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Regularity of a thermoelastic problem with variable parameters

Published online by Cambridge University Press:  19 October 2015

P. BARRAL ,
M. C. NAYA-RIVEIRO  and
P. QUINTELA
P. BARRAL
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain email: [email protected], [email protected]
M. C. NAYA-RIVEIRO
Affiliation:
Department of Pedagogy and Didactics, Faculty of Educational Studies Universidade da Coruña, 15071 A Coruña, Spain email: [email protected]
P. QUINTELA
Affiliation:
Department of Applied Mathematics, Faculty of Mathematics, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain email: [email protected], [email protected] ITMATI (Technological Institute for Industrial Mathematics) Campus Vida, 15782 Santiago de Compostela, Spain
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Abstract

This paper deals with a fully-coupled thermoelastic problem, in a heterogeneous medium, arising from the metallurgical industry. The aim is to prove regularity properties of the solution with respect to space and time. Regularity in space is obtained by means of regularity properties for elliptic operators. In order to prove regularity in time, a mathematical induction technique, together with an existence and uniqueness result for this type of problems, is applied.

Keywords

Thermoelasticityquasi-static problemheterogeneous materialsregularity35B6535D3035Q7474F05
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 2 , April 2016 , pp. 213 - 232
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Agmon, S., Douglis, A. & Nirenberg, L. (1964) Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17 (1), 35–92.CrossRefGoogle Scholar
[2] Barral, P., Naya-Riveiro, M. C. & Quintela, P. (2007) Mathematical analysis of a viscoelastic problem with temperature-dependent coefficients. I. Existence and uniqueness. Math. Meth. Appl. Sci. 30 (13), 1545–1568.CrossRefGoogle Scholar
[3] Barral, P., Naya-Riveiro, M. C. & Quintela, P. (2007) Mathematical analysis of a viscoelastic problem with temperature-dependent coefficients. II. Regularity. Math. Meth. Appl. Sci. 30 (13), 1569–1592.CrossRefGoogle Scholar
[4] Barral, P., Naya-Riveiro, M. C. & Quintela, P. (2015) Existence and uniqueness of a thermoelastic problem with variable parameters. Eur. J. Appl. Math. 26 (04), 497–520.CrossRefGoogle Scholar
[5] Barral, P. & Quintela, P. (1999) A numerical method for simulation of thermal stresses during casting of aluminium slabs. Comput. Meth. Appl. Mech. Eng. 178 (1–2), 69–88.CrossRefGoogle Scholar
[6] Bermúdez, A., Muñiz, M. C. & Quintela, P. (1993) Numerical solution of a three-dimensional thermoelectric problem taking place in an aluminum electrolytic cell. Comput. Meth. Appl. Mech. Eng. 106 (1–2), 129–142.CrossRefGoogle Scholar
[7] Copetti, M. I. M. & Elliott, C. M. (1993) A one-dimensional quasi-static contact problem in linear thermoelasticity. Eur. J. Appl. Math. 4 (2), 151–174.CrossRefGoogle Scholar
[8] Duvaut, G. & Lions, J. L. (1972) Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46 (4), 241–279.CrossRefGoogle Scholar
[9] Gawinecki, J. (1981) Uniqueness and regularity of the solution of the first boundary-initial value problem for thermal stresses equations of classical and generalized thermomechanics. Bull. Acad. Polon. Sci. Sér. Sci. Tech. 29 (11–12), 231–238.Google Scholar
[10] Gawinecki, J. (1983) Existence, uniqueness and regularity of the first boundary-initial value problem for thermal stresses equations of classical and generalized thermomechanics. J. Tech. Phys. 24 (4), 467–479.Google Scholar
[11] Gawinecki, J. (1986) Existence, uniqueness and regularity of the solution of the first boundary-initial value problem for the equations of linear thermomicroelasticity. Bull. Polish Acad. Sci. Tech. Sci. 34 (7–8), 447–460.Google Scholar
[12] Gawinecki, J. (1987) Existence, uniqueness and regularity of the first boundary-initial value problem for hyperbolic equations system of the thermal stresses theory for temperature-rate-dependent solids. Bull. Polish Acad. Sci. Tech. Sci. 35 (7–8), 411–419.Google Scholar
[13] Gawinecki, J. (1987) The faedo-galerkin method in thermal stresses theory. Comment. Math. Prace Mat. 27 (1), 83–107.Google Scholar
[14] Gawinecki, J., Kowalski, T. & Litewska, K. (1982) Existence and uniqueness of the solution of the mixed boundary-initial value problem in linear thermoelasticity. Bull. Acad. Polon. Sci. Sér. Sci. Tech. 30 (11–12), 173–178.Google Scholar
[15] Jiang, S. & Racke, R. (2000) Evolution Equations in Thermoelasticity, Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
[16] Kačur, J. & Ženíšek, A. (1986) Analysis of approximate solutions of coupled dynamical thermoelasticity and related problems. Apl. Mat. 31 (3), 190–223.Google Scholar
[17] Lions, J. & Magenes, E. (1968) Problèmes aux limites non-homogènes et applications, Dunod, Paris.Google Scholar
[18] Marzocchi, A., Muñoz, Rivera, J. E. & Naso, M. G. (2003) Transmission problem in thermoelasticity with symmetry. IMA J. Appl. Math. 68 (1), 23–46.CrossRefGoogle Scholar
[19] Mizohata, S. (1973) The Theory of Partial Differential Equations, Cambridge University Press, New York.Google Scholar
[20] Muñoz Rivera, J. E. & Racke, R. (1998) Multidimensional contact problems in thermoelasticity. SIAM J. Appl. Math. 58 (4), 1307–1337.CrossRefGoogle Scholar
[21] Nečas, J. (1967) Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris.Google Scholar
[22] Zhelezovskii, S. E. (2010) On the high smoothness of the solution of an abstract hyperbolic-parabolic system of equations of thermoelasticity system type. Differ. Equ. 46 (6), 840–852.CrossRefGoogle Scholar
[23] Zhelezovskii, S. E. (2010) On the smoothness of the solution of an abstract coupled problem of thermoelasticity type. Comput. Math. Math. Phys. 50 (7), 1178–1194.CrossRefGoogle Scholar