Two-Phase Stefan Problems in Non-Cylindrical Domains T. FUKAO, N. KENMOCHI and I. PAWLOW Technical Reports of Mathematical Sciences Chiba University, Vol. 17 (2001), No. 3 TWO-PHASE STEFAN PROBLEMS IN NON-CYLINDRICAL DOMAINS

T. Fukao, N. Kenmochi and I. Pawlow

Abstract. In this paper we discuss a two-phase Stefan problem in a non-cylindrical (time-dependent) domain. This work is motivated by the phase change arising in the Czochralski crystal growth process. The time-dependence of domain is a mathematical description of the situation in which the material domain changes its shape with time by the crystal growth. We consider the so-called enthalpy formulation for it and give its solvability, assuming that the time-dependence of the material domain is prescribed and smooth enough in time. Our main idea is to apply the theory of quasi-linear equations of parabolic type.

1. Introduction

Czochralski pulling method is widely used for the production of a column of single silicon crystal from the melt. The idea of pulling method due to Czochralski is quite simple. A crucible, equipped with heating system, contains the melt substance and a pul-rod with seed crystal, which moves vertically and rotates flexibly, is positioned above the crucible (see Fig.1). The rod is dipped into the melt, and then lifted slowly with an appropriate speed v_{p} so that a meniscus surface is formed below the seed crystal and the melt attached to the crystal solidifies continuously. By controlling some thermal situations in the process one obtains the growth of a single crystal column with a desired radius as well as a desired growth pattern of the solid-liquid interface and temperature pattern in the crystal in order to improve the crystal quality.

[figure: Rod; Seed; Gas; Crystal; Trijunction; Meniscus; Crucible (Heater); Melt; ν; n; Ω_{ℓ}(t); S(t); Ω_{s}(t); S(t); Γ(t); Fig.1; Fig.2]

In such a model of crystal growth the shape of crystal is determined by three kinetic equations of three interfaces between crystal-melt, melt-gas and gas-crystal. But, in this paper we suppose that the crystal radius is controlled to be constant and the trijunction curve on which three interfaces meet is prescribed, too. This might be designed by a good choice of the pulling velocity. As a consequence we may assume that the movement of the material domain is prescribed.

We use the following notation (see Fig.2): For 0 < T < ∞ and t ∈ [0, T], Ω_{ℓ}(t) : liquid (melt) region, Ω_{s}(t) : solid (crystal) region, S(t) : solid-liquid interface, Ω(t) := Ω_{ℓ}(t) ∪ Ω_{s}(t) ∪ S(t), Γ(t) := ∂Ω(t), ν = ν(t, x): 3-dimensional unit vector normal to Γ(t) at x ∈ Γ(t), n = n(t, x): 3-dimensional unit vector normal to S(t) at x ∈ S(t),

Q := ⋃_{t∈(0,T)}{t} × Ω(t),

Σ := ⋃_{t∈(0,T)}{t} × Γ(t),

S := ⋃_{t∈(0,T)}{t} × S(t).

Next, we denote by v_{Σ} := v_{Σ}(t, x) the normal speed of Γ(t) at (t, x) ∈ Σ. With this v_{Σ} the 4-dimensional unit vector outward normal to Σ at each (t, x) ∈ Σ is given by

ν⃗ := (ν⃗_{t}, ν⃗_{x}) = \frac{1}{(|v_{Σ}|^{2} + 1)^{\frac{1}{2}}} (−v_{Σ}, ν).

Similarly, with the normal speed v_{S} := v_{S}(t, x) of S(t) at (t, x) ∈ S, the 4-dimensional unit vector normal to S, pointing to the liquid region, is given by

n⃗ := (n⃗_{t}, n⃗_{x}) = \frac{1}{(|v_{S}|^{2} + 1)^{\frac{1}{2}}} (−v_{S}, n).

