IJPAM: Volume 117, No. 3 (2017)




José Luiz Boldrini$^1$, Patrıcia Nunes da Silva$^2$
P.O. Box 6065, Campinas, SP, 13083-859, BRAZIL
Office 6016D, São Francisco Xavier Street
Rio de Janeiro, RJ, 20550-900, BRAZIL


We analyse a family of orientation dependent systems consisting of a Cahn-Hilliard and several Allen-Cahn type equations. These systems are similar to one proposed by Fan, L.-Q. Chen, S. Chen and Voorhees (1998) for modelling Ostwald ripening of anisotropic crystals in a two-phase systems. They describe Ostwald ripening by taking several crystallographic orientations into account, considering both the evolution of the compositional field and of the crystallographic orientations. Fan et al. presented several numerical experiments to validate their modelling of the coarsening dynamics of one physical phase dispersed in the matrix of another The aim of the present article is to rigorously prove the existence and the uniqueness of solutions for such systems; for this, we firstly consider a suitable family of auxiliary approximate problems; we then deduce certain estimates for their corresponding solutions, and, by using compactness arguments, we extract subsequences that converge to a solution of the original problem.


Received: 2017-05-27
Revised: 2017-09-09
Published: January 15, 2018

AMS Classification, Key Words

AMS Subject Classification: 47J35, 35K57, 35Q99
Key Words and Phrases: Cahn-Hilliard/Allen-Cahn systems, phase fields, ostwald ripening, orientation dependent systems

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.


S.M. Allen, J.W. Cahn,A microscopic theory of domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics,J. Phys. Colloques, 38 (1977), C7-51-C7-54, doi: https://doi.org/10.1051/jphyscol:1977709.

J.W. Barrett, H. Garcke, R. Nürnberg,A phase-field model for the electromigration of intergranular voids,Interface Free Bound., 9 (2007), 171-210, doi: https://doi.org/10.4171/IFB/161.

J.W. Barrett, H. Garcke, R. Nürnberg,On sharp interface limits of Allen-Cahn/Cahn-Hilliard variational inequalities,Discret. Contin. Dyn. S., 1 (2008), 1-14, doi: https://doi.org/10.3934/dcdss.2008.1.1.

J. Beaucourt, F. Rioual, T. Séon, T. Biben, C. Misbah, Steady to unsteady dynamics of a vesicle in a flow,Phys. Rev. E, 69 (2004), 011906-011922, doi: https://doi.org/10.1103/PhysRevE.69.011906.

J.L. Boldrini, B.M.C. Caretta, E. Fernández-Cara,Analysis of a two-phase field model for the solidification of an alloy, J. Math. Anal. Appl., 357 (2009), 25-44, doi: https://doi.org/10.1016/j.jmaa.2009.03.063.

R. Borcia, M. Bestehorn, Phase-field model for Marangoni convection in liquid-gas systems with a deformable interface,Phys. Rev. E, 67 (2003), 066307-066316, doi: https://doi.org/10.1103/PhysRevE.67.066307.

G. Boussinot, Y. Le Bouar, A. Finel,Phase-field simulations with inhomogeneous elasticity: comparison with an atomic-scale method and application to superalloys,Acta Mater., 58 (2010), 4170-4181, doi: https://doi.org/10.1016/j.actamat.2010.04.008.

D. Brochet, D. Hilhorts, A. Novick-Cohen, Finite dimensional exponential attractor for a model for order-disorder and phase separation,Appl. Math. Lett., 7 (1994), 83-87, doi: https://doi.org/10.1016/0893-9659(94)90118-X.

J.W. Cahn, J.E. Hilliard,Free energy of a nonuniform system,J. Chem. Phys., 28 (1958), 258-267, doi: https://doi.org/10.1063/1.1744102.

J.W. Cahn, A. Novick-Cohen, Limiting motion for an Allen-Cahn/Cahn-Hilliard system,Free boundary problems, theory and applications, Pitman, Boston (1993), 89-97.

J.W. Cahn, A. Novick-Cohen, Motion by curvature and impurity drag: resolution of a mobility paradox,Acta Mater., 48 (2000), 3425-3440, doi: https://doi.org/10.1016/S1359-6454(00)00144-0.

L.Q. Chen, D. Fan,Computer simulation for coupled grain growth and Ostwald ripening-application to ${\rm Al}_{2}{\rm O}_{3}$- ${\rm Zr}{\rm }_{2}$ two-phase systems,J Am Ceram Soc, 79 (1996), 1163-1168, doi: https://doi.org/10.1111/j.1151-2916.1996.tb08568.x.

