Existence and Uniqueness of Solution for Cahn Hilliard Hyperbolic Phase Field System with Dirichlet Boundary Condition and Regular Potentials

http://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385

Existence and Uniqueness of Solution for

Cahn-Hilliard Hyperbolic Phase-Field System

with Dirichlet Boundary Condition and Regular

Potentials

Jean De Dieu Mangoubi, Daniel Moukoko, Fidele Moukamba, Franck Davhys Reval Langa

G.R.A.F.E.D.P, Faculté des Sciences et Techniques, Université Marien NGOUABI, Brazzaville, Congo

Abstract

Our aim in this paper is to study the existence and the uniqueness of the solutions for hyperbolic Cahn-Hilliard phase-field system, with initial conditions, Dirichlet boun-dary condition and regular potentials.

Keywords

Cahn-Hilliard Hyperbolic Phase-Field System, Regular Potential, Dirichlet Boundary Conditions

1. Introduction

G. Caginalp introduced in [1] the following phase-field system

( )

2

u

u f u

t θ

∂ _{− ∆ − ∆ = −∆}

∂ (1)

u

t t

θ _θ

∂ ∂

− ∆ = −

∂ ∂ (2) where u is the order parameter and θ is the (relative) temperature. These equations model phase transition processes such as melting-solidification processes and have been studied, see [2]-[6], for a similar phase-field model with a nonlinear term.

These Cahn-Hilliard phase-fiel system are known as the conserved phase-field sys-tem (see [7]-[9]) based on type III heat conduction and with two temperatures (see

[10]). The authors have proved the existence and the uniqueness of the solutions, the existence of global attractor and of exponential attractors with singularly or regular

How to cite this paper: De Dieu Mangou-bi, J., Moukoko, D., Moukamba, F. and Lan-ga, F.D.R. (2016) Existence and Uniqueness of Solution for Cahn-Hilliard Hyperbolic Phase-Field System with Dirichlet Boundary Condition and Regular Potentials. Applied Mathematics, 7, 1919-1926.

http://dx.doi.org/10.4236/am.2016.716157

Received: August 20, 2016 Accepted: October 11, 2016 Published: October 14, 2016

Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

potentials.

In [11], Ntsokongo and Batangouna have studied the following Cahn-Hilliard phase- field system

( )

2

u

u f u

t t t

α _β α

∂ _{+ ∆ − ∆ = −∆}∂ _{− ∆}∂ 

 

∂  ∂ ∂  (3)

2 2

2 2

u

t t

t t

α α α _α

∂ _{− ∆}∂ _{− ∆}∂ _{− ∆ = −}∂

∂ ∂

∂ ∂ (4)

where β =1, u is the order parameter and α is the (relative) temperature, they have

proved the existence and the uniqueness solution with Dirichlet boundary condition and regular potentials.

In this paper, we consider the following Cahn-Hilliard hyperbolic phase-fiel system

( )

2 2

( )

2 ,

u u

u f u

t t

t

α

∂ ∂ ∂

−∆ + + ∆ − ∆ = −∆

∂ ∂

∂

 (5)

2 2

2 2 ,

u

t t

t t

α α α _α

∂ _{− ∆}∂ _{− ∆}∂ _{− ∆ = −}∂

∂ ∂

∂ ∂ (6)

0,

u_∂Ω =α_∂Ω= ∆u_∂Ω = (7)

0 1 0 1

0 0

0 0

, , , ,

t t

t t

u

u u u

t t

α

α α α

= =

= =

∂ ∂

= = = =

∂ ∂ (8)

which is the perturbed phase-field system of Cahn-Hilliard phase-field system (3)-(4) with β =0. In the above hyperbolic system Ω is a bounded and regular domain of

n

 with n=2 or 3 and f is the nonlinear regular potentials.

The hyperbolic system has been extensively studied for Dirichlet boundary condi-tions and regular or singular potentials (see [12]-[14]). Whose certain have to end at existence of global attractor or at the existence of exponential attractors (see [15]).

