Classical Solvability of a Free Boundary Problem for an Incompressible Viscous Fluid with a Surface Density Equation

Abstract

We investigate a mathematical model introduced by Shikhmurzaev to remove singularities that arise when classical hydrodynamic models are applied to certain physical phenomena. The model is described as a free boundary problem consisting of the Navier-Stokes equations and a surface mass balance equation. We prove the local-in-time solvability in Hölder spaces.

1. Introduction

Let a time-dependent bounded domain Ω_𝑡⊂𝐑³ with the outer boundary Γ_𝑡≡𝜕Ω_𝑡 be filled with an incompressible viscous fluid, and let Γ_𝑡 represent the interface. In Ω_𝑡, we assume that the flow is governed by the Navier-Stokes equations: 𝜕𝐯+()𝜕𝑡𝐯⋅∇𝐯+∇𝑝−𝜈Δ𝐯=𝟎,∇⋅𝐯=0, (1) where 𝐯 is the velocity, 𝑝 is the pressure, and 𝜈, which is assumed to be a positive constant, is the kinematic viscosity.

On Γ_𝑡, we assume the following equations: Π𝐓𝐯,𝑝𝐧=∇𝜎,𝐧⋅𝐓𝐯,𝑝𝐧=𝜎𝐻,𝐷𝜌^𝑠𝐷𝑡+𝜌^𝑠∇⋅𝐯^𝑠=𝜌𝐯−𝐯^𝑠⋅𝐧,𝐯−𝐯^𝑠𝜌⋅𝐧=−^𝑠𝑒−𝜌^𝑠𝜌𝜏,Π𝐯−𝐯^𝑠=−𝜒∇𝜎,𝜎=𝛾𝜌−𝜌^𝑠. (2)(3)(4)(5) Here 𝐯^𝑠 and 𝜌^𝑠 are the velocity and the density of surface layer, respectively. 𝐓(𝐯,𝑝)=𝜈𝐃(𝐯)−𝑝𝐼 is the stress tensor, where 𝐃(𝐯)=((𝜕𝑣_𝑖/𝜕𝑥_𝑗)+(𝜕𝑣_𝑖/𝜕𝑥_𝑗))_{𝑖,𝑗=1,2,3} is the velocity deformation tensor. 𝐻 is the twice mean curvature of Γ_𝑡 at the point 𝑥, which is negative if Ω_𝑡 is convex in the neighborhood of 𝑥. 𝐧 is the unit outward normal to Γ_𝑡 at the point 𝑥. Π is the projection operator onto the tangent plane at the point 𝑥 on Γ_𝑡. 𝐷/𝐷𝑡 denotes the derivative along the trajectory of particle on Γ_𝑡. ∇ is the gradient restricted to the surface. 𝜌, 𝜌^𝑠_𝑒, 𝜏, 𝜒, 𝛾, 𝜌 are positive constants; in particular, 𝜌 is the density of the bulk and 𝜏 is the characteristic time scale over which the surface density 𝜌^𝑠 relaxes to its equilibrium value 𝜌^𝑠_𝑒.

Finally, to complete the problem, we give the initial conditions: 𝐯|_𝑡=0=𝐯₀onΩ≡Ω₀,𝜌^𝑠_𝑡=0=𝜌₀onΓ≡Γ₀. (6)

It is known that singularities arise when the the classical hydrodynamic equations and modeling assumptions are applied to certain physical phenomena. For example, the application of the classical no-slip boundary condition to the spreading of a drop on a plate gives rise to a nonintegrable shear stress, and the application of the classical kinematic condition at the free boundary to the formation of a cusp on a free surface of a viscous fluid leads to an infinite energy dissipation in the fluid (e.g., refer to [1] and the references therein).

To remove the above mentioned singularities, we are required to modify classical boundary conditions by taking into account molecular interaction near interfaces. The molecule in the liquid region which is very close to another phases experiences an asymmetric force due to the presence of another materials. This gives rise to the variation in the density in the liquid region near to the adjacent phase, and the surface tension occurs as a result of this variation in density. The thin layer in the liquid region in which the above mentioned density variation occurs is called the surface layer.

Through [1–4], Shikhmurzaev developed a theory to remove the above mentioned singularities by introducing a surface layer which is treated as a separate phase. In this theory, the no-slip condition assumed in classical models for dynamic wetting processes is modified as the Navier-slip condition through thermodynamic considerations on the surface layer (refer to [2, 3]). The formation of a free surface cusp associated with fluid flow is also investigated in [4]. In [4], the cusp formation is modeled as an interface disappearance process. In this model, an internal surface stretching from the cusp, which is referred to as “the surface-tension-relaxation tail”, is introduced. The above mentioned singularity associated with the modeling of cusp formation arises owing to the absence of viscous stress at the cusp with which the surface tension acting from the liquid surface is balanced. In this model, the surface tension at the cusp can be balanced by shear stresses acting on this tail.

The problem (1)–(6) is a model describing the behavior of an isolated liquid drop in which the interface is modeled as a surface layer based on Shikhmurzaev’s theory. The dynamics of the liquid in this layer are governed by (3) which represents conservation of mass. The right-hand side of (3) represents the source consisting of a flow of molecules from the bulk. Equations in (4) are conditions that minimize the rate of entropy production in the surface layer. Equation (5) represents a linearized state equation in the surface layer (refer to [1] for details). In Shikhmurzaev’s theory, the surface layer is modeled as a sharp interface as a result of a continuum approximation. Thus, in the above problem, the surface layer is described by the equations given in (3)–(5) defined on a geometric surface, and the behavior of the surface layer is related to (1) in the bulk through the boundary conditions given in (2).

In the present paper, we prove the local-in-time classical solvability of problem (1)–(6). As is mentioned above, this model is important as a basic model to describe the above mentioned physical phenomena; however, as far as the author knows, any rigorous proofs on its solvability have not been given. In the present paper, we consider the case where the mass exchange between the interface and the bulk does not occur. As is seen in Section 2, under such an assumption, we can reformulate our problem as a problem defined in a domain with a fixed known boundary by introducing Lagrangian coordinates, and in Section 3, we construct a unique solution of the reformulated problem in Hölder spaces with the aid of the method of successive approximations.

2. Reformulation of the Problem

In this section, we reformulate our problem in Lagrangian coordinates. By Lagrangian coordinates we mean the initial coordinates of the fluid particles. In the case where no exchange of molecules occurs between the surface and the bulk, (4)¹ is reduced to 𝐯⋅𝐧=𝐯^𝑠⋅𝐧. This relation indicates that the following kinematic condition at the interface is satisfied: the interface consists of the particles located at the interface at the initial time. This circumstance enables us to relate each point 𝑥∈Ω_𝑡 to its initial point 𝜉∈Ω by relation (10) given below.

