Identifiability of the Landau-Ginzburg potential in a mathematical model of shape memory alloys
type: Item
instance of: scholarly article
title: Identifiability of the Landau-Ginzburg potential in a mathematical model of shape memory alloys
zbMATH Open document ID: 0893.35128
author: Pedro Morin
author: R. D. Spies
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-08-28T00:00:00Z
review text: The authors consider an initial boundary value problem (IBVP) that involves a pair of coupled nonlinear partial differential equations. These equations arise from conservation laws of linear momentum and energy, and describe structural phase transitions in one-dimensional shape memory solids with non-convex Landau-Ginzburg potentials. As in previous work by one of the authors [\textit{R. D. Spies}, J. Math. Anal. Appl. 190, 58-100 (1995; Zbl 0829.35132)], the IBVP is first expressed as a semilinear Cauchy problem of the form \[ \dot z(t)= A(q)z(t) +F(q,t,z), \quad z(0) =z_0, \tag{1} \] set in an appropriate Hilbert space \(Z_q\), where \(q\) is a vector of admissible parameters. In this abstract formulation, \(A(q)\) is a differential operator associated with the linear part of the coupled partial differential equations, while \(F(q,t,z)\) corresponds to the nonlinear part of the system. The theories of analytic semigroups and real interpolation spaces for maximal accretive operators are then used to show that solutions of (1) depend continuously on the admissible parameters. In addition, the non-physical parameters that define the free energy are shown to be identifiable from the model.
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 80A22
zbMATH DE Number: 1053565
zbMATH Keywords: equations of thermo-viscoelasto-plasticity
zbMATH Keywords: accretive operators
zbMATH Keywords: semigroups
zbMATH Keywords: phase transitions
zbMATH Keywords: Landau-Ginzburg potentials
reviewed by: Wilson Lamb
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5502
OpenAlex ID: W2059033839
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DOI: 10.1006/JMAA.1997.5502
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description: scientific article; zbMATH DE number 1053565