ice formation in the arctic during summer: false-bottoms

Applied Mathematics and Computation 227 (2014) 857–870

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Applied Mathematics and Computation

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Ice formation in the Arctic during summer: False-bottoms

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.11.003

⇑ Corresponding author.E-mail addresses: [email protected], [email protected] (P.N. Dinh Alain).

Phan Thanh Nam a,b, Pham Ngoc Dinh Alain a,⇑, Dang Duc Trong b, Pham Hoang Quan b

a Mathematics Department, MAPMO UMR 6628, BP 67-59, 45067 Orleans cedex, Franceb Department of Mathematics, HoChiMinh City National University, Viet Nam

a r t i c l e i n f o

Keywords:False-bottomsFree boundary problemGreen’s functionContraction Mapping Principle

a b s t r a c t

The only source of ice formation in the Arctic during summer is a layer of ice called false-bottoms between an under-ice melt pond and the underlying ocean. Of interest is to give amathematical model in order to determine the simultaneous growth and ablation of false-bottoms, which is governed by both of heat fluxes and salt fluxes. In one dimension, thisproblem may be considered mathematically as a two-phase Stefan problem with two freeboundaries. Our main result is to prove the existence and uniqueness of the solution fromthe initial condition.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

There are some different reservoirs of fresh water in the Arctic during summer (see, e.g., Eicken et al. [2]). First, melt watercollects in surface melt pond (melting under the sun) which is the most important reservoir. Second, this melt water canpercolate into the ice matrix to form an under-ice melt pond (see [5] for more detail). At the interface between this fresh waterand the underlying salt water, double-diffusive convection of heat and salt leads to the formation of a layer of ice called false-bottoms (see Fig. 1 below). Very early, Nansen [8] in 1897 noted that this is the only source of forming new ice in the Arcticduring the summer. This phenomenon has been considered for a long time by many authors (see, e.g., [1,2,5–7,9–11]). How-ever, it has been considered in geophysical view-point based on practical experiments rather than rigorously mathematicalformulations.

One of the most interesting ones is the simultaneous growth and ablation of false-bottoms, which is governed by both ofheat fluxes and salt fluxes. The ablation of the sea-ice interface is caused by dissolution rather than by melting. Note that saltwater has the double properties: it does not freeze even for temperature less than 0 �C, and it dissolves ice when it is incontact with ice. McPhee et al. [7] emphasized that properly describing heat and salt flux at the ice-ocean interface is essen-tial for understanding and modeling the false-bottoms, and in particular without the double diffusion at this interface falsebottoms would be so short-lived. The growth of the upper interface between a false bottom and a under-ice melt pond isgoverned by the purely thermodynamic condition at the interface.

Recently, in 2003, Notz et al. [9] gave a model simulating successfully the simultaneous growth and ablation of false-bottoms. They formulated mathematically the problem by a system of partial differential equations and solved themnumerically by using a numerical routine in Mathematica. Although their numerical result fits quite well to early experimen-tal data from Martin and Kauffman [6], a rigorous proof of the existence and uniqueness of the solution for the system ofequations is still unavailable. Our aim in this paper is to give a such a mathematical proof. More precisely, we shall representthe problem explicitly by a system of partial differential equations associated with free boundary conditions similar to [9],and then show that the system has a unique solution from given initial conditions.

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Fig. 1. Ice formation in the Arctic during summer.

858 P.T. Nam et al. / Applied Mathematics and Computation 227 (2014) 857–870

Now we consider a one-dimensional model describing the simultaneous growth and ablation of the ice of false-bottoms.Here we have three environments: the ocean (Oc), the ice of false-bottoms (Fb) and the fresh water (Wa). Denote byTðx; tÞ; Sðx; tÞ the temperature and the salinity, and denote by h0ðtÞ;huðtÞ the free boundaries at the interfaces ice-ocean(Fb–Oc) and ice-water (Fb–Wa), respectively (see Fig. 2 below).

At the interface Fb–Oc, we apply the first principle of thermodynamics, (i.e. variation of energy = variation of heat fluxthrough the interface)

DU ¼ DU:

The net amount of heat transferred through the interface Fb–Oc in a section s is equal to

DU ¼ �dh0qILf s;

where qI is the density of the ice and Lf is the latent heat of fusion. On the other hand, the difference of the heat fluxesthrough a section s in the ice and the ocean during a time dt is

DU ¼ ð�kITxðh0ðtÞþ; tÞ þ kOTxðh0ðtÞ�; tÞÞsdt;

where kI; kO are thermal conductivities of the ice and the ocean. Here the notations h0ðtÞþ and h0ðtÞ�stand for the right limitand the left limit at x ¼ h0ðtÞ. Thus the law of conservation of energy mentioned above, i.e. DU ¼ DU, leads to the Stefancondition for the heat balance at the interface

h00ðtÞ ¼ ekITxðh0ðtÞþ; tÞ � ekOTxðh0ðtÞ�; tÞ; ð1:1Þ

where

ekI ¼kI

qILf> 0; ekO ¼

kO

qILf> 0:

For simplicity, we can neglect the salt of the ice of false-bottoms. The water near the interface Fb–Oc is a mixture of meltwater, which melts from the ice of false-bottoms, and sea water. This water freshens at the rate S0ðtÞh00ðtÞ, while salt diffusesinto this water at the rate �DSxðh0ðtÞ�; tÞ, where S0ðtÞ ¼ Sðh0ðtÞ�; tÞ is the salinity of the ocean at the interface and D > 0 isthe molecular diffusivity of salt in sea water. The balance of salt at this interface leads to the conservation condition

Fig. 2. One-dimensional model.

P.T. Nam et al. / Applied Mathematics and Computation 227 (2014) 857–870 859

S0ðtÞh00ðtÞ ¼ �DSxðh0ðtÞ�; tÞ: ð1:2Þ

Moreover, the interface temperature T0 ¼ Tðh0ðtÞ; tÞ and salinity S0 are connected via freezing-point relationship

T0 þ n0T 00 ¼ �m0S0; ð1:3Þ

where m0;n0 are non-zero constants. The establishment of Eqs. (1.1) and (1.2) bases on Martin and Kauffman [6] and Notzet al. [9]. In (1.3), if n0 ¼ 0 and m0 ¼ 0:054 �C psu�1 then we obtain the approximation T0 � �m0S0, which was used in [6,9].However, as we shall show later, it is reasonable to assume that the salinity is dependent on the gradient of the temperatureat the freezing point.