In this paper we prove the existence and the uniqueness of solutions of (5)-(8). We consider the regular potential

( )

3

f s =s −s which satisfies the following properties:

( )

2

is of class ; 0 = 0,

f C f (9)

( )

0 , 0 0, ,

c f′ s c s

− ≤ ≥ ∀ ∈ (10)

( )

( )

( )

( )

1 2, 1, 2 0, where ₀ d .

s

c F s f s s c c c s F s f τ τ

− ≤ ≤ + ≥ ∀ ∈_ =

_∫

₍₁₁₎

2. Notations

We denote by . the usual L2-norm (with associated product scalar (.,.)) and set

( )

1

2 1

. .

−

− = −∆ , where −∆ denotes the minus Laplace operator with Dirichlet

boundary conditions. More generally, . _X denote the norm of Banach space X.

Throughout this paper, the same letters c c1, 2 and c3 denote (generally positive)

3. A Priori Estimates

We multiply (5) by

( )

1 u

t

− ∂

−∆

∂ and (6) by t

α

∂

∂ , integrate over Ω and add the two resulting differential equalities. We find

2 2

1

1 d

2 2 0,

d

E u

t t t

α

−

∂ ∂

+ + ∇ =

∂ ∂

where

( )

2 2 2

2 2

1 2 d ,

u

E u F u x

t t t

α α _α

Ω

∂ ∂ ∂

= + ∇ + + + ∇ + ∇

∂

∫

∂ ∂



satisfies

1 1

1

2 2 2

2 2

1 H H , 0.

H

u

E C u C C

t t t

α α _α

 _{∂ ∂ ∂} 

′ ≥ _ + + + + _+ >

∂ ∂ ∂

 

Finaly, we conclude that

(

1

( )

)

0

, ,

u α∈L∞ R+ H Ω ,

( )

(

2

)

2

(

1

( )

)

, 0, ;

u

L L L T H

t

∞ + −

∂ _{∈ Ω ∩ Ω}

∂ R

and

( )

(

1

)

2

(

1

( )

)

0 0

, 0, ;

L H L T H

t

α ∞ +

∂ _{∈ Ω ∩ Ω}

∂ R

for all T >0.

Multiply (6) by 22

t

α

∂

∂ and integrate over Ω. We get.

2 2 ₂

2 2 2

2 2 2

2 2

2 2

d

2 2 2 , 2 ,

d

2 2

u

t t t t

t t t

u t

t t

α α α α _α α

α _α α

 

∂ _{+ ∇}∂ _{+ ∇}∂ _{= −} ∂ ∂ ₋ _{∇ ∇}∂ 

   

∂ ∂ ∂

∂ ∂  ∂   

∂ ∂ ∂

≤ + ∇ ∇

∂

∂ ∂

2 2 2 2

2 2

2

2 2

d

. d

u

t t t

t t

α α α _α

∂ ∂ ∂ ∂

+ ∇ + ∇ ≤ + ∇ ∂ ∂ ∂ ∂

Then 2 2

(

1

( )

)

0 2 L 0, ;T H

t

α

∂ _{∈ Ω}

∂ .

In this study, we have three main results; existence theorem, uniqueness theorem and existence theorem with more regularity.

4. Existence and Uniqueness of Solutions

Theorem 4.1. (Existence) We assume

(

)

( )

( )

(

( )

)

2

1 2 1

0, 1, 0, 1 0 0

u u α α ∈H Ω ×L Ω × H Ω then the system (5) - (8) possesses at least one solution

(

u,α

)

such that

( )

(

1

)

0

, ,

uα∈L∞ R+ H Ω ,

( )

(

2

)

2

(

1

( )

)

, 0, ;

u

L L L T H

t

∞ + −

∂ _{∈ Ω ∩ Ω}

( )

(

1

)

2

(

1

( )

)

0 0

, 0, ;

L H L T H

t

α ∞ +

∂ _{∈ Ω ∩ Ω}

∂ R

and 2 2

(

1

( )

)

0 2 L 0, ;T H

t

α

∂ _{∈ Ω}

∂ , for all T >0.

The proof is based on a priori estimates obtained in the previous section and on a standard Galerkin scheme.

