Oscillations in a dynamical model of phase transitions
type: Item
instance of: scholarly article
title: Oscillations in a dynamical model of phase transitions
zbMATH Open document ID: 0982.74052
DOI: 10.1017/S0308210500000792
author: Deborah Brandon
author: Pieter Swart
author: Irene Fonseca
publication date: 2001-07-02T00:00:00Z
Mathematics Subject Classification ID: 74N20
Mathematics Subject Classification ID: 74N15
Mathematics Subject Classification ID: 35Q72
zbMATH DE Number: 1590130
zbMATH Keywords: evolution of microstructure
zbMATH Keywords: viscoelastic damping
zbMATH Keywords: propagation of discontinuities
zbMATH Keywords: uniqueness of Young measure
zbMATH Keywords: one-dimensional continuum
zbMATH Keywords: non-monotone stress-strain relation
zbMATH Keywords: oscillation
zbMATH Keywords: weakly convergent sequence
zbMATH Keywords: periodically modulated case
MaRDI profile type: Publication
full work available at URL: s0308210500000792
OpenAlex ID: W1968783505
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published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics
review text: The authors consider the evolution of microstructure in a one-dimensional continuum modelled by the equation \(u_{tt} = \sigma(u_x)_x +\beta u_{xxt} -\alpha u\), \(a \geq 0\), \(\beta >0\), \(x \in (0,1)\), \(t >0\), with boundary conditions \(u(0,t)=u_{xx}(0,t) = 0\), \(u(1,t)=u_{xx}(1,t)=0 \), and the initial conditions \(u(x,0)=a(x)\), \(u_t(x,0)=b(x)\). Here the term \(\sigma(u_x)=u_x^3-u_x\) is the non-monotone stress-strain relation, \(\alpha\) measures the strength of bonding of the continuum to the substrate, and \(\beta\) measures the strength of viscoelastic damping. The paper concentrates on the class of initial conditions \(\{ a^\varepsilon, b^\varepsilon \}\) with finer and finer oscillations as \(\varepsilon \rightarrow 0\), and examines the evolution of Young measures associated with weakly convergent sequences \(\{ u_t^\varepsilon, u_x^\varepsilon \}\). NEWLINENEWLINENEWLINEThe main results of the paper are contained in theorems 2.5 and 3.7. In the first of these, the authors show that as \(\varepsilon \rightarrow 0\), \(\{ u_t^\varepsilon \}\) converge strongly in \(L^2_{\text{loc}} ((0,1) \times (0,T))\), which implies that oscillations in \(u_t\) do not persist for any positive time. Theorem 3.7 gives precise statements on the failure of \(u_x^\varepsilon\) to converge strongly, and this on the propagation of stationary strain discontinuities. In particular, the authors show that oscillations in \(u_x\) cannot migrate into regions originally free of oscillations. The paper also contains results on the uniqueness of the Young measure, and further characterization of the Young measure in the periodically modulated case.
description: scientific article