Before reformulating our problem, we rewrite (3) as a nonlinear parabolic equation on Γ_𝑡 with the time derivative 𝐷𝜌^𝑠/𝐷𝑡, where 𝐷𝜌^𝑠/𝐷𝑡 denotes the derivative along the trajectory of particle on the interface with velocity 𝐯. Noting the following relation (e.g., see [5]): 𝐷𝜌^𝑠𝐷𝑡_𝑛=𝐷𝜌^𝑠𝐷𝑡−𝐯^𝑠⋅∇𝜌^𝑠=𝐷𝜌^𝑠𝐷𝑡−𝐯⋅∇𝜌^𝑠, (7) where (𝐷𝜌^𝑠/𝐷𝑡)_𝑛 represents the derivative along the trajectory which is normal to the interface, (3) can be written as 𝐷𝜌^𝑠𝐷𝑡+𝜌^𝑠∇⋅𝐯^𝑠=𝐯−𝐯^𝑠⋅∇𝜌^𝑠. (8) Then eliminating 𝐯^𝑠 from the above equation with the aid of the relation (4)², we obtain the following equation: 𝐷𝜌^𝑠𝐷𝑡−𝜒𝛾𝜌^𝑠₀∇²𝜌^𝑠=−𝜌^𝑠𝜌∇⋅𝐯+𝜒𝛾^𝑠−𝜌^𝑠₀∇²𝜌^𝑠+∇𝜌^𝑠⋅∇𝜌^𝑠. (9)

Now let us reformulate our problem. The Lagrangian and Eulerian coordinates are related by 𝑥=𝑋_𝑢(𝜉,𝑡)≡𝜉+^𝑡₀𝐮(𝜉,𝜏)𝑑𝜏, (10) where 𝐮(𝜉,𝜏) is the velocity at time 𝑡 of the particle which was located at 𝜉 at 𝑡=0. By changing the variables from 𝑥 to 𝜉 by relation (10), problem (1)–(6) is reformulated as the following problem defined in the cylindrical domain Ω_0𝑇=Ω×(0,𝑇) with the lateral boundary Γ_0𝑇≡Γ×(0,𝑇): 𝜕𝐮𝜕𝑡−𝜈∇²_𝑢𝐮+∇_𝑢𝑞=𝟎,∇_𝑢⋅𝐮=0inΩ_0𝑇,ΠΠ_𝑢𝐓_𝑢𝐮,𝑞𝐧_𝑢=Π∇_Γ𝑡𝜃,𝐧⋅𝐓_𝑢𝐮,𝑞𝐧_𝑢=𝜃∇²_Γ𝑡𝑋_𝑢(𝜉,𝑡)_𝜉∈Γ⋅𝐧+𝐧⋅∇_Γ𝑡𝜃,𝜕𝑟^𝑠𝜕𝑡−𝜒𝛾𝜌^𝑠₀∇²_Γ𝑡𝑟^𝑠=−𝑟^𝑠∇_Γ𝑡⋅𝐮+𝜒𝛾𝑟^𝑠−𝜌^𝑠₀∇²_Γ𝑡𝑟^𝑠+∇_Γ𝑡𝑟^𝑠⋅∇_Γ𝑡𝑟^𝑠,𝜃=𝛾𝜌−𝑟^𝑠onΓ_0𝑇,𝐮|_𝑡=0=𝐯₀onΩ,𝑟^𝑠_𝑡=0=𝜌^𝑠₀onΓ. (11)(12)(13) In (11)–(13), 𝐮, 𝑞 and 𝑟^𝑠 are 𝐯(𝑋_𝑢(𝜉,𝑡),𝑡),𝑝(𝑋_𝑢(𝜉,𝑡),𝑡), and 𝜌^𝑠(𝑋_𝑢(𝜉,𝑡),𝑡), respectively. Consider ∇_𝑢=(𝒥_𝑢⁻¹)^𝑡∇≡𝒥^∗_𝑢∇; here 𝒥_𝑢 denotes the Jacobian matrix of 𝑋_𝑢, and the notation 𝐴^𝑡 means the transpose of the matrix 𝐴. 𝐧 is the outward unit normal to Γ at the point 𝜉, 𝐧_𝑢=𝒥^∗_𝑢𝐧/|𝒥^∗_𝑢𝐧|, and Π and Π_𝑢 are the operators defined by Π𝐟=𝐟−(𝐟⋅𝐧)𝐟 and Π_𝑢𝐟=𝐟−(𝐟⋅𝐧_𝑢)𝐟, respectively. 𝐓_𝑢(𝐮,𝑞) is the tensor with the elements 𝜈∑³_𝑘=1(𝐴_𝑗𝑘(𝜕𝑢_𝑖/𝜕𝜉_𝑘)+𝐴_𝑖𝑘(𝜕𝑢_𝑗/𝜕𝜉_𝑘))−𝑝𝛿_𝑖𝑗, where 𝐴_𝑖𝑗 is the (𝑖,𝑗)-element of 𝒥^∗_𝑢, and 𝛿_𝑖𝑗 is Kronecker’s delta. The operators ∇_Γ𝑡𝑓 and ∇_Γ𝑡⋅𝐀 are defined by ∇_Γ𝑡𝑓=_{𝛼,𝛽=1,2}𝑔^𝛼𝛽𝜕𝑓𝜕𝑠_𝛽𝜕𝑋_𝑢(𝑠)𝜕𝑠_𝛼,∇_Γ𝑡1⋅𝐀=√𝑔_𝛼=1,2𝜕𝜕𝑠_𝛼√𝑔𝐴_𝛼, (14) where 𝑔=det(𝑔_𝛼𝛽)_{𝛼,𝛽=1,2}, 𝑔_𝛼𝛽=(𝜕𝑋_𝑢(𝑠)/𝜕𝑠_𝛼)⋅(𝜕𝑋_𝑢(𝑠)/𝜕𝑠_𝛽), 𝑔^𝛼𝛽 denote the components of the inverse matrix of (𝑔_𝛼𝛽)_{𝛼,𝛽=1,2}, 𝑋_𝑢(𝑠)=𝑋_𝑢(𝜉(𝑠),𝑡), 𝑠=(𝑠₁,𝑠₂) denotes the local coordinates on Γ, and 𝐴_𝛼 denotes the components of the vector 𝐀 with respect to the basis (𝜕𝑋_𝑢(𝑠)/𝜕𝑠_𝛼), 𝛼=1,2. Finally, the operator ∇²_Γ𝑡𝑓 is defined as ∇²_Γ𝑡𝑓=∇_Γ𝑡⋅∇_Γ𝑡1𝑓=√𝑔_{𝛼,𝛽=1,2}𝜕𝜕𝑠_𝛼√𝑔𝑔^𝛼𝛽𝜕𝑓𝜕𝑠_𝛽. (15)

Note that in derivation of (12)², we have used the formula 𝐻𝐧=∇²𝑥≡∇²𝑋_𝑢(𝜉,𝑡). Note also that although (12)^1,2 are different from the following formulas which are obtained directly from (2): Π_𝑢𝐓_𝑢𝐮,𝑞𝐧_𝑢=∇_Γ𝑡𝐧𝜃,_𝑢⋅𝐓_𝑢𝐮,𝑞𝐧_𝑢=𝜃∇²_Γ𝑡𝑋_𝑢(𝜉,𝑡)_𝜉∈Γ⋅𝐧_𝑢, (16) problem (11)–(13) is equivalent to problem (1)–(6) as far as the condition 𝐧⋅𝐧_𝑢>0, which is valid for sufficiently small 𝑡, is satisfied.

We now introduce some function spaces. Let 𝐷 be a domain in 𝐑^𝑛, let 𝑇 be a positive constant, let 𝐷_𝑇 be a cylindrical domain 𝐷×(0,𝑇), let 𝑙 be a nonnegative integer, and let 𝛼, 𝛾∈(0,1).

By 𝐶^𝑙+𝛼(𝐷), we define the space of functions 𝑓(𝑥), 𝑥∈𝐷, with the norm 𝑓_𝐷^(𝑙+𝛼)≡_{|𝑚|≤𝑙}𝜕^𝑚_𝑥𝑓_𝐷+𝑓_𝐷^(𝑙+𝛼),𝑓_𝐷≡sup_𝑥∈𝐷𝑓(𝑥,𝑓_{𝐷(𝑙+𝛼)})≡_|𝑚|=𝑙𝜕^𝑚_𝑥𝑓_𝐷^(𝛼)≡sup_{𝑥,𝑦∈𝐷,𝑥≠𝑦𝑚=|𝑙|}𝜕^𝑚_𝑥𝑓(𝑥)−𝜕^𝑚_𝑦𝑓𝑦𝑥−𝑦^𝛼,|𝑚|=^𝑛_𝑖=1𝑚_𝑖,𝜕^𝑚_𝑥=𝜕^|𝑚|𝜕^𝑚1_𝑥1⋯𝜕^𝑚𝑛_𝑥𝑛, (17) for a multi-index 𝑚=(𝑚_𝑖)(𝑚_𝑖≥0,𝑖=1,…,𝑛).

By 𝐶^{𝑙+𝛼,((𝑙+𝛼)/2)}(𝐷_𝑇) we denote an anisotropic Hölder space of functions whose norm is defined by 𝑓_𝐷^{(𝑙+𝛼,((𝑙+𝛼)/2))}_𝑇≡^𝑙_{2𝑟+|𝑚|=0}𝜕^𝑟_𝑡𝜕^𝑚_𝑥𝑓_𝐷𝑇+𝑓_𝐷^{(𝑙+𝛼,((𝑙+𝛼)/2))}_𝑇, (18) where 𝑓_𝐷𝑇≡sup_{(𝑥,𝑡)∈𝐷𝑇}𝑓(,𝑓𝑥,𝑡)_{𝐷(𝑙+𝛼,((𝑙+𝛼)/2))𝑇}≡^𝑙_{2𝑟+|𝑚|=𝑙−1}𝜕^𝑟_𝑡𝜕^𝑚_𝑥𝑓_𝐷^{(0,((𝑙+𝛼−(2𝑟+|𝑚|))/2))}_𝑇+_{|𝑚|2𝑟+=𝑙}𝜕^𝑟_𝑡𝜕^𝑚_𝑥𝑓_𝐷^(𝛼,0)_𝑇. (19) Here, 𝑓_𝐷^(0,(𝛼/2))_𝑇≡sup_{()𝑥,𝑡,(𝑥,𝑡′)∈𝐷𝑇,𝑡≠𝑡′}𝑓(𝑥,𝑡)−𝑓𝑥,𝑡^′𝑡−𝑡^′(𝛼/2),𝑓_𝐷^(𝛼,0)_𝑇≡sup_{()𝑥,𝑡,(𝑥′,𝑡)∈𝐷𝑇,𝑥≠𝑥′}𝑓(𝑥,𝑡)−𝑓𝑥^′,𝑡𝑥−𝑥^′𝛼. (20)

Finally, we introduce the function space 𝐶^1+𝛼,𝛾(𝐷_𝑇) equipped with the norm 𝑓_𝐷^{(1+𝛼,𝛾)}_𝑇≡𝑓_𝐷𝑇+∇𝑓_𝐷^{(𝛼,(𝛼/2))}_𝑇+𝑓_𝐷^{(1+𝛼,𝛾)}_𝑇, (21) where 𝑓_𝐷^{(1+𝛼,𝛾)}_𝑇≡sup_{𝜏,𝑡∈(0,𝑇),𝜏≠𝑡}𝑓(⋅,𝑡)−𝑓(⋅,𝜏)_𝐷^(𝛾)|𝑡−𝜏|^{((1+𝛼−𝛾)/2)}. (22)

Now, let us state our main result.

Theorem 1. Let 𝛼, 𝛾 be constants satisfying 0<𝛼, 𝛾<1. Assume that 𝐯₀∈𝐶^2+𝛼Ω,𝜌^𝑠₀∈𝐶^2+𝛼(Γ),Γ∈𝐶^3+𝛼. (23) Assume that there exist positive constants Δ₁ and Δ₂ such that 𝜌−𝜌^𝑠₀≥Δ₁>0 and 𝜌^𝑠₀≥Δ₂>0 on Γ. In addition, assume that the following compatibility conditions are satisfied: ∇⋅𝐯₀=𝟎,𝜈Π𝐃𝐯₀𝐧+𝛾∇_Γ𝜌^𝑠_0Γ=𝟎, (24) where ∇_Γ is the operator corresponding to ∇_Γ𝑡 with 𝑡=0; namely, ∇_Γ is given by the formula in (14) with 𝑋_𝑢(𝑠)=𝜉(𝑠).
Then, for a positive constant 𝑇, problem (11)–(13) has a unique solution (𝐮,𝑞,𝑟^𝑠) with the following smoothness: 𝐮∈𝐶^{2+𝛼,1+(𝛼/2)}Ω_0𝑇,𝑞∈𝐶^1+𝛼,𝛾Ω_0𝑇∩𝐶^{1+𝛼,((1+𝛼)/2)}Γ_0𝑇,𝑟^𝑠∈𝐶^{2+𝛼,1+(𝛼/2)}Γ_0𝑇. (25)

3. Proof of the Main Result

In this section, we will prove Theorem 1.

We begin with preparing estimates of solutions to some linear problems. For the following problem: 𝜕𝐮𝜕𝑡−𝜈∇²_𝑤𝐮+∇_𝑤𝑞=𝐅₁,∇_𝑤⋅𝐮=𝐹₂inΩ_0𝑇,𝜈ΠΠ_𝑤𝐃_𝑤(𝐮)𝐧_𝑤=𝐅₃,𝐧⋅𝐓_𝑤𝐮,𝑞𝐧_𝑤−Θ𝐧⋅∇²Γ_𝑡^𝑡₀𝐮𝑑𝜏=𝑏+^𝑡₀𝐵𝑑𝜏onΓ_0𝑇,𝐮|_𝑡=0=𝐯₀onΩ, (26) the following result is given in [6]. In (26), ∇_𝑤, Π_𝑤, 𝐓_𝑤(𝐮,𝑞)(=𝜈𝐃_𝑤(𝐮)−𝑞𝐼), 𝐧_𝑤 are defined for a given vector 𝐰 in the same manner as ∇_𝑢, Π_𝑢, 𝐓_𝑢(𝐮,𝑞), 𝐧_𝑢 are defined, and ∇Γ_𝑡 is defined by (14) with 𝑋_𝑢=𝑋_𝑤.

