Degenerate Parabolic Equations with Convection in Non-Cylindrical Domains

Takesi FUKAO and Nobuyuki KENMOCHI

Technical Reports of Mathematical Sciences Chiba University, Vol. 19 (2003), No. 4

Degenerate parabolic equations with convection in non-cylindrical domains

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. In this paper we consider a class of weak variational formulations for nonlinear parabolic equations with convection in non-cylindrical (namely,timedependent) domains. In our equation the main differential term is of the second order, possibly degenerate, and a convective effect is described by the first order term. Our purpose is to show the existence of a solution in a class of global bounded functions in the non-cylindrical domain. Moreover we discuss the continuous dependence of solutions upon convective vectors.

Keywords. degenerate parabolic equation, Stefan problem, convection, global boundedness.

1 Introduction

Let 0 < T < +∞ and Ω_{m}(t) ⊂ R^{3} be a time dependent bounded domain with smooth boundary Γ_{m}(t) := ∂Ω_{m}(t) for all t ∈ [0, T]. We consider the following degenerate parabolic partial differential equation:

\frac{∂u}{∂t} + v · ∇u − Δβ(u) = f in Q_{m} := ⋃_{t∈(0,T)}\{t\} × Ω_{m}(t),  (1.1)

where β : R → R is a non-decreasing and Lipschitz continuous function. In our setting β admits the case when β′ = 0 in some interval, roughly speaking, diffusion term −Δβ(u) vanishes on this interval. Such a mathematical situation occurs in many physical models. For instance, equation (1.1) can be considered as the Stefan problem, which is one of the most typical problems related to phase transition phenomena. In this context, the function β(u) is interpreted as temperature field and the vector function v := (v_{1}, v_{2}, v_{3}) as convection. Also, when the crystal growth is taken into account, it is quite natural to assume that the domain, where the problem is formulated, changes its configuration smoothly in time and hence some transport-convection effect appears inside the timedependent domain. In the solid-liquid phase transition accompanying deformation of the domain, the vector field v can be interpreted not only as the velocity of deformation but also as internal force in the liquid. So far as the Stefan problem is concerned, we refer for the literature of various mathematical works for instance to the book of Visintin [12], to the papers of Pawłow [9] and of Fukao and Kenmochi [5].

2000 AMS Subject Classification 35K65, 35K20, 35R35,

2 Weak formulation and main theorems

In this section, we propose an initial-boundary value problem for a degenerate parabolic partial differential equation, and define its weak formulation. We give an existence result and prove a sort of continuous dependence of weak solutions upon convective vectors.

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

(A1) There exists a function y ∈ C^{2}(\overline{Q_{m}}) := C^{2}(\overline{Q_{m}})^{3} which gives a C^{2}-diffeomorphism y(t, ·) := (y_{1}(t, ·), y_{2}(t, ·), y_{3}(t, ·)) from \overline{Ω_{m}(t)} onto \overline{Ω_{m0}} for all t ∈ [0, T], that is,

y(t, \overline{Ω_{m}(t)}) = \overline{Ω_{m0}} for all s, t ∈ [0, T],

with y(0, ·) = I (identity) on \overline{Ω_{m0}}, where Ω_{m0} := Ω_{m}(0); we denote the inverse of y(t, ·) by x(t, ·) : \overline{Ω_{m0}} → \overline{Ω_{m}(t)} for all t ∈ [0.T].

We treat a function β : R → R of the following form

β(r) := \{ \begin{matrix}k_{s}r & if r < 0, \\ 0 & if 0 ≤ r ≤ L, \\ k_{ℓ}(r − L) & if r > L, \end{matrix}  (2.1)

where k_{s}, k_{ℓ} and L are positive constants. Clearly, β is a non-decreasing Lipschitz continuous function and β(0) = 0. Now we consider the following initial-boundary value problem (SP; v):= {(2.2)-(2.4)}:

\frac{∂u}{∂t} + v · ∇u − Δβ(u) = f in Q_{m},  (2.2)
\frac{∂β(u)}{∂ν} + n_{0}β(u) = q on Σ_{m} := ⋃_{t∈(0,T)}\{t\} × Γ_{m}(t),  (2.3)
u(0) = u_{0} in Ω_{m0},  (2.4)

where ν := ν(t, x) is the 3-dimensional unit vector outward normal to Γ_{m}(t); n_{0} is a positive constant; f, q and u_{0} are given functions on Q_{m}, Σ_{m} and Ω_{m0}, respectively. The vector field v is prescribed on Q_{m} such that (A2) v ∈ L^{2}(Q_{m}) := L^{2}(Q_{m})^{3} and v(t) ∈ H_{σ}^{1}(Ω_{m}(t)) for a.e. t ∈ [0, T],

∫_{0}^{T} |v(t)|_{H_{σ}^{1}(Ω_{m}(t))}^{2}dt < +∞,
v · ν = v_{Σ_{m}} on Σ_{m},  (2.5)

where H_{σ}^{1}(Ω_{m}(t)) is the closure of D_{σ}(Ω_{m}(t)) := {z ∈ C_{0}^{∞}(Ω_{m}(t)); divz = 0} with respect to the topology of H^{1}(Ω_{m}(t)) := H^{1}(Ω_{m}(t))^{3} and v_{Σ_{m}}(t, ·) is the normal speed of Γ_{m}(t).