At the interface Fb–Wa, we use the simplified scheme presented by Grenfell and Maykut [4], in which the temperature ofthe water in the under-ice melt pond is kept at 0 �C. In particular, it leads to the boundary condition at the interface

TðhuðtÞ�; tÞ ¼ 0: ð1:4Þ

Furthermore, due to the neglect of heat fluxes from the fresh water above the false-bottoms, a thermodynamic conditionsimilar to (1.1) reduces to the Stefan condition at the upper surface

h0uðtÞ ¼ ekITxðhuðtÞ�; tÞ: ð1:5Þ

Finally, we have the following diffusion equations for salt and heat in the ocean and in the ice

Tt ¼ DITxx; huðtÞ > x > h0ðtÞ; ð1:6Þ

Tt ¼ DOTxx; h0ðtÞ > x > �1; ð1:7Þ

St ¼ DSxx; h0ðtÞ > x > �1; ð1:8Þ

where DI > 0 and DO > 0 are the thermal diffusivity. Here all mentioned constants are given.Assume that the initial conditions h0ð0Þ ¼ h0

0; huð0Þ ¼ h0u; Tðx;0Þ ¼ T0ðxÞ in huð0Þ > x > �1 and Sðx;0Þ ¼ S0ðxÞ in

h0ð0Þ > x > �1 are given. The problem is of finding from the initial conditions ðh00;h

0u; T

0; S0Þ a solution ðh0;hu; T; SÞ of thesystem (1.1)–(1.8), where T ¼ Tðx; tÞ in t > 0;huðtÞ > x > �1 and S ¼ Sðx; tÞ in t > 0;h0ðtÞ > x > �1. This is a one-dimen-sional two-phase Stefan problem with two free boundaries.

The main result of the present paper is to prove the existence and uniqueness a local solution for the problem. Rigorously,we suppose that the initial conditions ðh0

0;h0u; T

0; S0Þ satisfy the following three assumptions

(H1) h00 < h0

u;(H2) ðT0Þx is continuous and bounded in x 2 ð�1;h0

0� and x 2 ½h00;h

0u�; T0 is continuous at x ¼ h0

0 and T0ðh0u�Þ ¼ 0;

(H3) S0 is continuous and bounded in x 2 ð�1;h00�.

Remark 1.1. The condition (H1) means the ice layer of the false-bottom has already existed. It is of course a much moreinteresting problem to consider the behavior of the false-bottom at the starting time, which corresponds to conditionh0

0 ¼ h0u, but it is extremely difficult. We mention that the difficulty does not only come from mathematical computation

but also be recognized by physical experiments. For example, Notz et al. [9] simulated the experiment of Martin and Kauff-man [6], in which they put fresh water at 0 �C on top of salt water at its freezing point in order to simulate the evolution of afalse-bottoms, but this model does not take salt transport through the false-bottom into account. Therefore they then startedwith a 5-cm layer of ice.

Remark 1.2. It is implicit from (1.1) and (1.5) that Txð:; tÞ is continuous at x ¼ h0ðtÞ�;h0ðtÞþ and huðtÞ�. Therefore, it isreasonable to assume that ðT0Þx is continuous also at x ¼ h0

0�;h00þ and h0

u� in (H2). Similarly, it is natural to assume thatT0 is continuous at x ¼ h0

0 and T0ðh0u�Þ ¼ 0 due to (1.3) and (1.4).

Let r > 0. We say that ðh0;hu; T; SÞ is a solution of the system (1.1)–(1.8) in ð0;rÞ corresponding to the initial conditionsðh0

0;h0u; T

0; S0Þ if the following four conditions holds

(C1) h0ðtÞ and huðtÞ are continuously differentiable in t 2 ½0;rÞ; h0ð0Þ ¼ h00 < h0

u ¼ huð0Þ and h0ðtÞ < huðtÞ in t 2 ð0;rÞ;(C2) Tt ; Txx is continuous in t 2 ð0;rÞ; x 2 ð�1;h0ðtÞÞ [ ðh0ðtÞ;huðtÞÞ; Txð:; tÞ is continuous at x ¼ h0ðtÞ�; x ¼ h0ðtÞþ and

x ¼ huðtÞ�; Tð:; tÞ is continuous at x ¼ h0ðtÞ; Tðx; :Þ is continuous at t ¼ 0þ and Tðx;0þÞ ¼ T0ðxÞ in x 2 ð�1;h0uÞ;

(C3) St ; Sxx is continuous in t 2 ð0;rÞ; x 2 ð�1;h0ðtÞÞ; Sxð:; tÞ is continuous at x ¼ h0ðtÞ�; Sðx; :Þ is continuous at t ¼ 0 andSðx;0þÞ ¼ S0ðxÞ in x 2 ð�1; h0

0Þ;(C4) Eqs. (1.1)–(1.8) hold in t 2 ð0;rÞ.

In general, the conditions (C1)–(C4) assure a classical solution for the system of differential Eqs. (1.1)–(1.8). Our mainresult is as follows.


Theorem 1.1. Assume that ðh00;h

0u; T

0; S0Þ satisfy (H1)–(H3). Then there is a unique solution ðh0; hu; T; SÞ of the system (1.1)–(1.8) in ð0;rÞ corresponding to the initial conditions ðh0

0;h0u; T

0; S0Þ for some r > 0. Moreover, this solution can be extendeduniquely whenever the condition huðrÞ > h0ðrÞ still holds.

Our proof follows the approach of Friedman [3, Chapter 8], which dealt with a classical one-phase Stefan problem withone free boundary. More precisely, we first reduce the problem to solving a system of nonlinear Volterra integral equationsof the second kind and then solve this system by contraction principle. The remainder of the paper is divided into there sec-tions. Section 2 is devoted to some preliminary results on Green functions and Volterra integral equations. In Section 3 weshall reformulate the problem to a system of nonlinear integral equations. In Section 4 we apply the contraction principle toprove the existence and uniqueness of a local solution for this system.

2. Preliminaries

Let aðtÞ; bðtÞ be continuously differentiable functions and bðtÞ > aðtÞ for all t P 0. Let j > 0 be a constant and let uðx; tÞ bea solution of the diffusion equation

@u@t¼ j

@2u@x2 ; t > 0; bðtÞ > x > aðtÞ: ð2:1Þ

We introduce the Green’s function of Eq. (2.1),

Gðx; t; n; sÞ ¼ Hðt � sÞ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipjðt � sÞ

p exp � ðx� nÞ2

4jðt � sÞ

!;

where H is the Heaviside function,

HðtÞ ¼1; if t > 0;0; if t < 0:

�
The following lemma is useful to reformulate the differential Eq. (2.1) to a Volterra integral equations of the second kind.
Lemma 2.1. If u is a solution of (2.1) then for t > 0 and aðtÞ < x < bðtÞ we have

uðx; tÞ ¼Z t

0Gðx; t; bðsÞ; sÞ½junðbðsÞ�; sÞ þ uðbðsÞ; sÞb0ðsÞ�ds�

Z t

0Gðx; t; aðsÞ; sÞ½junðaðsÞþ; sÞ þ uðaðsÞ; sÞa0ðsÞ�ds

� jZ t

0Gnðx; t; bðsÞ; sÞuðbðsÞ; sÞdsþ j

Z t

0Gnðx; t; aðsÞ; sÞuðaðsÞ; sÞdsþ

Z bð0Þ

að0ÞGðx; t; n;0Þuðn;0Þdn:

Proof. Note that

Gs þ jGnn ¼ 0 for all s < t; and Gðx; t; n; t�Þ ¼ dðx� nÞ; ð2:2Þ

where d ¼ H0 is Dirac delta function.Integrating the Green’s identity, here u ¼ uðn; sÞ,

j@

@nG@u@n� u

@G@n

� �� @

@s ðuGÞ ¼ 0

over the domain aðsÞ < n < bðsÞ;0 < s < t, we will obtain the desired result because
Z t
0

