Stefan Problems with Convection Governed by Navier-Stokes Equations

Takesi FUKAO and Nobuyuki KENMOCHI

Technical Reports of Mathematical Sciences Chiba University, Vol. 19 (2003), No. 5

Stefan problems with convection governed by Navier-Stokes equations

Takesi Fukao

Department of Mathematics, Faculty of Education, Chiba University 1-33 Yayoi-chō, Inage-ku, Chiba, 263-8522 Japan

E-mail: fukao@faculty.chiba-u.jp

and

Nobuyuki Kenmochi

Department of Mathematics, Faculty of Education, Chiba University 1-33 Yayoi-chō, Inage-ku, Chiba, 263-8522 Japan

E-mail: kenmochi@faculty.chiba-u.jp

Abstract. We consider a material which has two physical phases, liquid and solid. we are interested in the dynamics of their interface, taking account of the flow caused by convection in the melt (liquid region) and by the crystal (solid region) growth; therefore the shape of material domain changes smoothly in time. Our idea is to describe this phenomenon as a Stefan problem with convection, governed by the Navier-Stokes equation in the (unknown) liquid region, in a prescribed time dependent material domain. In this paper we first propose an approximation to the original problem and prove the existence of an approximate solution in a weak sense. Next we construct a solution of the original problem as a limit of weak approximate solutions.

Keywords. degenerate parabolic equation, Stefan problem, convection, Navier-Stokes equation.

1 Introduction

The Stefan problem is one of the most interesting mathematical models which describe phase transitions. In 1889, J. Stefan, a physicist in Austria, proposed a simplified model for one phase transition, which is called the one phase Stefan problem. Subsequently many kinds of Stefan problems have been proposed from various points of view along with their weak (variational) formulations. We refer for the literature of various Stefan problems for instance to the book of Visintin [23]. In [4], DiBenedetto and O’Leary treated a heat conduction problem with convective effect due to the flow of material governed by the Stokes equation in the liquid region.

In recent 15 years, this has been one of the main research subjects in nonlinear PDEs. We refer for the research papers on Stefan problems with convection, for instance, to Cannon, DiBenedetto and Knightly [2], Rodrigues [20] and Casella and Giangi [3]. In these models the movement of material configuration is neglected, namely the problems are formulated in a fixed material domain.

2000 AMS Subject Classification 35K65, 35R35.

We are interested in the mathematical modeling of the Czochralski pulling method for crystal growth process; see [19] for the details. In such a case the solidification (crystallization) of melt substances yields a smooth change of the material configuration in time and a smooth flow in the material domain. When this is taken into account, the modeling should be made in a time-dependent domain (i.e. non-cylindrical domain). From this point of view a model was also suggested by Fukao, Kenmochi and Pawłow [7] under the assumption that the convective vector is prescribed, and an existence-uniqueness result for its weak formulation was obtained.

In the present paper, we consider the case when the convective vector in the liquid domain is unknown, too; it will be found as a part of solution of our problem. Our model is a Stefan problem formulated in a time-dependent domain, coupled with the Navier-Stokes equation governing the flow in the (unknown) liquid region. Our mathematical question is to determine the temperature field and the location of the melt-crystal interface in the material. The main difficulty in our mathematical treatment arises from the timedependence of the material domain and lack of regularity properties of temperature and convective vector.

For 0 < T < +∞, let Ω_{m}(t) ⊂ R^{3} be a time dependent bounded domain with smooth boundary Γ_{m}(t) := ∂Ω_{m}(t) such that Ω_{m}(t) is occupied by a material having two phases of liquid and solid states; Ω_{ℓ}(t) (resp. Ω_{s}(t)) is the liquid (resp. solid) region at time t in which temperature is positive (resp. negative) and these regions are separated by an unknown interface S(t), namely Ω_{m}(t) := Ω_{ℓ}(t) ∪ S(t) ∪ Ω_{s}(t) for each t ∈ [0, T]. Now we denote by θ := θ(t, x) the temperature field, by v := v(t, x) := (v_{1}(t, x), v_{2}(t, x).v_{3}(t, x)) the velocity of the flow of material and by p_{ℓ} := p_{ℓ}(t, x) the pressure field. In the above setting we consider the classical formulation (CZ):= {(1.1)-(1.11)} of the Czochralski process with convection v governed by the Navier-Stokes equation. Let Q_{m} := ⋃_{t∈(0,T)}{t}×Ω_{m}(t). From the usual balance laws we derive

\frac{∂θ}{∂t} + v · ∇θ − k_{ℓ}Δθ = f in Q_{ℓ} := ⋃_{t∈(0,T)}\{t\} × Ω_{ℓ}(t),  (1.1)

\frac{∂θ}{∂t} + v · ∇θ − k_{s}Δθ = f in Q_{s} := ⋃_{t∈(0,T)}\{t\} × Ω_{s}(t),  (1.2)

and the Stefan condition

θ = 0, k_{ℓ} \frac{∂θ}{∂n^{−}} + k_{s} \frac{∂θ}{∂n^{+}} = L(v · n^{−} − v_{S}) on S := ⋃_{t∈(0,T)}\{t\} × S(t),  (1.3)

where and hereafter ∇θ := (∂θ/∂x_{1}, ∂θ/∂x_{2}, ∂θ/∂x_{3}), Δθ := ∑_{i=1}^{3}(∂^{2}θ/∂x_{i}^{2}). n^{+} = n^{+}(t, x) := (n_{1}(t, x)^{+}, n_{2}(t, x)^{+}, n_{3}(t, x)^{+}) is the 3-dimensional unit vector normal to S(t) at x ∈ S(t) pointing to the liquid region Ω_{ℓ}(t) and n^{−} := −n^{+}; k_{ℓ}, k_{s} and L are positive constants, f is a given function on Q_{m} and v_{S} is the normal speed of S(t). In the liquid region Q_{ℓ}, an incompressible Navier-Stokes equation

\frac{∂v}{∂t} + (v · ∇)v − ν_{ℓ}Δv + ∇p_{ℓ} = g_{ℓ}(θ) in Q_{ℓ},  (1.4)
divv = 0 in Q_{ℓ},  (1.5)

is considered as an equation governing the velocity of the material flow, where ν_{ℓ} is a positive constant and g_{ℓ} := (g_{1}, g_{2}, g_{3}) : R → R^{3} is a given Lipschitz continuous function, for example, the buoyancy force g_{ℓ}(θ) := (0, 0, g_{∗}(1 − θ)) with a positive constant g_{∗}; but, in general g_{i} are nonlinear functions for i = 1, 2, 3. In the solid region and on the interface, v coincides with a designed vector field v_{D} := (v_{D1}, v_{D2}, v_{D3}), namely,

v = v_{D} in Q_{s} ∪ S.  (1.6)

Moreover the following boundary and initial conditions are considered:

k_{ℓ} \frac{∂θ}{∂ν} + n_{0}k_{ℓ}θ = q on Σ_{mℓ} := ⋃_{t∈(0,T)}\{t\} × Γ_{mℓ}(t),  (1.7)

k_{s} \frac{∂θ}{∂ν} + n_{0}k_{s}θ = q on Σ_{ms} := ⋃_{t∈(0,T)}\{t\} × Γ_{ms}(t),  (1.8)
v = v_{D} on Σ_{m} := ⋃_{t∈(0,T)}{t} × Γ_{m}(t),  (1.9)
θ(0) = θ_{0}, S(0) = S_{0} on Ω_{m0} := Ω_{m}(0),  (1.10)
v(0) = v_{0} on Ω_{m0},  (1.11)

where Γ_{mi}(t) := ∂Ω_{i}(t) ∩ Γ_{m}(t) for i = ℓ, s; ν = ν(t, x) := (ν_{1}(t, x), ν_{2}(t, x), ν_{3}(t, x)) is the 3-dimensional unit vector outward normal to Γ(t) at x ∈ Γ(t); n_{0} is a positive constant, q and θ_{0} are given functions on Σ_{m} and Ω_{m0}, respectively.