Now we introduce a class of weak solutions as follows:

Definition 2.1. A function u ∈ L^{∞}(Q_{m}) is called a weak solution of (SP; v), if ∫_{0}^{T} |β(u(t))|_{H^{1}(Ω_{m}(t))}^{2}dt < +∞ and the function

t ↦ ∫_{Ω_{m}(t)} u(t, x)ξ(x)dx is continuous on [0, T] for all ξ ∈ L^{2}(R^{3}).  (2.6)

Moreover u satisfies the following variational identity:

− ∫_{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,  (2.7)

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

As is easily checked, (2.6) is equivalent to the fact that the 0-extension of u onto (0, T)×R^{3}, denoted by the same notation again, is weakly continuous in L^{2}(R^{3}) uniformly on [0, T].

Here we recall some known results. In the paper of Fukao, Kenmochi and Pawłow [6], the existence and uniqueness results for problem (2.7) were proved, in case the convective vector v is of C^{2}-class and satisfies (2.5). One of main purposes of this paper is to show the existence of a weak solution of (2.7) under assumption (A2) for v, which is a generalization of the existence result in [6], although the uniqueness is lost. For our proof the key idea, which was already employed in [6], is to transform the non-cylindrical domain Q_{m} onto a cylindrical one.

Our existence result is stated as follows:

Theorem 2.1. Assume that (A1) and (A2) hold, and that f ∈ L^{∞}(Q_{m}), q ∈ L^{∞}(Σ_{m}) and u_{0} ∈ L^{∞}(Ω_{m0}). Then there exist at least one solution u of (SP; v) such that u satisfies the following estimates.

|u|_{L^{∞}(Q_{m})} + ∫_{0}^{T} |∇β(u(t))|_{L^{2}(Ω_{m}(t))}^{2}dt ≤ M_{0},  (2.8)

where M_{0} := M_{0}(|f|_{L^{∞}(Q_{m})}, |q|_{L^{∞}(Σ_{m})}, |u_{0}|_{L^{∞}(Ω_{m0})}) is a positive constant such that M_{0}(·, ·, ·) is bounded when its arguments are bounded; the other data, such as constants n_{0}, L and the length T of the time interval, are fixed.

The next theorem give a continuous dependence property of a weak solution of (SP;

v) upon vector field v.

Theorem 2.2. Under the same assumption of Theorem 2.1. For each δ > 0, let v_{δ} ∈ L^{2}(Q_{m}) be any function satisfying (A2) and u_{δ} be a weak solution of (SP; v_{δ}) which is obtained by Theorem 2.1; assume as well that u_{δ} satisfies the boundedness (2.8) for a common positive constant M_{0}. Suppose that

v_{δ} → v weakly in L^{2}(Q_{m}) as δ → 0,  (2.9)
∫_{0}^{T} |v_{δ}(t)|_{H_{σ}^{1}(Ω_{m}(t))}dt is bounded for δ > 0.  (2.10)

Then, any weak cluster point u of {u_{δ}} in L^{2}(Q_{m}) is a weak solution of (SP; v), moreover, when u_{δ_{n}} → u in L^{2}(Q_{m}) as n → +∞ for a sequence {δ_{n}} with δ_{n} → 0, we have

β(u_{δ_{n}}) → β(u) in L^{2}(Q_{m}) as n → +∞,  (2.11)

and

\{ \begin{matrix}the 0-extension of u_{δ_{n}}(t) onto R^{3} converges weakly \\ to that of u(t) in L^{2}(R^{3}) uniformly in t ∈ [0, T]. \end{matrix}  (2.12)

In the next section, we shall prove these theorems.

3 L^{∞}-boundedness for Stefan problems

We use the same notation as in the previous section. The global L^{∞}-estimate of weak solutions is a key for the existence proof of weak solutions and the weak continuous dependence of them upon the convective vector v. The idea of our existence proof is essentially due to Fukao, Kenmochi and Pawłow [6].