Theorem 4.2. (Uniqueness) Let the assumpptions of Theorem 4.1 hold. Then, the system (5) - (8) possesses a unique solution

(

u,α

)

such that

( )

(

1

)

0

, ,

uα∈L∞ R+ H Ω ,

( )

(

2

)

2

(

1

( )

)

, 0, ;

u

L L L T H

t

∞ + −

∂ _{∈ Ω ∩ Ω}

∂ R ,

( )

(

1

)

2

(

1

( )

)

0 0

; 0, ;

L H L T H

t

α ∞ +

∂ _{∈ Ω ∩ Ω}

∂ R

and 2 2

(

1

( )

)

0 2 L 0, ;T H

t

α

∂ _{∈ Ω}

∂ for all T>0. Proof. Let

(

( )1 ( )1

)

,

u α and

(

u( )2,α( )2

)

be two solutions of the system (5)-(8) with

initial data

(

( )1 ( )1 ( )1 ( )1

)

0 , 1 , 0 , 1

u u α α and

(

( )2 ( )2 ( )2 ( )2

)

1

( )

2

( )

(

1

( )

)

2

0 , 1 , 0 , 1 0 0

u u α α ∈H Ω ×L Ω × H Ω ,

respectively. We set ( )1 ( )2

u=u −u and α α= ( )1 −α( )2 , then

(

u,α

)

is solution of the

following system

( )

2 2

(

( )

( )1

( )

( )2

)

2 ,

u u

u f u f u

t t

t

α

∂ ∂ ∂

−∆ + + ∆ − ∆ − = −∆

∂ ∂

∂

 (12)

2 2

2 2 ,

u

t t

t t

α α α _α

∂ ∂ ∂ ∂

− ∆ − ∆ − ∆ = −

∂ ∂

∂ ∂ (13)

0,

u_∂Ω = ∆u_∂Ω =α_∂Ω =

( )1 ( )2 ( )1 ( )2

0 0 0 1 1 1

0

0 , t

t

u

u u u u u u u

t

=

=

∂

= = − = = −

∂

( )1 ( )2 ( )1 ( )2

0 0 0 1 1 1

0

0

, .

t

t

t

α

α ₌ α α α α α α

=

∂

= = − = = −

∂

We multiply (12) by

( )

1 u

t

− ∂

−∆

∂ and integrate over Ω. We find

( )

( )

( )

( )

2 2

2 1 2

1

d

2 2 , 2 , .

d

u u u u

u f u f u

t t t t t t

α

−

 _∂  _{∂  ∂  ∂ ∂ }

+ ∇ + + − =

  _{   }

 _∂  _{∂  ∂  ∂ ∂ }

  (14)

Multiplying (13) by

t

α

∂

∂ and integrating over Ω, we get

2 2 2

2

d

2 2 , .

d

u

t t t t t t

α α _α α α

 _{∂ ∂}  _{∂ ∂ ∂ } + ∇ + ∇ + ∇ = −

  _{ }

 _{∂ ∂}  _{∂ ∂ ∂ }

  (15)

( )

( )

( )

( ) ( )

( )

( )

( )

2 2

1 2

2

1

2 2

1 2

d

2 2 2 ,

d

,

E u u

f u f u

t t t t

u f u f u

t

α

−

∂ ∂  ∂ 

+ + ∇ = − _ − _

∂ ∂  ∂ 

∂

≤ − +

∂

(16)

where

2 2 2

2 2

2 .

u

E u

t t t

α α _α

∂ ∂ ∂

= + ∇ + + ∇ + ∇

∂ ∂ ∂



Lagrange theorem gives a estimates

( )

( )

( )

( )

(

( )

(

( ) ( )

)

)

( )

₍

₎

( )

(

)

1

1 2 2 1 2

0

2

1 _{1 2}

0

d

3 1 1 d ,

f u f u f u s u u su

su s u s u

′

− = + −

 

= _ + − − _

 

∫

∫

which implies

( )

( )

( )

( )

( )

( )

( )

( ) ( ) ( )

( ) ( )

6

6 6

2

2 2 2 ₂

1 2 2 1

4 4 ₂

2 1

4 4 ₂

2 1

36 1 d

36 1

1 .