Throughout this paper, assume that the domain Ω_{m}(t) smoothly changes in time in the following sense:

(A1) There exists a bounded domain Ω ⊂ R^{3} with smooth boundary Γ := ∂Ω such that Q_{m} ⊂ Q := (0, T) × Ω. Moreover there exists a function y ∈ C^{3}(\overline{Q}) := C^{3}(\overline{Q})^{3} which gives a C^{3}-diffeomorphism y(t, ·) := (y_{1}(t, ·), y_{2}(t, ·), y_{3}(t, ·)) from \overline{Ω} onto itself for all t ∈ [0, T] such that

y(t, \overline{Ω_{m}(t)}) = \overline{Ω_{m0}} for all t ∈ [0, T], y(0, ·) = I (identity) on \overline{Ω_{m0}}.

Furthermore, assume

(A2) A vector function v_{D} ∈ C^{2}(\overline{Q}) is given so that

divv_{D}(t, ·) = 0 in Ω_{m}(t) for all t ∈ [0, T],
v_{D} · ν = v_{Σ_{m}} on Σ_{m},

where v_{Σ_{m}}(t, ·) is the normal speed of Γ_{m}(t).

For any vector function v := (v_{1}, v_{2}, v_{3}), we denote the composition (v_{1}(t, y(t, x)), v_{2}(t, y(t, x)), v_{3}(t, y(t, x))) by [v ∘ y](t, x) for all (t, x) ∈ Q_{m}. Also, unless otherwise stated, for any function u and vector function w defined on Q_{m} (resp. Ω_{m}(t)) we denote their 0-extensions onto (0, T) × R^{3} (resp. R^{3}) by the same notation u and w. Therefore, given a function u on Q_{m} or Ω_{m}(t), the convolution ρ_{ε} ∗ u is well defined as a function on (0, T) × R^{3} or R^{3} for every ε > 0, where ρ_{ε} := ρ_{ε}(x) is the usual mollifier with respect to the space variable in R^{3}.

2 Weak formulation and main theorems

In this section, we consider (CZ) in a weak variational sense, and it is a system of the weak variational formulation of the Stefan problem and of the Navier-Stokes equation.

We use the following notations:

H := L^{2}(Ω), Y := L^{4}(Ω), V := H^{1}(Ω) (:= W^{1,2}(Ω)), X := W^{1,4}(Ω)

with the usual norms, and Y^{∗}, V^{∗} and X^{∗} are the dual spaces of Y, V and X; we denote by ⟨·, ·⟩_{Y^{∗},Y}, ⟨·, ·⟩_{V^{∗},V} and ⟨·, ·⟩_{X^{∗},X} the duality pairs between Y^{∗} and Y, V^{∗} and V and X^{∗} and X, respectively. Especially H is a Hilbert space with standard inner product (·, ·)_{H} and we have the following relations:

X ⊂ V ↪ Y ⊂ H ⊂ Y^{∗} ↪ V^{∗} ⊂ X^{∗},

where ↪ means that the imbedding is compact. Moreover we use the following notations for vector valued function spaces:

D_{σ}(Ω) := {z ∈ C_{0}^{∞}(Ω); divz = 0 in Ω},
H := L_{σ}^{2}(Ω), Y := L_{σ}^{4}(Ω), V := H_{σ}^{1}(Ω), X := W_{σ}^{1,4}(Ω),

where L_{σ}^{2}(Ω), L_{σ}^{4}(Ω), H_{σ}^{1}(Ω) and W_{σ}^{1,4}(Ω) are the closures of D_{σ}(Ω) in spaces L^{2}(Ω) := H^{3}, L^{4}(Ω) := Y^{3}, H^{1}(Ω) := V^{3} and W^{1,4}(Ω) := X^{3}, respectively. They are equipped with the usual product norms, and Y^{∗}, V^{∗} and X^{∗} are the dual spaces of Y, V and X with duality pairs ⟨·, ·⟩_{Y^{∗},Y} and so on. We see that H is a Hilbert space with the usual inner product (·, ·)_{H} and the following relations hold:

X ⊂ V ↪ Y ⊂ H ⊂ Y^{∗} ↪ V^{∗} ⊂ X^{∗}.

Now we give a weak variational formulation of (CZ), introducing a new parameter u, which is called the enthalpy, and a function β : R → R defined by

u := \{ \begin{matrix}θ & if θ < 0, \\ r ∈ [0, L] & if θ = 0, \\ θ + L & if θ > 0, \end{matrix} β(r) := \{ \begin{matrix}k_{s}r & if r < 0, \\ 0 & if 0 ≤ r ≤ L, \\ k_{ℓ}(r − L) & if r > L. \end{matrix}

With these u and β, the Stefan problem in (CZ) can be rewritten in the following initial boundary value problem for a degenerate parabolic partial differential equation in the non-cylindrical domain Q_{m} (see Fukao, Kenmochi and Pawłow [7]):

\frac{∂u}{∂t} + v · ∇u − Δβ(u) = f in Q_{m},  (2.1)

\frac{∂β(u)}{∂ν} + n_{0}β(u) = q on Σ_{m},  (2.2)
u(0) = u_{0} := θ_{0}X_{Ω_{s}(0)} + (θ_{0} + L)X_{Ω_{ℓ}(0)} in Ω_{m0},  (2.3)

where X_{Ω_{i}(0)} is the characteristic function of Ω_{i}(0) for i = ℓ, s. Moreover, it is easy to derive the following variational identity from (2.1)-(2.3):

− ∫_{Q_{m}} u\frac{∂η}{∂t} dxdt − ∫_{Q_{m}}(v · ∇η)udxdt + ∫_{Q_{m}} ∇β(u) · ∇ηdxdt + n_{0} ∫_{Σ_{m}} β(u)ηdΓ_{m}(t)dt

= ∫_{Q_{m}} fηdxdt + ∫_{Σ_{m}} qηdΓ_{m}(t)dt + ∫_{Ω_{m0}} u_{0}η(0)dx for all η ∈ W,  (2.4)

where W := {η ∈ H^{1}(Q_{m}); η(T) = 0}.

In the paper [7], problem (2.4) was treated, when the convective vector v is prescribed. In fact, under the same assumption as (A1), the existence and uniqueness results are proved. Also the papers of [20] and [21] gave the well-posedness for this problem in the case when the domain Ω_{m}(t) does not depend on time t.