For each t ∈ [0, T], put Q_{m}(0, t) := ⋃_{τ∈(0,t)}{τ} × Ω_{m}(τ) and Σ_{m}(0, t) := ⋃_{τ∈(0,t)}{τ} × Γ_{m}(τ). We begin with the proof of the L^{∞}-estimate for weak solutions in the case of smooth v; note that problem (2.7) has a unique weak solution (cf. [6]).

Lemma 3.1. Assume that (A1) and (A2) hold, and v is in C^{2}(\overline{Q_{m}}). Let u be the weak solution of problem (SP; v). Then, putting

M_{1} := max \{L, |f|_{L^{∞}(Q_{m})}, |\frac{q}{n_{0}} |_{L^{∞}(Q_{m})} , |β(u_{0})|_{L^{∞}(Ω_{m0})}\} ,  (3.1)

we have

|β(u)|_{L^{∞}(Q_{m})} ≤ M_{1}(1 + T),  (3.2)

and hence

|u|_{L^{∞}(Q_{m})} ≤ max \{\frac{L}{k_{s}} , \frac{L}{k_{ℓ}} + L\} M_{1}(1 + T) =: M′_{1}(T).  (3.3)

Proof. As was checked in [6], the unique weak solution u of problem (2.7) can be approximated by classical solutions of (2.2)-(2.4) with smooth strictly increasing β and smooth data f, q, u_{0}. Therefore it is enough to check the L^{∞}-estimate for the classical solution u. We know that u satisfies for each t ∈ [0, T]

∫_{Q_{m}(0,t)} \frac{∂u}{∂τ}ηdxdτ + ∫_{Q_{m}(0,t)}(v · ∇u)ηdxdτ + ∫_{Q_{m}(0,t)} ∇β(u) · ∇ηdxdτ

+ n_{0} ∫_{Σ_{m}(0,t)} β(u)ηdΓ_{m}(τ)dτ = ∫_{Q_{m}(0,t)} fηdxdτ + ∫_{Σ_{m}(0,t)} qηdΓ_{m}(τ)dτ,  (3.4)

for all η ∈ H^{1}(Q_{m}). Now take the test function η := [β(u) − m]^{+} on Q_{m} where m(τ) := M_{1}τ + M_{1} for all τ ∈ [0, T]; for a moment let M_{1} be any positive constant. Since β(u) = k_{ℓ}(u − L) on the domain {(τ, x) ∈ Q_{m}; [β(u) − m]^{+} ≠ 0}, we have with the help of Stokes’ formula

∫_{Q_{m}(0,t)} \frac{∂u}{∂τ}[β(u) − m]^{+}dxdτ

= \frac{1}{2k_{ℓ}} ∫_{Q_{m}(0,t)} \frac{∂}{∂τ} |[β(u) − m]^{+}|^{2}dxdτ + \frac{1}{k_{ℓ}} ∫_{Q_{m}(0,t)} \frac{∂m}{∂τ}[β(u) − m]^{+}dxdτ
= \frac{1}{2k_{ℓ}} ∫_{Ω_{m}(t)} |[β(u(t)) − m(t)]^{+}|^{2}dx − \frac{1}{2k_{ℓ}} ∫_{Ω_{m0}} |[β(u_{0}) − m(0)]^{+}|^{2}dx

+ \frac{1}{2k_{ℓ}} ∫_{Σ_{m}(0,t)} |[β(u) − m]^{+}|^{2}(−v_{Σ_{m}})dΓ_{m}(τ)dτ + \frac{1}{k_{ℓ}} ∫_{Q_{m}(0,t)} M_{1}[β(u) − m]^{+}dxdτ.

Next, by (2.5) we have

∫_{Q_{m}(0,t)}(v · ∇u)[β(u) − m]^{+}dxdτ
= \frac{1}{2k_{ℓ}} ∫_{Q_{m}(0,t)} v · ∇([β(u) − m]^{+})^{2}dxdτ + \frac{1}{k_{ℓ}} ∫_{Q_{m}(0,t)}(v · ∇m)[β(u) − m]^{+}dxdτ
= \frac{1}{2k_{ℓ}} ∫_{Σ_{m}(0,t)} |[β(u) − m]^{+}|^{2}(v · ν)dΓ_{m}(τ)dτ
= \frac{1}{2k_{ℓ}} ∫_{Σ_{m}(0,t)} |[β(u) − m]^{+}|^{2}v_{Σ_{m}}dΓ_{m}(τ)dτ.

Moreover

∫_{Q_{m}(0,t)} ∇β(u) · ∇[β(u) − m]^{+}dxdτ ≥ 0,

and

n_{0} ∫_{Σ_{m}(0,t)} β(u)[β(u) − m]^{+}dΓ_{m}(τ)dτ ≥ ∫_{Σ_{m}(0,t)} n_{0}M_{1}[β(u) − m]^{+}dΓ_{m}(τ)dτ.

