American Institute of Mathematical Sciences

Advanced
October  2010, 14(3): 793-816. doi: 10.3934/dcdsb.2010.14.793

Minimum free energy in the frequency domain for a heat conductor with memory

Giovambattista Amendola 1, , Mauro Fabrizio 2, and  John Murrough Golden 3,

1. 

Department of Applied Mathematics "U. Dini”, via Diotisalvi 2, 56126-Pisa, Italy
2. 

Department of Mathematics, Piazza di Porta S. Donato 5, 40127-Bologna, Italy
3. 

School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Received  February 2009 Revised  January 2010 Published  July 2010

An explicit expression is given in the frequency domain for the minimum free energy related to a particular state of a linear rigid heat conductor with memory effects in the constitutive equations, using the fact that this quantity coincides with the maximum recoverable work obtainable from that state. The constitutive equations for the internal energy and the heat flux are expressed as linear functionals of the histories of temperature and its gradient, respectively, together with the present value of the latter quantity. Another equivalent expression for the minimum free energy is also deduced and used to derive explicit formulae for a discrete spectrum model.
Keywords: heat conduction., Memory effects.
Mathematics Subject Classification: 80A20, 74F0.
Citation: Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden. Minimum free energy in the frequency domain for a heat conductor with memory. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 793-816. doi: 10.3934/dcdsb.2010.14.793
[1]

Sandra Carillo, Vanda Valente, Giorgio Vergara Caffarelli. Heat conduction with memory: A singular kernel problem. Evolution Equations & Control Theory, 2014, 3 (3) : 399-410. doi: 10.3934/eect.2014.3.399
[2]

Corrado Mascia. Stability analysis for linear heat conduction with memory kernels described by Gamma functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3569-3584. doi: 10.3934/dcds.2015.35.3569
[3]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[4]

Fabio Camilli, Raul De Maio. Memory effects in measure transport equations. Kinetic & Related Models, 2019, 12 (6) : 1229-1245. doi: 10.3934/krm.2019047
[5]

Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic & Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165
[6]

Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078
[7]

Claudio Giorgi, Diego Grandi, Vittorino Pata. On the Green-Naghdi Type III heat conduction model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2133-2143. doi: 10.3934/dcdsb.2014.19.2133
[8]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[9]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221
[10]

Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations & Control Theory, 2014, 3 (1) : 35-58. doi: 10.3934/eect.2014.3.35
[11]

Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525
[12]

M. Carme Leseduarte, Ramon Quintanilla. Phragmén-Lindelöf alternative for an exact heat conduction equation with delay. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1221-1235. doi: 10.3934/cpaa.2013.12.1221
[13]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145
[14]

Zhiqiang Yang, Junzhi Cui, Qiang Ma. The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 827-848. doi: 10.3934/dcdsb.2014.19.827
[15]

Xin Zhong. Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019209
[16]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147
[17]

Shuji Yoshikawa, Irena Pawłow, Wojciech M. Zajączkowski. A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1093-1115. doi: 10.3934/cpaa.2009.8.1093
[18]

Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden, Adele Manes. Energy stability for thermo-viscous fluids with a fading memory heat flux. Evolution Equations & Control Theory, 2015, 4 (3) : 265-279. doi: 10.3934/eect.2015.4.265
[19]

Luciano Pandolfi. Riesz systems, spectral controllability and a source identification problem for heat equations with memory. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 745-759. doi: 10.3934/dcdss.2011.4.745
[20]

Tomás Caraballo, I. D. Chueshov, Pedro Marín-Rubio, José Real. Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 253-270. doi: 10.3934/dcds.2007.18.253

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences