On formation of singularities in one-dimensional nonlinear thermoelasticity
type: Item
instance of: scholarly article
title: On formation of singularities in one-dimensional nonlinear thermoelasticity
zbMATH Open document ID: 0712.73023
DOI: 10.1007/BF00375405
published in: Archive for Rational Mechanics and Analysis
publication date: 1990-01-01T00:00:00Z
review text: In this paper the authors studied the formation of singularities in one- dimensional nonlinear thermoelastic materials. Their constitutive relations are similar to those used by \textit{C. M. Dafermos} and \textit{L. Hsiao} [Q. Appl. Math. 44, 463-474 (1986; Zbl 0661.35009)]. Although their constitutive equations may not be appropriate for any real material, they satisfy the assumptions used to establish the global existence of smooth solutions when the data are close to the equilibrium. Moreover, they are fully compatible with the second law of thermodynamics; in particular there is a free energy. They considered the Cauchy problem in which the body occupies the entire real line and the initial values of the strain, velocity, and temperature are prescribed. As their main result they show that there are smooth initial data for which the solution will develop singularities at finite time. Their proof is similar to that of Dafermos and Hsiao. The primary difference is that their equations of evolution contain an additional term that prevents them from using the maximum principle to obtain an important a priori bound. This difficulty is overcome by exploiting some relations associated with the second law of thermodynamics. This work might be useful for those researchers working in the area of propagation of singularities.
reviewed by: Hilmi Demiray
Mathematics Subject Classification ID: 74J99
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74A15
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 4172522
zbMATH Keywords: global existence of smooth solutions
zbMATH Keywords: free energy
zbMATH Keywords: Cauchy problem
zbMATH Keywords: smooth initial data
zbMATH Keywords: singularities at finite time
author: William J. Hrusa
author: Salim A. Messaoudi
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00375405
OpenAlex ID: W2100196382
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description: scientific article; zbMATH DE number 4172522