Therefore it follows from (3.4) that

∫_{Ω_{m}(t)} |[β(u(t)) − m(t)]^{+}|^{2}dx
≤ 2k_{ℓ} ∫_{Q_{m}(0,t)}(f − M_{1})[β(u) − m]^{+}dxdτ
+ 2k_{ℓ} ∫_{Σ_{m}(0,t)}(q − n_{0}M_{1})[β(u) − m]^{+}dΓ_{m}(τ)dτ + ∫_{Ω_{m0}} |[β(u_{0}) − m(0)]^{+}|^{2}dx
≤ 2k_{ℓ}(|f|_{L^{∞}(Q_{m})} − M_{1})|[β(u) − m]^{+}|_{L^{1}(Q_{m}(0,t))}
+ 2k_{ℓ}(|q|_{L^{∞}(Σ_{m})} − n_{0}M_{1})|[β(u) − m]^{+}|_{L^{1}(Σ_{m}(0,t))}
+ vol(Ω_{m0}) |[β(u_{0}) − M_{1}]^{+}|_{L^{∞}(Ω_{m0})}^{2} for all t ∈ [0, T].

If M_{1} is chosen so that

M_{1} ≥ |f|_{L^{∞}(Q_{m})}, M_{1} ≥ |\frac{q}{n_{0}} |_{L^{∞}(Σ_{m})} , M_{1} ≥ |β(u_{0})|_{L^{∞}(Ω_{m0})},

then

∫_{Ω_{m}(t)} |[β(u(t)) − m(t)]^{+}|^{2}dx ≤ 0 for all τ ∈ [0, T],

namely,

β(u) ≤ M_{1}(1 + T) a.e. on Q_{m}.

We obtain an estimate for −β(u) similar to the above. Thus we have (3.2) and hence (3.3) for a constant M_{1} satisfying (3.1). □

Next, putting β̂(r) := ∫_{0}^{r} β(τ)dτ we see from (2.1) that

c_{1}|r|^{2} − c_{2} ≤ β̂(r) ≤ c_{3}(|r|^{2} + 1) for all r ∈ R,  (3.5)

for some positive constants c_{1}, c_{2}, c_{3}.

By an argument similar to Lemma 3.2 of [6], we have the H^{1}-estimate for the function β(u) in the following lemma.

Lemma 3.2. Under the same assumptions as in Lemma 3.1, let u be the weak solution of (SP; v). Putting

M_{2} := \frac{1}{2}|f|_{L^{2}(Q_{m})}^{2} + \frac{1}{4n_{0}} |q|_{L^{2}(Σ_{m})}^{2} + c_{2}vol(Q_{m}),  (3.6)

we have

∫_{0}^{T} |∇β(u)|_{L^{2}(Ω_{m}(τ))}^{2}dτ ≤ M_{2} + \frac{1 + 2c_{3}}{2} M′_{1}(T)^{2} =: M′_{2}(T).  (3.7)

Proof. We assume again that u is a classical solution of (2.2)-(2.4). Take any t ∈ [0, T] and η := β(u_{ϱ}) in (3.4). With the help of (2.5) and the Stokes’ formula we have

∫_{Q_{m}(0,t)} \frac{∂u}{∂τ}β(u)dxdτ + ∫_{Q_{m}(0,t)}(v · ∇u)β(u)dxdτ

= ∫_{Q_{m}(0,t)} \frac{∂}{∂τ} β̂(u)dxdτ + ∫_{Q_{m}(0,t)} v · ∇β̂(u)dxdτ
= ∫_{Ω_{m}(t)} β̂(u(t))dx − ∫_{Ω_{m0}} β̂(u_{0})dx + ∫_{Σ_{m}(0,t)} β̂(u)(−v_{Σ_{m}})dΓ_{m}(τ)dτ
+ ∫_{Q_{m}(0,t)} div (β̂(u)v) dxdτ
= ∫_{Ω_{m}(t)} β̂(u(t))dx − ∫_{Ω_{m0}} β̂(u_{0})dx.

Hence, (3.4) with η = β(u) yields that

∫_{Ω_{m}(t)} β̂(u(t))dx + ∫_{0}^{t} |∇β(u(τ))|_{L^{2}(Ω_{m}(τ))}^{2}dτ + n_{0} ∫_{0}^{t} |β(u(τ))|_{L^{2}(Γ_{m}(τ))}^{2}dτ

= ∫_{Q_{m}(0,t)} fβ(u)dxdτ + n_{0} ∫_{Σ_{m}(τ)} qβ(u)dΓ_{m}(τ)dτ + ∫_{Ω_{m0}} β̂(u_{0})dx.

