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

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1. Introduction

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

(1)

(2)

where u is the order parameter and 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 system (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 potentials.

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

(3)

(4)

where, u is the order parameter and 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

(5)

(6)

(7)

(8)

which is the perturbed phase-field system of Cahn-Hilliard phase-field system (3)-(4) with. In the above hyperbolic system is a bounded and regular domain of with or 3 and f is the nonlinear regular potentials.

The hyperbolic system has been extensively studied for Dirichlet boundary conditions 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 which satisfies the following properties:

(9)

(10)

(11)

2. Notations

We denote by the usual L^{2}-norm (with associated product scalar (.,.)) and set

, where denotes the minus Laplace operator with Dirichlet

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

Throughout this paper, the same letters and denote (generally positive) constants which may change from line to line, or even a same line.

3. A Priori Estimates

We multiply (5) by and (6) by, integrate over and add the two resulting differential equalities. We find

where

satisfies

Finaly, we conclude that,

and

for all.

Multiply (6) by and integrate over. We get.

Then.

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 then the system (5) - (8) possesses at least one solution such that

,

,

and, for all.

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 such that

,

,

and for all.

Proof. Let and be two solutions of the system (5)-(8) with initial data and, respectively. We set and, then is solution of the following system

(12)

(13)

We multiply (12) by and integrate over. We find

(14)

Multiplying (13) by and integrating over, we get

(15)

Now summing (14) and (15) we obtain

(16)

where

Lagrange theorem gives a estimates

which implies

Inserting the above estimate into (16), we have

Applying Gronwall’s lemma, we obtain for all

We deduce the continuous dependence of the solution relative to the initial conditions, 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

,

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

,

,

,

and, for all.

Proof. Following theorems 4.1 and 4.2, the system (5)-(8) possesses the unique solution such that

,

,

and, for all.

Multiply (2.1) by and integrate over. We have

we deduce the following inequality

(17)

Thanks to use, we find the following estimate

Since, then the estimate (17) implies

(18)

Multiplying (6) by and integrating over, we get

(19)

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

where

Appling the Gronwall’s lemma, we deduce that,

and

.

Multiplying (5) by and integrating ovre, we obtain

(20)

Thanks to use and the fact that, we get

Inserting the above estimate into (20), we obtain

which implies that.

Multiplying (6) by and integrating over, we find

that implies.

5. Conclusion

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

References

[1] Caginalp, G. (1988) Conserved-Phase Field System: Implications for Kinetic Undercooling. Physical Review B, 38, 789-791.

http://dx.doi.org/10.1103/PhysRevB.38.789

[2] Brochet, D., Hilhorst, D. and Novick-Cohen, A. (1996) Maximal Attractor and Inertial Sets for a Conserved Phase-Field Model. Advances in Differential Equations, 1, 547-578.

[3] Brochet, D. (1993) Maximal Attractor and Inertial Sets for Some Second and Fourth Order Phase-Field Models. In: Pitman Res. Notes Math. Ser, Vol. 296, Longman Sci. Tech., Harlow, 77-85.

[4] Colli, P., Gilardi, G., Grasselli, M. and Schimperna, G. (2001) The Conserved Phase-Field System with Memory. Adv. Math. Sci Appl., 11, 265-291.

[5] Gatti, S. and Pata, V. (2004) Exponential Attractor for a Conserved Phase-Field System with Memory. Physica D: Nonlinear Phenomena, 189, 31-48.

http://dx.doi.org/10.1016/j.physd.2003.10.005

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http://dx.doi.org/10.1016/j.jmaa.2012.11.038

[8] Caginalp, G. (1990) The Dynamic of Conserved Phase Field System: Stefan-Like, Hele-Shaw and Cahn-Hilliard Models as Asymptotic Limits. IMA Journal of Applied Mathematics, 44, 77-94.

[9] Colli, P., Gilardi, G., Laurenot, Ph. and Novick-Cohen, A. (1999) Uniqueness and Long-Time Behavior for the Conserved Phase-Field System Memory. Discrete and Continuous Dynamical Systems—Series A, 5, 375-390.

http://dx.doi.org/10.3934/dcds.1999.5.375

[10] Miranville, A. and Quintanilla, R. (2011) A Type III Phase-Field System with a Logarithmic Potential. Applied Mathematics Letters, 24, 1003-1008.

http://dx.doi.org/10.1016/j.aml.2011.01.016

[11] Ntsokongo, A.J. and Batangouna, N. (2016) Existence and Uniqueness of Solutions for a Conserved Phase-Field Type Model. AIMS Mathematics, 1, 144-155.

http://dx.doi.org/10.3934/Math.2016.2.144

[12] Goyaud, M.E.I., Moukamba, F., Moukoko, D. and Langa, F.D.R. (2015) Existence and Uniqueness of Solution for Caginalp Hyperbolic Phase Field System with Polynomial Growth Potential. International Mathematical Forum, 10, 477-486.

[13] Moukoko, D. (2014) Well-Posedness and Longtime Behaviors of a Hyprebolic Caginalp System. Journal of Applied Analysis and Computation, 4, 151-196.

[14] Moukoko, D. (2015) Etude de Modeles Hyperboliques de champ de phase de Caginalp, These unique, Falculté des Sciences et Techniques, Université Marien NGOUABI.

[15] Moukoko, D., Moukamba, F. and Reval, L.F.D. (2015) Global Attractor for Caginalp Hyperbolics Field-Phase System with Singular Potential. Journal of Mathematics Research, 7, 165-177.