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Electronic Journal of Differential Equations , Vol. 2003(2003), No. 109, pp. 125.ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE AND REGULARITY OF SOLUTIONS OF A PHASEFIELD MODEL FOR SOLIDIFICATION WITH CONVECTION OF

PURE MATERIALS IN TWO DIMENSIONS

JOS E LUIZ BOLDRINI & CRISTINA L UCIA DIAS VAZ

Abstract. We study the existence and regularity of weak solutions of a phaseeld type model for pure material solidication in presence of natural convec-

tion. We assume that the non-stationary solidication process occurs in a twodimensional bounded domain. The governing equations of the model are thephase eld equation coupled with a nonlinear heat equation and a modiedNavier-Stokes equation. These equations include buoyancy forces modelled byBoussinesq approximation and a Carman-Koseny term to model the ow inmushy regions. Since these modied Navier-Stokes equations only hold in thenon-solid regions, which are not known a priori, we have a free boundary-valueproblem.

1. Introduction

One of the rst papers to consider phase eld models applied to change of phaseswas written by Fix [ ? ]. This article fostered many other studies in this subject;see for instance the sequence of papers [ ? ], [? ], [? ], [? ]. Caginalp and others tookover the task of understanding the phase eld approach, both in its mathematicalaspects and in its relationship to the classical approach of using sharp interfaces toseparate the phases, which gives rise to what is known by Stefan type problems.We remark that, for the derivation of the kinetic equation for the phase eld,Caginalp and others used the free energy functional as a basis of the argument(see also Hoffman and Jiang [ ? ]). An alternative derivation, suggested by Peronseand Fife [? , ?], uses an entropy functional which gives a kinetic equation for thephase eld ensuring monotonic increase of the entropy in time. Peronse and Fifeexhibit a specic choice of entropy density which essentially recovers the phaseeld model employed by Caginalp [ ? ] by linearizing the heat ux. Thus, phase eld

models have a sound physical basis and provide simple and elegant descriptionsof phase transition processes. Moreover, it is more versatile than the enthalpymethod because it allows effects such as supercooling to be included. An important

2000 Mathematics Subject Classication. 76E06, 80A22, 82B26, 76D05.Key words and phrases. Phase-eld, phase transition, solidication, convection,Navier-Stokes equations.c 2003 Texas State University-San Marcos.

Submitted September 14, 2001. Published November 3, 2003.J. L. B. was partially supported by grant 300513/87-9 from CNPq, Brazil.

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2 JOS E LUIZ BOLDRINI & CRISTINA L UCIA DIAS VAZ EJDE2003/109

example of the utility of the phase eld approach is its use for the numerical studyof dendritic growth (see Caginalp [ ? ] and Kobayshi [ ? ]).

One point of importance is that many papers interested in the mathematicalanalysis of these models, whatever the approach used to model phase change, haveneglected the possibility of ow occurring in non solidied portions of the material.In many practical situations, however, this assumption is not satisfactory becausethe existence of such motions may affect in important ways the outcome of theprocess of phase change. In fact, melt convection adds new length and time scalesto the problem and results in morphologies that are potentially much different fromthose generated by purely diffusive heat and solute transport. Moreover, not onlydoes convection inuence the solidication pattern, but the evolving microstructurecan also trigger unexpected and complicated phenomena. For instance, Coriell et all [? ] and Davis [? ] studied in detail the coupled convective and morphologicalinstabilities at a growth front. This suggests that models that do not consider meltconvection may have some limitations.

One must realize, however, that the inclusion of this possibility brings anothervery difficult aspect to an already difficult problem. For this, it is enough to observethat such a ow must occur only in an a priori unknown non-solid region, and thusone may be left with a rather difficult free boundary-value problem.

In recent years, some authors have considered convective effects; for instance:Cannon et al [? , ?], DiBenedetto and Friedman [ ? ], DiBenedetto and OLeary[ ? ]and OLeary [ ? ], who addressed such questions by using weak formulations of theStefan type approach. Blanc et al. [? ], Pericleouns et al. [? ] and Voller et al. [? , ? ]considered convective effects in phase change problems by using the enthalpy ap-proach to describe change of phases, together with modied Navier-Stokes equationsto model the ow. In these works, the phases may be distinguished by the valuesof a variable corresponding to the solid fraction that is associated to the enthalpy;

this same variable is used in a term that is added to the Navier-Stokes equationsto cope with the inuence of the mushy zones in the ow. Particular expressionsfor this term may be obtained by modelling such mushy zones as porous media.Beckermann et al. [? ] and Diepers et al. [? ] used the phase eld methodology toobtain models including phase change and melt convection. By using numericalsimulations they studied the inuence of the convection in the melt on phenomenalike dendritic growth and coarsening. An interesting discussion of the applicationof diffusive-interface methods (phase eld being one of them) to uid mechanicscan be found in Anderson and McFadden [ ? ].

In this paper we are interested in the mathematical analysis of a model problemhaving some of the main aspects that a reasonable model for a solidication processwith convection should have. We will consider a rather simple situation of this

sort in the hope to obtain a better understanding of the mathematical difficultiesbrought by the coupling of terms describing phase change and the terms describingconvection. We also restrict the subject to the analysis of solidication of purematerials (the corresponding mathematical analysis for alloys will be consideredelsewhere.)

As in [? ] and [? ], we employ a phase eld methodology to model phase change;we also assume the solid phase to be rigid and stationary. However, differentlyfrom their models, convective effects will be included by using the ideas suggested


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EJDE2003/109 A PHASE FIELD MODEL FOR SOLIDIFICATION 3

by Blanc et al [? ] and Voller et al [? ]. Since the indicator of phase in these last pa-pers is the solid fraction, we relate the two approaches by postulating a functional

relationship between the solid fraction and the phase eld. The governing equationsof the model are the following: the phase eld equation is as in Hoffman and Jiang[? ]; it is coupled with equations for the temperature and velocity that are based onusual conservation principles. These last equations become respectively a nonlin-ear heat equation and modied a Navier-Stokes equations which include buoyancyforces modelled by Boussinesq approximation and a Carman-Koseny type term tomodel the ow in mushy regions. Since these modied Navier-Stokes equations onlyhold in a priori unknown non-solid regions, we actually have a free boundary valueproblem.

We remark that the phase eld model that we consider here is rather simpleand does not take care of several important transition events such as nucleation orspinodal decomposition. However, more complete and complex phase eld modelsfor phase change could be similarly considered. Our choice in this paper was justguided by mathematical simplicity. We do not present a detailed derivation of themodel equations directly from physical background because this would be basicallya repetition of the arguments of the references (they use just the usual balance of internal energy and linear momentum arguments arguments suitably adapted tothe situation) and they would be lengthy in an already large paper. We give theidea of such adaptations for the balance of internal energy in Section ?? ; for thebalance of linear momentum the argument is exactly as in Voller et al [? ] (see also[? ].) The details of the model problem can be found in Section ?? , equations ( ?? );the corresponding weak formulation can be found in Denition ?? .