L

L L

f u f u u u u x

u u u

C u u u

Ω

 

− ≤ _ + + _

 

 

≤ _ + + _

 

 

≤ _ ∇ + ∇ + _ ∇

 

∫

Inserting the above estimate into (16), we have

2 2 2

2 2

1

d

2 2 , 0.

d

E u u

K u K

t t t t

α

−

 

∂ ∂ ∂

+ + ∇ ≤ _ ∇ + _ > ∂ ∂ _  ∂ _

Applying Gronwall’s lemma, we obtain for all t∈

( )

0,T

( )

( )

2

( )

2 ( )

_{( )}

2 ₀ 2

1

2 t u ek t d 0 e .kT

E t E

t t

τ

α

τ τ − τ

−

 _{∂ ∂} 

+ _ + ∇ _ ≤

∂ ∂

 

∫

We deduce the continuous dependence of the solution relative to the initial condi-tions, hence the uniqueness of the solution.

The existence and uniqueness of the solution of problem (5)-(8) being proven in a larger space, we will seek the solution with more regularity. 

Theorem 4.3. Assume

(

)

(

( )

( )

)

( )

(

( )

( )

)

2

2 1 1 2 1

0, 1, 0, 1 0 0 0

u u α α ∈ H Ω ∩H Ω ×H Ω × H Ω ∩H Ω ,

then the system (5)-(8) possesses a unique solution

(

u,α

)

such that

( )

( )

(

2 1

)

0

, 0, ;

uα∈L∞ T H Ω ∩H Ω ,

( )

(

1

)

2

(

2

( )

)

0

0, ; 0, ;

u

L T H L T L

t

∞

∂

∈ Ω ∩ Ω

∂ ,

( )

( )

(

2 1

)

2

(

2

( )

1

( )

)

0 0

, 0, ;

L H H L T H H

t

α ∞ +

∂ _{∈ Ω ∩ Ω ∩ Ω ∩ Ω}

( )

( )

(

)

2

2 2 1

0

2 L 0, ;T H H

t

α

∂ _{∈ Ω ∩ Ω}

∂

and 2 2

(

2

( )

)

2 0, ;

u

L T L

t

∂ _{∈ Ω}

∂ , for all T > 0.

Proof. Following theorems 4.1 and 4.2, the system (5)-(8) possesses the unique solu-tion

(

u,α

)

such that

( )

(

1

)

0

, 0, ;

uα∈L∞ T H Ω ,

( )

(

2

)

2

(

1

( )

)

, 0, ;

u

L L L T H

t

∞ + −

∂ _{∈ Ω ∩ Ω}

∂ R ,

( )

(

1

)

2

(

1

( )

)

0 0

, 0, ;

L H L T H

t

α ∞ +

∂ _{∈ Ω ∩ Ω}

∂ R

and 2 2

(

1

( )

)

0 2 L 0, ;T H

t

α

∂ _{∈ Ω}

∂ , for all T >0. Multiply (2.1) by u

t

∂

∂ and integrate over Ω. We have

( )

2 2

2

d

2 2 , 2 ,

d

u u u u

u f u

t t t t t t

α  _∂  _{∂  ∂ ∂   ∂ } ∇ + ∆ + = ∇ ∇ − ∇ ∇   _{   }  _∂  _{∂  ∂ ∂   ∂ }  

we deduce the following inequality

( )

2 2

2

d

2 2 , 2 d .

d

u u u u

u f u u x

t t t t t t

α Ω  _∂  _{∂  ∂ ∂  ∂} ′ ∇ + ∆ + ≤ ∇ ∇ + ∇ ∇   _{ }  _∂  _{∂  ∂ ∂  ∂}

 

∫

(17)

Thanks to use f′

( )

s , we find the following estimate

( )

(

)

(

6

)

6

2

2

2

4 2

2 d 3 1 d

3 1 d

1 .