It is easily understood that by the crystal growth the shape of material domain Ω(t) changes with time and it yields a 3-dimensional convective vector field v := v(t, x) in Q. The determination of v is also one of the important questions in the mathematical modeling of the Czochralski crystal growth process. It is reasonable to postulate that v is nothing but the pulling velocity v_{p} in the crystal and may be a solution of the incompressible Navier-Stokes (or simply Stokes) equation in the melt (see Crowley [1], DiBenedetto and O’Leary [3]). Nevertheless, in this paper, we assume that the convective field v is prescribed, too, satisfying that

divv = 0 in Ω(t), 0 < t < T,  (1.1)
v · ν = v_{Σ} on Γ(t), 0 < t < T.  (1.2)

Now, from the usual energy balance lows we derive the following system to determine the temperature field θ := θ(t, x) and interface S(t); note that θ(t, x) is the solution of a Stefan problem with prescribed convection v formulated in the non-cylindrical domain Q,

(SPC) \{ \begin{matrix}θ_{t} − c_{ℓ}Δθ + v · ∇θ = f in Q_{ℓ} := ⋃_{t∈(0,T)}\{t\} × Ω_{ℓ}(t),  (1.3) \\ θ_{t} − c_{s}Δθ + v · ∇θ = f in Q_{s} := ⋃_{t∈(0,T)}\{t\} × Ω_{s}(t),  (1.4) \\ θ = 0, (c_{ℓ} \frac{∂θ}{∂n} − c_{s} \frac{∂θ}{∂n}) = L(v · n − v_{S}) on S,  (1.5) \\ c_{ℓ} \frac{∂θ}{∂ν} + n_{0}c_{ℓ}θ = p on Σ_{ℓ} := ⋃_{t∈(0,T)}\{t\} × \{∂Ω_{ℓ}(t) \ S(t)\},  (1.6) \\ c_{s} \frac{∂θ}{∂ν} + n_{0}c_{s}θ = p on Σ_{s} := ⋃_{t∈(0,T)}\{t\} × \{∂Ω_{s}(t) \ S(t)\},  (1.7) \\ θ(0, ·) = θ_{0} on Ω(0), S(0) = S_{0},  (1.8) \end{matrix}

where we suppose that the phase change temperature is 0 for simplicity; c_{ℓ}, c_{s} and L are positive constants which are the heat conductivities and latent heat, respectively; f is a given heat source on Q, p is a boundary datum prescribed on Σ and n_{0} is a positive constant; θ_{0} is the initial temperature on Ω(0) and S_{0} is the initial location of the solidliquid interface, satisfying that

θ_{0} > 0 on Ω_{ℓ}(0), θ_{0} < 0 on Ω_{s}(0), θ_{0} = 0 on S_{0}.  (1.9)

As is well known, by using the enthalpy we reformulate this problem as a weak variational form. In this paper we prove its well-posedness.

2. Weak formulation

The enthalpy u is defined as follows:

u := \{ \begin{matrix}θ + L & if θ > 0, \\ [0, L] & if θ = 0, \\ θ & if θ < 0. \end{matrix}

Moreover we define a function β : R → R by

β(r) := \{ \begin{matrix}c_{s}r & if & r < 0, \\ 0 & if & 0 ≤ r ≤ L, \\ c_{ℓ}(r − L) & if & r > L. \end{matrix}

Then β is a non-decreasing Lipschitz continuous function on R, and its Lipschitz constant is L_{β} := max{c_{ℓ}, c_{s}}.

By using the enthalpy u our problem (SPC) is reformulated as an initial-boundary value problem for a degenerate parabolic equation of the following form

(E) \{ \begin{matrix}u_{t} − Δβ(u) + v · ∇u = f in Q, \\ \frac{∂β(u)}{∂ν} + n_{0}β(u) = p on Σ, \\ u(0) = u_{0} on Ω(0), \end{matrix}

where u_{0} := θ_{0} + Lχ_{Ω_{ℓ}(0)} with the characteristic function χ_{Ω_{ℓ}(0)} of Ω_{ℓ}(0). In fact, multiply equations (1.3) and (1.4) by any test function η ∈ C^{2}(Q̄) with η = 0 on Ω(T), and then integrate them over Q_{ℓ} and Q_{s}, respectively, and add these two resultants. Then, with the help of the Green-Stokes’ formula and (1.1), (1.2), (1.5), (1.6) and (1.7) as well as the relations dΣ = (|v_{Σ}|^{2} + 1)^{\frac{1}{2}} dΓ(t)dt and dS = (|v_{S}|^{2} + 1)^{\frac{1}{2}} dS(t)dt, we arrive at the following variational identity