We derive easily from this with (3.5) that for each ε_{1} > 0

c_{1}|u(t)|_{L^{2}(Ω_{m}(t))}^{2} + ∫_{0}^{t} |∇β(u(τ))|_{L^{2}(Ω_{m}(τ))}^{2}dτ + (n_{0} − ε_{1}) ∫_{0}^{t} |β(u(τ))|_{L^{2}(Γ_{m}(τ))}^{2}dτ

≤ \frac{1}{2}|f|_{L^{2}(Q_{m})}^{2} + \frac{1}{2}|β(u)|_{L^{2}(Q_{m})}^{2} + \frac{1}{4ε_{1}} |q|_{L^{2}(Σ_{m})}^{2} + c_{3}|u_{0}|_{L^{2}(Ω_{m0})}^{2} + c_{2}vol(Q_{m}),

By choosing ε_{1} = n_{0} we get (3.7) with (3.6). □

It should be noticed that the estimates for weak solutions obtained by Lemmas 3.1 and 3.2 do not depend on the vector field v, and this is a remarkable aspect for weak solutions of (SP; v) along with the next lemma.

Given any positive number R, we consider the set

U_{R} := \{u ∈ L^{∞}(Q_{m}); \begin{matrix}u is a weak solution of (SP; v) for v ∈ C^{2}(\overline{Q_{m}}) with \\ ∫_{0}^{T} |v(t)|_{H_{σ}^{1}(Ω_{m}(t))}^{2}dt ≤ R \end{matrix} \} .

Also, by using the coordinate transformation x(t, ·) : \overline{Ω_{m0}} → \overline{Ω_{m}(t)} given in assumption (A1), any function w on Q_{m} is transformed into a function w̄ on the cylindrical domain Q_{T} := (0, T) × Ω_{m0} as follows:

w̄(t, y) = w(t, x(t, y)) for (t, y) ∈ Q_{T} .

With this notation we define the set

\overline{U}_{R} := \{ū; u ∈ U_{R}\}.

For simplicity we use the following notation: H := L^{2}(Ω_{m0}), V := H^{1}(Ω_{m0}), V^{∗} is the dual space of V with ⟨·, ·⟩_{V^{∗},V} for the duality pair between V^{∗} and V.

Lemma 3.3. For any positive number R, the set \overline{U}_{R} is relatively compact in C([0, T]; V^{∗}).

Proof. The variational identity (2.7) is written in the coordinate t = t, y = y(t, x) as follows:

∫_{0}^{T} ⟨ū′(τ), ξ⟩_{V^{∗},V} dτ + ∫_{0}^{T} ⟨K(τ, β(ū(τ))), ξ⟩_{V^{∗},V} dτ + ∫_{0}^{T} ⟨G(t, ū(τ), v̄(τ)), ξ⟩_{V^{∗},V} dτ

= ∫_{0}^{T} ⟨f̃^{∗}(τ), ξ⟩_{V^{∗},V} dτ for all ξ ∈ H^{1}(Q_{T}),  (3.8)

where for each τ ∈ [0, T], the operator K(τ, ·) : V → V^{∗} is defined by

⟨K(τ, z_{1}), z⟩_{V^{∗},V} := ∫_{Ω_{m0}} ∑_{i,j=1}^{3} a_{ij}(τ)\frac{∂z_{1}}{∂y_{j}} \frac{∂z}{∂y_{i}} dy + ∫_{Ω_{m0}} ∑_{j=1}^{3} b_{j}(τ)\frac{∂z_{1}}{∂y_{j}} zdy

+ n_{0} ∫_{Γ_{m0}} z_{1} \frac{|R_{x}(τ)|}{|detJ_{x(τ)}|}zdΓ_{m0} for all z, z_{1} ∈ V,

with coefficients

a_{ij}(τ, y) := ∑_{k=1}^{3} \frac{∂y_{i}}{∂x_{k}} (τ, x(τ, y))\frac{∂y_{j}}{∂x_{k}} (τ, x(τ, y)) for (τ, y) ∈ Q_{T} , i, j = 1, 2, 3,

b_{j}(τ, y) := ∑_{k,i=1}^{3} \frac{∂}{∂y_{i}} (\frac{∂y_{i}}{∂x_{k}} (τ, x(τ, y))) \frac{∂y_{j}}{∂x_{k}} (τ, x(τ, y)) for (τ, y) ∈ Q_{T} , j = 1, 2, 3,

