Phase-field models with hysteresis
type: Item
instance of: scholarly article
title: Phase-field models with hysteresis
zbMATH Open document ID: 0983.74048
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-04-14T00:00:00Z
review text: The authors consider the system of equations \(\mu(\theta)w_t+f_1[w]+\theta f_2[w]=0\), \((\theta+F_1[w])_t-\Delta\theta=\Psi(x,t,\theta)\) with \(x\in\Omega\subset\mathbb{R}^N\), where \(\Omega\) is a bounded domain with Lipschitz boundary, \(t\in(0,T)\), and \(f_1, f_2, F_1\) are hysteresis operators. In the context of the main application, i.e. phase-field equations modelling the phase transitions, \(\theta\) is the absolute temperature, \(\mu\) the phase relaxation coefficients, \(\Psi\) is a heat source, and the definition of hysteresis operators is linked to the definition of free energy. The system is supplemented by initial and boundary conditions \(\theta(x,0)=\theta^0(x)\), \(w(x,0)=w^0(x)\) in \(\Omega\), and \(\frac{\partial\theta}{\partial n}=0\) on \(\partial\Omega\times(0,T)\). The main result is an existence and uniqueness theorem, under suitable assumptions on \(\mu, f_1, f_2, F_1, \Psi\), and on the data \(\theta^0, w^0\). The solution satisfies the above system of equations a.e., and additionally an estimate of the type \(\theta \geq \delta e^{-\beta t}\) is proved. The technical part of the paper is preceded by a useful introduction in which the authors clearly illustrate the physical relevance of the problem. The paper extends in a significant way previous results by the same authors.
Mathematics Subject Classification ID: 74N30
Mathematics Subject Classification ID: 35Q72
zbMATH DE Number: 1563456
zbMATH Keywords: phase-field equations
zbMATH Keywords: variational inequality
zbMATH Keywords: decay estimate
zbMATH Keywords: phase transition
zbMATH Keywords: phase relaxation coefficient
zbMATH Keywords: heat source
zbMATH Keywords: hysteresis operator
zbMATH Keywords: free energy
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
author: Pavel Krejč{í }
author: Jürgen Sprekels
reviewed by: Antonio Fasano
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6974
OpenAlex ID: W2055139851
cites work: Hysteresis and phase transitions
cites work: An analysis of a phase field model of a free boundary
cites work: Q3704963
cites work: Q3994579
cites work: Q3543379
cites work: Q4263209
cites work: A hysteresis approach to phase-field models
cites work: Hysteresis operators in phase-field models of Penrose-fife type
cites work: Q5562267
cites work: Thermodynamically consistent models of phase-field type for the kinetics of phase transitions
cites work: Global smooth solutions to a thermodynamically consistent model of phase- field type in higher space dimensions
cites work: Differential models of hysteresis
cites work: Models of phase transitions
DOI: 10.1006/JMAA.2000.6974
description: scientific article; zbMATH DE number 1563456