Our objective is to present a result on the existence and regularity of solutionsof these model equations corresponding to a nonstationary phase change process ina bounded domain, which for technical reasons in this paper is assumed to be two

dimensional.Existence will be obtained by using a regularization technique similar to the one

used by Blanc et al in [? ]: an auxiliary positive parameter will be introduced inthe equations in such way that the original free boundary value problem will betransformed in a more standard (penalized) one. We say that this is the regularizedproblem. By solving this, one hopes to recover the solution of the original prob-lem as the parameter approaches zero. To accomplish such program, we will solvethe regularized problem by using Leray-Schauder degree theory (see Section 8.3, p.56 in Deimling [? ]); and use results for certain modied Navier-Stokes equationspresented in Vaz [ ? ]. Then, by taking a sequence of values of the parameter ap-proaching zero, we will correspondingly have a sequence of approximate solutions.By obtaining suitable uniform estimates for this sequence, we will then be able to

take the limit along a subsequence and, by compactness arguments, to show thatwe have in fact a solution of the original problem. The stated regularity of this so-lution will be obtained by applying the L p-theory of the parabolic linear equationstogether with bootstrapping arguments.

We should stress that the ideas presented in this mathematical analysis, in par-ticular the penalization used for obtaining the approximate solutions, suggest aconvenient discretization scheme for numerical simulations of phase change prob-lems with melt convection. Such scheme would be similar to, but different from,the ones in [? ] and [? ]. Moreover, such methods would not rely on specifying a


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variable viscosity across the diffusive interface regions that tend to a large value inthe rigid solid, as several other methods propose. This would be realistic only for

certain classes of materials, and certainly difficult to specify for rigid solids. Wealso remark that the mathematical analysis corresponding to the models presentedin [? ] and [? ] are presently under investigation.

This paper is organized as follows. In Section 2, we describe the mathematicalmodel and its variables. In Section 3, we x the notation and describe the thebasic functional spaces to be used; we recall certain results and present auxiliaryproblems; we also state assumptions holding throughout the paper and dene theconcept of generalized solution. In Section 4, we consider the question of existence,uniqueness and regularity of solutions of the regularized problem. Section 5 isdedicated to the proof of the existence of a solution of the original free boundaryvalue problem.

Finally, as it is usual in papers of this sort, C will denote a generic constantdepending only on a priori known quantities.

2. Model Equations

The model problem presented here has aspects of the models studied in the worksof Blanc [? ], Caginalp [? ] and Voller et al [? , ? ]. As we said in the Introduction, thephase of the material will be described by using the phase eld methodology, whichin its simplest approach assumes that there is a scalar eld (x, t ), the phase eld,depending on the spatial variable x and time t and real values s < such that if

(x, t ) s then the material at point x at time t is in solid state; if (x, t )then the material at point x at time t is in liquid state; if s < (x, t ) < then,at time t the point x is in the mushy region. We follow Caginalp [ ? ] and Hoffmanand Jiang[ ? ] and take the phase eld equation as

t = a + b2 3 + ,

where is the temperature; is a (small) xed positive constant, and a and b areknown functions which regularity will be described later on.

We observe that the function g(s) = as + bs2 s3 used at the right hand sideof the above equation is the classical possibility coming from the classical double-well potential (see Hoffman and Jiang [ ? ]). Other possibilities for the double-wellpotential can be found for instance in Caginalp [ ? ] and Penrose [ ? ].

To obtain a equation for the temperature, we observe that when there is phasechange, the thermal energy has the following expression:

e = + 2

(1 f s ),

where and / 2 represent respectively the sensible heat (for simplicity of notation,we took the specic heat coefficient to be one) and latent heat. f s is the solidfraction (1 f s is the non-solid fraction), which for simplicity we assume to be aknown function only of the phase eld (obviously dependent on the material beingconsidered.)

Then, the energy balance in pure material solidication process may be written(see Vaz [? ]) as follows:

t

+ v. = 2

f s

( )t


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where v represent the velocity of the material.We will assume that only non solid portions of the material can move, and this is

done as an incompressible ow. Consequently, in non-solid regions Navier- Stokestype equations are required. According to Voller et al [? ] and Blanc et al [? ] theseequations can be taken as

vt

v + ( v. )v + p = G(f s , v) + F ()

div v = 0

where v is velocity, p is pressure, is viscosity and G(f s , v) and F () are sourceterms which will be dened below.

Assuming the Boussinesq treatment to be valid, natural convection effects canbe accounted for by dening the buoyancy source term to be

F () = Cg ( r )

where is the mean value of the density, g is the gravity, C is a constant andr is a reference temperature. In order to simplify the calculations let us considerF () = .

The source term G(f s , v) is used to modify the Navier-Stokes equations in themushy regions, and according to [ ? , ?], can be taken of form G(f s , v) = k(f s )v.Usually the function k(f s ) is taken as the Carman-Koseny expression (see again[? , ? ]), which is

k(f s ) = f 2s

(1 f s )3 .

As in Blanc et al [? ], we will consider a more general situation including theprevious one. We will assume that assuming that k is a nonnegative function inC 0( , 1), k = 0 in R and lim y 1 k(y) = + , and in this case, we will refer toG as the Carman-Kosen type term.

To complete the description of the model problem, we must dene the regionswhere the above equations are valid. By using the solid fraction, the followingsubsets of Q, denoted by Ql , Qm and Qs and corresponding respectively to theliquid, mushy and solid regions, are dened as:

Q l = {(x, t ) Q : f s ( (x, t )) = 0 }Qs = {(x, t ) Q : f s ( (x, t )) = 1 }

Qm = {(x, t ) Q : 0 < f s ( (x, t )) < 1}

In the following, Qml = Q \ Qs will denote the non-solid part of Q. Moreover, foreach time t [0, T ], we dene s (t) = {x : f s ( (x, t )) = 1 }, ml (t) = \ s (t)and S ml = {(x, t ) Q : x ml (t)}.

We must emphasize that this model is the free boundary problem since that Q l ,Qm and Qs are a priori unknown. Now, we can now summarize the formulation of


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the problem to be analyzed as:t = a + b

2

3

+ in Q,t

+ v. = 2

f s

( )t

in Q,

vt

v + ( v. )v + p + k(f s ( ))v = in Qml ,

div v = 0 in Qml ,

v = 0 inQs ,

(2.1)

subject to the boundary conditionsn

= 0 on S,

= 0 on S,v = 0 on S ml .