Here, J_{x(τ)} is the Jacobian matrix of x(τ, ·) with its determinant |detJ_{x(τ)}|, |R_{x(τ)}| is the ratio between the surface elements dΓ_{m}(τ) and dΓ_{m0} which is determined by the restriction of x(τ, ·); hence for each (τ, x) ∈ Q_{m} and y = y(τ, x) ∈ Ω_{m0} we have the relations dx = |detJ_{x(τ)}|dy and dΓ_{m}(τ) := |R_{x(τ)}|dΓ_{m0}. For each τ ∈ [0, T], the operator G(τ, ·, v̄(τ)) : H → V^{∗} is defined by

⟨G(τ, z_{1}, v̄(τ)), z⟩_{V^{∗},V} := − ∫_{Ω_{m0}} z_{1}(ṽ(τ) · ∇_{y}z)dy − ∫_{Ω_{m0}} c(τ)z_{1}zdy

+ ∫_{Ω_{m0}} ∑_{i,j=1}^{3} \{\frac{∂}{∂y_{j}} (\frac{∂x_{i}}{∂t} (τ, ·)) \frac{∂y_{j}}{∂x_{i}} (τ, x(τ, ·))\} z_{1}zdy for all z ∈ V, z_{1} ∈ H, (3.9)

with the vector field ṽ := (ṽ_{1}, ṽ_{2}, ṽ_{3}) defined by

ṽ_{j}(τ, y) := ∑_{i=1}^{3} (v̄_{i}(τ, y) − \frac{∂x_{i}}{∂τ} (τ, y)) \frac{∂y_{j}}{∂x_{i}} (τ, x(τ, y)) for j = 1, 2, 3,

c(τ, y) := ∑_{i,j=1}^{3} (v̄_{i}(τ, y) − \frac{∂x_{i}}{∂τ} (τ, y)) (τ, y)\frac{∂}{∂y_{i}} \{\frac{∂y_{i}}{∂x_{j}} (τ, x(τ, y))\} ,

for all (τ, y) ∈ Q_{T}. As a remark, the last term in (3.9) vanishes in the case when the mass of the time dependent domain is conserved. f̃^{∗} ∈ L^{2}(0, T; V^{∗}) is defined by

⟨f̃^{∗}(τ), z⟩_{V^{∗},V} := ∫_{Ω_{m0}} f̄(τ)zdy + ∫_{Γ_{m0}} q̄(τ)\frac{|R_{x(τ)}|}{|detJ_{x(τ)}|}zdΓ_{m0} for all z ∈ V.

By the above expressions of K(·, ·), G(·, ·, ·) and f̃^{∗}, there exists a positive constant M_{3} independent of v such that

|⟨K(τ, β(ū(τ))), z⟩_{V^{∗},V} | ≤ M_{3}|β(ū(τ))|_{V} |z|_{V} ,
|⟨G(τ, ū(τ), v̄(τ)), z⟩_{V^{∗},V} | ≤ M_{3}(|v̄(τ)|_{L^{2}(Ω_{m0})} + 1)|ū(τ)|_{L^{∞}(Ω_{m0})}|z|_{V} ,
|⟨f̃^{∗}(τ), z⟩_{V^{∗},V} | ≤ M_{3}(|f̄(τ)|_{H} + |q̄(τ)|_{L^{2}(Γ_{m0})})|z|_{V} for all z ∈ V and τ ∈ [0, T].

Now, on account of these estimates and those of Lemmas 3.1 and 3.2 we derive from (3.8) that

∫_{0}^{T} ⟨ū′, ξ⟩_{V^{∗},V} dτ ≤ M_{3}(M′_{2}(T) + M′_{1}(T)|v̄|_{L^{2}(0,T;L^{2}(Ω_{m0}))} + M′_{1}(T)T
+ |f̄|_{L^{2}(0,T;H)} + |q̄|_{L^{2}(0,T;L^{2}(Γ_{m0}))})|ξ|_{L^{2}(0,T;V)}.  (3.10)

This shows that \{ū′; ū ∈ \overline{U}_{R}\} is bounded in L^{2}(0, T; V^{∗}). Since L^{∞}(Ω_{m0}) is embbeded compactly in V^{∗} and \overline{U}_{R} is bounded in C([0, T]; V^{∗}), it follows from the Ascoli-Arzela theorem that \overline{U}_{R} is relatively compact in C([0, T]; V^{∗}). □

Corollary 3.1. For any positive number R, the set {β(u); u ∈ U_{R}} is relatively compact in L^{2}(Q_{m}). Proof. Let \{ū_{n}\} be any sequence in \overline{U}_{R}. By Lemma 3.3, we may assume that ū_{n}(t) converges in V^{∗} uniformly in t ∈ [0, T]. Also, it is easy to see from Lemmas 3.1 and 3.2 that {β(ū_{n})} is bounded in L^{2}(0, T; V). Therefore

lim_{n,m→∞} ∫_{Q_{T}} (ū_{n} − ū_{m})(β(ū_{n}) − β(ū_{m}))dydt = lim_{n,m→∞} ∫_{0}^{T} ⟨ū_{n} − ū_{m}, β(ū_{n}) − β(ū_{m})⟩_{V^{∗},V} dt
= 0.  (3.11)

