝜕/𝜕𝑡)𝑓^(𝜁,𝑡)𝑓^(𝜁,𝑡)−Φ(𝑓(𝜁Φ,𝑡()𝑓)((𝜁𝜕,/𝜕𝑡)𝑡))𝑓(𝜁,𝑡)||||_{𝜁=𝜁0=𝑒0𝑖𝜃,𝑡=𝑡0}=𝑄2𝜋−𝛾𝐻𝑖𝜕𝜕𝑘𝜃𝑒^𝑖𝜃×Im𝜁𝑓_,𝑡𝑓^−Φ^(𝑓)Φ(𝑓)𝜁𝑓^+2ReΦ^(𝑓)Im𝜁𝑓^Φ(𝑓)||||_{𝜁=𝜁0,𝑡=𝑡0,𝜃=𝜃0}−𝛾𝐻[𝑖𝐴](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(𝑒^𝑖𝜃𝑓^(𝑒^𝑖𝜃,𝑡)/Φ(𝑓(𝑒^𝑖𝜃,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 𝜁∈𝑈, 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)≡𝑒^−𝑖𝛼𝑤 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+𝑎₋₁/𝜁+⋯, where 𝑎≠0. Let Φ be a holomorphic function on 𝐹(𝑈⁻) such that lim_𝜁→∞Φ(𝜁)=∞ and lim_𝜁→∞Φ^(𝜁)>0. We say that 𝐹 is Φ-like on 𝑈⁻ if Re𝜁𝐹^(𝜁)Φ(𝐹(𝜁))>0,𝜁∈𝑈⁻.(3.1)

Remark 3.2. (a) If 𝐹 is a Φ-like function on 𝑈⁻, then the function 𝑓∶𝑈→𝐂 given by 𝑓(𝑧)=1/𝐹(1/𝑧), 𝑧≠0, and 𝑓(0)=0, is Φ-like on 𝑈, where Φ∶𝑓(𝑈)→𝐂, Φ(𝑤)=𝑤²Φ(1/𝑤), for all𝑤∈𝑓(𝑈)⧵{0} and Φ(0)=0.
(b) If 𝑓 is a Φ-like function on 𝑈, then the function 𝐹∶𝑈⁻→𝐂, 𝐹(𝜁)=1/𝑓(1/𝜁) is Φ-like on 𝑈⁻, where Φ∶𝐹(𝑈⁻)→𝐂, Φ(𝜔)=𝜔²Φ(1/𝜔), for all 𝜔∈𝐹(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=⋃_{0≤𝑡<𝑇}Ω(𝑡)=⋃_{0≤𝑡<𝑇}𝐹(𝑈⁻,𝑡), and Φ is a holomorphic function on Ω which satisfies the following conditions: ReΦ(𝜔)𝜔>0,ReΦ^(𝜔)<2ReΦ(𝜔)𝜔,∀𝜔∈Ω.(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: Re𝑓̇(𝜁,𝑡)𝜁𝑓^(𝜁,𝑡)=−𝑄||𝑓(𝜁,𝑡)||⁴2𝜋,||𝜁||=1.(3.3) Due to the previous remark, the function 𝐹(𝜁,𝑡), 𝜁∈𝑈⁻, is Φ-like if and only if 𝑓(𝜁,𝑡), 𝜁∈𝑈, is Φ-like, where the relationship between Φ and Φ is Φ(𝜔)=𝜔²Φ(1/𝜔), for all 𝜔∈𝐹(𝑈⁻)(orΦ(𝑤)=𝑤²Φ(1/𝑤), for all 𝑤∈𝑓(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 𝑡₀≥0 and 𝜁₀=𝑒^𝑖𝜃0 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 𝜕𝜕𝑡argΦ𝜁(𝑓𝑓′((𝜁𝜁,,𝑡𝑡)))||||_{𝜁=𝜁0,𝑡=𝑡0}=𝑄||𝑓||⁴2𝜋||𝑓^||²Im𝜁𝑓^𝑓^−Φ^(𝑓)𝜁𝑓^Φ(𝑓)+2ImΦ𝜁𝑓(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(𝑒^𝑖𝜃𝑓^(𝑒^𝑖𝜃,𝑡)/Φ(𝑓(𝑒^𝑖𝜃,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=⋃_{0≤𝑡<𝑡(𝛾)}Ω(𝑡)=⋃_{0≤𝑡<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),𝜁∈𝑈, 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 Re𝑓̇(𝜁,𝑡)𝜁𝑓^(𝜁,𝑡)=−||𝑓||⁴2𝑄𝜋−𝛾𝐻𝑖𝜕𝜕𝑘𝜃(𝜃),||𝜁||=1.(3.5) Suppose by contrary that the conclusion of Theorem 3.4 is not true. Then there exist 𝑡₀≥0 and 𝜁₀=𝑒^𝑖𝜃0 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^I||²m2𝑄𝜋𝜁𝑓−_𝛾𝐻𝑖𝜕𝜕𝑘𝜃(𝜃)𝑓^−Φ^(𝑓)𝜁𝑓^Φ(𝑓)+ImΦ𝜁𝑓(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(𝑒^𝑖𝜃𝑓^(𝑒^𝑖𝜃,𝑡)/Φ(𝑓(𝑒^𝑖𝜃,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 𝑈⁻and univalent on 𝑈⁻. If 𝐹₀ satisfies the condition (3.1) for each 𝜁∈𝑈⁻, 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.