− ∫_{Q} uη_{t}dxdt + ∫_{Q} ∇β(u) · ∇ηdxdt − ∫_{Σ} \frac{∂β(u)}{∂ν}ηdΓ(t)dt − ∫_{Q} u(v · ∇η)dxdt

= ∫_{Q} fηdxdt + ∫_{Ω(0)} u_{0}η(0)dx for all η ∈ C^{2}(Q̄), η = 0 on Ω(T).  (2.1)

Next, in order to consider a weak formulation of the boundary conditions, for each t ∈ [0, T], we take a harmonic function g(t, ·) such that

\{ \begin{matrix}−Δg(t) = 0 in Ω(t), \\ \frac{∂g(t)}{∂ν} + n_{0}g(t) = p(t) on Γ(t), \end{matrix}

in fact, g(t) ∈ H^{1}(Ω(t)) is a unique solution of the variational problem

∫_{Ω(t)} ∇g(t) · ∇ξdx + n_{0} ∫_{Γ(t)} g(t)ξdΓ(t) = ∫_{Γ(t)} p(t)ξdΓ(t) for all ξ ∈ C^{2}(\overline{Ω(t)}).

Also, we define the class W of test functions as follows:

W := {w ∈ H^{1}(Q); w = 0 on Ω(T) (in the trace sense)}.

Then (2.1) can be rewritten in the form

− ∫_{Q} uw_{t}dxdt + ∫_{Q} ∇(β(u) − g) · ∇wdxdt + ∫_{Σ} n_{0}(β(u) − g)wdΓ(t)dt − ∫_{Q} u(v · ∇η)dxdt

= ∫_{Q} fwdxdt + ∫_{Ω(0)} u_{0}w(0, ·)dx for all w ∈ W,  (2.2)

and as usual, this is regarded as a weak formulation of (E).

As to the solvability of two-phase Stefan problems without convection in cylindrical domains, the time-dependent subdifferential operator theory was skillfully applied by Damlamian [2]. The case of non-cylindrical domains was treated by Kenmochi and Pawlow [7] and only the existence result was there obtained, but the uniqueness question has been left open.

Now we formulate our main result. First of all we define the weak solution of our problem.

Definition 2.1 u is called a weak solution of (SPC) if u ∈ L^{2}(Q), β(u(t)) ∈ H^{1}(Ω(t)) for a.e. t ∈ [0, T] with

∫_{0}^{T} |β(u)|_{H^{1}(Ω(t))}^{2}dt < ∞,

u(t, ·) ∈ L^{2}(Ω(t)) for all t ∈ [0, T], the function

t ⟼ ∫_{Ω(t)} u(t, x)ξ(x)dx is continuous on [0, T] for each ξ ∈ L_{loc}^{2}(R^{3}),

and u satisfies the variational identity (2.2).

We suppose that the material domain Ω(t) depends smoothly on time t in the sense that there is a C^{3}-diffeomorphism y = X(t, x) from Q̄ onto Q̄_{0}, with Q_{0} := (0, T) × Ω(0), satisfying properties

(1) X(t, ·) := (X_{1}(t, x), X_{2}(t, x), X_{3}(t, x)) maps \overline{Ω(t)} onto \overline{Ω(0)} for all t ∈ [0, T];

(2) X(0, ·) = I (identity) on \overline{Ω(0)}.

We use the following notation:

Ω_{0} := Ω(0), Γ_{0} := ∂Ω(0), Σ_{0} := (0, T) × Γ_{0}, y = (y_{1}, y_{2}, y_{3}) ∈ \overline{Ω_{0}};

and the inverse of y = X(t, x) is denoted by x = Y (t, y) := (Y_{1}(t, y), Y_{2}(t, y), Y_{3}(t, y)).

Under some assumptions on the data v, f, p and u_{0}, we prove: Theorem 2.1 Assume that f ∈ H^{1}(Q), p ∈ C^{1}(Σ̄), u_{0} ∈ L^{2}(Ω(0)) and β(u_{0}) ∈ H^{1}(Ω(0)). Also, assume that v ∈ C^{1}(Q̄)^{3} and (1.1)-(1.2) are satisfied. Then there is one and only one weak solution u of (SPC).