Next we consider the Navier-Stokes equation (1.4). We put w := v − v_{D} on Q_{m} and w_{0} := v − v_{D}(0) on Ω_{m0}. Then, by (1.6) and (1.9) we have that

w = 0 on Q_{s} ∪ S and on Σ_{m}.  (2.5)

For simplicity, we denote by w and w_{0} again the 0-extensions of them onto Q and Ω, respectively, so that w(t) ∈ H for all t ∈ [0, T] as well as w_{0} ∈ H. A weak variational formulation of Navier-Stokes equation in (CZ) is described in terms of w and w_{0} as follows:

− ∫_{0}^{T} (η′, w)_{H}dt + ∫_{0}^{T} a(w, η)dt − ∫_{0}^{T} b(t; w, η, w)dt − ∫_{0}^{T} c(t; w, η)dt
= ∫_{0}^{T} (g(β(u)), η)_{H}dt + (w_{0}, η(0))_{H} for all η ∈ W(u),  (2.6)

subject to the constraint

w = 0 a.e. on Q_{s}(u),

where

W(u) := \{η ∈ L^{4}(0, T; X); \begin{matrix}η′ ∈ L^{2}(0, T; H), η(T, ·) = 0 a.e. on Ω, \\ η = 0 a.e. on Q \ Q_{ℓ}(u) \end{matrix} \};

here η′ := ∂η/∂t. In order to emphasize the dependence of the class of test functions η upon u we denote it by W(u) as well as Ω_{i}(t) and Q_{i} by Ω_{i}(u; t) and Q_{i}(u) for i = ℓ, s; moreover a(·, ·) : V × V → R and for each t ∈ [0, T], b(t; ·, ·, ·) : Y × V × Y → R, c(t; ·, ·) : H × H → R are defined by

a(z, η) := ν_{ℓ} ∑_{i=1}^{3} ∫_{Ω} ∇z_{i} · ∇η_{i}dx for all z, η ∈ V,

b(t; z, η, z̄) := ∑_{i=1}^{3} ∫_{Ω}((z + v_{D}(t)) · ∇η_{i})z̄_{i}dx for all z, z̄ ∈ Y, η ∈ V,

c(t; z, η) := ∑_{i=1}^{3} ∫_{Ω}(z · ∇v_{Di}(t))η_{i}dx for all z, η ∈ H,

and g(β(u)) ∈ L^{2}(0, T; H) is defined by g(β(u(t))) := P_{L} [g_{L}(β(u(t)))] with

g_{L}(β(u)) := \{ \begin{matrix}g_{ℓ}(β(u)) − \frac{∂v_{D}}{∂t} − (v_{D} · ∇)v_{D} + ν_{ℓ}Δv_{D} on Q_{m}, \\ 0 otherwise, \end{matrix}

where P_{L} : L^{2}(Ω) → H is the Leray projector.

The system {(2.4)-(2.6)} is a weak variational formulation of our problem (CZ). However, from the mathematical point of view it is extremely difficult to handle this system, especially (2.5) because of the lack of regularity of the enthalpy u. In order to avoid this difficulty we replace the liquid region Ω_{ℓ}(u; t) and solid region Ω_{s}(u; t) by their approximations

Ω_{ℓ,ε}(u; t) := \{x ∈ Ω_{m}(t); (ρ_{ε} ∗ u)(t, x) > \frac{L}{2}\} ,

and

Ω_{s,ε}(u; t) := \{x ∈ Ω_{m}(t); (ρ_{ε} ∗ u)(t, x) < \frac{L}{2}\} ,

respectively, where ρ_{ε} is the usual ε-mollifier in R^{3} with respect to the space variable x, and the class W(u) of test functions by

W_{ε}(u) := \{η ∈ L^{4}(0, T; X); \begin{matrix}η′ ∈ L^{2}(0, T; H), η(T, ·) = 0 a.e. on Ω, \\ η = 0 a.e. on Q \ Q_{ℓ,ε}(u) \end{matrix} \} ,

with

Q_{ℓ,ε}(u) := ⋃_{t∈(0,T)}{t} × Ω_{ℓ,ε}(u, t).

Now we introduce a class of weak solutions, which is a reasonable approximation to the original one, as follows:

Definition 2.1. For a fixed ε > 0, the pair {u_{ε}, w_{ε}} of functions u_{ε} ∈ L^{∞}(Q_{m}) and w_{ε} ∈ L^{2}(Q) is called a weak solution of (CZ)_{ε}, if (D1)-(D3) are satisfied:

(D1) β(u_{ε}) ∈ L^{∞}(Q_{m}), ∫_{0}^{T} |β(u_{ε}(t))|_{H^{1}(Ω_{m}(t))}^{2}dt < +∞ and the function

t ↦ ∫_{Ω_{m}(t)} u_{ε}(t, x)ξ(x)dx is continuous on [0, T] for all ξ ∈ H.

(D2) w_{ε} ∈ L^{∞}(0, T; H) ∩ L^{2}(0, T; V), w_{ε} is weakly continuous from [0, T] into H and

w_{ε} = 0 a.e. on Q_{s,ε}(u_{ε}) := ⋃_{t∈(0,T)}{t} × Ω_{s,ε}(u_{ε}, t).

(D3) u_{ε} and w_{ε} satisfy the following variational identities:

− ∫_{Q_{m}} u_{ε} \frac{∂η}{∂t} dxdt−∫_{Q_{m}}(v_{ε}·∇η)u_{ε}dxdt+∫_{Q_{m}} ∇β(u_{ε})·∇ηdxdt+n_{0} ∫_{Σ_{m}} β(u_{ε})ηdΓ_{m}(t)dt

= ∫_{Q_{m}} fηdxdt + ∫_{Σ_{m}} qηdΓ_{m}(t)dt + ∫_{Ω_{m0}} u_{0}η(0)dx for all η ∈ W,  (2.7)

where v_{ε} := w_{ε} + v_{D};

− ∫_{0}^{T} (η′, w_{ε})_{H}dt + ∫_{0}^{T} a(w_{ε}, η)dt − ∫_{0}^{T} b(t; w_{ε}, η, w_{ε})dt − ∫_{0}^{T} c(t; w_{ε}, η)dt
= ∫_{0}^{T} (g(β(u_{ε})), η)_{H}dt + (w_{0}, η(0))_{H} for all η ∈ W_{ε}(u_{ε}).  (2.8)

Our main theorem is now stated as follows.

Theorem 2.1. Assume (A1) and (A2) hold. Moreover f ∈ L^{∞}(Q_{m}), q ∈ L^{∞}(Σ_{m}), u_{0} ∈ L^{∞}(Ω_{m0}) and v_{0} ∈ L^{2}(Ω_{m0}) := L^{2}(Ω_{m0})^{3} with divv_{0} = 0 in Ω_{m0}. Then, for each (small) positive number ε, there exist at least one solution {u_{ε}, w_{ε}} of (CZ)_{ε} such that the following uniform estimates in ε ∈ (0, 1]

|u_{ε}|_{L^{∞}(Q_{m})} + ∫_{0}^{T} |∇β(u_{ε}(t))|_{L^{2}(Ω_{m}(t))}^{2}dt ≤ R_{0},  (2.9)

and

|w_{ε}|_{L^{∞}(0,T;H)} + |w_{ε}|_{L^{2}(0,T;V)} ≤ R_{1},  (2.10)

where R_{0} and R_{1} are positive constants which depend on |u_{0}|_{L^{∞}(Ω_{m0})}, |f|_{L^{∞}(Q_{m})}, |q|_{L^{∞}(Σ_{m})} and |v_{D}|_{C^{2}(\overline{Q})}, independent of ε ∈ (0, 1].

As to the limit of solutions {u_{ε}, w_{ε}} we have the following result.

Theorem 2.2. Under the same assumption of Theorem 2.1, let {u_{ε}, w_{ε}}_{ε>0} be the family of approximate solutions constructed by Theorem 2.1. Then there exists a sequence {ε_{n}} of positive numbers with ε_{n} → 0 as n → +∞ such that

u_{ε_{n}} → u weakly in L^{2}(Q_{m}),
w_{ε_{n}} → w weakly in L^{2}(0, T; V) as n → +∞,

and u satisfies (2.4) with v := w + v_{D}.

As to the vector function w obtained in Theorem 2.2 as a limit of {w_{ε_{n}}}, it is not clear whether w is a solution of the variational form (2.6) of the Navier-Stokes equation, because the class W(u) of test functions for the limit u of {u_{ε_{n}}} is not able to be defined without regularity of u.

These theorems will be proved in the next sections.