Noting that

(r_{1} − r_{2})(β(r_{1}) − β(r_{2})) ≥ min{k_{ℓ}, k_{s}}|β(r_{1}) − β(r_{2})|^{2} for all r_{1}, r_{2} ∈ R,

we conclude by (3.11) that

lim_{n,m→∞} ∫_{Q_{T}} |β(ū_{n}) − β(ū_{m})|^{2}dydt = 0,

or equivalently

lim_{n,m→∞} ∫_{Q_{m}} |β(u_{n}) − β(u_{m})|^{2}dxdt = 0.

Thus {β(u); u ∈ U_{R}} is relatively compact in L^{2}(Q_{m}). □

4 Proofs of the main theorems

In this section we prove our main theorems.

Proof of Theorem 2.1. Given v satisfying assumption (A2), we take a sequence {v_{n}} ⊂ C^{2}(\overline{Q_{m}}) such that each v_{n} satisfies the same properties as in assumption (A2) and

lim_{n→∞} ∫_{0}^{T} |v_{n}(t) − v(t)|_{H_{σ}^{1}(Ω_{m}(t))}^{2}dt = 0.  (4.1)

We then observe from Lemmas 3.1, 3.2 and Corollary 3.1 that {u_{n}} is bounded in L^{∞}(Q_{m}), {β(u_{n})} is relatively compact in L^{2}(Q_{m}) and {∫_{Q_{m}} |∇β(u_{n})|^{2}dxdt} is bounded. Therefore, we may assume that

u_{n} → u weakly- ∗ in L^{∞}(Q_{m}),  (4.2)
β(u_{n}) → β(u) in L^{2}(Q_{m}),  (4.3)

and

∇β(u_{n}) → ∇β(u) weakly in L^{2}(Q_{m}) as n → +∞,  (4.4)

for a certain function u ∈ L^{∞}(Q_{m}). For each n the following variational identity holds:

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

(4.5)

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

Letting n → ∞ in (4.5) and using convergences (4.1)-(4.4), we have immediately (2.7) in Definition 2.1. By virtue of Lemmas 3.1 and 3.2, the required estimate of the form (2.8) is satisfied. □

Proof of Theorem 2.2. We put v̄(t, y) = v(t, x(t, y)) and ū(t, y) = u(t, x(t, y)) for (t, y) ∈ Q_{T} as well as v̄_{δ} and ū_{δ}, as before. By our assumption (2.9) and (2.10) we observe that

v̄_{δ} → v̄ weakly in L^{2}(0, T; H^{1}(Ω_{m0})) as δ → 0.  (4.6)

Here, recall the proof of Lemma 3.3 and especially (3.10). Then it is easy to check that the boundedness of {v̄_{δ}} in L^{2}(0, T; H^{1}(Ω_{m0})) implies that {ū_{δ}} is relatively compact in C([0, T]; V^{∗}), and hence by Corollary 3.1 that {β(u_{δ})} is relatively compact in L^{2}(Q_{m}). Accordingly, there exist a sequence {δ_{n}} with δ_{n} → 0 as n → +∞ and a function u ∈ L^{∞}(Q_{m}) such that

u_{n} := u_{δ_{n}} → u weakly- ∗ in L^{∞}(Q_{m}),  (4.7)
β(u_{n}) → β(u) in L^{2}(Q_{m}) as n → +∞.  (4.8)

Also, thanks to Lemma 3.2, we have

∇β(u_{n}) → ∇β(u) weakly in L^{2}(Q_{m}) as n → +∞.  (4.9)

Moreover, we have

∫_{Q_{m}} u_{n}(v_{n} · ∇η)dxdt → ∫_{Q_{m}} u(v · ∇η)dxdt for all η ∈ H^{1}(Q_{m}),  (4.10)

where v_{n} := v_{δ_{n}}. In fact, in order to get (4.10) it is enough to show that

∫_{Q_{T}} ū_{n}(v̄_{n} · ∇_{y}ξ)dydt → ∫_{Q_{T}} ū(v̄ · ∇_{y}ξ)dydt for all ξ ∈ H^{1}(Q_{T}).  (4.11)