We give the sketch of the proof of Theorem 2.1 in the rest of this paper. For the detail proof see the forthcoming paper Fukao, Kenmochi and Pawlow [4].

3. Regular approximation for (SPC)

In this section, let us consider an approximate problem (SPC)_{δ}, with parameter δ ∈ (0, 1], for (SPC):

(SPC)_{δ} \{ \begin{matrix}u_{δ,t} − Δβ_{δ}(u_{δ}) + v · ∇u_{δ} = f_{δ} in Q,  (3.1) \\ \frac{∂(β_{δ}(u_{δ}))}{∂ν} + n_{0}β_{δ}(u_{δ}) = p_{δ} on Σ,  (3.2) \\ u_{δ}(0) = u_{0,δ} on Ω(0),  (3.3) \end{matrix}

where β_{δ}, f_{δ}, p_{δ} and u_{δ,0} are regular approximations of β, f, p and u_{0}, respectively, as follows.

(1) β_{δ} is a smooth, increasing and Lipschitz continuous function on R such that

δ ≤ β′_{δ}(r) (= \frac{d}{dr}β_{δ}(r)) ≤ C_{0} for all r ∈ R,

for a positive constant C_{0}, and such that

β_{δ} → β uniformly on R as δ → 0;

we put β̂_{δ}(r) := ∫_{0}^{r} β_{δ}(s)ds as well as β̂(r) := ∫_{0}^{r} β(s)ds for all r ∈ R.

(2) f_{δ} is a smooth function on Q̄ such that

f_{δ} → f in H^{1}(Q) as δ → 0.

(3) p_{δ} is a smooth function on Σ̄ such that

p_{δ} → p in C^{1}(Σ̄) as δ → 0.

(4) u_{0,δ} is a smooth function on \overline{Ω(0)} such that u_{0,δ} → u_{0} in L^{2}(Ω(0)), β_{δ}(u_{0,δ}) → β(u_{0}) in H^{1}(Ω(0)) as δ → 0 and the compatibility condition

\frac{∂β_{δ}(u_{0,δ})}{∂ν} + n_{0}β_{δ}(u_{0,δ}) = p_{δ} on \overline{Ω(0)}  (3.4)

holds.

We give first an existence-uniqueness result for the approximate problem (SPC)_{δ}.

Lemma 3.1 (SPC)_{δ} has one and only one solution u_{δ} such that u_{δ} and all the derivatives u_{δ,t}, u_{δ,x_{i}}, u_{δ,x_{i}x_{k}} and u_{δ,tx_{i}}, i, k = 1, 2, 3, are Hölder continuous on Q̄. Proof. By y = X(t, x), we transform (SPC)_{δ} to a problem (\overline{SPC})_{δ} formulated in the cylindrical domain Q_{0}:

(\overline{SPC})_{δ} \{ \begin{matrix}ū_{δ,t} − ∑_{i,k=1}^{3} \frac{∂}{∂y_{i}} (a_{ik} \frac{∂}{∂y_{k}} β_{δ}(ū_{δ})) + w_{1} · ∇β_{δ}(ū_{δ}) + w_{2} · ∇ū_{δ} = f̄_{δ} in Q_{0}, (3.5) \\ \frac{∂(β_{δ}(ū_{δ}))}{∂ν_{A}} + n_{0}β_{δ}(ū_{δ}) = p̄_{δ} on Σ_{0},  (3.6) \\ ū_{δ}(0) = u_{0,δ} on Ω(0),  (3.7) \end{matrix}

where ū_{δ}(t, y) := u_{δ}(t, Y (t, y)), f̄_{δ}(t, y) := f_{δ}(t, Y (t, y)), p̄_{δ}(t, y) := p_{δ}(t, Y (t, y)),

a_{ik}(t, y) := ∑_{j=1}^{3} \frac{∂X_{i}}{∂x_{j}} \frac{∂X_{k}}{∂x_{j}} , i, k = 1, 2, 3,

