Transmission Problems for Degenerate Parabolic Equations

T. FUKAO Technical Reports of Mathematical Sciences Chiba University, Vol. 17 (2001), No. 5 TRANSMISSION PROBLEMS FOR DEGENERATE PARABOLIC EQUATIONS

Takeshi Fukao

Abstract. In this paper, we consider a transmission problem for nonlinear two degenerate parabolic PDEs, in such a setting that a fixed bounded domain in R^{3} is separated by a moving interface into two subdomains. The degenerate parabolic PDEs are respectively formulated on one of these subdomains and moreover the so-called transmission condition is imposed on the moving interface. The purpose of this paper is to show the existence and uniqueness of a solution.

1. Introduction

In this paper we consider the following initial-boundary value problem for degenerate parabolic equations in a cylindrical domain Q := (0, T) × Ω, 0 < T < ∞, Ω ⊂ R^{3}, where Ω is a bounded domain with smooth boundary Γ and it is separated by a time dependent interface Γ_{m}(t) into two subdomains Ω_{1}(t) and Ω_{2}(t), that is, Ω := Ω_{1}(t) ∪ Ω_{2}(t) ∪ Γ_{m}(t):

(TP) \{ \begin{matrix}u_{t} − Δβ_{i}(u) + v · ∇u = f in Q_{i} := ⋃_{t∈(0,T)}\{t\} × Ω_{i}(t), i = 1, 2,  (1.1) \\ β_{1}(u) = β_{2}(u), \frac{∂β_{1}(u)}{∂ν^{+}} + \frac{∂β_{2}(u)}{∂ν^{−}} = 0 on Σ_{m} := ⋃_{t∈(0,T)}\{t\} × Γ_{m}(t),  (1.2) \\ \frac{∂β_{i}(u)}{∂ν} + n_{0}β_{i}(u) = p on Σ_{i} := ⋃_{t∈(0,T)}\{t\} × \{∂Ω_{i}(t) \ Γ_{m}(t)\}, i = 1, 2, (1.3) \\ u(0) = u_{0} on Ω.  (1.4) \end{matrix}

Here u_{t} := ∂u/∂t, Δu := ∑_{i=1}^{3}(∂^{2}u/∂x_{i}) and ∇u := (∂u/∂x_{1}, ∂u/∂x_{2}, ∂u/∂x_{3}). We denote by ν^{+} := ν^{+}(t, x) the 3-dimensional unit vector normal to Γ_{m}(t) at x ∈ Γ_{m}(t), pointing to Ω_{2}(t), by ν^{−} := ν^{−}(t, x) the vector −ν^{+} and by ν := ν(x) the 3-dimensional unit vector normal to Γ at x ∈ Γ. n_{0} is a positive constant, f and p are given functions prescribed on Q and Σ, respectively. u_{0} is the initial datum on Ω. β_{1} and β_{2} are functions on R which satisfy the following conditions:

• β_{i} : R → R is a Lipschitz continuous and non-decreasing function for i = 1, 2.

• β_{i}(0) = 0 and

|β̂_{i}(r)| ≥ c_{β_{i}}|r|^{2} − c′_{β_{i}} for all r ∈ R and i = 1, 2,  (1.5)

where β̂_{i}(r) := ∫_{0}^{r} β_{i}(s)ds, i = 1, 2 and c_{β_{i}}, c′_{β_{i}} are positive constants.

In (TP), (1.2) describes a relation between β_{1}(u) and β_{2}(u) on the moving interface Γ_{m}(t). It is called a transmission condition.

In this paper we assume that the following conditions are satisfied:

(A1) For each t ∈ [0, T] there exists a C^{2}-diffeomorphism Θ := Θ(t, ·) on \overline{Ω}, such that

Θ ∈ C^{2}(\overline{Q}) and

Θ(t, \overline{Ω_{i}(t)}) = \overline{Ω_{i}(0)}, i = 1, 2,
Θ(t, Γ_{m}(t)) = Γ_{m}(0), for all t ∈ [0, T].

This assumption says that the movement of Γ_{m}(t) is prescribed and smooth in time. Next, for convection vector field v we assume that

(A2) v ∈ C^{1}(\overline{Q})^{3} and the following properties are satisfied:

divv = 0 in Q,  (1.6)
v · ν^{+} = v_{Σ_{m}} on Σ_{m},  (1.7)
v · ν = 0 on Σ := (0, T) × Γ,  (1.8)

where v_{Σ_{m}} := v_{Σ_{m}}(t, x) is the normal speed of Γ_{m}(t) at (t, x) ∈ Σ_{m}.

Here we define a time-dependent function β : Q × R → R

β(t, x, r) := β_{1}(r)χ_{Ω_{1}(t)} + β_{2}(r)χ_{Ω_{2}(t)},

where χ_{Ω_{i}(t)} is the characteristic function of Ω_{i}(t) for i = 1, 2 and t ∈ [0, T]. Then the problem (TP) is reformulated as the following form (E):

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

As to the solvability of (E), it seems that the time-dependent subdifferential operator theory is useful (cf. [1, 4, 7, 8, 9, 10, 16]). In sections 3 and 4 we prove the existence and uniqueness of a solution.

2. Definition of the weak solution and the main result

We begin with the definition of a weak solution of (E) and then formulate our main result. To this end we introduce some function spaces.

We denote by V the Sobolev space H^{1}(Ω) with norm | · |_{V} given by

|u|_{V} := \{|∇u|_{L^{2}(Ω)}^{2} + n_{0}|u|_{L^{2}(Γ)}^{2}\}^{\frac{1}{2}} for all u ∈ V,

which is an equivalent norm to the standard one of H^{1}(Ω). The dual space of V is denoted by V^{∗} and the duality pairing between V^{∗} and V is denoted by ⟨·, ·⟩_{V^{∗},V}. The duality mapping F : V → V^{∗} is defined by

⟨F u_{1}, u_{2}⟩_{V^{∗},V} := ∫_{Ω} ∇u_{1} · ∇u_{2}dx + n_{0} ∫_{Γ} u_{1}u_{2}dΓ for all u_{1}, u_{2} ∈ V.

Moreover, define an inner product (·, ·)_{V^{∗}} in V^{∗} by

(u_{1}^{∗}, u_{2}^{∗})_{V^{∗}} := ⟨u_{1}^{∗}, F^{−1}u_{2}^{∗}⟩_{V^{∗},V} for all u_{1}^{∗}, u_{2}^{∗} ∈ V^{∗}.

Then V^{∗} is a Hilbert space with the following dense and compact imbeddings:

V ↪ L^{2}(Ω) ↪ V^{∗}.

Now we give the definition of a weak solution of (E). Definition 2.1 u : [0, T] → L^{2}(Ω) is called a weak solution of (E) if u ∈ W^{1,2}(0, T; V^{∗})∩ L^{∞}(0, T; L^{2}(Ω)), β(u) := β(t, x, u) ∈ L^{2}(0, T; V), and u satisfies (E) in the variational sense that

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

= ∫_{Q} fηdxdt + ∫_{Σ} pηdΓdt + ∫_{Ω} u_{0}η(0)dx for all η ∈ W,  (2.1)

where W := {η ∈ H^{1}(Q); η(T) = 0 on Ω}.

Our main result is stated as follows: Theorem 2.1 Assume that (A1) and (A2) hold. Let f ∈ L^{2}(Q), p ∈ C^{1}(\overline{Σ}), u_{0} ∈ L^{2}(Ω) and β(u_{0}) ∈ H^{1}(Ω). Then there exists one and only one weak solution u of (E).

The existence proof for (E) will be given in section 3 by using the abstract theory of time-dependent subdifferential operators. The uniqueness proof will be given in section 4.

3. Existence proof

Let us show the existence of a weak solution of (E). Now we are ready to apply the existence result in [16] (or [8]) for a nonlinear evolution equations including timedependent subdifferential operators and perturbation of the form (K):

(K) \{ \begin{matrix}u_{t}(t) + ∂ϕ^{t}(u(t)) + g(t, u(t)) ∋ f^{∗}(t) in V^{∗} for a.e. t ∈ [0, T], \\ u(0) = u_{0} in V^{∗}. \end{matrix}

Here we define the functions ϕ^{t}, g(t, ·) and f^{∗}(t) for each t ∈ [0, T] as follows:

• ϕ^{t} : L^{2}(Ω) → (−∞, ∞] is defined by

ϕ^{t}(z) := \{ \begin{matrix}∫_{Ω_{1}(t)} β̂_{1}(z(x))dx + ∫_{Ω_{2}(t)} β̂_{2}(z(x))dx & if & z ∈ L^{2}(Ω), \\ +∞ & if & z ∈ V^{∗} \ L^{2}(Ω). \end{matrix}

Clearly ϕ^{t} is a proper, lower semi-continuous and convex function on V^{∗}.

• g(t, ·) : L^{2}(Ω) → V^{∗} is defined by

⟨g(t, z), η⟩_{V^{∗},V} := − ∫_{Ω} z(x)(v(t) · ∇η)(x)dx for all η ∈ V and z ∈ L^{2}(Ω).

• f^{∗} ∈ L^{2}(0, T; V^{∗}) is defined by

⟨f^{∗}(t), η⟩_{V^{∗},V} := ∫_{Ω} f(t, x)η(x)dx + ∫_{Γ} p(t, x)η(x)dΓ for all η ∈ V.

Then it is easily to see that the problem (E) is equivalent to (K). Therfore it is enough to solve the problem (K) with the above functions ϕ^{t}, g(t, ·) and f^{∗}(t).

The characterization of ∂ϕ^{t} is given by the following lemma. Lemma 3.1 Let z ∈ D(∂ϕ^{t}) and β(z) := β(t, x, z(x)). Then z^{∗} ∈ ∂ϕ^{t}(z) in V^{∗} if and only if

(F^{−1}z^{∗}, w)_{L^{2}(Ω)} = (β(z), w)_{L^{2}(Ω)} for all w ∈ L^{2}(Ω),  (3.1)

where (·, ·)_{L^{2}(Ω)} denotes the inner product in L^{2}(Ω). Moreover, this is equivalent to the statement that β(z) ∈ V and

⟨z^{∗}, η⟩_{V^{∗},V} = ∫_{Ω} ∇β(z) · ∇ηdx + n_{0} ∫_{Γ} β(z)ηdΓ for all η ∈ V.  (3.2)

The next lemma is concerned with the demi-closedness for g(t, ·). Lemma 3.2 The operator g(t, ·) : D(g(t, ·)) = L^{2}(Ω) → V^{∗}, t ∈ [0, T], satisfies that

|g(t, z)|_{V^{∗}}^{2} ≤ C_{0}ϕ^{t}(z) + C_{1} for all t ∈ [0, T] and all z ∈ L^{2}(Ω),  (3.3)

where C_{0} and C_{1} are positive constants, and that if {ϕ^{t_{n}}(z_{n})} is bounded and z_{n} → z in V^{∗} and t_{n} → t, as n → ∞ for {z_{n}} ⊂ L^{2}(Ω) and {t_{n}} ⊂ [0, T], then g(t_{n}, z_{n}) → g(t, z) weakly in V^{∗}. Furthermore the set L_{r} := ⋃_{0≤t≤T} {z ∈ V^{∗}; ϕ^{t}(z) + |z|_{V^{∗}}^{2} ≤ r} is bounded in L^{2}(Ω) and hence relatively compact in V^{∗} for each r ≥ 0.

The proofs of these lemmas are quite standard, so we omit them.

In the following lemma we check the time-dependence of ϕ^{t} and the conditions required in order to apply the abstract result in [16] (or [8]) to problem (K).

Lemma 3.3 There exists a positive constant C_{2} with the following property: for each s, t ∈ [0, T], s ≤ t and z ∈ D(ϕ^{s}) there exists z_{1} ∈ D(ϕ^{t}) such that

|z_{1} − z|_{V^{∗}} ≤ C_{2}|t − s|(1 + |ϕ^{s}(z)|^{\frac{1}{2}} ),  (3.4)
|ϕ^{t}(z_{1}) − ϕ^{s}(z)| ≤ C_{2}|t − s|(1 + |ϕ^{s}(z)|).  (3.5)

Proof. For each z ∈ D(ϕ^{s}) := L^{2}(Ω), we take

z_{1}(x) := z(Θ^{−1}(s, Θ(t, x))) for all x ∈ Ω.

Then, by geometric assumption (A1), z_{1} ∈ L^{2}(Ω) = D(ϕ^{t}) and the mapping Θ_{t,s} := Θ(s, ·)^{−1} ∘ Θ(t, ·) is a C^{2}-diffeomorphism on \overline{Ω} for any s, t ∈ [0, T] such that Θ_{t,s}(\overline{Ω_{i}(t)}) = \overline{Ω_{i}(s)}, i = 1, 2. Moreover there exist positive constants C_{3} and C_{4} such that

|det.J_{Θ_{t,s}}(x) − 1| ≤ C_{3}|t − s| for all s, t ∈ [0, T], and x ∈ Ω,
|η ∘ Θ_{t,s} − η|_{L^{2}(Ω)} ≤ C_{4}|t − s||η|_{V} for all s, t ∈ [0, T] and η ∈ V,

(cf. Kenmochi [9; Lemma 3.2.2]), where J_{Θ_{t,s}} is the Jacobian of Θ_{t,s} and det.J_{Θ_{t,s}} is its determinant. By using these properties we immediately obtain (3.4) and (3.5). □

By Lemma 3.1-3.3 and an existence result of [16] (or [8]) we see that there exists at least one solution u of (K) in W^{1,2}(0, T; V^{∗}) ∩ L^{∞}(0, T; L^{2}(Ω)) and this solution u is a weak solution of (E) in the sense of Definition 2.1.

4. Uniqueness proof

In this section we prove the uniqueness of a weak solution of (E) obtained in the previous section. Let u_{1} and u_{2} be two weak solutions. Then by taking the difference of their weak variational forms (2.1) we have

− ∫_{Q}(u_{1} − u_{2})η_{t}dxdt + ∫_{Q}(∇β(u_{1}) − ∇β(u_{2})) · ∇ηdxdt − ∫_{Q}(u_{1} − u_{2})(v · ∇η)dxdt

+ n_{0} ∫_{Σ}(β(u_{1}) − β(u_{2}))ηdΓdt = 0 for all η ∈ W.

Moreover define a function b : Q → R by

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. With this function b, by the Green’s formula

− ∫_{Q}(u_{1} − u_{2})\{η_{t} + bΔη + v · ∇η\}dxdt + ∫_{Σ}(β(u_{1}) − β(u_{2})) \{\frac{∂η}{∂ν} + n_{0}η\} dΓdt = 0 (4.1)

for all η ∈ W.

Now we take smooth and strictly positive approximations b_{ε} of b such that

b ≤ b_{ε} a.e. on Q, ε ≤ b_{ε} ≤ M_{0} a.e. on Q,  (4.2)
b_{ε} → b a.e. on Q as ε → 0,

where M_{0} is a positive constant. In order to prove the uniqueness we consider the following backward problem for the auxiliary linear parabolic equation for any given ℓ ∈ D(Q):

\{ \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. \end{matrix}  (4.3)

By using Ladyženskaja, Solonnikov and Ural’ceva [11;Chapter 4, Theorem 5.3] problem (4.3) has a unique solution η_{ε} ∈ H^{2+α,1+α/2}(Q̄) for a certain number 0 < α < 1.

In the next two lemmas we give the L^{∞} estimate of {η_{ε}(t)} in V and boundedness of {b_{ε}^{1/2}Δη_{ε}} in L^{2}(Q). Lemma 4.1 There exists a positive constant M_{1} such that

sup_{0≤s≤T} |η_{ε}(s)|_{V}^{2} + ∫_{s}^{T} ∫_{Ω} b_{ε}|Δη_{ε}|^{2}dxdt ≤ M_{1} ∫_{s}^{T} ∫_{Γ} v · ∇(η_{ε}^{2})dΓdt + M_{1}|ℓ|_{L^{2}(0,T;V)}

for all s ∈ [0, T] and ε ∈ (0, 1]. Proof. Multiplying the first equation in (4.3) by Δη_{ε} and integrating over Ω with respect to x, with the help of the Green’s formula we have

−\frac{d}{dt}|η_{ε}|_{V}^{2} + 2 ∫_{Ω} |b_{ε}||Δη_{ε}|^{2}dx ≤ (6|v|_{C^{1}(\overline{Q})^{3}} + 1)|η_{ε}|_{V}^{2} + |ℓ|_{V}^{2} + 2n_{0} ∫_{Γ} v · ∇(η_{ε}^{2})dΓ (4.4)

for all t ∈ [0, T]. In fact, in order get (4.4) we use the following equations and inequalities.

− ∫_{Ω}(v · ∇η_{ε})Δη_{ε}dx
= ∫_{Ω} ∇(v · ∇η_{ε}) · ∇η_{ε}dx − ∫_{Γ}(v · ∇η_{ε})\frac{∂η_{ε}}{∂ν} dΓ
≤ 3|v|_{C^{1}(\overline{Q})^{3}} ∫_{Ω} |∇η_{ε}|^{2}dx + ∫_{Ω} v · ∇ (\frac{1}{2}|∇η_{ε}|^{2}) dx + \frac{n_{0}}{2} ∫_{Γ} v · ∇(η_{ε}^{2})dΓ

and by virtue of (1.6) and (1.8)

∫_{Ω} v · ∇ (\frac{1}{2}|∇η_{ε}|^{2}) dx = ∫_{Ω} div (\frac{1}{2}|∇η_{ε}|^{2}v) dx
= ∫_{Γ} \frac{1}{2}|∇η_{ε}|^{2}(v · ν)dΓ
= 0.

Finally integrate (4.4) over [s, T] in time and apply the Gronwall’s inequality to get the conclusion. □ Lemma 4.2 There exist a positive constant M_{2} such that

∫_{Γ} v(t) · ∇(η_{ε}^{2}(t))dΓ ≤ M_{2}|η_{ε}(t)|_{L^{2}(Γ)}^{2} for all t ∈ [0, T] and ε ∈ (0, 1].

Proof. Our geometric condition (A1) ensures that without loss of generality we may assume by taking an approximate local coordinate transformation and a partition of unity on Γ that

• Ω ⊂ {x = (x_{1}, x_{2}, x_{3}); x_{3} < 0} and Γ contains a relatively open set Γ_{0} in R^{2} ≅ {x; x_{3} = 0}.

• supp η_{ε} is compact in Ω ∪ Γ_{0} for all ε ∈ (0, 1] and

\frac{∂η_{ε}}{∂ν} = \frac{∂η_{ε}}{∂x_{3}} on Γ_{0} for all ε ∈ (0, 1].

In the above situation we observe for any t ∈ [0, T] that

∫_{Γ} v · ∇(η_{ε}^{2})dΓ = ∫_{R^{2}} v · ∇(η_{ε}^{2})dx_{1}dx_{2}
= − ∫_{R^{2}}(v_{1,x_{1}} + v_{2,x_{2}})η_{ε}^{2}dx_{1}dx_{2} + ∫_{R^{2}} 2v_{3}η_{ε} \frac{∂η_{ε}}{∂x_{3}} dx_{1}dx_{2}
= − ∫_{R^{2}}(v_{1,x_{1}} + v_{2,x_{2}} + 2n_{0}v_{3})η_{ε}^{2}dx_{1}dx_{2},

where v := (v_{1}, v_{2}, v_{3}). Therefore, for any t ∈ [0, T],

∫_{Γ} v · ∇(η_{ε}^{2})dΓ ≤ (2n_{0} + 2)|v|_{C^{1}(\overline{Q})^{3}}|η_{ε}|_{L^{2}(Γ)}^{2}.

Thus we have the conclusion of the lemma. □

Now, on account of Lemmas 4.1 and 4.2 we have the uniform estimate

sup_{0≤t≤T} |η_{ε}(t)|_{V}^{2} + ∫_{Q} b_{ε}|Δη_{ε}|^{2}dxdt ≤ M_{3},  (4.5)

where M_{3} is a positive constant independent of ε ∈ (0, 1].

By (4.5) there exist a sequence {ε_{n}} with ε_{n} → 0 (n → ∞) and a function η ∈ V such that

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

Take η_{ε_{n}} as a test function η in (4.2), and pass to the limit to obtain that

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

since

|∫_{Q}(u_{1} − u_{2})(b_{ε_{n}} − b)Δη_{ε_{n}}dxdt|
≤ \{∫_{Q} |u_{1} − u_{2}|^{2}|b_{ε_{n}} − b|dxdt\}^{\frac{1}{2}} \{∫_{Q} |b_{ε_{n}} − b||Δη_{ε_{n}}|^{2}dxdt\}^{\frac{1}{2}}
→ 0.