Theorem 2. Let 𝑇>0, and let 𝛼, 𝛾 be positive constants satisfying 0<𝛼, 𝛾<1. Assume that Γ∈𝐶^2+𝛼,𝐯₀∈𝐶^2+𝛼Ω,𝐅₁∈𝐶^{𝛼,(𝛼/2)}Ω_0𝑇,𝐹₂∈𝐶^{1+𝛼,((1+𝛼)/2)}Ω_0𝑇,𝐅₃∈𝐶^{1+𝛼,((1+𝛼)/2)}Γ_0𝑇,𝑏∈𝐶^{1+𝛼,((1+𝛼)/2)}Γ_0𝑇,𝐵∈𝐶^{𝛼,(𝛼/2)}Γ_0𝑇,Θ∈𝐶^𝛼(Γ). (27) Assume that there exists a positive constant Δ₁ such that Θ≥Δ₁>0 on Γ. Assume that the following compatibility conditions are satisfied: ∇⋅𝐯₀=𝐹₂(𝜉,0),𝜈Π𝐃𝐯₀𝐧_Γ=𝐅₃(𝜉,0). (28) Assume that there exist functions 𝐡∈𝐶^{𝛼,(𝛼/2)}(Ω_0𝑇), 𝐇_𝑘, 𝑘=1,2,3, with a finite norm |𝐇_𝑘|_Ω^{(1+𝛼,𝛾)}_0𝑇 satisfying the relation 𝜕𝐹₂𝜕𝑡−∇_𝑤⋅𝐅₁=∇⋅𝐡,𝐡=³_𝑘=1𝜕_𝜉𝑘𝐇_𝑘, (29) in the sense of distribution. Furthermore, assume that 𝐰∈𝐶^{2+𝛼,1+(𝛼/2)}(Ω_0𝑇) satisfies the inequality 𝑇+𝑇^1/2|𝐰|_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑇^{((1−𝛼+𝛾)/2)}𝜕_𝜉𝐰_Ω0𝑇≤𝛿, (30) for a sufficiently small positive constant 𝛿.
Then problem (26) has a unique solution (𝐮,𝑞) satisfying the following inequality: |𝐮|_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑞_Ω^{(1+𝛼,𝛾)}_0𝑇+𝑞_Γ^{(1+𝛼,((1+𝛼)/2))}_0𝑇≤𝐶₁(𝑇)𝐅_1Ω^{(𝛼,(𝛼/2))}_0𝑇+𝐹_2Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇+𝐅_3Γ^{(1+𝛼,((1+𝛼)/2))}_0𝑇+|𝑏|_Γ^{(1+𝛼,((1+𝛼)/2))}_0𝑇+|𝐵|_Γ^{(𝛼,(𝛼/2))}_0𝑇+𝐯_0Ω^(2+𝛼)+|𝐡|_Ω^{(𝛼,(𝛼/2))}_0𝑇+³_𝑘=1𝐇_𝑘Ω^{(1+𝛼,𝛾)}_0𝑇+𝑃_𝑇(𝐰)𝐯_0Ω⁽¹⁾+|𝑏(⋅,0)|_Γ, (31) where 𝑃_𝑇(𝐰)=𝑇^{((1−𝛼)/2)}|𝐰|_Ω^(1,0)_0𝑇+|𝜕_𝜉𝐰|_Ω^{(𝛼,(𝛼/2))}_0𝑇+[𝜕_𝜉𝐰]_Ω^{(0,((1+𝛼−𝛾)/2))}_0𝑇 and 𝐶₁(𝑇) is a nondecreasing function of 𝑇.

For the following problem: 𝜕𝑟^𝑠𝜕𝑡−𝜒𝛾𝜌^𝑠₀∇²Γ_𝑡𝑟^𝑠=𝐺onΓ_0𝑇,𝑟^𝑠_𝑡=0=𝜌^𝑠₀onΓ, (32) we have the following theorem. The assertion of the theorem immediately follows from the Hölder estimates for linear parabolic equations (e.g., see [7]).

Theorem 3. Let 𝑇>0, and let 𝛼 be a positive constant satisfying 0<𝛼<1. Assume that Γ∈𝐶^2+𝛼,𝜌^𝑠₀∈𝐶^2+𝛼(Γ),𝐺∈𝐶^{𝛼,(𝛼/2)}Γ_0𝑇. (33) Assume that there exists a positive constant Δ₂ such that 𝜌^𝑠₀≥Δ₂>0 on Γ. Further assume the same assumptions for 𝐰 stated in Theorem 2.
Then, problem (32) has a unique solution 𝑟^𝑠 satisfying the following inequality: 𝑟^𝑠_Γ^{(2+𝛼,1+(𝛼/2))}_0𝑇≤𝐶₂(𝑇)|𝐺|_Γ^{(𝛼,(𝛼/2))}_0𝑇+𝜌^𝑠_0Γ^(2+𝛼), (34) where 𝐶₂(𝑇) is a nondecreasing function of 𝑇.

Combining the above results, we can easily obtain Theorem 4 given below for the following problem: 𝜕𝐮𝜕𝑡−𝜈∇²_𝑤𝐮+∇_𝑤𝑞=𝐅₁,∇_𝑤⋅𝐮=𝐹₂inΩ_0𝑇,𝜈ΠΠ_𝑤𝐃_𝑤(𝐮)𝐧_𝑤+𝛾Π∇Γ_𝑡𝑟^𝑠=𝐅₃,𝐧⋅𝐓_𝑤𝐮,𝑞𝐧_𝑤−Θ𝐧⋅∇²Γ_𝑡^𝑡₀Γ𝐮𝑑𝜏+𝛾𝐧⋅∇_𝑡𝑟^𝑠=𝑏+^𝑡₀𝐵𝑑𝜏onΓ_0𝑇,𝐮|_𝑡=0=𝐯₀onΩ,𝜕𝑟^𝑠𝜕𝑡−𝜒𝛾𝜌^𝑠₀∇²Γ_𝑡𝑟^𝑠=𝐺onΓ_0𝑇,𝑟^𝑠_𝑡=0=𝜌^𝑠₀onΓ. (35) The estimate given in the theorem will be essentially used in the later argument to prove Theorem 1.

Theorem 4. Under the same assumptions given in Theorem 2 where only compatibility condition is replaced by ∇⋅𝐯₀=𝐹₂(𝜉,0),𝜈Π𝐃𝐯₀𝐧+𝛾Π∇_Γ𝜌^𝑠_0Γ=𝐅₃(𝜉,0), (36) and Theorem 3, problem (35) has a unique solution (𝐮,𝑞,𝑟^𝑠) satisfying the following inequality: |𝐮|_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑞_Ω^{(1+𝛼,𝛾)}_0𝑇+𝑞_Γ^{(1+𝛼,((1+𝛼)/2))}_0𝑇+𝑟^𝑠_Γ^{(2+𝛼,1+(𝛼/2))}_0𝑇≤𝐶₃(𝑇)𝐅_1Ω^{(𝛼,(𝛼/2))}_0𝑇+𝐹_2Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇+𝐅_3Γ^{(1+𝛼,((1+𝛼)/2))}_0𝑇+|𝑏|_Γ^{(1+𝛼,((1+𝛼)/2))}_0𝑇+|𝐵|_Γ^{(𝛼,(𝛼/2))}_0𝑇+|𝐡|_Ω^{(𝛼,(𝛼/2))}_0𝑇+³_𝑘=1𝐇_𝑘Ω^{(1+𝛼,𝛾)}_0𝑇+𝐯_0Ω^(2+𝛼)+|𝐺|_Γ^{(𝛼,(𝛼/2))}_0𝑇+𝜌^𝑠_0Γ^(2+𝛼)+𝑃_𝑇(𝐰)𝐯_0Ω⁽¹⁾+|𝑏(⋅,0)|_Γ, (37) where 𝐶₃(𝑇) is a nondecreasing function of 𝑇.

In addition, we prepare estimates for 𝒥^∗_𝑢, which are used later.

Lemma 5. Let 𝒥_𝑢 and 𝒥_𝑢′ be the Jacobian matrices of the mappings 𝑋_𝑢 and 𝑋_𝑢′, respectively. Let us assume that 𝐮 and 𝐮^′ satisfy condition (30) for sufficiently small 𝛿>0. Then, the following inequalities hold: 𝒥^∗_𝑢Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇,𝒥^∗_𝑢Ω^{(1+𝛼,𝛾)}_0𝑇≤𝐶, (38) where 𝐶 is a positive constant independent of 𝛿, and 𝒥^∗_𝑢−𝒥^∗_𝑢′Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇,𝒥^∗_𝑢−𝒥^∗_𝑢′Ω^{(1+𝛼,𝛾)}_0𝑇≤𝜖𝐮−𝐮^′_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝐶(𝜖)^𝑇₀𝐮−𝐮^′_Ω^{(2+𝛼,1+(𝛼/2))}_0𝜏𝑑𝜏, (39) for arbitrary 0<𝜖<1, where 𝐶(𝜖) is a positive constant depending only on 𝜖.