(2.2)

and to the initial conditions(x, 0) = 0(x) in ,

(x, 0) = 0(x) in ,v(x, 0) = v0(x) in ml (0) ,

(2.3)

where 0 , 0 and v0 are suitably given functions such that for compatibility v0 isidentically zero outside ml (0).

3. Preliminaries and Main Result

3.1. Notation, functional spaces and auxiliary results. Let R 2 be anopen and bounded domain with a sufficiently smooth boundary and Q = [0, T ] the space-time cylinder with lateral surface S = [0, T ]. For t [0, T ],we denote Qt = [0, t ].

We denote by W pq () the usual Sobolev space and W 2,1q (Q) the Banach spaceconsisting of functions u(x, t ) in Lq(Q) whose generalized derivatives Dx u, D 2x u, u tare Lq-integrable ( q 1). The norm in W 2,1q (Q) is dened by

u (2)q,Q = u q,Q + D x u q,Q + D2x u q,Q + u t q,Q (3.1)

where Dsx denotes any partial derivatives with respect to variables x1 , x2 ,...,x n of order s=1,2 and q the usual norm in the space Lq(Q).

Moreover, W 1,02 (Q) is a Hilbert space for the scalar product

(u, v )W 1 , 02 (Q ) = Q uv + u. vdxdtand

0W 1,12 (Q) is a Hilbert space for the scalar product

(u, v)W 1 , 12 (Q ) = Q uv + u. v + ut vt dxdtwhose functions vanish on S in the sense of traces.


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We also denote by V 2(Q) the Banach space consisting of function u(x, t ) inW 1,02 (Q) having the following nite norm

|u |V 2 (Q ) = ess sup0 t T

u(x, t ) 2, + u(x, t ) 2,Q . (3.2)

Let0V 2(Q) be the Banach space consisting of those elements of V 2(Q) that vanish

on S in the sense of traces.We now dene spaces consisting of functions that are continuous in the sense of

holder. We say that a function u(x, t ) dened in Q is holder continuous in x and t ,respectively with exponents and (0, 1), if following quantities, called holderconstants, are nite:

u ( )x = sup(x 1 ,t ) ,(x 2 ,t ) Q, x 1 = x 2

|u(x1 , t ) u(x2 , t ) ||x1 x2 |

u( )t = sup(x,t 1 ) ,(x,t 2 ) Q, t 1 = t 2

|u(x, t 1) u(x, t 2)||t1 t2 |

Then, we dene the holder space H ,/ 2(Q), with 0 < 1, (see Ladyzenskajaet al [? ]), as the Banach space of functions u(x,t) that are continuous in Q, havingnite norm given by:

|u | ( )Q = maxQ

|u | + Dx u ( )x + u( / 2)t . (3.3)

For the functional spaces associated to the velocity eld, we denote D = {u C ()2 : supp u } and V = {u D : div u = 0}. The closure of V in L2()2

is denoted by H and the closure of V in0

W 12()2 is denoted by V. These functionalspaces appear in the mathematical theory of the Navier-Stokes equations; theirproperties can be found for instance in Temam [ ? ].

The following two lemmas are particular case of Lemma 3.3 in Ladyzenskaja et al ([? ]). They are stated here for ease of reference. The rst lemma is immediateconsequence of Lemma 3.3 in [ ? , p. 80], by taking there l = 1, n = 2 and r = s = 0.

Lemma 3.1. Let and Q as in the beginning of this section. Then for any function u W 2 ,1q (Q) we also have u L p(Q), and it is valid the following inequality

u p,Q C u(2)q,Q , (3.4)

provided that

p = if 1q

12 < 0

p 1 if 1q 12 = 0

1q

12

1 if 1q 12 > 0 .

The constant C > 0 depends only on T, , p and q .

The second lemma is immediately obtained from Lemma 3.3 in [ ? , p. 80], bytaking there l = 1, n = 2, r = s = 0 and q = 3.

Lemma 3.2. Let and Q be as in the beginning of this section. Then for any function u W 2 ,13 (Q) we also have u H 2/ 3,1/ 3(Q) satisfying the estimate

|u | (2 / 3)Q C u(2)3,Q . (3.5)

The constant C > 0 depends only on T and .


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In the following we will consider two auxiliary problems, respectively related tothe phase eld and the velocity equations.

The rst problem ist

= a + b 2 3 + g(x, t ) in Q,

= 0 on S,

(x, 0) = 0(x) in .

(3.6)

where in a positive constant. This problem was treated by Hoffman and Jaing [ ? ]when the initial date satises 0 W 2 (). Since we will need an existence resultfor 0 W

2 2/qq () W 3/ 2 2 (), with (0, 1), we restate the result of [ ? ]. We

remark that exactly the same proof presented in [ ? ] holds in this situation (see alsoVaz [? ] for details, where some other specic results concerning ( ?? ) are proved.)

Proposition 3.3. Let and Q be as in the beginning of this section. Assume that a(x, t ) and b(x, t ) in L (Q), g Lq(Q), 0 W

2 2/qq () W 3/ 2 2 (), where q

2, (0, 1) and 0 = 0 in . Then there exists an unique solution W 2,1q (Q)

of problem ( ?? ), which satises the estimate

(2)q,Q C 0 W 2 2 /qq () + g q,Q , (3.7)

where C depends only on T, , , a(x, t ) ,Q and on b(x, t ) ,Q .

The second auxiliary problem is

vt v + ( v. )v + p + k(x, t )v = f (x, t ) in Q,

div v = 0 in Q,v = 0 on S,

v(x, 0) = v0(x) in .

(3.8)

Proposition 3.4. Let and Q be as in the beginning of this section. Assume that k(x, t ) C 0(Q), k(x, t ) 0, f (x, t ) L2(Q)2 and v0(x) H . Then there exists an unique solution v(x, t ) L2(0, T ; V ) L (0, T ; H ) of problem (?? ) which satises the estimate

v L (0 ,T,H ) + v L 2 (0 ,T,V ) C v0 H + f 2,Q (3.9)

Moreover, by interpolation results v L4(Q)2 and

v 4,Q C v0 H + f 2,Q , (3.10)

where C depends only on T and on .

The proof of Proposition ?? is done by using the same arguments used in theclassical theory of weak solutions of the Navier-Stokes equations. As in this clas-sical situation, the fact that the domain is two dimensional is important to obtainuniqueness of solutions (see Temam [ ? ], p.282, for instance.)


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3.2. Technical Hypotheses and Generalized Solution. For the rest of thisarticle we will be using the following technical hypotheses:

(H1) R 2 is an open and bounded domain with sufficiently smooth boundary ; T is a nite positive number; Q = (0, T ).