L L

u u

f u u x u u x

t t

u

u u x

t u

C u u

t Ω Ω Ω ∂ ∂ ′ ∇ ∇ ≤ − ∇ ∇ ∂ ∂ ∂ ≤ + ∇ ∇ ∂ ∂ ≤ + ∇ + ∇ ∂

∫

∫

∫

Since

(

1

( )

)

0

0, ;

u∈L∞ T H Ω , then the estimate (17) implies

2 2 2

2 2

d

2 2 , .

d

u u u u

u C u

t t t t t t

α

 _∂  _{∂  ∂ ∂  ∂} ∇ + ∆ + ≤ ∇ ∇ + ∆ + ∇

  _{ }

 _∂  _{∂  ∂ ∂  ∂}

  (18)

Multiplying (6) by

t

α

∂ −∆

∂ and integrating over Ω, we get

2 2 2

2

d

2 2 , .

d

u

t t t t t t

α α _α α α

 _{∂ ∂}  _{∂  ∂ ∂ } ∇ + ∆ + ∆ + ∆ = − ∇ ∇

  _{ }

 _{∂ ∂}  _{∂  ∂ ∂ }

  (19)

Now summing (18) and (19), we obtain

2 2 2

2 3

d

2 2

d

E u u

C u

t t t t

α

∂ ∂ ∂

+ + ∆ ≤ ∆ + ∇

∂ ∂ ∂

2 2 2

2 2

3 .

u

E u

t t t

α α _α

∂ ∂ ∂

= ∇ + ∆ + ∇ + ∆ + ∆

∂ ∂ ∂



Appling the Gronwall’s lemma, we deduce that

(

2

( )

1

( )

)

0

, 0, ;

u α∈L∞ T H Ω ∩H Ω ,

( )

(

1

)

2

(

2

( )

)

0

0, ; 0, ;

u

L T H L T L

t

∞

∂

∈ Ω ∩ Ω

∂ and

( )

( )

(

2 1

)

2

(

2

( )

1

( )

)

0 0

0, ; 0, ;

L T H H L T H H

t

α ∞

∂ _{∈ Ω ∩ Ω ∩ Ω ∩ Ω}

∂ .

Multiplying (5) by

( )

1 2 2

u t

− ∂

−∆

∂ and integrating ovre Ω, we obtain

( )

( )

2 2

2 2 2 2

2 2 2 2

1

2 2 2

2 2 2

d

2 2 , 2 , 2 , ,

d

2 2 2 d .

u u u u u

u f u

t t t

t t t t

u u u

u f u x

t t t t

α

α

−

Ω

     

∂ ₊ ∂ ₌ ∂ ∂ _{+ ∆} ∂ ₋ ∂

     

∂ ∂

∂ _ ∂ _{ } ∂ _{ } ∂ _

∂ ∂ ∂ ∂

≤ + ∆ +

∂ ∂ ∂

∫

∂



(20)

Thanks to use f s

( )

and the fact that u L

(

0, ;T H2

( )

H10

( )

)

∞

∈ Ω ∩ Ω , we get

( )

2 2 2 2

2 2 2

2 2 2

2

d d d

. 3 L

u u u

f u x u u x u x

t t t

u

C u

t

∞

Ω Ω Ω

∂ ∂ ∂

≤ +

∂ ∂ ∂

∂

≤ ∇ +

∂

∫

∫

∫



Inserting the above estimate into (20), we obtain

2 2 2

2

2 2

1 2 3 1 2 3

2

1

d

, , , 0

d

u u

C C u C u C C C

t t t

t

α

−

∂ ∂ ∂

+ ≤ + ∆ + ∇ >

∂ ∂

∂



which implies that 2 2

(

2

( )

)

2 0, ;

u

L T L

t

∂ _{∈ Ω}

∂ .

Multiplying (6) by 22

t

α

∂ −∆

∂ and integrating over Ω, we find

2 2 2 2

2 2

2

2 2

d

2 2 2 ,

d

u

t t t

t t

α α α _α

∂ ∂ ∂ ∂

∇ + ∆ + ∆ ≤ + ∆ ∂ ∂

∂ ∂

that implies 2 2

(

2

( )

1

( )

)

0

2 L 0, ;T H H

t

α

∂

∈ Ω ∩ Ω

∂ . 

5. Conclusion

We have just shown the theorems of existence and uniqueness of the solutions for per-turbed Cahn-Hilliard hyperbolic phase-field system with regular potentials.

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