Proof. In the following proof, 𝑐₁, 𝑐₂, and 𝑐₃ are positive constants independent of 𝛿 and 𝜖. Let 𝑎_𝑖𝑗 be the (𝑖,𝑗)-component of 𝒥_𝑢.
Then, we have 𝛿_𝑖𝑗−^𝑡₀𝜕_𝜉𝑗𝑢_𝑖𝑑𝜏≤𝑎_𝑖𝑗≤𝛿_𝑖𝑗+^𝑡₀𝜕_𝜉𝑗𝑢_𝑖𝑑𝜏, (40) where 𝛿_𝑖𝑗 denotes Kronecker’s delta. Then, using the inequality ^𝑡₀𝜕_𝜉𝑗𝑢_𝑖𝑑𝜏_Ω0𝑇≤𝑇|𝐮|_Ω^(1,0)_0𝑇≤𝛿, (41) from (40), we have 𝛿_𝑖𝑗−𝛿≤𝑎_{𝑖𝑗Ω0𝑇}≤𝛿_𝑖𝑗+𝛿. (42) This inequality implies that det𝒥_𝑢>0 holds for sufficiently small 𝛿>0.
Now, let 𝑎^′_𝑖𝑗 be the (𝑖,𝑗)-components of 𝒥_𝑢′, and let 𝐴_𝑖𝑗 and 𝐴^′_𝑖𝑗 be the cofactors of 𝑎_𝑖𝑗 and 𝑎^′_𝑖𝑗, respectively.
Then from the inequalities 𝑎_𝑖𝑗Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇,𝑎^′_𝑖𝑗Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇≤𝑐₁(1+𝛿), (43) we have 𝐴_𝑖𝑗Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇,𝐴_𝑖𝑗Ω^{(1+𝛼,𝛾)}_0𝑇≤𝑐₂. (44)
On the other hand, from the inequality 𝑎_𝑖𝑗−𝑎^′_𝑖𝑗Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇≤𝜖𝐮−𝐮^′_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑐₃1+𝜖^{−((1+𝛼)/(2−𝛼))𝑇}₀𝐮−𝐮^′_Ω^{(2+𝛼,1+(𝛼/2))}_0𝜏𝑑𝜏, (45) which are obtained with the aid of the following inequality which holds for arbitrary 0<𝜖<1: ^𝑡₀𝑓(𝜉,𝜏)𝑑𝜏_Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇≤^𝑇₀𝑓_Ω^(1+𝛼,0)_0𝜏𝑑𝜏+sup_{|𝑡′−𝑡′′|<𝜖(2/(1−𝛼))}𝑡^′−𝑡^{′′−((1+𝛼)/2)𝑡′}_𝑡′′𝑓(𝜉,𝜏)_Ω+𝑡𝑑𝜏_′−𝑡^{′′−(𝛼/2)𝑡′}_𝑡′′𝜕_𝜉𝑓(𝜉,𝜏)_Ω𝑑𝜏+sup_{𝜖(2/(1−𝛼))<|𝑡′−𝑡′′|≤𝑇}𝑡^′−𝑡^{′′−((1+𝛼)/2)𝑡′}_𝑡′′𝑓(𝜉,𝜏)_Ω+𝑡𝑑𝜏_′−𝑡^{′′−(𝛼/2)𝑡′}_𝑡′′𝜕_𝜉𝑓(𝜉,𝜏)_Ω𝑓𝑑𝜏≤𝜖_Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇+𝑐₃1+𝜖^{−((1+𝛼)/(1−𝛼))𝑇}₀𝑓_Ω^{(1+𝛼,((1+𝛼)/2))}_0𝜏𝑑𝜏, (46) we have 𝐴_𝑖𝑗−𝐴^′_𝑖𝑗Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇,𝐴_𝑖𝑗−𝐴^′_𝑖𝑗Ω^{(1+𝛼,𝛾)}_0𝑇≤𝜖𝐮−𝐮^′_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝐶₁(𝜖)^𝑇₀𝐮−𝐮^′_Ω^{(2+𝛼,1+(𝛼/2))}_0𝜏𝑑𝜏, (47) where 𝐶₁(𝜖) is a positive constant depending only on 𝜖. From (44) and (47), estimates (38) and (39) immediately follow. Thus, the proof is completed.

Now, let us prove Theorem 1 by the method of successive approximations. We take (𝐮₀,𝑞₀,𝑟^𝑠₀)=(𝟎,0,0), and, for the known 𝑛th approximation, we define the (𝑛+1)th approximation by the solutions of the following problem: 𝜕𝐮_𝑛+1𝜕𝑡−𝜈∇²_𝑢𝑛𝐮_𝑛+1+∇_𝑢𝑛𝑞_𝑛+1∇=𝟎,_𝑢𝑛⋅𝐮_𝑛+1=0inΩ_0𝑇,𝜈ΠΠ_𝑢𝑛𝐃_𝑢𝑛𝐮_𝑛+1𝐧_𝑢𝑛+𝛾Π∇_Γ𝑛,𝑡𝑟^𝑠_𝑛+1=𝟎,𝐧⋅𝐓_𝑢𝑛𝐮_𝑛+1,𝑞_𝑛+1𝐧_𝑢𝑛−Θ𝐧⋅∇²_Γ𝑛,𝑡^𝑡₀𝐮_𝑛+1𝑑𝜏+𝛾𝐧⋅∇_Γ𝑛,𝑡𝑟^𝑠_𝑛+1=𝑏𝐮_𝑛,𝑟^𝑠_𝑛+^𝑡₀𝐵𝐮_𝑛,𝑟^𝑠_𝑛𝑑𝜏onΓ_0𝑇,𝐮_𝑛+1𝑡=0=𝐯₀onΩ,𝜕𝑟^𝑠_𝑛+1𝜕𝑡−𝜒𝛾𝜌^𝑠₀∇²_Γ𝑛,𝑡𝑟^𝑠_𝑛+1=𝐺𝑟^𝑠_𝑛,𝐮_𝑛onΓ_0𝑇,𝑟^𝑠_𝑛+1𝑡=0=𝜌^𝑠₀onΓ, (48) where 𝑏𝐮_𝑛,𝑟^𝑠_𝑛=𝜃_𝑛𝐻₀+𝐧⋅^𝑡₀∇²_{Γ𝑛,𝜏𝜏},𝐻𝜉𝑑𝜏₀=𝐧⋅∇²_Γ𝜉𝐵𝐮_𝑛,𝑟^𝑠_𝑛=𝜕𝜃𝜕𝜏^𝑛−Θ𝐧⋅∇²_Γ𝑛,𝜏^𝜏₀𝐮_𝑛()+𝜃𝜉,𝑠𝑑𝑠_𝑛−Θ𝐧⋅∇²_{Γ𝑛,𝜏𝜏}^𝜏₀𝐮_𝑛()+𝜃𝜉,𝑠𝑑𝑠_𝑛−Θ𝐧⋅∇²_Γ𝑛,𝜏𝐮_𝑛,𝜃_𝑛=𝛾𝜌−𝑟^𝑠_𝑛,Θ=𝛾𝜌−𝜌^𝑠₀,𝐺𝐮_𝑛,𝑟^𝑠_𝑛≡−𝑟^𝑠_𝑛∇_Γ𝑛,𝑡⋅𝐮_𝑛+𝜒𝛾𝑟^𝑠_𝑛−𝜌^𝑠₀∇²_Γ𝑛,𝑡𝑟^𝑠_𝑛+∇_Γ𝑛,𝑡𝑟^𝑠_𝑛⋅∇_Γ𝑛,𝑡𝑟^𝑠_𝑛. (49) In the above formulas, ∇_Γ𝑛,𝑡 is the operator corresponding to ∇_Γ𝑡 with 𝑋_𝑢=𝑋_𝑢𝑛, and (∇²_Γ𝑛,𝜏)_𝜏 denotes the operator obtained by differentiating the coefficients of ∇²_Γ𝑛,𝜏 with respect to 𝜏.