3 Auxiliary results

In this section we recall some important results on Stefan problems with prescribed convections and Navier-Stokes equations formulated in non-cylindrical domains. Throughout this section assumptions (A1) and (A2) are always made and the same notation as in the previous section is used. Furthermore, given a ∈ [0, T] and T_{0} > 0 with a + T_{0} ≤ T, we use the following notation:

Q(a, a + T_{0}) := (a, a + T_{0}) × Ω, Q_{m}(a, a + T_{0}) := ⋃_{t∈(a,a+T_{0})}{t} × Ω_{m}(t),

Σ(a, a + T_{0}) := (a, a + T_{0}) × Γ, Σ_{m}(a, a + T_{0}) := ⋃_{t∈(a,a+T_{0})}{t} × Γ_{m}(t).

3.1 Auxiliary results for Stefan problems with convections

[A](cf.[6]) Let ṽ be a 3-dimensional vector field defined on Q such that ṽ ∈ L^{∞}(0, T; H) ∩L^{2}(0, T; V) and ṽ = v_{D} a.e. on Q \ Q_{m}, let f ∈ L^{∞}(Q_{m}) as well as q ∈ L^{∞}(Σ_{m}). Then, for each a ∈ (0, T), T_{0} > 0 with a + T_{0} ≤ T and ũ_{0} ∈ L^{∞}(Ω_{m}(a)), there exists at least one function ũ on Q_{m}(a, a + T_{0}) such that

(i) ũ ∈ L^{∞}(Q_{m}(a, a + T_{0})), β(ũ(t)) ∈ H^{1}(Ω_{m}(t)) for a.e. t ∈ (a, a + T_{0}) with

∫_{a}^{a+T_{0}} |β(ũ(t))|_{H^{1}(Ω_{m}(t))}^{2}dt < +∞,

and the 0-extension of ũ onto Q(a, a + T_{0}), denoted by ũ again, is weakly continuous from [a, a + T_{0}] into H.

(ii) ũ satisfies the variational identity

− ∫_{Q_{m}(a,a+T_{0})} ũ\frac{∂η}{∂t} dxdt − ∫_{Q_{m}(a,a+T_{0})}(ṽ · ∇η)ũdxdt + ∫_{Q_{m}(a,a+T_{0})} ∇β(ũ) · ∇ηdxdt

+ n_{0} ∫_{Σ_{m}(a,a+T_{0})} β(ũ)ηdΓ_{m}(t)dt  (3.1)
= ∫_{Q_{m}(a,a+T_{0})} fηdxdt + ∫_{Σ_{m}(a,a+T_{0})} qηdΓ_{m}(t)dt + ∫_{Ω_{m}(a)} ũ_{0}η(a)dx,

for all η ∈ H^{1}(Q_{m}(a, a + T_{0})) with η(a + T_{0}, ·) = 0 a.e. on Ω(a + T_{0}).

(iii) Putting

M_{1} := max \{L, |f|_{L^{∞}(Q_{m})}, |\frac{q}{n_{0}} |_{L^{∞}(Σ_{m})} , |β(ũ_{0})|_{L^{∞}(Ω_{m}(a))}\} ,

we have

|β(ũ)|_{L^{∞}(Q_{m}(a,a+T_{0}))} ≤ M_{1}(1 + T_{0}),

and hence

|ũ|_{L^{∞}(Q_{m}(a,a+T_{0}))} ≤ max \{\frac{M_{1}}{k_{s}} , \frac{M_{1}}{k_{ℓ}} + L\} (1 + T_{0}) =: M_{2}.

(iv) The following inequality holds:

∫_{Ω_{m}(a+T_{0})} β̂(ũ(a + T_{0}))dx + c_{1} ∫_{a}^{a+T_{0}} |β(ũ)|_{H^{1}(Ω_{m}(t))}^{2}dt ≤ ∫_{Ω_{m}(a)} β̂(ũ_{0})dx + M_{3}T_{0}, (3.2)

where M_{3} is a positive constant depending only on M_{1}, M_{2}, |Ω| (the volume of Ω) and the maximum of |Γ_{m}(t)| (the area of Γ_{m}(t)) for t ∈ [0, T], β̂ is the primitive of β with β̂(0) = 0, and c_{1} is a positive constant satisfying that

c_{1}|z|_{H^{1}(Ω_{m}(t))}^{2} ≤ |∇z|_{L^{2}(Ω_{m}(t))}^{2} + n_{0}|z|_{L^{2}(Γ_{m}(t))}^{2}
for all z ∈ H^{1}(Ω_{m}(t)) and t ∈ [0.T].

For simplicity, we denote by (SPC; ṽ, ũ_{0}) on [a, a + T_{0}] the variational problem (3.1), and any function ũ satisfying the above conditions (i)-(iv) is called a solution of (SPC; ṽ, ũ_{0}) on [a, a + T_{0}].

Remark 3.1. It is clear to check that (3.1) is equivalent to the following variational identity:

∫_{Ω_{m}(t)} ũ(a + T_{0})η(t)dx − ∫_{Q_{m}(a,t)} ũ\frac{∂η}{∂t} dxdt − ∫_{Q_{m}(a,t)}(ṽ · ∇η)ũdxdt

+ ∫_{Q_{m}(a,t)} ∇β(ũ) · ∇ηdxdt + n_{0} ∫_{Σ_{m}(a,t)} β(ũ)ηdΓ_{m}(t)dt
= ∫_{Q_{m}(a,t)} fηdxdt + ∫_{Σ_{m}(a,t)} qηdΓ_{m}(t)dt + ∫_{Ω_{m}(a)} ũ_{0}η(a)dx,
for all t ∈ [a, a + T_{0}] and η ∈ H^{1}(Q_{m}(a, a + T_{0})).

[B] (cf. [6]) We have a sort of continuous dependence of solutions ũ, obtained by the above results, upon the convection vector field ṽ. Now, assume that {ṽ_{n}} is a bounded sequence of vector fields in L^{∞}(0, T; H) and L^{2}(0, T; V) such that

ṽ_{n} = v_{D} a.e. on Q \ Q_{m} for all n ∈ N,
ṽ_{n} → ṽ weakly- ∗ in L^{∞}(0, T; H) weakly in L^{2}(0, T; V) as n → ∞.

Let ũ_{n} be any solution of (SPC; ṽ_{n}, ũ_{0}) on [a, a + T_{0}], and let ũ is the weak-∗ limit of a subsequence {ũ_{n_{k}}} in L^{∞}(Q_{m}). Then, ũ is a solution of (SPC; ṽ, ũ_{0}) on [a, a + T_{0}]. Moreover the 0-extension of ũ_{n_{k}}(t) onto R^{3} weakly converges in L^{2}(R^{3}) to that of ũ(t) uniformly in t ∈ [a, a + T_{0}] and β(ũ_{n_{k}}) → β(ũ) in L^{2}((a, a + T_{0}) × R^{3}) as k → ∞.

3.2 Auxiliary results for Navier-Stokes equations

We now recall an existence result for the variational problem associated with the Navier-Stokes equation, which is applied to the construction of approximate solutions of (CZ)_{ε}, formulated in non-cylindrical domains.

The solvability for the Navier-Stokes equation in non-cylindrical domains was discussed by many authors, for example Fujita and Sauer [5], Inoue and Wakimoto [8], Inoue and Ôtani [9], Kenmochi [11], Morimoto [16], Ôtani and Yamada [18] and Yamada [24]. In the existence proofs of [5] and [11], one of main points is an extensive use of a compactness theorem of Aubin’s type [1] and its extension of Kenmochi [10].