w_{1} := (w_{11}, w_{12}, w_{13}) with w_{1k} := ∑_{i,j=1}^{3} \frac{∂}{∂y_{j}} (\frac{∂X_{j}}{∂x_{i}} ) \frac{∂X_{k}}{∂x_{i}} , k = 1, 2, 3,

w_{2} := \frac{∂X}{∂t} + vB with B = ( \begin{matrix}\frac{∂X_{1}}{∂x_{1}} & \frac{∂X_{2}}{∂x_{1}} & \frac{∂X_{3}}{∂x_{1}} \\ \frac{∂X_{1}}{∂x_{2}} & \frac{∂X_{2}}{∂x_{2}} & \frac{∂X_{3}}{∂x_{2}} \\ \frac{∂X_{1}}{∂x_{3}} & \frac{∂X_{2}}{∂x_{3}} & \frac{∂X_{3}}{∂x_{3}} \end{matrix} )

and

\frac{∂(·)}{∂ν_{A}} := ∑_{i,k=1}^{3} a_{ik} \frac{∂(·)}{∂y_{i}} ν̄_{k} on Γ_{0},

where ν̄ = (ν̄_{1}, ν̄_{2}, ν̄_{3}) is the unit outward normal vector to Γ_{0}.

Since X(0, ·) = I on Ω̄_{0}, the matrix a_{ik}(0, y) is the unit on Ω̄_{0} and hence a_{ik}(t, y) is strictly positive definite on Ω̄_{0} for t ∈ [0, T′] with a certain positive T′(≤ T). Therefore (\overline{SPC})_{δ} is (uniformly) parabolic quasi-linear equation with smooth coefficients on Q_{0}(T′) := (0, T′) × Ω_{0}, and by (3.4) the compatibility condition for initial and boundary data is satisfied. Now, apply the general existence-uniqueness theorem due to Ladyženskaja, Solonnikov and Ural’ceva [8;Chapter 5, section 7] to (\overline{SPC})_{δ}. Then we see that (\overline{SPC})_{δ} has a unique solution ū_{δ} in the Hölder space H^{2+α,1+α/2}(\overline{Q_{0}(T′)}) for a certain exponent α ∈ (0, 1). It is also easy to check that u_{δ}(t, x) := ū_{δ}(t, X(t, x)) is a solution of (SPC)_{δ} on Q(T′) := ⋃_{t∈(0,T′)}{t} × Ω(t), satisfying the required regularities. If T′ < T, then the solution u_{δ} can be extended beyond time T′ by repeating the same argument as above with initial time T′. Finally we can construct a unique solution u_{δ} of (SPC)_{δ} on Q in the Hölder class. □

Lemma 3.2 (Uniform estimate) There exists a positive constant M_{0}, independent of parameter δ ∈ (0, 1], such that

sup_{t∈[0,T]} |u_{δ}(t)|_{L^{2}(Ω(t))}^{2} + sup_{t∈[0,T]} |β_{δ}(u_{δ}(t))|_{H^{1}(Ω(t))}^{2} + ∫_{Q} |\frac{∂}{∂t}β_{δ}(u_{δ})|^{2} dxdt ≤ M_{0}  (3.8)
for all δ ∈ (0, 1].

Proof. We use essentially conditions (1.1) and (1.2) in order to get the uniformly estimates (3.8).

First, multiplying (3.1) by β_{δ}(u_{δ}) and integrating over Q(t) := ⋃_{s∈(0,t)}{s} × Ω(s), we have by (1.1) and (1.2)

∫_{Ω(t)} β̂_{δ}(u_{δ}(t))dx + ∫_{Q(t)} |∇β_{δ}(u_{δ})|^{2}dxds + n_{0} ∫_{0}^{t} ∫_{Γ(s)} |β_{δ}(u_{δ})|^{2}dΓ(s)ds

= ∫_{Q(t)} f_{δ}β_{δ}(u_{δ})dxds + ∫_{0}^{t} ∫_{Γ(s)} p_{δ}β_{δ}(u_{δ})dΓ(s)ds + ∫_{Ω(0)} β̂_{δ}(u_{0,δ})dx  (3.9)
for all t ∈ [0, T].