Now, let us verify that all terms of the sequence {(𝐮_𝑛,𝑞_𝑛,𝑟^𝑠_𝑛)} are defined on some time interval independent of 𝑛. We begin with the following lemma.

Lemma 6. Let 𝑇_𝑛 be a constant satisfying 0<𝑇_𝑛≤1. Then there exist positive constants 𝛿 and 𝛽 such that if 𝐮_𝑛 and 𝑟^𝑠_𝑛 satisfy the following conditions: 𝑇_𝑛+𝑇_𝑛^1/2𝐮_𝑛Ω^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇+𝑇_𝑛^{((1−𝛼+𝛾)/2)}𝜕_𝜉𝐮_{𝑛Ω𝑛0𝑇}𝑇≤𝛿,_𝑛+𝑇^𝛽_𝑛𝐮_𝑛Ω^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇+𝑟^𝑠_𝑛Γ^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇≤𝛿, (50) then the following inequality holds for a positive constant 𝐶 independent of 𝐮_𝑛, 𝑟^𝑠_𝑛 and 𝑇_𝑛: 𝑏𝐮_𝑛,𝑟^𝑠_𝑛Γ^{(1+𝛼,((1+𝛼)/2))}_𝑛0𝑇+𝐵𝐮_𝑛,𝑟^𝑠_𝑛Γ^{(𝛼,(𝛼/2))}_𝑛0𝑇+𝐺𝐮_𝑛,𝑟^𝑠_𝑛Γ^{(𝛼,(𝛼/2))}_𝑛0𝑇+𝑃_𝑇𝑛𝐮_𝑛≤𝐶. (51)

Proof. In the proof, 𝑐₁,…,𝑐₉ are positive constants independent of 𝐮_𝑛, 𝑟^𝑠_𝑛, and 𝑇_𝑛.
Choosing 𝑇_𝑛^{((1−𝛼)/2𝛼)} as 𝜖 in the following interpolation inequality: 𝜃_𝑛−𝜃_𝑛𝑡=0Γ^{(1+𝛼,((1+𝛼)/2))}_𝑛0𝑇≤𝜖𝑟^𝑠_𝑛−𝜌^𝑠_0Γ^(2,1)_𝑛0𝑇+𝑐₁𝜖^{−(𝛼/(1−𝛼))}𝑟^𝑠_𝑛−𝜌^𝑠_0Γ𝑛0𝑇≤𝜖𝑟^𝑠_𝑛−𝜌^𝑠_0Γ^(2,1)_𝑛0𝑇+𝑐₁𝜖^{−(𝛼/(1−𝛼))𝑇𝑛}₀𝜕_𝜏𝑟^𝑠_𝑛−𝜌^𝑠_0Γ0𝜏𝑑𝜏, (52) we have the following estimate: 𝜃_𝑛Γ^{(1+𝛼,((1+𝛼)/2))}_𝑛0𝑇≤𝑐₂𝜌^𝑠_0Γ^(1+𝛼)+𝜃_𝑛−𝜃_𝑛𝑡=0Γ^{(1+𝛼,((1+𝛼)/2))}_𝑛0𝑇≤𝑐₃𝜌^𝑠_0Γ^(1+𝛼)+𝑇_𝑛^{((1−𝛼)/2𝛼)}+𝑇_𝑛^(1/2)𝑟^𝑠_𝑛−𝜌^𝑠_0Γ^(2,1)_𝑛0𝑇≤𝑐₄(1+𝛿). (53) With the aid of the above estimate, we can easily obtain the desired estimate for 𝑏(𝐮_𝑛,𝑟^𝑠_𝑛).
𝐵(𝐮_𝑛,𝑟^𝑠_𝑛) is estimated as follows. With the aid of the inequality ^𝑡₀𝜕²_𝜉𝑓(𝜉,𝜏)𝑑𝜏_Γ^{(𝛼,(𝛼/2))}_0𝑇≤𝑐₅𝑇+𝑇^1−(𝛼/2)𝑓_Γ^{(2+𝛼,1+(𝛼/2))}_0𝑇, (54) we have 𝜕𝜃𝜕𝑡^𝑛−Θ𝐧⋅∇²_Γ𝑛,𝑡^𝑡₀𝐮_𝑛()+𝜃𝜉,𝜏𝑑𝜏_𝑛−Θ𝐧⋅∇²_{Γ𝑛,𝑡𝑡}^𝑡₀𝐮_𝑛(𝜉,𝜏)𝑑𝜏_Γ^{(𝛼,(𝛼/2))}_𝑛0𝑇≤𝑐₆𝑇_𝑛+𝑇_𝑛^1−(𝛼/2)𝑟^𝑠_𝑛−𝜌^𝑠_0Γ^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇𝐮_𝑛Γ^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇≤𝑐₇1+𝛿². (55) On the other hand, with the aid of 𝜃_𝑛−Θ_Γ^{(𝛼,(𝛼/2))}_𝑛0𝑇≤^𝑇𝑛₀𝜕_𝜏𝜃_𝑛−Θ_Γ^{(𝛼,(𝛼/2))}_0𝜏𝑑𝜏≤𝑇_𝑛𝑟^𝑠_𝑛−𝜌^𝑠_0Γ^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇, (56) we have 𝜃_𝑛−Θ𝐧⋅∇²_Γ𝑛,𝑡𝐮_𝑛Γ^{(𝛼,(𝛼/2))}_𝑛0𝑇≤𝑐₈𝑇_𝑛𝑟^𝑠_𝑛−𝜌^𝑠_0Γ^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇𝐮_𝑛Γ^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇≤𝑐₉1+𝛿². (57) From (55) and (57), we have the desired estimate for 𝐵(𝐮_𝑛,𝑟^𝑛_𝑠). 𝐺(𝐮_𝑛,𝑟^𝑠_𝑛) and 𝑃_𝑇𝑛(𝐮_𝑛) are estimated in a similar manner. Thus, we have proved the lemma.

From this lemma, if 𝐮_𝑛 and 𝑟^𝑠_𝑛 satisfy conditions (50), by applying Theorem 4 to problem (48) we have the following estimate of 𝐮_𝑛+1 and 𝑟^𝑠_𝑛+1: 𝐮_𝑛+1Ω^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇+𝑟^𝑠_𝑛+1Γ^{(2+𝛼,1+(𝛼/2))}_𝑛0𝑇≤𝐶₃(1)𝐶, (58) where 𝐶₃(⋅) is the function given in Theorem 4.