Z bðsÞ

aðsÞ

@

@nG@u@n� u

@K@n

� �dnds ¼

Z t

0G@u@n� u

@G@n

� �n¼bðsÞ

n¼aðsÞds

¼Z t

0Gðx; t; bðsÞ; sÞunðbðsÞ�; sÞds�

Z t

0Gðx; t; aðsÞ; sÞunðaðsÞþ; sÞds

�Z t

0Gnðx; t; bðsÞ; sÞuðbðsÞ; sÞdsþ

Z t

0Gnðx; t; aðsÞ; sÞuðaðsÞ; sÞds

and
Z t
0

Z bðsÞ

aðsÞ

@

@sðuGÞdnds¼

Z t

0

@

@s

Z bðsÞ

aðsÞuGdn

!�½uG�n¼bðsÞb

0ðsÞþ ½uG�n¼aðsÞa0ðsÞ

( )ds

¼Z bðsÞ

aðsÞuGdn

" #s¼t�

s¼0

�Z t

0½uG�n¼bðsÞb

0ðsÞdsþZ t

0½uG�n¼aðsÞa

0ðsÞds

¼uðx; tÞ�Z bð0Þ

að0ÞGðx;t;n;0Þuðn;0Þdn�

Z t

0Gðx;t;bðsÞ;sÞuðbðsÞ;sÞb0ðsÞdsþ

Z t

0Gðx;t;aðsÞ;sÞuðaðsÞ;sÞa0ðsÞds:


Here we have made use the following equality, with v ¼ uG,
Z bðsÞ
aðsÞ

@

@sðvðn; sÞÞdn ¼ @

@s

Z bðsÞ

aðsÞvðn; sÞdn

!� vðbðsÞ; sÞb0ðsÞ þ vðaðsÞ; sÞa0ðsÞ:

In fact, in the case aðtÞ � 0 we have
Z bðsÞ
0

@

@s ðvðn; sÞÞdn ¼Z 1

0HðbðsÞ � nÞ @

@s ðvðn; sÞÞdn ¼Z 1

0

@

@s ½HðbðsÞ � nÞvðn; sÞ�dn�Z 1

0

@

@s ½HðbðsÞ � nÞ�vðn; sÞdn

¼ @

@s

Z 1

0HðbðsÞ � nÞvðn; sÞdn

� ��Z 1

0dðbðsÞ � nÞb0ðsÞvðn; sÞdn

¼ @

@s

Z bðsÞ

0vðn; sÞdn

!� vðbðsÞ; sÞb0ðsÞ:

If aðtÞ is not constant, we can write
Z bðsÞ
aðsÞ

@

@sðvðn; sÞÞdn ¼

Z bðsÞ

0

@

@sðvðn; sÞÞdn�

Z aðsÞ

0

@

@sðvðn; sÞÞdn

and apply the above result. This completes the proof. h

Remark 2.1. The result in Lemma 2.1 still holds for bðtÞ � þ1 or aðtÞ � �1. For example, if aðtÞ � �1 then the formula inLemma 2.1 reduces to

uðx; tÞ ¼Z t

0Gðx; t; bðsÞ; sÞ½junðbðsÞ�; sÞ þ uðbðsÞ; sÞb0ðsÞ�ds� j

Z t

0Gnðx; t; bðsÞ; sÞuðbðsÞ; sÞds

þZ bð0Þ

�1Gðx; t; n;0Þuðn; 0Þdn:

We shall need also an useful lemma giving the jump relation at the boundary (see Friedman [3, p. 217, Lemma 1]).

Lemma 2.2. Let pðtÞ be continuous and let sðtÞ > 0 satisfy the Lipschitz condition, 0 6 t 6 r. Then, for 0 < t 6 r,

limx!sðtÞ�

jZ t

0pðsÞGxðx; t; sðsÞ; sÞds ¼ 1

2pðtÞ þ j

Z t

0pðsÞGxðsðtÞ; t; sðsÞ; sÞds:

Remark 2.2. In applications of the above lemma, sometimes we need to note that Gn ¼ �Gx. Moreover, for the right limit wehave

limx!sðtÞþ

jZ t

0pðsÞGxðx; t; sðsÞ; sÞds ¼ �1

2pðtÞ þ j

Z t

0pðsÞGxðsðtÞ; t; sðsÞ; sÞds:

Finally, we state a simple version of the uniqueness for a system of linear Volterra integral equations of the second kind.

Lemma 2.3. Let n 2 N; r > 0; p > 1; q > 1; 1=pþ 1=q ¼ 1. Assume that s # Wjðt; sÞ is measurable in ð0; tÞ for all t 2 ð0;r�and
Z t
0jWjðt; sÞjpds 6 const:; 8t 2 ð0;r�; j ¼ 1;n:

Then the system

WiðtÞ ¼Xn

j¼1

Z t

0Wjðt; sÞWjðsÞds

� �; 8t 2 ð0;r�; j ¼ 1;n;

has a unique solution fWjgnj¼1 ¼ 0 in Lqð0;rÞ.

Proof. Using Holder inequality one has

jWiðtÞj 6Xn

j¼1

Z t

0Wjðt; sÞWjðsÞds

�� 6Xn

j¼1

Z t

0jWjðt; sÞjpds

� �1=p Z t

0jWjðsÞjqds

� �1=q

6 const:Xn

j¼1

Z t

0jWjðsÞjqds

� �1=q

:


Therefore,
Xn
j¼1

jWjðsÞjq 6 const:Xn

j¼1

Z t

0jWjðsÞjqds

� �¼ const:

Z t

0

Xn

j¼1

jWjðsÞjq !

ds

and it follows from Gronwall’s Lemma thatPn

j¼1jWjðsÞjq ¼ 0. Thus fWjgnj¼1 ¼ 0. h

Remark 2.3. If, for example,

jWjðt; sÞj 6const:ffiffiffiffiffiffiffiffiffiffiffi

t � sp ; j ¼ 1;n:

then we can choose any p 2 ð1;2Þ in order to apply Lemma 2.3.

3. Reduction to integral equations

Denote by G1;G2;G3 the Green function G in Lemma 2.1 corresponding to j ¼ D;DO;DI , respectively. We shall reformulatethe problem (1.1)–(1.8) to a system of integral equations of time-depending functions ðv0;v1;v2;v3Þ where

v0ðtÞ ¼ T 00ðtÞ; v1ðtÞ ¼ Txðh0ðtÞ�; tÞ; v2ðtÞ ¼ Txðh0ðtÞþ; tÞ; v3ðtÞ ¼ TxðhuðtÞ�; tÞ: ð3:1Þ

Assume that ðh0;hu; T; SÞ is a solution of the system (1.1)–(1.8) in ð0;rÞ corresponding to the initial conditions ðh00;h

0u; T

0; S0Þfor some r > 0. Applying Lemma 2.1 to Eq. (1.8) with h0ðtÞ > x > �1 and using condition (1.2), one has

Sðx; tÞ ¼ �DZ t

0G1nðx; t; h0ðsÞ; sÞS0ðsÞdsþ

Z h00

�1G1ðx; t; n;0ÞS0ðnÞdn: ð3:2Þ

In particular, we see from (3.2) that Sðx; tÞ is determined completely by h0 and S0ðtÞ. Taking x! h0ðtÞ� in (3.2) and using thejump relation in Lemma 2.2, we get

S0ðtÞ ¼ �2DZ t

0G1nðh0ðtÞ; t; h0ðsÞ; sÞS0ðsÞdsþ 2

Z h00

�1G1ðh0ðtÞ; t; n;0ÞS0ðnÞdn: ð3:3Þ

Note that for v0ðtÞ ¼ T 00ðtÞ the condition (1.3) can be rewritten as

T0ðh00Þ þ

Z t

0v0ðsÞdsþ n0v0ðtÞ ¼ �m0S0ðtÞ: ð3:4Þ

Here we have used T0ð0Þ ¼ Tðh0ð0Þ;0Þ ¼ T0ðh00Þ. We deduce from (3.3) and (3.4) the first integral equation

v0ðtÞ ¼ �1n0

T0ðh00Þ �

1n0

Z t

0v0ðsÞds�

2m0

n0

Z h00

�1G1ðh0ðtÞ; t; n;0ÞS0ðnÞdn

� 2Dn0

Z t

0G1nðh0ðtÞ; t; h0ðsÞ; sÞ T0ðh0

0Þ þZ s

0v0ðsÞdsþ n0v0ðsÞ

� �ds: ð3:5Þ

We next apply Lemma 2.1 to Eq. (1.7) for h0ðtÞ > x > �1 to get

Tðx; tÞ ¼Z t

0G2ðx; t; h0ðsÞ; sÞ½DOv1ðsÞ þ T0ðsÞh00ðsÞ�ds� DO

Z t

0G2nðx; t; h0ðsÞ; sÞT0ðsÞds

þZ h0

0

�1G2ðx; t; n;0ÞT0ðnÞdn; ð3:6Þ

where v1ðtÞ ¼ Txðh0ðtÞ�; tÞ. We now differentiate both sides of (3.6) with respect to x, then take x! h0ðtÞ�. To go into thedetails, because DOG2nx ¼ �DOG2nn ¼ G2s, we have

�DO

Z t

0G2nxðx; t; h0ðsÞ; sÞT0ðsÞds ¼ �

Z t

0G2sðx; t; h0ðsÞ; sÞT0ðsÞds

¼ �Z t

0

@

@sðG2ðx; t; h0ðsÞ; sÞÞ � G2nðx; t; h0ðsÞ; sÞh00ðsÞ

� �T0ðsÞds

¼ G2ðx; t; h00;0ÞT0ð0Þ þ

Z t

0G2ðx; t; h0ðsÞ; sÞT 00ðsÞds�

Z t

0G2xðx; t; h0ðsÞ; sÞT0ðsÞh00ðsÞds:

Moreover,
Z h00
�1G2xðx; t; n;0ÞT0ðnÞdn ¼ �

Z h00

�1G2nðx; t; n;0ÞT0ðnÞdn ¼ �G2ðx; t; h0

0;0ÞT0ðh0

0Þ þZ h0

0

�1G2ðx; t; n;0ÞT0

nðnÞdn:


Thus it follows from (3.6) that

Txðx; tÞ ¼ DO

Z t

0G2xðx; t; h0ðsÞ; sÞv1ðsÞdsþ

Z t

0G2ðx; t; h0ðsÞ; sÞv0ðsÞdsþ

Z h00


nðnÞdn ð3:7Þ

for all h0ðtÞ > x > �1. Here we have used the compatible condition T0ð0Þ ¼ Tðh0ð0Þ;0Þ ¼ T0ðh00Þ and replaced T00ðsÞ by v0ðsÞ.

Taking x! h0ðtÞ� in (3.7) and using Lemma 2.1 for the first term of the right hand side, we have the second integral equation

v1ðtÞ ¼ 2DO

Z t

0G2xðh0ðtÞ; t; h0ðsÞ; sÞv1ðsÞdsþ 2

Z t

0G2ðh0ðtÞ; t; h0ðsÞ; sÞv0ðsÞdsþ 2

Z h00

�1G2ðh0ðtÞ; t; n;0ÞT0

nðnÞdn: ð3:8Þ

Now we consider the heat distribution in false-bottoms. Applying Lemma 2.1 to Eq. (1.6) for huðtÞ > x > h0ðtÞ one has

Tðx; tÞ ¼DI

Z t

0G3ðx; t; huðsÞ; sÞv3ðsÞds�

Z t

0G3ðx; t; h0ðsÞ; sÞ½DIv2ðsÞ þ T0ðsÞh00ðsÞ�ds

þ DI

Z t

0G3nðx; t; h0ðsÞ; sÞT0ðsÞdsþ

Z h0u

h00

G3ðx; t; n;0ÞT0ðnÞdn; ð3:9Þ

where v2ðtÞ ¼ Txðh0ðtÞþ; tÞ and v3ðtÞ ¼ TxðhuðtÞ�; tÞ ¼ 1ekI

h0uðtÞ. Let us differentiate both sides of (3.9) with respect to x. Wehave

DI

Z t

0G3nxðx; t; h0ðsÞ; sÞT0ðsÞds ¼

Z t

0G3sðx; t; h0ðsÞ; sÞT0ðsÞds

¼Z t

0

@

@sðG3ðx; t; h0ðsÞ; sÞÞ � G3nðx; t; h0ðsÞ; sÞh00ðsÞ

� �T0ðsÞds

¼ �G3ðx; t; h00; 0ÞT0ð0Þ �

Z t

0G3ðx; t; h0ðsÞ; sÞT 00ðsÞdsþ

Z t

0G3xðx; t; h0ðsÞ; sÞT0ðsÞh00ðsÞds

and
Z h0u
h00

G3xðx; t; n;0ÞT0ðnÞdn ¼ �Z h0

u

h00

G3nðx; t; n;0ÞT0ðnÞdn

¼ �G3ðx; t; h0u;0ÞT

0ðh0u�Þ þ G3ðx; t; h0

0;0ÞT0ðh0

0�Þ þZ h0

u

h00

G3ðx; t; n;0ÞT0nðnÞdn:

Using the compatible conditions T0ðh0u�Þ ¼ 0 and T0ð0Þ ¼ T0ðh0

0Þ and replacing T 00ðsÞ by v0ðsÞ, we find from (3.9) that

Txðx; tÞ ¼ DI

Z t

0G3xðx; t; huðsÞ; sÞv3ðsÞds� DI

Z t

0G3xðx; t; h0ðsÞ; sÞv2ðsÞds�

Z t

0G3ðx; t; h0ðsÞ; sÞv0ðsÞds

þZ h0

u

h00

G3ðx; t; n; 0ÞT0nðnÞdn ð3:10Þ

for all huðtÞ > x > h0ðtÞ. Taking x! h0ðtÞþ and x! huðtÞ� in (3.10) and using Lemma 2.2 again we obtain the last two inte-gral equations

v2ðtÞ ¼2DI

Z t

0G3xðh0ðtÞ; t; huðsÞ; sÞv3ðsÞds� 2DI

Z t

0G3xðh0ðtÞ; t; h0ðsÞ; sÞv2ðsÞds

� 2Z t

0G3ðh0ðtÞ; t; h0ðsÞ; sÞv0ðsÞdsþ 2

Z h0u

h00

G3ðh0ðtÞ; t; n;0ÞT0nðnÞdn ð3:11Þ

and

v3ðtÞ ¼2DI

Z t

0G3xðhuðtÞ; t; huðsÞ; sÞv3ðsÞds� 2DI

Z t

0G3xðhuðtÞ; t; h0ðsÞ; sÞv2ðsÞds

� 2Z t

0G3ðhuðtÞ; t; h0ðsÞ; sÞv0ðsÞdsþ 2

Z h0u

h00

G3ðhuðtÞ; t; n;0ÞT0nðnÞdn: ð3:12Þ

Note that due to the interface condition (1.1) and (1.5), one has

h0ðtÞ ¼ h00 þ ekI

Z t

0v2ðsÞds� ekO

Z t

0v1ðsÞds ð3:13Þ

and


huðtÞ ¼ h0u þ ekI

Z t

0v3ðsÞds: ð3:14Þ

Therefore the Eqs. (3.5), (3.8), (3.11) and (3.12) form a system of nonlinear integral equations of the form v ¼ Pv , wherev ¼ ðv0;v1;v2;v3Þ and Pv ¼ ðP0v ; P1v ; P2v; P3vÞ. Note that the time-depending function v ¼ ðv0; v1;v2;v3Þ is first only con-tinuous in ð0;rÞ by its definition in (3.1). However, because it is a solution of the system of integral Eqs. (3.5), (3.8), (3.11)and (3.12), it is indeed continuous in ½0;r�. We thus have proved the direct part of the following statement.

Proposition 3.1. The problem of finding a solution ðh0;hu; T; SÞ of (1.1)–(1.8) in ð0;rÞ is equivalent to the problem of finding acontinuous solution v ¼ ðv0;v1;v2;v3Þ in ½0;r� for the system (3.5), (3.8), (3.11) and (3.12) such that huðtÞ > h0ðtÞ in ð0;rÞ,where h0 and hu are defined by (3.13) and (3.14).

We of course just need to prove the converse part of the statement.

Proof. Suppose that v ¼ ðv0;v1;v2;v3Þ is a continuous solution in ½0;r� for the system (3.5), (3.8), (3.11) and (3.12) suchthat huðtÞ > h0ðtÞ in ð0;rÞ; h0 and hu are defined by (3.13) and (3.14). We shall now recover the solution ðh0;hu; T; SÞ of (1.1)–(1.8).

Since h0 and hu have already been defined by (3.13) and (3.14), it remains to determine T and S. We define T0ðtÞ by

T0ðtÞ ¼ T0ðh00Þ þ

Z t

0v0ðsÞds ð3:15Þ

and determine naturally T by (3.6) and (3.9). Similarly, we define S0 by (3.4) and determine S by (3.2). We need to prove thatðh0;hu; T; SÞ satisfies (C1)–(C4).

Step 1. We first prove (C1)–(C3) except the behavior of T and S at the interfaces x ¼ h0ðtÞ and x ¼ huðtÞ�. In fact, the condi-tion (C1) follows the definition of h0 in (3.13), the definition of hu in (3.14), and the condition huðtÞ > h0ðtÞ in defi-nition of the solution v ¼ ðv0;v1;v2;v3Þ.Due to definition of S in the integral forms (3.2), St and Sxx are continuous in t 2 ð0;rÞ; x 2 ð�1;h0ðtÞÞ. Moreover, get-ting t ! 0þ in (3.2) and using limt!0þG1ðx; t; n;0Þ ¼ dðx� nÞ, we obtain the initial condition Sðx;0þÞ ¼ S0ðxÞ for allx 2 ð�1;h0

0Þ.Similarly, due to definition of T in (3.6) and (3.9), Tt and Txx are continuous in t 2 ð0;rÞ; x 2 ð�1;h0ðtÞÞ [ ðh0ðtÞ;huðtÞÞ, and Tðx;0þÞ ¼ T0ðxÞ for all x 2 ð�1;h0

0Þ [ ðh00;h

0uÞ.

Step 2. We next check three diffusion equations (1.6)–(1.8). To prove (1.8) from the definition of S in (3.2), we simply verifythat each term in the right-hand side of (3.2) is a homogeneous solution of the operator ð@=@t � D@2=@x2Þ int 2 ð0;rÞ; x 2 ð�1;h0ðtÞÞ. In fact, for the second term one it follows from the properties of Green function that

@

@t� D

@2

@x2

! Z h00

�1G1ðx; t; n;0ÞS0ðnÞdn

!¼Z h0

0

�1

@

@t� D

@2

@x2

!ðG1ðx; t; n;0ÞÞS0ðnÞdn ¼ 0:

For the first term, we have to be more careful because in general differentiating with respect to t a function of the formt #

R t0 Kðt; sÞds may cause a jump,

ddt

Z t

0Kðt; sÞds

� �¼ lim

s!t�Kðt; sÞ þ

Z t

0Ktðt; sÞds:

However, in this case the jump lims!t�

Kðt; sÞ vanishes since

lims!t�

G1ðx; t; n; sÞ ¼ lims!t�

G1nðx; t; n; sÞ ¼ 0; 8x – n:

Therefore, the first term can be treat similarly to the second term. Thus (1.8) holds. The proofs for (1.6) and (1.7) are thesame.Step 3. We now prove that Sðh0ðtÞ�; tÞ ¼ S0ðtÞ, which in particular accomplish (C3), and verify the Stefan condition (1.2).

In fact, (3.3) holds due to (3.4) and (3.5). On the other hand, taking x! h0ðtÞ� in (3.2) and using Lemma 2.2 one has

Sðh0ðtÞ�; tÞ ¼12

S0ðtÞ � DZ t

0G1nðh0ðtÞ; t; h0ðsÞ; sÞS0ðsÞdsþ

Z h00

�1G1ðh0ðtÞ; t; n;0ÞS0ðnÞdn:

It follows from the latter equation and (3.3) that Sðh0ðtÞ�; tÞ ¼ S0ðtÞ.Applying Lemma 2.1 to Eq. (1.8) for h0ðtÞ > x > �1 and using Sðh0ðtÞ�; tÞ ¼ S0ðtÞ, we have

Sðx; tÞ ¼Z t

0G1ðx; t; h0ðsÞ; sÞ½DSxðh0ðsÞ�; sÞ þ S0ðsÞh00ðsÞ�ds� D

Z t

0G1nðx; t; h0ðsÞ; sÞS0ðsÞdsþ

Z h00

�1G1ðx; t; n;0ÞS0ðnÞdn:

Comparing the latter equation to the original definition of S in (3.2), we obtain

Z t
0G1ðx; t; h0ðsÞ; sÞW1ðsÞds ¼ 0; x 2 ð�1; h0ðtÞÞ; ð3:16Þ

where W1ðtÞ ¼ DSxðh0ðtÞ�; tÞ þ S0ðtÞh00ðtÞ. We shall deduce from (3.16) that W1 � 0, which is equivalent to (1.2). Indeed, dif-ferentiating (3.16) with respect to x, then taking x! h0ðtÞ� and using Lemma 2.2 we get

W1ðtÞ ¼ �2Z t

0G1xðh0ðtÞ; t; h0ðsÞ; sÞW1ðsÞds:

Note that this is a linear Volterra integral equation of the second kind and

jG1xðh0ðtÞ; t; h0ðsÞ; sÞj 6const:ffiffiffiffiffiffiffiffiffiffiffi

t � sp :

Therefore, it follows from Lemma 2.3 that W1ðtÞ ¼ 0.Step 4. We prove that Txðh0ðtÞ�; tÞ ¼ v1ðtÞ; Txðh0ðtÞþ; tÞ ¼ v2ðtÞ and TxðhuðtÞ�; tÞ ¼ v3ðtÞ, which in particular imply the Ste-

fan conditions (1.1) and (1.5). Note that we have already had from (3.15) that v0ðtÞ ¼ T 00ðtÞ and T0ð0Þ ¼ T0ðh00Þ.