[C](cf. [5], [11]) Given a ∈ (0, T) and T_{0} > 0 with a+T_{0} ≤ T, we consider the following variational problem associated with the Navier-Stokes equation in non-cylindrical domain Q_{m}(a, a + T_{0}). Let p̃ ∈ L^{∞}(Q) with p̃ ≥ 0 a.e. on Q, g̃ ∈ L^{2}(a, a + T_{0}; H) and w̃_{0} ∈ H. Then, there exists at least one function w̃ such that

(i) w̃ ∈ L^{∞}(a, a+T_{0}; H)∩L^{2}(a, a+T_{0}; V) with w̃ = 0 a.e. on Q(a, a+T_{0})\Q_{m}(a, a+T_{0}) and w̃ is weakly continuous from [a, a + T_{0}] into H.

(ii) w̃ satisfies the variational identity

− ∫_{a}^{a+T_{0}}(η′, w̃)_{H}dτ + ∫_{a}^{a+T_{0}} a(w̃, η)dτ − ∫_{a}^{a+T_{0}} b(τ; w̃, η, w̃)dτ + ∫_{a}^{a+T_{0}} c(τ; w̃, η)dτ

+ ∫_{a}^{a+T_{0}}(P_{L}(p̃w̃), η)_{H}dτ = ∫_{a}^{a+T_{0}}(g̃, η)_{H}dτ + (w̃_{0}, η(0))_{H} for all η ∈ W_{0}(a, a + T_{0}),

(3.3)

where for any a ∈ [0, T]

W_{0}(a, a + T_{0}) := \{η ∈ L^{4}(a, a + T_{0}; X); \begin{matrix}η′ ∈ L^{2}(a, a + T_{0}; H), \\ η(a + T_{0}, ·) = 0 a.e. on Ω, \\ η = 0 a.e. on \\ Q(a, a + T_{0}) \ Q_{m}(a, a + T_{0}) \end{matrix} \} .

(iii) The following inequality holds:

\frac{1}{2}|w̃(t)|_{H}^{2} + c_{2} ∫_{a}^{t} |w̃(τ)|_{V}^{2}dτ + ∫_{Q(a,t)} p̃|w̃|^{2}dxdτ ≤ \frac{1}{2}|w̃_{0}|_{H}^{2} + ∫_{a}^{t}(g̃, w̃)_{H}dτ,  (3.4)
for all t ∈ [a, a + T_{0}],

where c_{2} is a positive constant independent of w̃_{0}, g̃, p̃ and time interval [a, a + T_{0}].

We denote by (NS; p̃, g̃, w̃_{0}) on [a, a+T_{0}] the above variational problem associated with the Navier-Stokes equation on Q_{m}, and any function w̃ satisfying the above conditions (i)-(iii) is called a solution of (NS; p̃, g̃, w̃_{0}) on [a, a + T_{0}].

Remark 3.2. It is easy to check that (3.3) is equivalent to the following variational identity:

(w̃(t), η(t))_{H} − ∫_{a}^{t}(η′, w̃)_{H}dτ + ∫_{a}^{t} a(w̃, η)dτ − ∫_{a}^{t} b(τ; w̃, η, w̃)dτ
+ ∫_{a}^{t} c(τ; w̃, η)dτ + ∫_{a}^{t}(P_{L}(p̃w̃), η)_{H}dτ = ∫_{a}^{t}(g̃, η)_{H}dτ + (w̃_{0}, η(0))_{H}
for all t ∈ [a, a + T_{0}] and η ∈ W(a, a + T_{0}),

where for any a ∈ [0, T]

W(a, a + T_{0}) := \{η ∈ L^{4}(a, a + T_{0}; X); \begin{matrix}η′ ∈ L^{2}(a, a + T_{0}; H), \\ η = 0 a.e. on \\ Q(a, a + T_{0}) \ Q_{m}(a, a + T_{0}) \end{matrix} \} .

4 Approximate solutions

In this section, we first construct approximate solutions of our system (CZ)_{ε}, and then we prove Theorems 2.1 and 2.2 by discussing the convergence of approximate solutions.

4.1 Construction of approximate solutions

Let 0 = t_{0}^{N} < t_{1}^{N} < t_{2}^{N} < ⋯ < t_{N}^{N} = T, be the partition of [0, T] given by

t_{k}^{N} = kh_{N} for k = 0, 1, …, N with h_{N} = \frac{T}{N} .

We are now going to construct a sequence of approximate solutions by applying the existence results for (SPC; ṽ, ũ_{0}) and (NS; p̃, g̃, w̃_{0}) on each time interval [t_{k−1}^{N}, t_{k}^{N}] mentioned in the previous section. Let f, q, u_{0} and v_{0} be the data given for our original problem (CZ)_{ε}. Moreover for each s, t ∈ [0, T], Θ_{t,s}(·) be the C^{3}-diffeomorphism in Ω given by

Θ_{t,s}(x) = x(s, y(t, x)) for all x ∈ Ω,

where x(s, ·) is the inverse of y(s, ·); note that Θ_{t,s} maps Ω_{m}(t) onto Ω_{m}(s) for each s, t ∈ [0, T]. Now, for fixed positive parameters ε, δ ∈ (0, 1], let us define a set of functions {u_{εδ,k}^{N}, w_{εδ,k}^{N}}_{k=1}^{N}, in the following manner (1)-(4):

(1) w_{εδ,1}^{N} is a solution of (NS; p_{εδ,0}^{N}/δ, g̃_{εδ,0}^{N}, w_{0}) on [0, t_{1}^{N}], where

w_{0} := v_{0} − v_{D}(0),

p_{εδ,0}^{N}(t, x) := [ρ_{ε} ∗ (u_{0}(y(t, ·)) − \frac{L}{2})]^{−} (x),

and

g̃_{εδ,0}^{N}(t, x) := g(β(u_{0}(y(t, x))) for all (t, x) ∈ Q(0, t_{1}^{N});

(2) u_{εδ,1}^{N} is a solution of (SPC; v_{εδ,1}^{N}, u_{0}) on [0, t_{1}^{N}], where

v_{εδ,1}^{N} := w_{εδ,1}^{N} + v_{D} on Q(0, t_{1}^{N});

(3) for 2 ≤ k ≤ N, w_{εδ,k}^{N} is a solution of (NS; p_{εδ,k−1}^{N}/δ, g̃_{εδ,k−1}^{N}, w_{εδ,k−1}^{N}(t_{k−1}^{N})) on [t_{k−1}^{N}, t_{k}^{N}], where

p_{εδ,k−1}^{N}(t, x) := [ρ_{ε} ∗ (u_{εδ,k−1}^{N}(t − h_{N}, Θ_{t,t−h_{N}} (·)) − \frac{L}{2})]^{−} (x),

and

g̃_{εδ,k−1}^{N}(t, x) := g(β(u_{εδ,k−1}^{N}(t − h_{N}, Θ_{t,t−h_{N}} (x))) for all (t, x) ∈ Q(t_{k−1}^{N}, t_{k}^{N});

(4) for 2 ≤ k ≤ N, u_{εδ,k}^{N} is a solution of (SPC; v_{εδ,k}^{N}, u_{εδ,k−1}^{N}(t_{k−1}^{N})) on [t_{k−1}^{N}, t_{k}^{N}], where

v_{εδ,k}^{N} := w_{εδ,k}^{N} + v_{D} on Q(t_{k−1}^{N}, t_{k}^{N}).

Now, for each N ∈ N we define two functions u_{εδ}^{N} on Q_{m} and w_{εδ}^{N} on Q by

u_{εδ}^{N}(t, x) := u_{εδ,k}^{N}(t, x), if t ∈ [t_{k−1}^{N}, t_{k}^{N}) and x ∈ Ω_{m}(t),
w_{εδ}^{N}(t, x) := w_{εδ,k}^{N}(t, x) if t ∈ [t_{k−1}^{N}, t_{k}^{N}) and x ∈ Ω.

4.2 Auxiliary lemmas

On account of our construction of {u_{εδ,k}^{N}, w_{εδ,k}^{N}}, we see from Remarks 3.1 and 3.2 that

(a) u_{εδ}^{N} ∈ L^{∞}(Q_{m}), the 0-extension of u_{εδ}^{N}, denoted by u_{εδ}^{N} again, is weakly continuous from [0, T] into H, β(u_{εδ}^{N}(t)) ∈ H^{1}(Ω_{m}(t)) for a.e. t ∈ [0, T] and

∫_{0}^{T} |β(u_{εδ}^{N}(t))|_{H^{1}(Ω_{m}(t))}^{2}dt < +∞.

(b) w_{εδ}^{N} is weakly continuous from [0, T] into H, this implies that w_{εδ}^{N} ∈ L^{∞}(0, T; H), and w_{εδ}^{N} ∈ L^{2}(0, T; V).

(c) u_{εδ}^{N} satisfies the following variational identity:

− ∫_{Q_{m}} u_{εδ}^{N} \frac{∂η}{∂t} dxdt− ∫_{Q_{m}}(v_{εδ}^{N} ·∇η)u_{εδ}^{N}dxdt+ ∫_{Q_{m}} ∇β(u_{εδ}^{N})·∇ηdxdt+n_{0} ∫_{Σ_{m}} β(u_{εδ}^{N})ηdΓ_{m}(t)dt

= ∫_{Q_{m}} fηdxdt + ∫_{Σ_{m}} qηdΓ_{m}(t)dt + ∫_{Ω_{m0}} u_{0}η(0)dx for all η ∈ W,  (4.1)

where

v_{εδ}^{N} := w_{εδ}^{N} + v_{D}.

(d) w_{εδ}^{N} satisfies the following variational identity:

− ∫_{0}^{T} (η′, w_{εδ}^{N})_{H} dt + ∫_{0}^{T} a(w_{εδ}^{N}, η)dt − ∫_{0}^{T} b(t; w_{εδ}^{N}, η, w_{εδ}^{N})dt + ∫_{0}^{T} c(t; w_{εδ}^{N}, η)dt

+ \frac{1}{δ} ∫_{0}^{T} (P_{L}(p_{εδ}^{N}w_{εδ}^{N}), η)_{H}dt = ∫_{0}^{T} (g_{εδ}^{N}, η)_{H}dt + (w_{0}, η(0))_{H} for all η ∈ W_{0}(0, T), (4.2)

where

p_{εδ}^{N}(t, x) := \{ \begin{matrix}[ρ_{ε} ∗ (u_{0}(y(t, ·) − \frac{L}{2})]^{−} (x) & if (t, x) ∈ Q(0, t_{1}^{N}), \\ [ρ_{ε} ∗ (u_{εδ}^{N}(t − h_{N}, Θ_{t,t−h_{N}} (·)) − \frac{L}{2})]^{−} (x) & if (t, x) ∈ Q(t_{1}^{N}, T), \end{matrix}

and

g_{εδ}^{N}(t, x) := \{ \begin{matrix}g(β(u_{0}(y(t, x))) & if (t, x) ∈ Q(0, t_{1}^{N}), \\ g(β(u_{εδ}^{N}(t − h_{N}, Θ_{t,t−h_{N}} (x))) & if (t, x) ∈ Q(t_{1}^{N}, T). \end{matrix}

Furthermore, we have the uniform estimates for u_{εδ}^{N} and w_{εδ}^{N} with respect to ε, δ ∈ (0, 1] and N ∈ N which are given in the following lemmas.

Lemma 4.1. Put

M_{4} := max \{L, |f|_{L^{∞}(Q_{m})}, |\frac{q}{n_{0}} |_{L^{∞}(Σ_{m})} , |β(u_{0})|_{L^{∞}(Ω_{m0})}\} .

Then

|β(u_{εδ}^{N})|_{L^{∞}(Q_{m})} ≤ M_{4} (1 + \frac{T}{N} )^{N} < M_{4}e^{T} ,  (4.3)
|u_{εδ}^{N}|_{L^{∞}(Q_{m})} ≤ max \{\frac{M_{4}}{k_{s}} , \frac{M_{4}}{k_{ℓ}} + L\} e^{T} =: M_{5}  (4.4)
for all ε ∈ (0, 1], δ ∈ (0, 1] and N ∈ N,

and there is a positive constant M_{6} such that

∫_{0}^{T} |β(u_{εδ}^{N}(t))|_{H^{1}(Ω_{m}(t))}^{2}dt ≤ M_{6} for all ε ∈ (0, 1], δ ∈ (0, 1] and N ∈ N.  (4.5)

Moreover there is a positive constant M_{7} such that

sup_{0≤t≤T} |w_{εδ}^{N}(t)|_{H} ≤ M_{7}, |w_{εδ}^{N}|_{L^{2}(0,T;V)} ≤ M_{7}, \frac{1}{δ} ∫_{Q} p_{εδ}^{N}|w_{εδ}^{N}|^{2}dxdt ≤ M_{7},  (4.6)
for all ε ∈ (0, 1], δ ∈ (0, 1] and N ∈ N.

Proof. Estimate (4.3) is a consequence from the results [A], (iii)-(iv) with [a, a + T_{0}] = [t_{k−1}^{N}, t_{k}^{N}], of the previous section. In fact, with

M_{k}^{N} := max \{L, |f|_{L^{∞}(Q_{m})}, |\frac{q}{n_{0}} |_{L^{∞}(Σ_{m})} , |β(u_{εδ}^{N}(t_{k}^{N}))|_{L^{∞}(Ω_{m}(t_{k}^{N}))}\} ,

for k = 0, 1, ⋯, N − 1, we have

|β(u_{εδ}^{N})|_{L^{∞}(Q_{m}(t_{k}^{N} ,t_{k+1}^{N}))} ≤ M_{k}^{N} (1 + \frac{T}{N} ) ,

and

M_{k+1}^{N} ≤ M_{k}^{N} (1 + \frac{T}{N} ) ,

for all k = 0, 1, ⋯, N − 1. Therefore, M_{k}^{N} ≤ M_{4}e^{T} for all k = 0, 1, ⋯, N − 1. Estimate (4.3) immediately follows from these inequalities, and hence (4.4) does, too. Also, by (3.2) we get (4.5) from (4.4). Moreover by (4.3) or (4.4) functions g_{εδ}^{N} are uniformly bounded on Q_{m} with respect to N ∈ N, ε ∈ (0, 1] and δ ∈ (0, 1]. Since p_{εδ}^{N} are nonnegative, making use of (3.4) with [a, a + T_{0}] := [0, T], p := p_{εδ}^{N}/δ and g := g_{εδ}^{N} we obtain uniform estimates of the form (4.6). □

In the rest of this section we fix parameters ε, δ ∈ (0, 1]. On account of the uniform estimates in Lemma 4.1, there is a sequence {N_{n}} of positive integers with N_{n} ↑ +∞ (as n → +∞) such that

u_{εδ}^{N_{n}} → u_{εδ} weakly- ∗ in L^{∞}(Q_{m}),

as well as

w_{εδ}^{N_{n}} → w_{εδ} weakly in L^{2}(0, T; V) weakly- ∗ in L^{∞}(0, T; H) as n → +∞.