From (3.9) we obtain a uniform estimate of the form

sup_{t∈[0,T]} |u_{δ}(t)|_{L^{2}(Ω(t))}^{2} + ∫_{Q} |∇β_{δ}(u_{δ})|^{2}dxdt ≤ M_{1}  (3.10)

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

Next, just as (3.10), multiplying (3.1) by u_{δ}, we obtain a uniform estimate of the form

∫_{Q} β′_{δ}(u_{δ})|∇u_{δ}|^{2}dxdt ≤ M_{2}  (3.11)

for a positive constant M_{2}, independent of δ ∈ (0, 1].

The required estimate for ∂β_{δ}(u_{δ})/∂t is obtained from that of the solution ū_{δ} of (\overline{SPC})_{δ}. In fact, multiplying (3.5) by ∂β_{δ}(ū_{δ})/∂t and integrating the resultant over Q_{0}(t) := (0, t)× Ω_{0}, we have

∫_{Q_{0}(t)} β′_{δ}(ū_{δ})|ū_{δ,s}|^{2}dyds − ∑_{i,k=1}^{3} ∫_{Q_{0}(t)} \frac{∂}{∂y_{k}} (a_{ik} \frac{∂β_{δ}(ū_{δ})}{∂y_{i}}) \frac{∂β_{δ}(ū_{δ})}{∂s}dyds

+ ∫_{Q_{0}(t)}(w_{1} · ∇β_{δ}(ū_{δ}))β′_{δ}(ū_{δ})ū_{δ,s}dyds + ∫_{Q_{0}(t)}(w_{2} · ∇ū_{δ})β′_{δ}(ū_{δ})ū_{δ,s}dyds  (3.12)

= ∫_{0}^{t} ∫_{Γ_{0}} p̄_{δ} \frac{∂β_{δ}(ū_{δ})}{∂s}dΓ_{0}ds + ∫_{Q_{0}(t)} f̄_{δ} \frac{∂β_{δ}(ū_{δ})}{∂s}dyds

for all t ∈ [0, T].

Here, for the time-dependent convex functional

Φ_{δ}(t; v) := \frac{1}{2} ∑_{i,k=1}^{3} ∫_{Ω_{0}} a_{ik}(t, y)\frac{∂v}{∂y_{i}} \frac{∂v}{∂y_{k}} dy + \frac{n_{0}}{2} ∫_{Γ_{0}} |v|^{2}dΓ_{0} − ∫_{Γ_{0}} p̄_{δ}(t, y)vdΓ_{0}

for all v ∈ H^{1}(Ω_{0})

we observe (cf. Kenmochi [5], Kenmochi and Pawlow [6]) that if v ∈ W^{1,2}(0, T; L^{2}(Ω_{0})) ∩ L^{2}(0, T; H^{2}(Ω_{0})) and v(0, ·) ∈ H^{2}(Ω_{0}), then Φ_{δ}(t, v(t)) is absolutely continuous on [0, T] and

\frac{d}{dt}Φ_{δ}(t, v(t)) + ∑_{i,k=1}^{3} ∫_{Ω_{0}} a_{ik}(t, y)\frac{∂v(t, x)}{∂y_{i}} \frac{∂v(t, y)}{∂t}dy ≤ R_{0}(Φ_{δ}(t, v(t)) + r_{0})  (3.13)
a.e. on (0, T),

where R_{0} and r_{0} are positive constants independent of δ ∈ (0, 1]. Now, we take β_{δ}(ū_{δ}) as a function v in (3.13) to obtain from (3.12) with the help of estimates (3.10) and (3.11) that

sup_{t∈[0,T]} |Φ_{δ}(t, β_{δ}(ū_{δ}(t)))| + ∫_{Q} β′_{δ}(ū_{δ})|ū_{δ,t}|^{2}dydt ≤ M_{3}  (3.14)

for a certain positive constant M_{3} independent of δ ∈ (0, 1]. The estimates (3.10), (3.11) and (3.14) imply that (3.8) holds for some positive constant M_{0} independent of δ ∈ (0, 1]. □ 4. Proof of the theorem

Existence:

Let {u_{δ}}_{δ∈(0,1]} be the family of approximate solutions of (SPC)_{δ}. By Lemma 3.2 with the standard compactness argument we can find a sequence {δ_{n}} with δ_{n} → 0 as n → +∞ and a function u such that

u_{n} := u_{δ_{n}} → u weakly in L^{2}(Q),
β_{δ_{n}}(u_{n}) → β(u) in L^{2}(Q) and weakly in H^{1}(Q).