To prove TxðhuðtÞ�; tÞ ¼ v3ðtÞ from the definition of T in (3.9), we use the same process of getting (3.12) from (3.9). Infact, differentiating both sides of (3.9) with respect to x and then taking x! huðtÞ� we have

TxðhuðtÞ�; tÞ ¼12

v3ðtÞ þ DI

Z t

0G3xðhuðtÞ; t; huðsÞ; sÞv3ðsÞds� DI

Z t

0G3xðhuðtÞ; t; h0ðsÞ; sÞv2ðsÞds

�Z t

0G3ðhuðtÞ; t; h0ðsÞ; sÞT 00ðsÞdsþ

Z h0u

h00

G3ðhuðtÞ; t; n;0ÞT0nðnÞdn:

Comparing the latter equation to (3.12) and using v0ðsÞ ¼ T 00ðsÞ, we obtain TxðhuðtÞ�; tÞ ¼ v3ðtÞ. By the same way we findthat Txðh0ðtÞþ; tÞ ¼ v2ðtÞ and Txðh0ðtÞ�; tÞ ¼ v1ðtÞ.Step 5. Finally, we verify that Tðh0ðtÞ; tÞ ¼ T0ðtÞ and TðhuðtÞ�; tÞ ¼ 0 to accomplish (C2) and (C4).

Applying Lemma 2.1 to Eq. (1.7) for h0ðtÞ > x > �1 one has

Tðx; tÞ ¼Z t

0G2ðx; t; h0ðsÞ; sÞ½DOTxðh0ðsÞ�; sÞ þ Tðh0ðsÞ�; sÞh00ðsÞ�ds� DO

Z t

0G2nðx; t; h0ðsÞ; sÞTðh0ðsÞ�; sÞds

þZ h0

0

�1G2ðx; t; n;0ÞT0ðnÞdn:

Comparing the latter equation to the original definition of T in (3.6) and using Txðh0ðtÞ�; tÞ ¼ v1ðtÞ, we obtain
Z t
0½G2ðx; t; h0ðsÞ; sÞh00ðsÞ � DOG2nðx; t; h0ðsÞ; sÞ�W2ðsÞds ¼ 0; ð3:17Þ

where W2ðtÞ ¼ Tðh0ðtÞ�; tÞ � T0ðtÞ. We now prove W2 � 0 by using the same technique of dealing W1 � 0. Taking x! huðtÞ�in (3.17) and using Lemma 2.2, we find that

W2ðtÞ ¼ 2Z t

0½G2ðx; t; h0ðsÞ; sÞh00ðsÞ � DOG2nðx; t; h0ðsÞ; sÞ�W2ðsÞds:

Note that this is a linear Volterra integral equation of the second kind and

jG2ðx; t; h0ðsÞ; sÞh00ðsÞ � DOG2nðx; t; h0ðsÞ; sÞj 6const:ffiffiffiffiffiffiffiffiffiffiffi

t � sp :

Therefore, it follows from Lemma 2.3 that W2 � 0. Thus Tðh0ðtÞ�; tÞ ¼ T0ðtÞ.We shall use the same argument, in fact a little more complicated one, to deduce that Tðh0ðtÞþ; tÞ ¼ T0ðtÞ and

TðhuðtÞ; tÞ ¼ 0. Applying Lemma 2.1 to Eq. (1.6) for huðtÞ > x > h0ðtÞ and comparing to the original definition of T in (3.9), onehas
Z t
0½G3ðx; t; h0ðsÞ; sÞh00ðsÞ � DIG3nðx; t; h0ðsÞ; sÞ�W3ðsÞds ¼

Z t

0½G3ðx; t; huðsÞ; sÞh0uðsÞ

� DIG3nðx; t; huðsÞ; sÞ�W4ðsÞds; ð3:18Þ

where W3ðsÞ ¼ Tðh0ðtÞþ; tÞ � T0ðtÞ and W4ðtÞ ¼ TðhuðtÞ�; tÞ. Taking x! h0ðtÞþ and x! huðtÞ� in (3.18), we have

W3ðtÞ ¼ 2Z t

0½G3ðh0ðtÞ; t; h0ðsÞ; sÞh00ðsÞ � DIG3nðh0ðtÞ; t; h0ðsÞ; sÞ�W3ðsÞds� 2

Z t

0½G3ðh0ðtÞ; t; huðsÞ; sÞh0uðsÞ

� DIG3nðh0ðtÞ; t; huðsÞ; sÞ�W4ðsÞds ð3:19Þ


and

�W4ðtÞ ¼ �2Z t

0½G3ðhuðtÞ; t; h0ðsÞ; sÞh00ðsÞ � DIG3nðhuðtÞ; t; h0ðsÞ; sÞ�W3ðsÞdsþ 2

Z t

0½G3ðhuðtÞ; t; huðsÞ; sÞh0uðsÞ

� DIG3nðhuðtÞ; t; huðsÞ; sÞ�W4ðsÞds: ð3:20Þ

Note that Eqs. (3.19) and (3.20) form a system of linear Volterra integral equations of the second kind and

jG3ðhuðtÞ; t; h0ðsÞ; sÞh00ðsÞ � DIG3nðhuðtÞ; t; h0ðsÞ; sÞj 6const:ffiffiffiffiffiffiffiffiffiffiffi

t � sp ;

jG3ðh0ðtÞ; t; huðsÞ; sÞh0uðsÞ � DIG3nðh0ðtÞ; t; huðsÞ; sÞj 6const:ffiffiffiffiffiffiffiffiffiffiffi

t � sp :

We therefore deduce from Lemma 2.3 that W3 � W4 � 0. Thus TðhuðtÞ�; tÞ ¼ 0, i.e. (1.4), and Tðh0ðtÞ; tÞ ¼ T0ðtÞ, which accom-plishes (C2) and also implies (1.3) in viewing (3.4). The proof is completed. h

Remark 3.1. We learned the technique of proving W1 � 0 by (3.16) from Friedman [3]. However, there is a minor mistake in[3]. The formula (1.28) in page 221, i.e.
Z t
0Gnðx; t; sðsÞ; sÞuðsðsÞ; sÞds ¼ 0;

should be
Z t
0½Gnðx; t; sðsÞ; sÞ � Gðx; t; sðsÞ; sÞs0ðsÞ�uðsðsÞ; sÞds ¼ 0:

Fortunately, the conclusion uðsðtÞ; tÞ ¼ 0 still holds by the same argument.

4. Solving integral equations

Now we accomplish the proof of Theorem 1.1. We want to prove the system (3.5), (3.8), (3.11) and (3.12) has a uniquelocal solution v ¼ ðv0; v1;v2;v3Þ, and this solution can be extended uniquely by prolongation whenever the conditionhuðtÞ > h0ðtÞ still holds. Note that this system is of the form v ¼ Pv :¼ ðP0v ; P1v; P2v ; P3vÞ, where P0v ; P1v ; P2v and P3vare the right-hand sides of (3.5), (3.8), (3.11) and (3.12) respectively.