Thus u_{1} = u_{2} a.e. on Q. □

Remark 4.1 The idea of our uniqueness proof is due to Ladyženskaja, Solonnikov and Ural’ceva [11; Chapter 3, Section 3], and this was also extensively used in Niezgódka and Pawlow [12], Rodrigues and Yi [15] and Rodrigues [14] for the uniqueness proof of generalized Stefan problems and continuous casting problems. Also, a similar technique was recently applied for the uniqueness proof of Stefan problems arising in the Czochralski process of crystal growth (cf. Fukao, Kenmochi and Pawlow [6, 7]). Remark 4.2 In particular, if β_{1} and β_{2} are strongly monotone in R, namely

(β_{i}(r_{1}) − β_{i}(r_{2}))(r_{1} − r_{2}) ≥ m_{0}|r_{1} − r_{2}|^{2} for all r_{1}, r_{2} ∈ R and i = 1, 2  (4.6)

where m_{0} is a positive constant, then the uniqueness of a solution of (E) is a direct application of [16;Theorem 3.1]. The main point of the proof is to use the estimate

|∫_{Ω}(u_{1} − u_{2})v · ∇(F^{−1}(u_{1} − u_{2}))dx| ≤ m_{1}|v|_{C^{1}(\overline{Q})^{3}}|u_{1} − u_{2}|_{L^{2}(Ω)}|u_{1} − u_{2}|_{V^{∗}}  (4.7)

for all u_{1}, u_{2} ∈ L^{2}(Ω), where m_{1} is a positive constant so that

|∇F^{−1}z|_{L^{2}(Ω)^{3}} ≤ m_{1}|F^{−1}z|_{V} (= m_{1}|z|_{V^{∗}}) for all z ∈ V^{∗}.

In fact, for two solutions u_{1} and u_{2}, we observe that

⟨u′_{1}(t) − u′_{2}(t), η⟩_{V^{∗},V} + ⟨F (β(u_{1}(t)) − β(u_{2}(t))), η⟩_{V^{∗},V} − ∫_{Ω}(u_{1}(t) − u_{2}(t))(v(t) · ∇η)dx = 0

for all η ∈ V, a.e. t ∈ [0, T].

Now, take F^{−1}(u_{1}(t) − u_{2}(t)) as η and use (4.6) and (4.7). Then

\frac{1}{2} \frac{d}{dt}|u_{1}(t) − u_{2}(t)|_{V^{∗}}^{2} + (m_{0} − δ)|u_{1}(t) − u_{2}(t)|_{L^{2}(Ω)}^{2} ≤ \frac{m_{1}^{2}}{4δ} |v|_{C^{1}(\overline{Q})^{3}}^{2}|u_{1}(t) − u_{2}(t)|_{V^{∗}}^{2}

for a.e. t ∈ [0, T], where δ is an arbitrary positive number. Choosing m_{0}/2 as δ and applying the Gronwall’s inequality, we infer that u_{1} = u_{2}.

5. Application to Czochralski crystal growth process

This work is motivated by modeling the Czochralski process of crystal growth. The Czochralski pulling method is widely used for the production of a column of single silicon crystal from the melt. The idea for pulling method due to Czochralski is quite simple. A crucible, equipped with a heater, contains the melt substance and a pul-rod with seed crystal, which moves vertically and rotates flexibly, is positioned above the crucible. The rod is dipped into the melt, and then lifted slowly with an appropriate speed so that a meniscus surface is formed below the crystal seed 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 (see [3, 5, 6, 7, 13]).

In such a model of crystal growth, if we assume that the movement of the material (solid-liquid) domain is prescribed and a 3-dimensional convective vector field, which is caused by the deformation of the material domain, is prescribed, too, at least satisfying condition (A2), then our approach seems applicable to the Czochralski process. In this case, we consider two equations in the crucible. One of them is a degenerate parabolic equation which describe the solid-liquid phase transition in the material region. The other is an usual heat equation in the gas region. These two equations are combined by the transmission condition on the common boundary.

In this case the transmission condition implies that the temperature is possibly discontinuous on the solid-gas and liquid-gas interfaces. This is one of the important points which should be furthermore discussed from the mechanical point of view.

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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

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.

No. 4 Transmission problems arising in Czochralski process of crystal growth, by T. Fukao, N. Kenmochi and I. Pawlow.

No. 5 Transmission problems for degenerate parabolic equations, by T. Fukao.

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