(H2) a(x, t ), b(x, t ) are given functions in L (Q); f s C 1,1b (R ), 0 f s (z) 1 forall z R ; k(y) C 0( , 1), k(0) = 0, k(y) = 0 in R , k(y) is nonnegativeand lim y 1 k(y) = + .

(H3) v0 H; 0 W 12 (), 0 = 0 on ; 0 W 4/ 33 () W

3/ 2+ 2 (), for some

(0, 1), 0 = 0 on .

Now we explain in what sense we will understand a solution of ( ?? ), (?? ), (?? ).

Denition 3.5. By a generalized solution of the problem ( ?? ), (?? ), (?? ), we

mean a triple of functions ( ,,v ) such that V 2(Q), 0V 2(Q) and v

L2(0, T ; V ) L (0, T ; H ). Moreover, being

Qs = {(x, t ) Q : f s ( (x, t )) = 1 },s (t) = {x : f s ( (x, t )) = 1 },

Qml = Q \ Qs ,ml (0) = \ s (0)

we have v = 0 a.e inQs , and , and v satisfy the integral relations

Q t dxdt + Q dxdt=

Qa + b 2 dxdt +

Q dx dt +

0(x) (x, 0)dx ,

(3.11)

Q t dxdt + Q dxdt + Q v. dxdt= Q 2

f s

( ) t dx dt + 0(x) (x, 0)dx ,(3.12)

Q ml v t dxdt + Q ml v dxdt+ Q ml (v. )vdxdt + Q ml k(f s ( ))vdxdt= Q ml dx dt + ml (0) v0(x)(x, 0)dx ,

(3.13)

for all in W 1,12 (Q) such that (x, T ) = 0; for all in0

W 1 ,12 (Q) such that (x, T ) = 0,and for all C ([0, T ];W 12 (ml (t))) such that (., T ) = 0, div (., t ) = 0 for allt [0, T ] and supp (x, t ) Qml ml (0).

Note that due to our technical hypotheses and choice of functional spaces, all of the integrals in Denition ?? are well dened.

3.3. Existence of Generalized Solutions. The purpose of this paper is to provethe following result


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Theorem 3.6. Under the hypotheses (H1), (H2), (H3), there is a generalized so-lution of the problem ( ?? ), ( ?? ), ( ?? ) in the sense of the Denition ?? . More-

over, when 0 W 2 2/qq () W

3/ 2+ 2 () for some (0,1) and q 3, and

0 W 2 2/p

p () with 3 p < 4, then such solution satises W 2,1q (Q) L (Q), W 2,1 p (Q) L (Q), v L2(0, T ; V ) L (0, T ; H ).

The proof of the previous result is long and will be done in the following sec-tions. Here we want just to sketch it: existence of a solution of problem ( ?? ), (?? ),(?? ) will proved by using a regularization technique already used by Blanc et al in[? ]. The purpose this regularization is to deal with the Navier-Stokes equations inwhole domain instead of unknown regions. Thus, the problem will be adequatelyregularized with the help of a positive parameter, and the existence of solutions forthis regularized problem will obtained by using the Leray-Schauder degree theory(see Theorem ?? ). Then, as this parameter approaches zero, a sequence of regu-larized solutions is obtained. With the help of suitable estimates and compactnessarguments, a limit of a subsequence is then proved to exist and to be a solution of problem ( ?? ), (?? ), (?? ).

We also remark that the phase eld equation admits classic solution when 0 issufficiently smooth. In fact, its right hand side term satises a + b 2 3 + L (Q), and, in particular when 0 W

2 2/qq () W 3/ 2+ 2 (), with q 2, we

obtain a strong solution with the equation satised in the a.e-sense. The boundaryand initial conditions are also satised in the pontual sense because C 1(Q).When 0 W

2 2/p p (), with 3 p < 4, the same sort of arguments applies and

the solution is strong with C 0(Q); the temperature equation and the boundaryand initial conditions are valid in point wise sense. Unfortunately, we are not ableto improve the regularity of the corresponding solution even if the initial velocityis very regular. Thus, we only generalized solutions are obtained for the velocityequation.

4. Regularized Problem

In this section we regularize problem ( ?? ), (?? ), (?? ) by changing the termk(f s ( ))v in the velocity equation. We will obtain a result of the existence, unique-ness and regularity for this associated regularized problem.

Theorem 4.1. Fix (0, 1]. Under the hypotheses (H 1), (H 2), (H 3), there exists an unique solution ( , v , ) W 2 ,13 (Q) (L2(0, T ; V )L (0, T ; H )) W

2,12 (Q)

L6(Q) L2(0, T ; H ) L3(Q) of the following problem:

t = a + b 2 3 + ,

vt

v + ( v . )v + p + k(f s ( ) )v = ,

div v = 0,t

+ v . = 2

f s

( ) t

(4.1)

in Q; n

= 0 , = 0, v = 0 (4.2)


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on S ; and

(x, 0) = 0(x), (x, 0) = 0(x), v (x, 0) = v0(x) (4.3)in . Moreover, as varies in [0, 1], such solutions ( , v , ) are uniformly bounded with respect to in W 2,13 (Q) (L2(0, T ; V ) L (0, T ; H )) W

2,12 (Q).

This theorem will be proven the end of this section, after some preparation andauxiliary lemmas. The solvability of problem ( ?? ), (?? ), (?? ) will be proved byapplying the Leray-Schauder degree theory (see Deimling [ ? ]) as in Morosanu andMotreanu [ ? ]. For this, we will reformulate the problem as T (1, ,v, ) = ( ,v, ),where T (, ) is a compact homotopy depending on a parameter [0, 1] to bedescribed shortly.

Basic tools in our argument are L p theory of parabolic equations and Theorems?? and ?? in Section ?? . Moreover, we emphasize that the regularity of solutionof Navier-Stokes and phase eld equations plays an essential role in this proof.

Such connection is strictly related with a selection of the order of the equations inquasilinear problem, mainly in deriving a priori estimates for possible solutions.Moreover, since that the phase eld has smooth solution (classical solution), theregularity of Navier-Stokes equations becomes very important but this regularity isgoverned by the additional Carman-Koseny type term k(f s ( ))v that one not per-mits one to obtain uniform estimate in some different as L2(0, T ; V) L (0, T ; H ).

For simplicity of notation, we omit the subscript in the rest of this section.

Denition 4.2. Dene the homotopy T : [0, 1] L6(Q) L2(0, T ; H ) L3(Q) L6(Q) L2(0, T ; H ) L3(Q) as

T (,,u, ) = ( ,v, ) (4.4)where ( ,v, ) is the unique solution of the quasilinear problem:

t = a + b2 3 + ,

vt

v + ( v. )v + p + k(f s ( ) )v = ,

div v = 0 ,t

+ v. = 2

f s

( )t

(4.5)

in Q;n

= 0 , = 0 , v = 0 (4.6)

on S ; and(x, 0) = 0(x) , (x, 0) = 0(x) , v(x, 0) = v0(x) (4.7)

in .We observe that the homotopy T (, ) is well dened. In fact, for xed [0, 1],

by using Proposition ?? and Lemma ?? , we conclude that rst equation of problem(?? ), (?? ), (?? ) has a unique solution W 2,13 (Q) L (Q). Once is known,Proposition ?? implies that the modied Navier-Stokes equations has an uniquesolution v L2(0, T ; V ) L (0, T ; H ). By usual interpolation, it results thatv L4(Q)2 . Now that and v are known, the L p theory of parabolic equations, that also is valid for Neumann boundary condition (see Ladyzenskaja et al [? ,p.351]), Lemma ?? and the facts that t L

3(Q), v L4(Q)2 , and f s C 1,1b (R )


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imply that there is a unique solution W 2,13 (Q) L (Q) for the third equationof (?? ).

Lemma 4.3. Under assumptions (H1), (H2), (H3), the mapping T : [0, 1] L6(Q)L2(0, T ; H ) L3(Q) L6(Q) L2(0, T ; H ) L3(Q) is a compact mapping, i.e, it is continuous and maps bounded sets into relatively compact sets.

Proof. Let us check the continuity of T (, . ). For this, let n in [0,1] and(n , un , n ) (,u, ) in L6(Q) L2(0, T ; H) L3(Q). Denoting T (n , n , un , n )= ( n , vn , n ), from (?? ), we write

nt

n = a n + b 2n 3n + n n , (4.8)

vnt

vn + ( vn . )vn + pn + k(f s ( n ) )vn = n n , (4.9)

div vn = 0 , (4.10)

nt

n + vn . n = n 2f s

( n ) n

t (4.11)

in Q; n

= 0 , vn = 0 , n = 0 , (4.12)

on S ; and

n (x, 0) = 0(x) , vn (x, 0) = v0(x) , n (x, 0) = 0(x) (4.13)

in . By applying Proposition ?? with n L2(Q), we obtain the followingestimate for the phase-eld equation ( ?? )

n(2)2,Q C |n | n 2 ,Q + 0 W 12 () (4.14)

Now, by applying Proposition ?? , we obtain the following estimates for the velocityequation ( ?? )

vn L (0 ,T,H ) + vn L 2 (0 ,T,V ) C v0 H + |n | n 2,Q , (4.15)which by usual interpolation implies

vn 4,Q C v0 H + |n | n 2,Q . (4.16)For ( ?? ), the L p-theory of the parabolic equation (see Ladyzenskaja [ ? , p. 351])with the facts that nt L

2(Q), f s ( n ) L (Q), vn L4(Q)2 and 0 W 12 ()

provides the estimate

n(2)2,Q C vn 4,Q 0 W 12 () + |n |

nt 2,Q

+ 0 W 12 () . (4.17)

Since the sequences ( n ) and ( n ) are respectively bounded in L2(Q) and [0,1],from (?? ) and ( ?? ) we obtain for all n that

n(2)2,Q C and vn L (0 ,T,H ) + vn L 2 (0 ,T,V ) C . (4.18)

Consequently, from ( ?? ) we have for all n

n(2)2 ,Q C . (4.19)

From ( ?? ) and ( ?? ), it follows {T (n , n , u n , n )} = {( n , vn , n )} is uniformlybounded sequence with respect to n in the functional space W 2,12 (Q) (L2(0, T ; V )L (0, T ; H )) W 2,12 (Q). Moreover, we observe that for xed (0, 1], from the


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properties of k(y) (see the conditions stated in ( H 2)), there is a nite positiveconstant C depending only on such that sup {k(y )} C . By using this and

our previous estimates as in Lions [ ? , p. 71], we conclude that for all n ,(vn )t L 2 (0 ,T ;V ) C (). (4.20)

Thus, the previous estimates, and the Aubin-Lions Lemma (see Temam [ ? ] orLions [? ]), allow us to select a subsequence, which we denote {T (k , k , uk , k )} ={( k , vk , k )} such that

k in W 2,12 (Q) (4.21)

vk v in L2(0, T ; V ) (4.22)

vk v in L (0, T ; H ) (4.23)

k in W 2,12 (Q) (4.24)

(vk )t v t in L2(0, T ; V ) (4.25)

k in L6(Q) (4.26)

vk v in L2(0, T ; H ) (4.27)

k in L3(Q) (4.28)

Now, let us verify that T (,,u, ) = ( ,v, ), in other words, that ( ,v, ) issolution of ( ?? ), (?? ),( ?? ). For this, we are going to pass to the limit with respectto the above subsequence in equations ( ?? )-(?? ) together with the conditions ( ?? )-(?? ).

Let us prove that the equations are satised in the sense distribution. For this,x in the sequel g C c (Q), and let us describe the process of taking the limit onlyfor those terms of the equations that are neither trivial nor standard. We observe

that by using ( ?? ) and k , we obtain

Q k (a k + b 2k 3k )gdxdt Q (a + b 2 3)gdxdt g (4.29)Thus, passing to the limit in phase eld equation ( ?? ), using the convergence

(?? ), (?? ) and ( ?? ), we obtain the rst equation in ( ?? ).To verify the convergence

Q k(f s ( k ) )vk g dxdt Q k(f s ( ) )vgdxd t (4.30)we use (?? ), the fact that for xed (0, 1], k(f s () is bounded, and followingargument. Consider hk = |k(f s ( k ) ) k(f s ( ) )|2 . Since k(f s () ) iscontinuous and ( ?? ) is valid, passing to a subsequence if necessary, we know that

hk 0 almost everywhere in Q. Moreover, |hk | C f s ( )2 a.e and thereforehk 0 in L1(Q) by Lebesgue dominated convergence theorem. Thus, k(f s ( k )

) k(f s ( ) ) in L2(Q), what together with ( ?? ) implies (?? ). By passing tothe limit in velocity equation ( ?? ), using the convergence ( ?? ), (?? ) and ( ?? ) weobtain the second equation in ( ?? ).

Now, we use (?? ), (?? ), k and arguments similar to the ones previouslywith f s ( k ) in place of f s to obtain

Q kf s

( k ) kt

gdxdt Q f s

( )t

gdxdt (4.31)


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By passing to the limit in temperature equation ( ?? ), using the convergence ( ?? ),(?? ) and ( ?? ), we obtain the third equation in ( ?? )

The required boundary conditions are included in the denitions of the functionalspaces where ( ,v, ) is in. Also, with the estimates we have obtained, it is standardto prove that , v and satisfy the required initial conditions. Hence, ( ,v, ) issolution of ( ?? )-(?? ).