Denote by Cðr;MÞ the space of functions v ¼ ðv0;v1;v2;v3Þ continuous in ½0;r� and

kvk½0;r� :¼ maxt2½0;r�;j¼0;3

jv jðtÞj 6 M:

When v ¼ 0 then Pð0Þ depends only on the initial conditions ðh00;h

0u; T

0; S0Þ and is continuous on ½0;1Þ. Fix M > 0 large en-ough, says M P 2kPð0Þk½0;1�.

Proposition 4.1. There exists r > 0 depending on ðh00;h

0u; T

0; S0Þ such that P maps Cðt0;MÞ into itself and is a contraction.Note that Cðr;MÞ is a complete metric space. Therefore it follows from Proposition 4.1 and the Contraction Mapping Prin-

ciple that P has a unique fixed point v in Cðr;MÞ. It gives also the unique solution ðh0;hu; T; SÞ of (1.1)–(1.8) in ð0;rÞ due toProposition 3.1.

Proof. First at all, let us consider r > 0 small enough, says

r 6 h0u � h0

0

2ð2ekI þ ekOÞM:

Then for v 2 Cðr;MÞ, it follows from (3.13), (3.14) that

huðtÞ � h0ðsÞP h0u � h0

0 � jhuðtÞ � huð0Þj � jh0ðsÞ � h0ð0ÞjP h0u � h0

0 � ekIMt � ðekO þ ekIÞMs Ph0

u � h00

2> 0 ð4:1Þ

for all 0 6 0 6 t 6 r. In particular huðtÞ > h0ðtÞ for all t 2 ½0;r� and hence P is well-defined on Cðr;MÞ. We now estimatekPv � Pevk½0;r� for v; ev 2 Cðr;MÞ. In what follows, denote by ehu and eh0 the functions given by (3.13), (3.14) where v is replacedby ev . For simplicity, we shall always denote by C0 > 0 an universal constant depending only on the initial conditionsðh0

0;h0u; T

0; S0Þ. We shall go to the details for P3, and P0; P1; P2 can be treated by the same way. Recall that


ðP3vÞðtÞ ¼ 2DI

Z t


Z t

0G3xðhuðtÞ; t; h0ðsÞ; sÞv2ðsÞds� 2

Z t

0G3ðhuðtÞ; t; h0ðsÞ; sÞv0ðsÞds

þ 2Z h0

u

h00

G3ðhuðtÞ; t; n; 0ÞT0nðnÞdn:

We need some preliminary estimates related to the Green’s function G3. h

Lemma 4.1.For all v ; ev 2 Cðr;MÞ; 0 6 s < t 6 r and n 2 R, we have

jG3xðhuðtÞ; t; huðsÞ; sÞ � G3xðehuðtÞ; t; ehuðsÞ; sÞj 6C0ffiffiffiffiffiffiffiffiffiffiffit � sp kv � evk½0;r�; ð4:2Þ

jG3ðhuðtÞ; t; h0ðsÞ; sÞ � G3ðehuðtÞ; t; eh0ðsÞ; sÞj 6C0ffiffiffiffiffiffiffiffiffiffiffit � sp kv � evk½0;r�; ð4:3Þ

jG3xðhuðtÞ; t; h0ðsÞ; sÞ � G3xðehuðtÞ; t; eh0ðsÞ; sÞj 6C0ffiffiffiffiffiffiffiffiffiffiffit � sp kv � evk½0;r�; ð4:4Þ

jG3xðhuðtÞ; t; n; sÞ � G3xðehuðtÞ; t; n; sÞj 6 C0 exp �ðhuðtÞ � nÞ2

8DIt

!þ exp �ðhuðtÞ � nÞ2

8DIt

!" #kv � evk½0;r�: ð4:5Þ

Proof. Considering

G3xðhuðtÞ; t; huðsÞ; sÞ ¼ �huðtÞ � huðsÞ

4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipD3

I ðt � sÞ3q exp �ðhuðtÞ � huðsÞÞ2

4DIðt � sÞ

!

as a function of one-variable ðhuðtÞ � huðsÞÞ and using Lagrange formula one has

G3xðhuðtÞ; t; huðsÞ; sÞ � G3xðehuðtÞ; t; ehuðsÞ; sÞ ¼ �½ðhuðtÞ � huðsÞÞ � ðehuðtÞ � ehuðsÞÞ�


I ðt � sÞ3q

� exp � h2

4DIðt � sÞ

!1� 2h2

4DIðt � sÞ

!
for some h between ðhuðtÞ � huðsÞÞ and ðehuðtÞ � ehuðsÞÞ. Thus (4.2) follows
jðhuðtÞ � huðsÞÞ � ðehuðtÞ � ehuðsÞÞj 6 ekIðt � sÞkv � evk½0;r�
and the elementary inequality
e�zj1� 2zj 6 j1� 2zj1þ z

6 2; where z ¼ h2

4DIðt � sÞP 0:

To prove (4.3) we shall use the identity

G3ðhuðtÞ; t; h0ðsÞ; sÞ � G3ðehuðtÞ; t; eh0ðsÞ; sÞ ¼ �½ðhuðtÞ � h0ðsÞÞ � ðehuðtÞ � eh0ðsÞÞ�h1


I ðt � sÞ3q exp � h2

1

4DIðt � sÞ

!

for some h1 between ðhuðtÞ � h0ðsÞÞ and ðehuðtÞ � eh0ðsÞÞ. Note from (4.1) that jh1jP h0u�h0

02 . Therefore (4.3) follows

jðhuðtÞ � h0ðsÞÞ � ðehuðtÞ � eh0ðsÞÞj 6 ð2ekI þ ekOÞrkv � evk½0;r�
and the elementary inequality
e�zz 6 1; where z ¼ h21

4DIðt � sÞP 0:

We can also prove (4.4) by the same argument. In fact, we write


G3xðhuðtÞ; t; h0ðsÞ; sÞ � G3xðehuðtÞ; t; eh0ðsÞ; sÞ ¼ �½ðhuðtÞ � h0ðsÞÞ � ðehuðtÞ � eh0ðsÞÞ�


I ðt � sÞ3q

� exp � h22

4DIðt � sÞ

!1� 2h2

2

4DIðt � sÞ

!
for some h2 between ðhuðtÞ � h0ðsÞÞ and ðehuðtÞ � eh0ðsÞÞ and use
e�zzj1� 2zj 6 zj1� 2zj1þ zþ 1

2 z26 4; where z ¼ h2

2

4DIðt � sÞP 0:

Finally, for the last inequality (4.5) we write

G3ðhuðtÞ; t; n;0Þ � G3ðehuðtÞ; t; n; 0Þ ¼ �½huðtÞ � ehuðtÞ�h3

4ffiffiffiffiffiffiffiffiffiffiffiffiffipD3

I t3q exp � h2

3

4DIt

!

for some h3 between ðhuðtÞ � nÞ and ðehuðtÞ � nÞ. Note that

exp � h23

8DIt

!6 max exp �ðhuðtÞ � nÞ2

8DIt

!; exp �ð

ehuðtÞ � nÞ2

8DIt

0@ 1A8<:9=;:

