A rapidly converging phase field model
type: Item
instance of: scholarly article
title: A rapidly converging phase field model
zbMATH Open document ID: 1119.80009
author: Xinfu Chen
author: C. Eck
author: Gunduz Caginalp
published in: Discrete and Continuous Dynamical Systems
publication date: 2007-04-10T00:00:00Z
review text: The authors introduce asymptotic expansion of the solution of a phase field problem in order to simplify its resolution. The authors consider the coupled evolution problem \(\varepsilon ^{2}(\alpha _{\varepsilon }\varphi _{\varepsilon t}-\Delta \varphi _{\varepsilon })+W^{\prime }(\varphi _{\varepsilon })=\varepsilon mG^{\prime }(\varphi _{\varepsilon })u_{\varepsilon }\) ; \((u_{\varepsilon }+l\varphi _{\varepsilon }/2)_{t}=\Delta u_{\varepsilon }\), where \(\varphi _{\varepsilon }\) (resp. \(u_{\varepsilon }\)) is the phase indicator (between \(-1\) and \(+1\)) (resp. the temperature). \(W\) is a smooth double-well potential and \(G\) is a smooth function, both satisfying usual conditions. The authors assume that \(\alpha _{\varepsilon }=\alpha +\varepsilon \alpha _{1}\), where \( \alpha \) and \(\alpha_{1}\) are positive constants, \(\alpha _{1}\) being supposed to be linked to \(G\) and to the data of the problem. The main purpose of the paper is to introduce inner and outer asymptotic expansions of the solution, according to the proximity of \(\varphi _{\varepsilon }\) to \( \pm 1\), to compute the two first order terms of this asymptotic expansions and to prove, under further assumptions, that the difference between the solution and these two first-order terms is of order \(O(\varepsilon ^{2})\). The paper ends with a 1D-example.
Mathematics Subject Classification ID: 80A22
Mathematics Subject Classification ID: 80M35
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 5141118
zbMATH Keywords: phase field model
zbMATH Keywords: sharp interface limit
zbMATH Keywords: second order asymptotics
reviewed by: Alain Brillard
MaRDI profile type: Publication
full work available at URL: dcds.2006.15.1017
OpenAlex ID: W2162070653
DOI: 10.3934/DCDS.2006.15.1017
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description: scientific article; zbMATH DE number 5141118