We now show that u is a weak solution of (SPC). To do so, multiply (3.1) by any test function η ∈ C^{2}(Q̄) with η(T, ·) = 0 and integrate it over Q. Then we have by the Green-Stokes formula

− ∫_{Q} u_{n}η_{t}dxdt − ∫_{Σ} u_{n}ηv_{Σ}dΓ(t)dt + ∫_{Q} ∇β_{δ_{n}}(u_{n}) · ∇ηdxdt + n_{0} ∫_{Σ} β_{δ_{n}}(u_{n})ηdΓ(t)dt

− ∫_{Q} u_{n}(v · ∇η)dxdt + ∫_{Σ} u_{n}η(v · ν)dΓ(t)dt
= ∫_{Q} f_{δ_{n}}ηdxdt + ∫_{Σ} p_{δ_{n}}ηdΓ(t)dt + ∫_{Ω(0)} u_{0,δ_{n}}η(0, ·)dx.

Here, noting condition (1.2) again and passing to the limit in n yield

− ∫_{Q} uη_{t}dxdt + ∫_{Q} ∇β(u) · ∇ηdxdt + n_{0} ∫_{Σ} β(u)ηdΓ(t)dt − ∫_{Q} u(v · ∇η)dxdt

= ∫_{Q} fηdxdt + ∫_{Σ} pηdΓ(t)dt + ∫_{Ω(0)} u_{0}η(0, ·)dx,

which is the required variational identity. Thus u is a weak solution of (SPC).

Uniqueness:

The idea of our uniqueness proof is due to Ladyženskaja, Solonnikov and Ural’ceva [8;Chapter 5,section 8], and this is also extensively used in Niezgódka and Pawlow [9], Rodrigues [11] and Rodrigues and Yi [12] for the uniqueness proof of generalized Stefan problems and continuous casting problems.

Let u_{1} and u_{2} be two weak solutions. Then

− ∫_{Q}(u_{1} − u_{2})η_{t}dxdt − ∫_{Q}(β(u_{1}) − β(u_{2}))Δηdxdt + ∫_{Σ}(β(u_{1}) − β(u_{2}))\frac{∂η}{∂ν} dΓ(t)dt

+n_{0} ∫_{Σ}(β(u_{1}) − β(u_{2}))ηdΓ(t)dt − ∫_{Q}(u_{1} − u_{2})(v · ∇η)dxdt = 0  (4.1)
for all η ∈ C^{1}(Q̄) with η(T, ·) = 0.

As usual, consider the function

b(t, x) := \{ \begin{matrix}\frac{β(u_{1}(t, x)) − β(u_{2}(t, x))}{u_{1}(t, x) − u_{2}(t, x)} & if & u_{1}(t, x) ≠ u_{2}(t, x), \\ 0 & if & u_{1}(t, x) = u_{2}(t, x), \end{matrix}

which is non-negative and bounded on Q. Then, by (4.1)

− ∫_{Q}(u_{1} − u_{2})\{η_{t} + bΔη + v · ∇η\}dxdt + ∫_{Σ}(β(u_{1}) − β(u_{2})) \{\frac{∂η}{∂ν} + n_{0}η\} dΓ(t)dt = 0 (4.2)
for all η ∈ C^{1}(Q̄) with η(T, ·) = 0.