Note that for any given subsequence of {T (n , n , vn , n )}, the above argumentscan be applied to conclude that this subsequence admits another subsequence con-verging to a solution of ( ?? )-(?? ). Since (,u, ) is also xed and the solution of thislast problem is unique, we conclude that {T (n , n , vn , n )} is a sequence with theproperty that any one of its subsequences has by its turn a subsequence convergingto a limit that is independent of the chosen subsequence. Hence, {T (n , n , vn , n )}converges to this limit, and the continuity of T is proved.

The same arguments prove that mapping T is a compact mapping. In fact, if {(n , un , n )} is any bounded sequence in L6(Q) (L2(0, T ; V ) L (0, T ; H )) L3(Q), the above arguments can be applied to obtain exactly the same sort of estimates for T (n , n , un , n ). These imply that {( n , vn , n )} is relatively com-pact in L6(Q) L2(0, T ; H ) L3(Q), and thus there exists a subsequence of T (n , n , u n , n ) converging in L6(Q) L2(0, T ; H ) L3(Q). Therefore, the com-pactness is proved.

The next lemma give us an uniform estimate for any possible x point of T (, ).

Lemma 4.4. Under assumptions (H1), (H2), (H3), there exists a positive number , depending only on the given data of the problem and in particular independent of [0, 1], with the property any x point of T (, . ) is in the interior of the ball of radius in L6(Q) L2(0, T ; H ) L3(Q). That is,

T (, ,v, ) = ( ,v, ) ( ,v, ) < , (4.32)

where denotes the norm in L6(Q) L2(0, T ; H ) L3(Q).

Proof. Using (?? ), the condition T (, ,v, ) = ( ,v, ) is equivalent to

t

= a + b 2 3 + , (4.33)

vt

v + ( v. )v + p + k(f s ( ) )v = , (4.34)

div v = 0, (4.35)

t + v. =

2

f s

( )

t (4.36)

in Q;

= 0 , = 0 , v = 0 (4.37)

on S ;(x, 0) = 0(x), (x, 0) = 0(x), v(x, 0) = v0(x) (4.38)

in . To obtain estimates for ( ,v, ), we start by multiplying the rst equation(?? ) by . After integrating of the result over Qt (t (0, T ]), using Fubinis


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theorem, Greens formula and Youngs inequality, we get

2dx + t

0 | |2dxdt + 2

t

0 4dxdt C 0 22, +

T

0 ||2dxdt + T

0 | |2dxdt ,(4.39)

where C depends on and max (x,t ) Q a(x, t ) + b(x, t )s 12 s2 . Applying Gron-

walls inequality in ( ?? ), we get

T

0 | |2dxdt C 0 22, + T

0 ||2dxdt . (4.40)Thus, combining ( ?? ) and ( ?? ), we conclude

T

0 | |2dxdt C 0 22, + T

0 | 2dxdt (4.41)2

T

0 4dxdt C 0 22, + T

0 ||2dxdt (4.42)Now, we multiply equation ( ?? ) by and integrate over Qt . Then we use thefact that f s L

(R ), (?? ), Greens formula and also Poincares and Youngsinequalities to obtain

2dx + t

0 | |2 dxdt C 0 22, + T

0 | t |2 dxdt , (4.43)where C depends on , and f s .

Multiplying the rst equation ( ?? ) by t , integrating over Qt , using Greensformula and Youngs inequality, we obtain

t

0 t

2dxdt + t

0 | |2dxdt C 0 22, +

T

0 | |2dxdt + T

0 ||2dxdt ,(4.44)

where C depends on and max (x,t ) Q a(x, t ) + b(x, t )s s2 . Using (?? ) in (?? )and applying the resulting estimate in ( ?? ), we get

2dx +

t

0 | |2 dxdt C 0 22, + 0

2W 12 () +

T

0 ||2 dxdt (4.45)

Applying Gronwall inequality in ( ?? ), we obtain

2,Q C 0 2, + 0 W 12 () , (4.46)

and, consequently, 2,Q C 0 2,+ 0 W 12 () . Moreover, by interpolationresults (see Ladyzenskaja [ ? , p. 74]), we have

4,Q M 0 2, + 0 W 12 () . (4.47)


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Using (?? ) in (?? ), (?? ) and ( ?? ), we conclude that

2,Q C 0 2, + 0 W 1

2 () , (4.48)

2 ,Q C 0 2, + 0 W 12 () , (4.49)

4,Q C 0 2, + 0 W 12 () . (4.50)

Using (?? ) and ( ?? ) in (?? ), we have

t 2,Q

C 0 2, + 0 W 12 () . (4.51)

Multiplying the rst equation ( ?? ) equation by integrating over Qt , usingGreen formula and Young inequalities, we obtain

| |2

dx + t

0 | |2

dxdt + 3 t

0 2

| |2

dxdt

C 0 22, + T

0 ||2dxdt + T

0 | |2dxdt + T

0 | |4dxdt ,(4.52)

where C depends on , , a ,Q , b ,Q and f s . Using (?? ), (?? ) and ( ?? )

in (?? ), we obtain

2,Q C 0 2, + 0 W 12 () (4.53)

Combining estimates ( ?? ), (?? ), (?? ) and ( ?? ), using the imbedding in Lemma ?? ,we have

p,Q C (2)

2,Q C 0 2, + 0 W 1

2 () ( p 6). (4.54)Now, by multiplying the second equation ( ?? ) by v, integrating over Qt , using

Greens formula, and Poncares and Youngs inequalities, we get

12 v2dx +

2

t

0 | v|2 dxdt + t

0 k (f s ( ) ) v2dxdt C v0 H +

T

0 ||2 dxdt .(4.55)

Combining ( ?? ) and ( ?? ), using that k(f s ( ) ) 0, we conclude that

v L (0 ,T ;H) + v L 2 (0 ,T ;V) C v0 H + 0 2, + 0 W 21 () .

Finally, by the interpolation result given in Theorem ( ?? ),

v 4,Q C v0 H + 0 2, + 0 W 21 () . (4.56)

The next lemma tell us that there is an unique x point in the special case = 0.

Lemma 4.5. Under assumptions (H1), (H2), (H3), there exists an unique solution of the problem T (0, ,v, ) = ( ,v, ) ( T dened in (?? )).


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Proof. The equation T (0, ,v, ) = ( ,v, ) is equivalent to the nonlinear system

t = a + b

2

3,

vt

v + ( v. )v + p + k(f s ( ) )v = 0,

div v = 0,t

+ v. = 0

in Q;

= 0 , = 0 , v = 0

on S ;(x, 0) = 0(x), (x, 0) = 0(x), v(x, 0) = v0(x)

in . For these equations, Proposition ?? ensures the existence and uniqueness of ; then Proposition ?? gives the existence and uniqueness v. The L p theory of

the linear parabolic equations ensures then the existence and uniqueness of .

