Stability analysis for phase field systems associated with crystalline-type energies

Ken Shirakawa

doi:10.3934/dcdss.2011.4.483

Article Contents

2011, Volume 4, Issue 2: 483-504. Doi: 10.3934/dcdss.2011.4.483

This issue Previous Article On certain convex compactifications for relaxation in evolution problems Next Article

Stability analysis for phase field systems associated with crystalline-type energies

Ken Shirakawa^1,

1.
Department of Mathematics, Department of Mathematics Faculty of Education, Chiba University, 1-33 Yayoichō, Inage, Chiba, 263-8522

Received: January 31, 2009

Revised: September 30, 2009

Published: March 2011

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Abstract

In this paper, a mathematical model, to represent the dynamics of two-dimensional solid-liquid phase transition, is considered. This mathematical model is formulated as a coupled system of a heat equation with a time-relaxation diffusion, and an Allen-Cahn equation such that the two-dimensional norm, of crystalline-type, is adopted as the mathematical expression of the anisotropy. Through the structural observations for steady-state solutions, some geometric conditions to guarantee their stability will be presented in the main theorem of this paper.

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Mathematics Subject Classification: Primary: 35B35, 74A50; Secondary: 82B26.

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