We now take a smooth and strictly positive approximation b_{ε} of b such that

b ≤ b_{ε} a.e. on Q, ε ≤ b_{ε} ≤ C_{1} a.e. on Q
b_{ε} → b a.e. on Q as ε → 0,

where C_{1} is a positive constant, and consider the following auxiliary linear parabolic equation (P)_{ε} for any given ℓ ∈ D(Q):

(P)_{ε} \{ \begin{matrix}η_{ε,t} + b_{ε}Δη_{ε} + v · ∇η_{ε} = ℓ in Q, \\ \frac{∂η_{ε}}{∂ν} + n_{0}η_{ε} = 0 on Σ, \\ η_{ε}(T, ·) = 0 on Ω(T). \end{matrix}  (4.3)

By the general theory of linear parabolic equations this problem has a unique solution η_{ε} ∈ H^{2+α,1+α/2}(Q̄) and the following estimates are obtained:

sup_{t∈[0,T]} |η_{ε}(t)|_{L^{2}(Ω(t))}^{2} + ∫_{0}^{T} |∇η_{ε}(t)|_{L^{2}(Ω(t))}^{2}dt + ∫_{Q} |b_{ε}||Δη_{ε}|^{2}dxdt ≤ M_{4},  (4.4)

where M_{4} is a positive constant independent of ε ∈ (0, 1]. In fact, (4.3) is obtained by multiplying (4.2) by Δη_{ε}. By (4.4), there exists a sequence {ε_{n}} with ε_{n} → 0 and a function η ∈ L^{2}(Q) with ∇η ∈ L^{2}(Q)^{3} such that

η_{ε_{n}} → η weakly in L^{2}(Q),
∇η_{ε_{n}} → ∇η weakly in L^{2}(Q)^{3} as n → 0.

Taking η_{ε_{n}} as a test function η in (4.2) and passing to the limit in n, we see that

− ∫_{Q}(u_{1} − u_{2})ℓdxdt = − ∫_{Q}(u_{1} − u_{2})(b_{ε_{n}} − b)Δη_{ε_{n}}dxdt → 0.

Therefore

∫_{Q}(u_{1} − u_{2})ℓdxdt = 0 for all ℓ ∈ D(Q),

which implies that u_{1} = u_{2} a.e. on Q.

References

1. A. B. Crowley, Mathematical modelling of heat flow in Czochralski crystal pulling, IMA Journal of Applied Mathematics, 30(1983), 173–189.

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3. E. DiBenedetto and M. O’Leary, Three-dimensional conduction-convection problems with change of phase, Archive for Rational Mechanics and Analysis, 123 (1993), 99–116.

4. T. Fukao, N. Kenmochi and I. Pawlow, Two-phase Stefan problems in non-cylindrical domains arising in Czochralski crystal growth process, in preparation.

5. N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bulletin, Faculty of Education, Chiba University, 30(1981), 1–87.

6. N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-depedent constraints, Nonlinear Analysis, 10(1986), 1181–1202.

7. N. Kenmochi and I. Pawlow, Controlled Czochralski crystal growth, Journal of the Society of Instrument and Control Engineers, 35(1996), 944–950.

8. O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol.23, 1968.

9. M. Niezgódka and I.Pawlow, A generalized Stefan problem in several spaces variables, Applied Mathematics and Optimization, 9(1983), 193–224.

10. I. Pawlow, Analysis of the Czochralski crystal growth models designated for optimal control, Technical report, UCLA School of Engineering and Applied Science, (1995).

11. J. F. Rodrigues, Variational methods in the Stefan problem, pp.147–212, In A.Visintin editor, Phase Transitions and Hysteresis, volume 1584 of Lecture Notes in Mathematics, Springer-Verlag, 1994.

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Takeshi Fukao Department of Mathematics, Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-chō, Inage-ku, Chiba, 263-8522, Japan e-mail: t-fukao@math.e.chiba-u.ac.jp

Nobuyuki Kenmochi Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-chō, Inage-ku, Chiba, 263-8522, Japan e-mail: kenmochi@math.e.chiba-u.ac.jp

Irena Pawlow Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland e-mail: pawlow@ibspan.waw.pl

Technical Reports of Mathematical Sciences

Chiba University, vol. 17 (2001)

No. 1 A note on admissibility of the inverse estimator in multivariate calibration, by H. Tsukuma.

No. 2 Isomorphisms of C^{∗}-algebras after tensoring, by Y. Katabmi.

No. 3 Two-phase Stefan problems in non-cylindrical domains, by T. Fukao, N. Kenmochi and I. Pawlow.

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