L.Q. Chen, D. Fan,Computer simulation of grain growth using a continuum field model,Acta Mater., 45 (1997), 611-622, doi: https://doi.org/10.1016/S1359-6454(96)00200-5.

L.Q. Chen, D. Fan,Diffusion-controlled grain growth in two-phase solids,Acta Mater., 45 (1997), 3297-3310, doi: https://doi.org/10.1016/S1359-6454(97)00022-0.

L.Q. Chen, D. Fan, C. Geng, Computer simulation of topological evolution in 2-d grain growth using a continuum diffuse-interface field model, Acta Mater., 45 (1997), 1115-1126, doi: https://doi.org/10.1016/S1359-6454(96)00221-2.

J.B. Collins, H. Levine,Diffuse interface model of diffusion-limited crystal growth,Phys. Rev. B, 31 (1985), 6119-6122, doi: https://doi.org/10.1103/PhysRevB.31.6119.

R. Dal Passo, L. Giacomelli, A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility,Interface Free Bound., 1 (1999), 199-226, doi: https://doi.org/10.4171/IFB/9.

C.M. Elliott, H. Garcke,On the Cahn-Hilliard equation with degenerate mobility,SIAM J. Math. Anal., 27 (1996), 404-423, doi: https://doi.org/10.1137/S0036141094267662.

C.M. Elliott, S. Luckhaus,A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy,I.M.A. (Minnesota) Preprint Series, 887 (1991), 37-pages, Avaliable on: https://www.ima.umn.edu/sites/default/files/887.pdf.

D. Fan, L.Q. Chen, S. Chen, P.W. Voorhees, Phase field formulations for modeling the Ostwald ripening in two-phase systems,Comp. Mater. Sci., 9 (1998), 329-336, doi: https://doi.org/10.1016/S0927-0256(97)00158-4.

H. Garcke, B. Nestler, B. Stoth, On anisotropic order parameter models for multi-phase systems and their sharp interface limits,Physica D, 115 (1998), 87-108, doi: https://doi.org/10.1016/S0167-2789(97)00227-3.

G. Gilardi, E. Rocca,Well posedness and long time behaviour for a singular phase field system of conserved type,IMA J. Appl. Math., 72 (2007), 498-530, doi: https://doi.org/10.1093/imamat/hxm015.

R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth,Physica D, 63 (1993), 410-423, doi: https://doi.org/10.1016/0167-2789(93)90120-P.

A. Karma, W-J. Rappel, Numerical simulation of three-dimensional dendritic growth,Phys. Rev. Lett., 77 (1996), 4050-4053, doi: https://doi.org/10.1103/PhysRevLett.77.4050.

K. Kassner, C. Misbah, A phase field approach for stress-induced instabilities,Europhys. Lett., 46 (1999), 217-223, doi: https://doi.org/10.1209/epl/i1999-00247-9.

J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaries,Dunod, Paris (1969).

M. Mahadevan, R.M. Bradley, Phase-field model of surface electromigration in single crystal metal thin films,Physica D, 126 (1999), 201-213, doi: https://doi.org/10.1016/S0167-2789(98)00276-0.

T. Roths, C. Friedrich, M. Marth, J. Honerkamp,Dynamics and rheology of the morphology of immiscible polymer blends - on modeling and simulation,Rheol. Acta, 41 (2002), 211-222, doi: https://doi.org/10.1007/s003970100189.

P.N. Silva, J.L. Boldrini, A generalized solution to a Cahn-Hilliard/Allen-Cahn system,Electron. J. Diff. Equations., 126 (2004), 1-24.

P.N. Silva, J.L. Boldrini, Existence and approximate solutions of a model for Ostwald ripening,Numer. Func. Anal. Opt., 29 (2008), 883-904, doi: https://doi.org/10.1080/01630560802295906.

P.N. Silva, J.L. Boldrini, Error estimates for full discretization of a model for Ostwald ripening,Numer. Func. Anal. Opt., 29 (2008), 905-926, doi: https://doi.org/10.1080/01630560802295922.

J.D. van der Waals,The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, J. Stat. Phys., 20 (1979), 200-244, doi: https://doi.org/10.1007/BF01011513.

How to Cite?

DOI: 10.12732/ijpam.v117i3.10 How to cite this paper?

International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 117
Issue: 3
Pages: 447 - 465

Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).