ON AN ORIENTATION-DEPENDENT CAHN-HILLIARD/ALLEN-CAHN SYSTEM

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IJPAM: Volume 117, No. 3 (2017)

Title

ON AN ORIENTATION-DEPENDENT
CAHN-HILLIARD/ALLEN-CAHN SYSTEM

Authors

José Luiz Boldrini

, Patrıcia Nunes da Silva

UNICAMP-IMECC
P.O. Box 6065, Campinas, SP, 13083-859, BRAZIL

IME-UERJ
Office 6016D, São Francisco Xavier Street
Rio de Janeiro, RJ, 20550-900, BRAZIL

Abstract

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.

History

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

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How to Cite?

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

Source: 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

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