Now, let us take 𝑇 satisfying the following conditions: 0<𝑇≤1,𝐶₃(1)𝐶𝑇+𝑇^𝛽≤𝛿. (59) Here 𝛽 is assumed to be chosen, so that 𝛽<1/2 and 𝛽<(1−𝛼+𝛾)/2. Since (𝐮₀,𝑟^𝑠₀)=(𝟎,0), the zeroth approximation (𝐮₀,𝑟^𝑠₀) obviously satisfies conditions (50) for the above 𝑇. Hence, from (58), (𝐮₁,𝑟^𝑠₁) satisfies 𝐮_1Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑟^𝑠_1Γ^{(2+𝛼,1+(𝛼/2))}_0𝑇≤𝐶₃(1)𝐶. (60) From (59), this inequality indicates that (𝐮₁,𝑟^𝑠₁) satisfies conditions (50), and hence we can obtain the same estimate as (60) for (𝐮₂,𝑟^𝑠₂). Thus, repeating this argument, we can construct a sequence {(𝐮_𝑛,𝑞_𝑛,𝑟^𝑠_𝑛)} such that each term is defined on (0,𝑇).

Let us proceed to the proof of the convergence of the sequence {(𝐮_𝑛,𝑞_𝑛,𝑟^𝑠_𝑛)}. In the following argument, 𝐶₁,…,𝐶₁₀ denote positive constants independent of 𝑛 and 𝐶(𝜖) represents various positive constants depending only on 𝜖.

Let us set 𝑈^(𝑛+1)≡𝐮_𝑛+1−𝐮_𝑛,𝑄^(𝑛+1)≡𝑞_𝑛+1−𝑞_𝑛,𝑅^(𝑛+1)≡𝑟^𝑠_𝑛+1−𝑟^𝑠_𝑛. (61) Subtracting (48) with index 𝑛 from that with index 𝑛+1, we have 𝜕𝑈^(𝑛+1)𝜕𝑡−𝜈∇²_𝑢𝑛𝑈^(𝑛+1)+∇_𝑢𝑛𝑄^(𝑛+1)=ℱ₁^(𝑛),∇_𝑢𝑛⋅𝑈^(𝑛+1)=ℱ₂^(𝑛)inΩ_0𝑇,𝜈ΠΠ_𝑢𝑛𝐃_𝑢𝑛𝑈^(𝑛+1)𝐧_𝑢𝑛+𝛾Π∇_Γ𝑛,𝑡𝑅^(𝑛+1)=ℱ₃^(𝑛),𝐧⋅𝐓_𝑢𝑛𝑈^(𝑛+1),𝑄^(𝑛+1)𝐧_𝑢𝑛−Θ𝐧⋅∇²_Γ𝑛,𝑡^𝑡₀𝑈^(𝑛+1)𝑑𝜏+𝛾𝐧⋅∇_Γ𝑛,𝑡𝑅^(𝑛+1)=𝑏^(𝑛)+^𝑡₀ℬ^(𝑛)𝑑𝜏onΓ_0𝑇,𝑈^(𝑛+1)_𝑡=0=𝟎onΩ,𝜕𝑅^(𝑛+1)𝜕𝑡−𝜒𝛾𝜌^𝑠₀∇²_Γ𝑛,𝑡𝑅^(𝑛+1)=𝒢^(𝑛)inΓ_0𝑇,𝑅^(𝑛+1)_𝑡=0=0onΓ, (62) where ℱ₁^(𝑛)≡𝜈∇²_𝑢𝑛−∇²_𝑢𝑛−1𝐮_𝑛−∇_𝑢𝑛−∇_𝑢𝑛−1𝑞_𝑛,ℱ₂^(𝑛)≡−∇_𝑢𝑛−∇_𝑢𝑛−1⋅𝐮_𝑛,ℱ₃^(𝑛)≡−𝜈ΠΠ_𝑢𝑛𝐃_𝑢𝑛𝐮_𝑛𝐧_𝑢𝑛−Π_𝑢𝑛−1𝐃_𝑢𝑛−1𝐮_𝑛𝐧_𝑢𝑛−1−𝛾Π∇_Γ𝑛,𝑡−∇_{Γ𝑛−1,𝑡}𝑟^𝑠_𝑛,𝑏^(𝑛)≡−𝐧⋅𝐓_𝑢𝑛𝐮_𝑛,𝑞_𝑛𝐧_𝑢𝑛−𝐓_𝑢𝑛−1𝐮_𝑛,𝑞_𝑛𝐧_𝑢𝑛−1−𝛾𝐧⋅∇_Γ𝑛,𝑡−∇_{Γ𝑛−1,𝑡}𝑟^𝑠_𝑛+𝑏𝐮_𝑛,𝑟^𝑠_𝑛−𝑏𝐮_𝑛−1,𝑟^𝑠_𝑛−1,ℬ^(𝑛)≡Θ𝐧⋅∇²_Γ𝑛,𝜏−∇²_{Γ𝑛−1,𝜏𝜏}^𝜏₀𝐮_𝑛()+𝜉,𝑠𝑑𝑠Θ𝐧⋅∇²_Γ𝑛,𝜏−∇²_{Γ𝑛−1,𝜏}𝐮_𝑛+𝐵𝐮_𝑛,𝑟^𝑠_𝑛−𝐵𝐮_𝑛−1,𝑟^𝑠_𝑛−1,𝒢^(𝑛)≡𝜒𝛾𝜌^𝑠₀∇²_Γ𝑛,𝑡−∇²_{Γ𝑛−1,𝑡}𝑟^𝑠_𝑛+𝐺𝐮_𝑛,𝑟^𝑠_𝑛−𝐺𝐮_𝑛−1,𝑟^𝑠_𝑛−1. (63)

Now, noting that the relations ∑³_𝑗=1𝜕_𝜉𝑗𝐴_𝑖𝑗=0 hold for the cofactors 𝐴_𝑖𝑗 of the Jacobian matrix of any transformation from 𝜉 to 𝑥, by direct calculations, we can verify that the following relations hold: 𝜕ℱ₂^(𝑛)𝜕𝑡−∇_𝑢𝑛⋅ℱ₁^(𝑛)=∇⋅𝐡^(𝑛),𝐡^(𝑛)=³_𝑘=1𝜕_𝜉𝑘𝐇_𝑘^(𝑛), (64) where 𝐇_𝑘^(𝑛)=−𝒥_𝑢⁻¹_𝑛−𝒥_𝑢⁻¹_𝑛−1𝐋_𝑘^(𝑛)−𝒥_𝑢⁻¹_𝑛𝐌_𝑘^(𝑛)+1𝜕4𝜋^𝜉𝑘_Ω𝐍_𝑘^(𝑛)𝜂,𝑡𝜉−𝜂𝑑𝜂, (65) with 𝐋_𝑘^(𝑛)≡𝜈𝐴^(𝑛−1)_𝑖𝑘𝐴^(𝑛−1)_𝑖𝑙𝜕_𝜉𝑙𝐮_𝑛−𝒥^∗_𝑢𝑛−1𝐞_𝑘𝑞_𝑛,𝐌_𝑘^(𝑛)≡𝜈𝐴^(𝑛)_𝑖𝑘𝐴^(𝑛)_𝑖𝑙−𝐴^(𝑛−1)_𝑖𝑘𝐴^(𝑛−1)_𝑖𝑙𝜕_𝜉𝑙𝐮_𝑛−𝒥^∗_𝑢𝑛−𝒥^∗_𝑢𝑛−1𝐞_𝑘𝑞_𝑛,𝐍_𝑘^(𝑛)≡𝜕_𝑡𝒥_𝑢⁻¹_𝑛−𝒥_𝑢⁻¹_𝑛−1𝐮_𝑛−𝜕_𝜂𝑘𝒥_𝑢⁻¹_𝑛−𝒥_𝑢⁻¹_𝑛−1𝐋_𝑘^(𝑛)−𝜕_𝜂𝑘𝒥_𝑢⁻¹_𝑛𝐌_𝑘^(𝑛). (66) In (66), the Einstein summation convention is used, 𝐴^(𝑛)_𝑖𝑗 denotes the (𝑖,𝑗)-component of 𝒥^∗_𝑛, and 𝐞_𝑘, 𝑘=1,2,3, denote fundamental unit vectors in 𝐑³.