Now, put v_{εδ} := w_{εδ} + v_{D}. Then, by virtue of [B] in section 3, the 0-extensions of u_{εδ}^{N_{n}} and u_{εδ} onto (0, T) × R^{3}, denoted by the same notations, satisfy that

u_{εδ}^{N_{n}}(t) → u_{εδ}(t) weakly in L^{2}(R^{3}) and uniformly in t ∈ [0, T],  (4.7)

and

β(u_{εδ}^{N_{n}}) → β(u_{εδ}) in L^{2}((0, T) × R^{3}) as n → +∞.  (4.8)

Next, we observe from (4.2) with (4.3), (4.4) and (4.6) that {d/dtw_{εδ}^{N}} is bounded in L^{4/3}(0, T; X^{∗}) and {w_{εδ}^{N}}_{N∈N} is bounded in L^{2}(0, T; V). Since V ↪ H ⊂ X^{∗}, with compact injections, it follows from the Aubin’s compactness theorem (cf. [15], Chap.1, section 5) that {w_{εδ}^{N}} is relatively compact in L^{2}(0, T; H), whence

w_{εδ}^{N_{n}} → w_{εδ} in L^{2}(0, T; H) as n → ∞.  (4.9)

Furthermore we have:

Lemma 4.2. (i) Putting

ũ_{εδ}^{N_{n}}(t, x) = \{ \begin{matrix}u_{0}(y(t, x)) & if (t, x) ∈ Q_{m}(0, t_{1}^{N_{n}}) \\ u_{εδ}^{N_{n}}(t − h_{N_{n}}, Θ_{t,t−h_{Nn}}(x)) & if (t, x) ∈ Q_{m}(t_{1}^{N_{n}}, T), \end{matrix}

we have for their 0-extensions of ũ_{εδ}^{N_{n}} and u_{εδ}

ρ_{ε} ∗ ũ_{εδ}^{N_{n}} → ρ_{ε} ∗ u_{εδ} uniformly on [0, T] × R^{3},  (4.10)

p_{εδ}^{N_{n}} → p_{εδ} := [ρ_{ε} ∗ (u_{εδ} − \frac{L}{2})]^{−} uniformly on [0, T] × R^{3}  (4.11)

as n → ∞. (ii) g(β(u_{εδ}^{N_{n}})) → g(β(u_{εδ})) in L^{2}(0, T; H) as n → ∞. Proof. It is easy to see that ũ_{εδ}^{N_{n}}(t) weakly converges to u_{εδ}(t) in L^{2}(R^{3}) uniformly in t ∈ [0, T]. From this fact we obtain (4.10) and hence (4.11). Thus (i) holds. The convergence of (ii) immediately follows from (4.9), since g is Lipschitz continuous from R into R^{3}. □

Now it is easy to prove the following proposition.

Proposition 4.1. For any ε ∈ (0, 1] and δ ∈ (0, 1] the pair of functions {u_{εδ}, w_{εδ}} in L^{∞}(Q_{m}) × (L^{2}(0, T; V) ∩ L^{∞}(0, T; H)) satisfies the following (4.12)-(4.15):

− ∫_{Q_{m}} u_{εδ} \frac{∂η}{∂t} dxdt− ∫_{Q_{m}}(v_{εδ} ·∇η)u_{εδ}dxdt+ ∫_{Q_{m}} ∇β(u_{εδ})·∇ηdxdt+n_{0} ∫_{Σ_{m}} β(u_{εδ})ηdΓ_{m}(t)dt

= ∫_{Q_{m}} fηdxdt + ∫_{Σ_{m}} qηdΓ_{m}(t)dt + ∫_{Ω_{m0}} u_{0}η(0)dx for all η ∈ W,  (4.12)
− ∫_{0}^{T} (η′, w_{εδ})_{H} dt + ∫_{0}^{T} a(w_{εδ}, η)dt − ∫_{0}^{T} b(t; w_{εδ}, η, w_{εδ})dt + ∫_{0}^{T} c(t; w_{εδ}, η)dt

+ \frac{1}{δ} ∫_{0}^{T} (P_{L}(p_{εδ}w_{εδ}), η)_{H}dt = ∫_{0}^{T} (g(β(u_{εδ})), η)_{H}dt + (w_{0}, η(0))_{H} for all η ∈ W_{0}(0, T),

(4.13)

and for a positive constant M_{8}, independent of ε ∈ (0, 1] and δ ∈ (0, 1],

|β(u_{εδ})|_{L^{∞}(Q_{m})} ≤ M_{8}, |u_{εδ}|_{L^{∞}(Q_{m})} ≤ M_{8}, ∫_{0}^{T} |β(u_{εδ}(t))|_{H^{1}(Ω_{m}(t))}^{2}dt ≤ M_{8}, (4.14)

sup_{0≤t≤T} |w_{εδ}(t)|_{H} ≤ M_{8}, |w_{εδ}|_{L^{2}(0,T;V)} ≤ M_{8}, \frac{1}{δ} ∫_{Q} p_{εδ}|w_{εδ}|^{2}dxdt ≤ M_{8}.  (4.15)

Proof. The variational identities (4.12) and (4.13) are immediately obtained by letting n → ∞ in (4.1) and (4.2) with the help of convergences (4.7)-(4.9) as well as those in Lemma 4.2. Also, the uniform estimates (4.14) and (4.15) immediately follow from Lemma 4.1; in fact, it is enough to take

M_{8} = max{M_{4}e^{T} , M_{5}, M_{6}, M_{7}}.
□

5 Proofs of main theorems

In this section, by discussing the convergence in δ → 0 and ε → 0, we give proofs of Theorems 2.1 and 2.2.

Proof of Theorem 2.1. Fixing ε ∈ (0, 1], we discuss the convergence in δ. Let {u_{εδ}, w_{εδ}} be the same family as constructed in Proposition 4.1. By the uniform estimates (4.14) and (4.15), there are a sequence {δ_{n}} ⊂ (0, 1] converging to 0 and functions u_{ε} ∈ L^{∞}(Q_{m}) and w_{ε} ∈ L^{2}(0, T; V) ∩ L^{∞}(0, T; H) such that

u_{εδ_{n}} → u_{ε} weakly- ∗ in L^{∞}(Q_{m}),
w_{εδ_{n}} → w_{ε} weakly in L^{2}(0, T; V) as n → +∞.

We denote the 0-extensions of u_{εδ_{n}} and u_{ε} onto [0, T] × R^{3} by the same notation. In this case, just as in the previous section, it follows from the result in [B] of section 3 that

u_{εδ_{n}}(t) → u_{ε}(t) weakly in L^{2}(R^{3}) and uniformly in t ∈ [0, T],  (5.1)
β(u_{εδ_{n}}) → β(u_{ε}) in L^{2}((0, T) × R^{3}) as n → +∞,  (5.2)

and

− ∫_{Q_{m}} u_{ε} \frac{∂η}{∂t} dxdt − ∫_{Q_{m}}(v_{ε} · ∇η)u_{ε}dxdt + ∫_{Q_{m}} ∇β(u_{ε}) · ∇ηdxdt + n_{0} ∫_{Σ_{m}} β(u_{ε})ηdΓ_{m}(t)dt

= ∫_{Q_{m}} fηdxdt + ∫_{Σ_{m}} qηdΓ_{m}(t)dt + ∫_{Ω_{m0}} u_{0}η(0)dx for all η ∈ W,  (5.3)

where v_{ε} := w_{ε} + v_{D}. Moreover, by (5.1),

ρ_{ε} ∗ u_{εδ_{n}} → ρ_{ε} ∗ u_{ε} uniformly on [0, T] × R^{3},

and hence

p_{εδ_{n}} → p_{ε} := [ρ_{ε} ∗ (u_{ε} − \frac{L}{2})]^{−} uniformly on [0, T] × R^{3} as n → +∞.  (5.4)

Now, put

Q_{ℓ,ε}(u_{ε}) := \{(t, x) ∈ Q_{m}; (ρ_{ε} ∗ u_{ε})(t, x) > \frac{L}{2}\} (= ⋃_{t∈(0,T)}\{t\} × Ω_{ℓ,ε}(u_{ε}; t) ) .