Thus (4.5) follows

e�zffiffiffizp6

ffiffiffizp

1þ z6

12; where z ¼ h2

3

8DItP 0

and the estimate jhuðtÞ � ehuðtÞj 6 ekItkv � evk½0;r�. h

We now already to estimate kP3v � P3evk½0;r� for v ; ev 2 Cðr;MÞ. Recall that P3 is the sum of four terms

ðP3vÞðtÞ ¼ 2DI

Z t


Z t

0G3xðhuðtÞ; t; h0ðsÞ; sÞv2ðsÞds� 2

Z t

0G3ðhuðtÞ; t; h0ðsÞ; sÞv0ðsÞds

þ 2Z h0

u

h00

G3ðhuðtÞ; t; n; 0ÞT0nðnÞdn:

For the first term of P3, using (4.2) we have

jG3xðhuðtÞ; t; huðsÞ; sÞv3ðsÞ � G3xðehuðtÞ; t; ehuðsÞ; sÞev 3ðsÞj 6 jG3xðhuðtÞ; t; huðsÞ; sÞ � G3xðehuðtÞ; t; ehuðsÞ; sÞj:jv3ðsÞj

þ jG3xðehuðtÞ; t; ehuðsÞ; sÞj:jv3ðsÞ � ev 3ðsÞj

6C0Mffiffiffiffiffiffiffiffiffiffiffit � sp kv � evk½0;r� þ C0ffiffiffiffiffiffiffiffiffiffiffi

t � sp kv � evk½0;r�

¼ C0ðM þ 1Þffiffiffiffiffiffiffiffiffiffiffit � sp kv � evk½0;r�

for all 0 6 s 6 t 6 r. Therefore
Z t
0jG3xðhuðtÞ; t; huðsÞ; sÞv3ðsÞ � G3xðehuðtÞ; t; ehuðsÞ; sÞev 3ðsÞjds 6 C0ðM þ 1Þ

ffiffiffiffirpkv � evk½0;r�; t 2 ½0;r�:

We have also the same estimate for the second term and the third term of P3 due to (4.3) and (4.4). For the last term of P3, itfollows from (4.5) and the boundness of T0

x that
Z h0u
h00

jG3ðhuðtÞ; t; n; 0Þ � G3ðehuðtÞ; t; n; 0ÞjT0nðnÞdn 6 C0kv � evk½0;r� Z h0

u

h00

exp �ðhuðtÞ � nÞ2

8DIt

!þ exp �ð

ehuðtÞ � nÞ2

8DIt

0@ 1A24 35dn:

Note that by changing of variables
Z 1
�1exp �ðhuðtÞ � nÞ2

8DIt

!þ exp �ð

ehuðtÞ � nÞ2

8DIt

0@ 1A24 35dn ¼ 2Z 1

�1exp � n2

8DIt

!dn ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffip8DIt

p:


Thus
Z huð0Þ
h0ð0ÞjG3ðhuðtÞ; t; n;0Þ � G3ðehuðtÞ; t; n;0ÞjT0

nðnÞdn 6 2C0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip8DIr

pkv � evk½0;r�:

In summary, we have

kP3v � P3evk½0;r� 6 C0ffiffiffiffirpkv � evk½0;r�;

where C0 stands for a constant depending only on the initial conditions ðh00; h

0u; T

0; S0Þ. We have also the same estimates forP0; P1; P2. Thus for r > 0 small enough, says r 6 1

4C20, one has

kPv � Pevk½0;r� 6 12kv � evk½0;r�;8v; ev 2 Cðr;MÞ: ð4:6Þ

Note that M P 2kPð0Þk½0;1� P 2kPð0Þk½0;r�. Thus it follows from (4.6) that P maps from Cðr;MÞ into itself and is acontraction. h

To finish the proof of Theorem 1.1, we prove that the solution can be extended uniquely by prolongation wheneverhuðrÞ > h0ðrÞ. Proposition 3.1 allows us consider the problem of finding v ¼ ðv0; v1;v2;v3Þ instead of the problem of findingðh0;hu; T; SÞ. Assume that v ¼ ðv0;v1;v2;v3Þ is a solution of the system (3.5), (3.8), (3.11) and (3.12) in ½0;r� such thathuðtÞ > h0ðtÞ for all t 2 ½0;r�. From such solution v, we can extend naturally ðh0;hu; T; SÞ in ½0;r� by using the scheme ofthe proof of Proposition 3.1. We now want to check that ðh0ðrÞ;huðrÞ; Tð:;rÞ; Sð:;rÞÞ satisfy (H1)–(H3).

In fact, the condition (H1) is automatically satisfied. For (H2), the continuity of Txð:;rÞ in ð�1;h0ðrÞÞ [ ðh0ðrÞ;huðrÞÞ isguaranteed by (3.7) and (3.10), which still hold for t ¼ r. The behavior of T at the interface is also insured by the relationshipsTxðh0ðrÞ�;rÞ ¼ v1ðrÞ; Txðh0ðrÞþ;rÞ ¼ v2ðrÞ; TxðhuðrÞ�;rÞ ¼ v3ðrÞ, Tðh0ðrÞ;rÞ ¼ T0ðrÞ and TðhuðrÞ�;rÞ ¼ 0. Moreover, wededuce from (3.7) that

lim supx!�1

jTxðx;rÞj ¼ lim supx!�1

Z h0ð0Þ


nðnÞdn

�� 6 sup

h0ð0Þ>n>�1jT0

nðnÞj

and the boundness of Tx follows. Here we have used a property of the Green’s function
Z 1
�1jG2ðx; t; n;0Þjdn ¼ 1:

Thus (H2) is indeed true. Similarly, (H3) holds.By considering ðh0ðrÞ;huðrÞ; Tð:;rÞ; Sð:;rÞÞ as the new initial condition, we use Proposition 4.1 to extend uniquely the

solution v ¼ ðv0;v1; v2;v3Þ of the system (3.5), (3.8), (3.11) and (3.12) in ½0;r�� such that huðtÞ > h0ðtÞ for all t 2 ð0;r�Þ forsome r� > r. Finally, it follows from Proposition 3.1 that the extended solution v gives the extended solution ðh0;hu; T; SÞof the system (1.1)–(1.8) in ð0;r�Þ corresponding to the initial conditions ðh0

0; h0u; T

0; S0Þ. The proof is completed.

Remark 4.1. We may also replace the infinite domain huð0Þ > x > �1 of the initial conditions by a finite interval h0u > x > L

where L is a deep point in the ocean satisfying L < h0ðtÞ for 0 6 t 6 r. However, in this case it is necessary to require thehistory information ðTðL; :Þ; SðL; :ÞÞ;0 < t < r, at the point L. We can find the same result by using the Green’s function for thehalf-plane x > L,

Kjðx; t; n; sÞ ¼ Gjðx; t; n; sÞ � Gjð2L� x; t; n; sÞ; j ¼ 1;2;3:

Acknowledgments

The main part of the paper was accomplished when the first author was doing on the project of his Master thesis in Lab-oratory MAPMO, University d’Orleans in the summer 2008. He would like to express his heart-felt thanks to everybody of thelaboratory for the warm hospitality.

References

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ice formation in the arctic during summer: false-bottoms

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