For the terms in (63), 𝐡^(𝑛), and 𝐇_𝑘^(𝑛), we have the following estimate for arbitrary 0<𝜖<1: ℱ₁^(𝑛)_Ω^{(𝛼,(𝛼/2))}_0𝑇+ℱ₂^(𝑛)_Ω^{(1+𝛼,((1+𝛼)/2))}_0𝑇+ℱ₃^(𝑛)_Γ^{(1+𝛼,((1+𝛼)/2))}_0𝑇+𝑏^(𝑛)_Γ^{(1+𝛼,((1+𝛼)/2))}_0𝑇+ℬ^(𝑛)_Γ^{(𝛼,(𝛼/2))}_0𝑇+𝐡^(𝑛)_Ω^{(𝛼,(𝛼/2))}_0𝑇+³_𝑘=1𝐇_𝑘^(𝑛)_Ω^{(1+𝛼,𝛾)}_0𝑇+𝒢^(𝑛)_Γ^{(𝛼,(𝛼/2))}_0𝑇≤𝐶₁𝜖𝑈^(𝑛)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑅^(𝑛)_Γ^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝐶(𝜖)^𝑇₀𝑈^(𝑛)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝜏+𝑅^(𝑛)_Γ^{(2+𝛼,1+(𝛼/2))}_0𝜏𝑑𝜏. (67)

We will derive here the estimate only of 𝐇_𝑘^(𝑛) because the other terms can be estimated in a similar and simpler manner. Noting that the following estimates hold for 𝐮_𝑛 and 𝑞_𝑛: 𝐮_𝑛Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑞_𝑛Ω^{(1+𝛼,𝛾)}_0𝑇≤𝐶₂, (68) and with the aid of estimates (38) and (39), we have 𝒥_𝑢⁻¹_𝑛−𝒥_𝑢⁻¹_𝑛−1𝐋_𝑘^(𝑛)_Ω^{(1+𝛼,𝛾)}_0𝑇,𝒥_𝑢⁻¹_𝑛𝐌_𝑘^(𝑛)_Ω^{(1+𝛼,𝛾)}_0𝑇,𝐍_𝑘^(𝑛)_Ω^{(0,((1+𝛼)/2))}_0𝑇≤𝐶₃𝜖𝑈^(𝑛)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝐶(𝜖)^𝑇₀𝑈^(𝑛)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝜏𝑑𝜏. (69)

Noting also the following inequality: 𝜕_𝜉𝑘Ω𝐍_𝑘^(𝑛)𝜂,𝑡𝜉−𝜂𝑑𝜂_Ω^{(1+𝛼,𝛾)}_0𝑇≤𝐶₄𝐍_𝑘^(𝑛)_Ω^{(0,((1+𝛼−𝛾)/2))}_0𝑇, (70) from estimate (69) for 𝐍_𝑘^(𝑛), we have 𝜕_𝜉𝑘Ω𝐍_𝑘^(𝑛)𝜂,𝑡𝜉−𝜂𝑑𝜂_Ω^{(1+𝛼,𝛾)}_0𝑇≤𝐶₅𝜖𝑈^(𝑛)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝐶(𝜖)^𝑇₀𝑈^(𝑛)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝜏𝑑𝜏. (71)

Thus, from estimates (69) and (71), we have the desired estimate for 𝐇_𝑘^(𝑛).

Now, applying Theorem 4 to problem (62), we have the following estimate: 𝑈^(𝑛+1)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑄^(𝑛+1)_𝑇+𝑅^(𝑛+1)_Γ^{(2+𝛼,1+(𝛼/2))}_0𝑇≤𝐶₆𝜖𝑈^(𝑛)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑅^(𝑛)_Γ^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝐶(𝜖)^𝑇₀𝑈^(𝑛)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝜏+𝑅^(𝑛)_Γ^{(2+𝛼,1+(𝛼/2))}_0𝜏𝑑𝜏, (72) where the norm ‖𝑓‖_𝑇 is defined as ‖𝑓‖_𝑇≡|𝑓|_Ω^{(1+𝛼,𝛾)}_0𝑇+|∇𝑓|_Ω^{(𝛼,(𝛼/2))}_0𝑇+|𝑓|_Γ^{(1+𝛼,((1+𝛼)/2))}_0𝑇. Fixing 𝜖 so that 𝐶₆𝜖≤1/2 and summing the above inequalities from 𝑛=1 to 𝑛=𝑚, we have 𝑆_𝑚(𝑇)≤𝐶₇𝑆₁(𝑇)+^𝑇₀𝑆_𝑚(𝜏)𝑑𝜏, (73) where 𝑆_𝑚(𝑇)≡^𝑚_𝑛=1𝑈^(𝑛)_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝑄^(𝑛)_𝑇+𝑅^(𝑛)_Γ^{(𝛼,(𝛼/2))}_0𝑇. (74) Then, using Gronwall’s inequality, from the above inequality, we have 𝑆_𝑚(𝑇)≤𝐶₈1+𝑇𝑒^𝐶9𝑇. (75)

Noting that the right-hand side in (75) is independent of 𝑚, we can conclude that the sequence {(𝐮_𝑛,𝑞_𝑛,𝑟^𝑠_𝑛)} is convergent in 𝐶^{2+𝛼,1+(𝛼/2)}(Ω_0𝑇)×(𝐶^1+𝛼,𝛾(Ω_0𝑇)∩𝐶^{1+𝛼,((1+𝛼)/2)}(Γ_0𝑇))×𝐶^{2+𝛼,1+(𝛼/2)}(Γ_0𝑇).

Now, let us prove Theorem 1. Taking the limit as 𝑛 tends to infinity in problem (48), we can easily see that the limit of the sequence {(𝐮_𝑛,𝑞_𝑛,𝑟^𝑠_𝑛)} is a solution of problem (11)–(13). The uniqueness can be proved as follows. Let (𝐮,𝑞,𝑟^𝑠) and (𝐮,𝑞,𝑟^𝑠̃) be two solutions of problem (11)–(13). By subtracting one equation from the other, we obtain the equations for the differences 𝑉≡𝐮−𝐮, 𝑃≡𝑞−𝑞, and 𝑅≡𝑟^𝑠−𝑟^𝑠̃, the form of which is similar to (62). Then, in a similar manner to obtain (72), we can obtain the following estimate: |𝑉|_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+|𝑅|_Γ^{(2+𝛼,1+(𝛼/2))}_0𝑇≤𝐶₁₀𝜖|𝑉|_Ω^{(2+𝛼,1+(𝛼/2))}_0𝑇+|𝑅|_Γ^{(2+𝛼,1+(𝛼/2))}_0𝑇+𝐶(𝜖)^𝑇₀|𝑉|_Ω^{(2+𝛼,1+(𝛼/2))}_0𝜏+|𝑅|_Γ^{(2+𝛼,1+(𝛼/2))}_0𝜏𝑑𝜏. (76) This inequality implies that 𝑉=𝟎 and 𝑅=0, and as a consequence, 𝑃=0 follows. Thus, the proof of Theorem 1 is completed.