Proof of Theorem ?? . According to Lemma ?? , there exists a number satisfying(?? ). Let us consider the open ball

B = ( ,v, ) L6(Q) L2(0, T ; H ) L3(Q) : ( ,v, ) < ,

where is the norm in the space L6(Q) L2(0, T ; H ) L3(Q). Lemma ?? ensuresthat the mapping T : [0, 1] L6(Q) L2(0, T ; H) L3(Q) L6(Q) L2(0, T ; H) L3(Q) is a homotopy of compact transformations on the closed ball B and Lemma?? implies that

T (, ,v, ) = ( ,v, ) ( ,v, ) B , [0, 1]

The foregoing properties allow us consider the Leray-Schauder degree D(Id T (, . ), B , 0), [0, 1] (see Deimling [? ]). The homotopy invariance of Leray-Schauder degree shows that the equality below holds

D (Id T (0,. ), B , 0) = D(Id T (1,. ), B , 0) (4.57)

Moreover, the Lemma ?? ensures that the problem T (0, ,v, ) = ( ,v, ) has aunique solution in L6(Q) L2(0, T ; H) L3(Q). Hence we can choose a sufficientlylarge > 0 such that the ball B contains this solution, it turns out that D(Id T (0,. ), B , 0) = 1. Then relation ( ?? ) ensures that the equation T (1, ,v, ) ( ,v, ) = 0 has a solution ( ,v, ) B L6(Q) L2(0, T ; H) L3(Q). By (?? )with = 1, this is just a solution of the problem ( ?? )-(?? )-(?? ).

The uniqueness and regularity of problem ( ?? ), (?? ), (?? ) are consequence of the application of the Propositions ?? and ?? and L p-regularity theory for linearparabolic equations. To prove uniqueness let i , vi and i with i = 1, 2 be twosolutions of problem ( ?? ), ( ?? ), ( ?? ), with corresponding pressures pi (for simplicityof exposition, we omit the subscript ). We rst observe that by using the previouslyobtained estimates and arguments similar to the ones used to prove that T is welldened (Denition ?? ), we conclude that i W 2,13 (Q) L (Q), vi L2(0, T ; V )L (0, T ; H ) and W 2,12 (Q) L p(Q) (for any nite p 1).


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Let = 1 2 , v = v1 v2 , = 1 2 and p = p1 p2 . These functionssatisfy the following conditions:

t

= [a(x, t ) + b(x, t )( 1 + 2) ( 21 + 1 2 + 22)] + , (4.58)

vt

v + ( v1 . )v + p + k(f s ( 1) ) v

= (v. )v2 + {k(f s ( 1) ) k(f s ( 2) )}v,(4.59)

div v = 0, (4.60)

t

+ v1 . = 2

f s

( 1) t

(v. )2 + 2

f s

( 1) f s

( 2) 2

t ;

(4.61)

= 0 , = 0 , v = 0 (4.62)

on S ; and

(x, 0) = (x, 0) = 0 , v(x, 0) = 0 (4.63)

in . Multiplying equation ( ?? ) by and integrating on , after usual integrationby parts, using the fact that a(), b(), 1 , 2 L (Q) and Holders inequality, weobtain

ddt

(t) 22, + 2 (t)22, C 1 (t)

22, + (t)

22, . (4.64)

Multiply ( ?? ) by t and integrate on . Proceeding as before, we obtain

t

(t) 22, + 2

ddt

(t) 22, C 2 (t)22, + (t)

22, . (4.65)

Multiply ( ?? ) by v and proceed as usual with the help of the facts that div v1 = 0,k(f s ( 1) 0 and Holders inequality to obtain

12

ddt

v(t) 22, + v(t)22,

C (t) 22, + v(t) 22 , + (v(t). )v2(t)v(t)+

[k(f s ( 1(t)) ) k(f s ( 2(t)) )]|v(t)|2

(4.66)

The integral terms on the right hand side of the previous inequality can be estimatedas follows.

| (v(t). )v2(t)v(t)| C v2(t) 2, v(t) 24 , C v2(t) 2, v(t) 2 , v(t) 2,

C v2(t) 22, v(t)22, +

4

v(t) 22,


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Next, by using the facts that k() is a Lipschitz function on ( , 1 ) and f s ()is a L -function, we obtain

| [k(f s ( 1(t)) ) k(f s ( 2(t)) )]|v(t) |2dx | C |[f s ( 1(t)) )] [f s ( 1(t)) ] v(t)|2dx= C |f s ( 1(t)) f s ( 1(t)) v(t)|2dx C v(t) 22,.

Using the last two estimates in ( ?? ), we obtainddt

v(t) 22, + 32

v(t) 22, C 3 (t)22, + C 4(1 + v2(t)

22,) v(t)

22, (4.67)

We proceed by multiplying equation ( ?? ) by , integranting on . After integrationby parts and the use of the facts that div v

1 = 0, f s

L (R), with the help of

Holders inequality, we obtain12

ddt

(t) 22, + (t)22,

C ( (t) 22, + t

(t) 22,)

+ (v(t). )2 (t)dx + 2(f s

( 1(t)) f s

( 1(t))) 2t

(t) (t) dx

(4.68)

The last two integrals terms in the above inequality can be estimated as follows:

| (v(t). )2 (t)dx | = | div (v(t) 2) (t)dx |= | (v(t)2) (t)dx | v(t) 4, 2(t) 4, (t) 2,

4 v(t) 24, 2(t)24, +

14

(t) 22,

C v(t) 2, v(t) 2, 2(t) 24, + 14

(t) 22,

C 2(t) 44, v(t)22 , +

2

v(t) 22 , + 14

(t) 22,.

Using the fact that f s is a Lipschitz function, we obtain

|

2(

f s

( 1(t))

f s

( 1(t)))

2

t (t) (t) dx |

C | (t)| | 2t

(t)| | (t)| dx

C (t) 2 , 2t

(t) 3, (t) 6,

C (t) 2 , 2t

(t) 3, (t) 2,

C 2t

(t) 23, (t)22, +

14

(t) 22,.


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Using the last two inequalities in ( ?? ),

ddt (t)

22, + (t)

22,

C 5( (t) 22, + t

(t) 22,) + C 6 2(t)44, v(t)

22, + C 7

2t

(t) 23, 22,.

(4.69)Now, we multiply ( ?? ) by 2C 5 and add the result to ( ?? ), (?? ) and ( ?? ). Aftersome simplications, we obtain

ddt

(t) 22 , + ddt

v(t) 22, + ddt

(t) 22, + C 5ddt

(t) 22,

C 8(1 + 2(t)

t 23,) (t) 2,

+ C 9(1 + v2(t) 22, + 2(t))44,) v(t) 2, + C 10

22,.