This can be checked as follows: Given ξ ∈ H^{1}(Q_{T}), take a sequence {ξ_{m}} ⊂ C_{0}^{∞}(Q_{T}) such that ξ_{m} → ξ in L^{2}(0, T; V). Then, we have

|∫_{Q_{T}} ū_{n}(v̄_{n} · ∇_{y}ξ)dydt − ∫_{Q_{T}} ū(v̄ · ∇_{y}ξ)dydt|

≤ |∫_{Q_{T}} (ū_{n} − ū)(v̄_{n} · ∇_{y}(ξ − ξ_{m}))dydt − ∫_{Q_{T}} ū((v̄_{n} − v̄) · ∇_{y}(ξ − ξ_{m}))dydt
+ ∫_{Q_{T}} (ū_{n} − ū)(ṽ_{n} · ∇_{y}ξ_{m})dydt + ∫_{Q_{T}} ū((ṽ_{n} − ṽ) · ∇_{y}ξ_{m})dydt|
≤ {|ū_{n} − ū|_{L^{∞}(Q_{T})}|v̄_{n}|_{L^{2}(0,T;H^{1}(Ω_{m0}))} + |ū|_{L^{∞}(Q_{T})}|v̄_{n} − v̄|_{L^{2}(0,T;H^{1}(Ω_{m0}))}} |ξ_{m} − ξ|_{L^{2}(0,T;V)}
+ |∫_{0}^{T} ⟨ū_{n} − ū, v̄_{n} · ∇_{y}ξ_{m}⟩_{V^{∗},V} dt| + |∫_{0}^{T} ∑_{i=1}^{3} ⟨ū\frac{∂ξ_{m}}{∂y_{i}} , v̄_{ni} − v̄_{i}⟩_{V^{∗},V}dt| ,  (4.12)

where v̄_{ni} and v̄_{i} are the i-th component of vector fields v_{n} and v, respectively. Here we note that ū_{n} → ū in C([0, T]; V^{∗}) and v̄_{n} → v̄ weakly in L^{2}(0, T; H^{1}(Ω_{m0})), which is seen from (2.9) and (2.10). According to these convergences the last two integrals of (4.12) converge to 0 as n → +∞. Therefore, fix m and pass to the limit in n to get

lim sup_{u→∞} |∫_{Q_{T}} ū_{n}(v̄_{n} · ∇_{y}ξ)dydt − ∫_{Q_{T}} ū(v̄ · ∇_{y}ξ)dydt| ≤ M_{4}|ξ_{m} − ξ|_{L^{2}(0,T;V)},

where M_{4} is a positive constant. Next, letting m → ∞, we have (4.11) and thus (4.10). Finally, just as in the proof of Theorem 2.1, we see from (4.7)-(4.10) that u is a weak solution of (SP; v) satisfying estimate (2.8) and convergence (2.12) as well as (2.13). □

References

[1] J. P. Aubin, Un théorème de compacité. C. R. Acad. Sci. Paris, 256(1963), 5042– 5044.

[2] J. R. Cannon, E. DiBenedetto and G. H. Knightly, The bidimensional Stefan problem with convection: The time dependent case, Comm. Partial Differential Equations, 14(1983), 1549–1604.

[3] E. Casella and M. Giangi, An analytical and numerical study of the Stefan problem with convection by means of an enthalpy method, Math. Methods Appl. Sci., 24(2001), 623–639.

[4] E. DiBenedetto and M. O’Leary, Three-dimensional conduction-convection problems with change of phase, Arch. Ration. Mech. Anal., 123(1993), 99–116.

[5] T. Fukao and N. Kenmochi, Stefan problems with convection governed by Navier-Stokes equation, Technical Reports of Mathematical Sciences, Chiba Univ., No. 5, 19(2003).

[6] T. Fukao, N. Kenmochi and I. Pawłow, Stefan problems in non-cylindrical domains arising in Czochralski process of crystal growth, Control Cybernet., 32(2003), 201– 221.

[7] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Edu, Chiba Univ., 30(1981), 1–87.

[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, Amer. Math. Soci., 1968.

[9] I. Pawłow, Three-phase boundary Czochralski model, pp.203–227, in Mathematical Aspects of Modeling Structure Formation Phenomena, GAKUTO Internat. Ser. Math. Sci. Appl., Vol.17, Gakkōtosho, Tokyo, 2002.

[10] J. F. Rodrigues, Variational methods in the Stefan problem, pp.147–212 in Phase Transitions and Hysteresis, Lecture Notes Math., Vol.1584, Springer-Verlag, 1994.

[11] J. F. Rodrigues and F. H. Yi, On a two-phase continuous casting Stefan problem with nonlinear flux, European J. Appl. Math., 1(1990), 259–278.

[12] A. Visintin, Models of phase transitions, PNLDE, Birkhäuser, Boston, 1996.

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