Clearly Q_{ℓ,ε}(u_{ε}) is an open subset of Q_{m}, which may be empty. Let (s_{1}, s_{2}) × ω be any relatively compact and open cylindrical subdomain of Q_{ℓ,ε}(u_{ε}). Then, it follows for all n sufficiently large that w_{εδ_{n}} satisfies

− ∫_{0}^{T} (η′, w_{εδ_{n}})_{H} dt + ∫_{0}^{T} a(w_{εδ_{n}}, η)dt − ∫_{0}^{T} b(t; w_{εδ_{n}}, η, w_{εδ_{n}})dt  (5.5)

+ ∫_{0}^{T} c(t; w_{εδ_{n}}, η)dt = ∫_{0}^{T} (g(β(u_{εδ_{n}})), η)_{H}dt,

where the test function η is taken as follows

η ∈ W_{0}(0, T) with suppη ⊂ (s_{1}, s_{2}) × ω.  (5.6)

In fact, on account of (5.3) we have p_{εδ_{n}}η = 0 on [0, T] × R^{3} for all large n, so that (5.5) is derived from (4.13) for any η satisfying (5.6). This together with estimates (4.14) and (4.15) shows that {d/dtw_{εδ_{n}}} is bounded in L^{4/3}(s_{1}, s_{2}; W_{σ}^{−1,4/3}(ω)) and {w_{εδ_{n}}} is bounded in L^{2}(s_{1}, s_{2}; V_{1}(ω)), where V_{1}(ω) is the closure of {z ∈ C^{∞}(ω); divz = 0 in ω} with respect to the topology of H^{1}(ω). Since V_{1}(ω) ↪ L_{σ}^{2}(ω) ⊂ W_{σ}^{−1,4/3}(ω), with compact injections, it follows from Aubin’s compactness result again that {w_{εδ_{n}}} is relatively compact in L^{2}(s_{1}, s_{2}; L_{σ}^{2}(ω)), which implies

w_{εδ_{n}} → w_{ε} in L^{2}(s_{1}, s_{2}; L_{σ}^{2}(ω)) as n → +∞.

Noting that this is valid for every relatively compact and open cylindrical subdomain of the form (s_{1}, s_{2}) × ω in Q_{ℓ,ε}(u_{ε}), we can conclude that

w_{εδ_{n}} → w_{ε} in L_{loc}^{2}(Q_{ℓ,ε}(u_{ε})) as n → +∞,  (5.7)

since any compact subset of Q_{ℓ,ε}(u_{ε}) can be covered by a finite number of subdomains of the form (s_{1}, s_{2}) × ω. Furthermore, letting n → ∞ in (4.13) for any η ∈ W_{0}(0, T) with supp η(t) ⊂ Ω_{ℓ,ε}(u_{ε}; t) for all t ∈ [0, T], we obtain with the help of the convergences (5.1), (5.2), (5.4) and (5.7) that

− ∫_{0}^{T} (η′, w_{ε})_{H} dt + ∫_{0}^{T} a(w_{ε}, η)dt − ∫_{0}^{T} b(t; w_{ε}, η, w_{ε})dt + ∫_{0}^{T} c(t; w_{ε}, η)dt

= ∫_{0}^{T} (g(β(u_{ε})), η)_{H}dt + (w_{0}, η(0))_{H} for all η ∈ W_{ε}(u_{ε}).

Also, we have uniform estimates

|β(u_{ε})|_{L^{∞}(Q_{m})} ≤ M_{8}, |u_{ε}|_{L^{∞}(Q_{m})} ≤ M_{8}, ∫_{0}^{T} |β(u_{ε}(t))|_{H^{1}(Ω_{m}(t))}^{2}dt ≤ M_{8},

sup_{0≤t≤T} |w_{ε}(t)|_{H} ≤ M_{8}, |w_{ε}|_{L^{2}(0,T;V)} ≤ M_{8},

for all ε ∈ (0, 1], where M_{8} is the same positive constants in Proposition 4.1. Especially, the last estimate of (4.15) implies that

[ρ_{ε} ∗ (u_{ε} − \frac{L}{2})]^{−} |w_{ε}|^{2} = 0 a.e. on Q,

namely

w_{ε} = 0 a.e. on Q_{s,ε}(u_{ε}).

Thus the pair {u_{ε}, w_{ε}} is a solution of (CZ)_{ε} such that (2.9) and (2.10) hold for R_{0} = R_{1} = M_{8}. □

Remark 5.1. As is easily checked, the similar statements to Theorems 2.1 and 2.2 are valid, when the mollifier ρ_{ε} is replaced by any nonnegative smooth kernel α(t, x) with compact support and the convolution ρ_{ε} ∗ (u − L/2) is replaced by

α ∗ (u − \frac{L}{2}) (t, x) := ∫_{0}^{t} ∫_{R^{3}} α(t − τ, x − ξ) (u(τ, ξ) − \frac{L}{2}) dξdτ.

Proof of Theorem 2.2. We discuss finally the convergence in ε. Let {u_{ε}, w_{ε}} be the family constructed in Theorem 2.1. By the uniform estimates (2.7) and (2.8) there are a sequence {ε_{n}} converging to 0 and functions u ∈ L^{∞}(Q_{m}) and w ∈ L^{∞}(0, T; H) ∩ L^{2}(0, T; V) such that

u_{ε_{n}} → u weakly* ∗ in L^{∞}(Q_{m}), (hence, weakly in L^{2}(Q_{m}) ),
w_{ε_{n}} → w weakly in L^{2}(0, T; V), weakly- ∗ in L^{∞}(0, T; H) as n → +∞.

By applying the result [B] in section 3, we see that

u_{ε_{n}}(t) → u(t) weakly in L^{2}(R^{3}) and uniformly in t ∈ [0, T],

as well as

β(u_{ε_{n}}) → β(u) in L^{2}((0, T) × R^{3}) as n → +∞,

and the limit u satisfies

− ∫_{Q_{m}} u\frac{∂η}{∂t} dxdt − ∫_{Q_{m}}(v · ∇η)udxdt + ∫_{Q_{m}} ∇β(u) · ∇ηdxdt + n_{0} ∫_{Σ_{m}} β(u)ηdΓ_{m}(t)dt

= ∫_{Q_{m}} fηdxdt + ∫_{Σ_{m}} qηdΓ_{m}(t)dt + ∫_{Ω_{m0}} u_{0}η(0)dx for all η ∈ W,

where v := w + v_{D}. Thus we obtain the conclusion. □

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Technical Reports of Mathematical Sciences

Chiba University, vol. 19 (2003)

[1] Numerical radius norm for bouded module maps and Schur multipliers, by T. Itoh and M. Nagisa.

[2] Some remarks on the Picard curves over a finite field, by Y. Takizawa

[3] Hypothesis testing based on the maximum of two statistics from weighted and unweighted estimating equations, by T. Emura and H. Tsukuma.

[4] Degenerate parabolic equations with convection in non-cylindrical domains, by T. Fukao and N. Kenmochi.

[5] Stefan problems with convection governed by Navier-Stokes equations, by T. Fukao and N. Kenmochi.

Correspondences concerning these Technical Reports should be addressed to Koji Nishida Graduate School of Science and Technology Chiba University Yayoi-cho 1-33, Inage-ku, Chiba-shi, 263-8522 JAPAN. e-mail: nishida@math.s.chiba-u.ac.jp