By denoting z(t) = (t) 22 , + v(t) 22, + (t) 22, + C 5 (t) 22,, the lastinequality implies

ddt

z(t) C 1 + 2t

(t) 23,) + v2(t)22, + 2(t))

44, z(t).

This inequality implies that for t [0, T ],

0 z(t) z(0)exp C (T )[1 + 2t

(t) 23,Q ) + v2(t)2L 2 (0 ,T ;V ) + 2(t))

44,Q ] .

Since 2t (t)2L 3 (Q ) ) + v2(t)

2L 2 (0 ,T ;V ) + 2(t))

4L 4 (Q ) is nite, due to the known

regularity of the involved functions, and z(0) = 0, we conclude that z(t) 0, andtherefore 0, v 0, 0, which imply the uniqueness of the solutions.

Next, we show that the solutions ( , v , ) L6

(Q) L2

(0, T ; H) L3

(Q) of the problem ( ?? ), (?? ), (?? ) are uniformly bounded with respect to in the spaceW 2,13 (Q) L2(0, T ; V )L (0, T ; H ) W

2,12 (Q). For this, note rst that L3(Q);

the L p-theory of parabolic linear equation combined with Theorem ?? and Lemma?? allow us to conclude that there exists an unique W 2,13 (Q) L (Q) suchthat

,Q C (2)3 ,Q C (a + b

2 ) 3,Q + 3,Q + 0 W 4 / 33 () .

(4.70)Since max (x,t ) Q a(x, t ) + b(x, t )s s2 is nite, from ( ?? ), we have

,Q C (2)3 ,Q C 6,Q + 3,Q + 0 W 4 / 33 () . (4.71)

Combining ( ?? ), (?? ) and ( ?? ) and using usual Sobolev imbeddings, we concludethat

(2)3,Q C 0 2, + 0 W 4 / 33 () . (4.72)

Moreover, Lemma ?? gives us that H 2/ 3,1/ 3(Q) such that

| |(2 / 3)Q C

(2)3,Q C 0 2, + 0 W 4 / 33 () . (4.73)

We consider then the equation for the temperature. By applying the L p-theoryof parabolic linear equations (see Ladyzenskaja [ ? ]) together with the facts that


21/25


22/25


23/25


i = 1 , . . . , , we can take the compact set i [a i , bi ] as K ml in (?? ) and concludethat there is i (0, 1] and C i > 0 independent of (0, i ] such that for such

we havek(f s ( ) ) L ( i [a i ,b i ]) C i .

This and our previous estimates allow us to work with the second equation in ( ?? )restricted to i (a i , bi ) to obtain that there is C i > 0 independent of (0, i ]such that for such we have

vt L 2 (a i ,b i ;V ( i )

C i ,

where V (i ) is the topological dual of the Banach space

V (i ) = {u 0

W 12(i )2;div u = 0},

considered with the norm of 0

W 12(i )2 .Also, our previous estimates tell us in particular that {v } for is uniformly

bounded with respect to (0, i ] in L2(a i , bi ; W (i )), where W (i ) ={u W 12 (i )2 ;div u = 0} is a Banach space with the W 12 (i )2-norm.

Consider the Banach space

H (i ) = {u L2(i )2 ;div u = 0 , and null normal trace }

with the L2(i )2-norm (see Temam [ ? ] for properties of this and the previous Ba-nach spaces). We observe that W (i ) H (i ) V (i ), and the rst imbeddingis compact, we can use Corolary 4, p. 85, in Simon [ ? ] to conclude that there is asubsequence of {v } converging to v in L2(a i , bi ; H (i )). In particular, this impliesthat along such subsequence v v in L2(i (a i , bi )).

Proceeding as above for each i = 1 , . . . , , with the help of the usual diagonalargument, we obtain a subsequence such that

v v in L2loc (Qml ).

Thus, along such subsequence, we can pass to the limit as 0 in (?? ) by pro-ceeding exactly as in the case of the classical Navier-Stokes equations and concludethat ( ?? ) is satised.

To obtain the other equations in Denition ?? , we multiply the rst and third

equations of ( ?? ) respectively by W 1,12 (Q) with (. , T ) = 0 and 0

W 1,12 (Q)with

(., T

) = 0, and proceed as before. Using arguments similar to the ones in(?? ) and ( ?? ), we conclude that

Q a + b 3 dxdt Q a + b 3 dx dt, Q

f s

( ) t

dxdt Qf s

( )t

dxdt,

as 0. With these results, it is easy to to pass to the limit as 0 and concludethat equations ( ?? ) and ( ?? ) are also satised.


24/25


Regularity of the Solution. Now we have to examine the regularity of ( ,,v ).For this, we remark that by interpolation (see Ladyzenskaja [ ? ] p. 74), L4(Q).

Thus, applying Proposition ?? with L3(Q), we conclude that W 2,13 (Q)

L (Q). Also, Proposition ?? give us that v L4(Q)2 .Applying the L p theory of parabolic equations together with the facts that

f s C 1 ,1b (R ), v L4(Q)2 ,

t L2(Q) and Lemma the result of ?? , we concludethat W 2,12 (Q) L p(Q) ( p 2). Therefore, by using a bootstrapping argumentwith Lq(Q) where q 3 and smoothness of the data 0 we conclude that

W 2,1q (Q) L (Q).Applying again the L p theory of parabolic equations with f s C 1,1b (R ), v

L4(Q)2 , t L p(Q), with 2 p < 4, recalling the given smoothness of 0 and the

result of Lemma ?? , we conclude that W 2,1 p (Q) L (Q), with 2 p < 4. Thiscompletes the proof of Theorem ?? .

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[4] G. Caginalp and J. Jones; A derivation and analysis of phase eld models of thermal alloys,Annal. Phys. 237, pp. 66-107, 1995.

[5] G. Caginalp, Phase eld computations of single-needle crystals, crystal growth and motionby mean curvature, SIAM J. Sci. Comput. 15 (1), pp. 106-126, 1994.

[6] G. Caginalp and W. Xie, Phase-eld and sharp-interface alloy models, Phys. Rev. E, 48 (03),pp. 1897-1999, 1993.

[7] G. Caginalp, Stefan and Hele-Shaw type models as asymption limits of the phase-eld equa-tions, Phys. Rev. A 39 (11), pp. 5887-5896, 1989.

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Jos e Luiz BoldriniDepartment of Mathematics, UNICAMP-IMECC, Brazil

E-mail address : [email protected]

Cristina L ucia Dias VazDepartment of Mathematics, Universidade Federal do Par a, Brazil

E-mail address : [email protected]

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