Thermodynamic properties and stability for the heat flux equation with linear memory

Giorgi, C.; Gentili, G.

doi:10.1090/qam/1218373

Authors: C. Giorgi and G. Gentili
Journal: Quart. Appl. Math. 51 (1993), 343-362
MSC: Primary 80A20
DOI: https://doi.org/10.1090/qam/1218373
MathSciNet review: MR1218373
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Abstract: Within the linearized theory of heat conduction with fading memory, some restrictions on the constitutive equations are found as a direct consequence of thermodynamic principles. Such restrictions allow us to obtain existence, uniqueness, and stability results for the solution to the heat flux equation. Both problems, which respectively occur when the instantaneous conductivity ${k_0}$ is positive or vanishes, are considered.

Bernard D. Coleman and Victor J. Mizel, Thermodynamics and departures from Fourier’s law of heat conduction, Arch. Rational Mech. Anal. 13 (1963), 245–261. MR 151119, DOI https://doi.org/10.1007/BF01262695
Bernard D. Coleman and Victor J. Mizel, Norms and semi-groups in the theory of fading memory, Arch. Rational Mech. Anal. 23 (1966), 87–123. MR 210343, DOI https://doi.org/10.1007/BF00251727
Bernard D. Coleman and Morton E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18 (1967), 199–208 (English, with German summary). MR 214334, DOI https://doi.org/10.1007/BF01596912
Morton E. Gurtin, On the thermodynamics of materials with memory, Arch. Rational Mech. Anal. 28 (1968), no. 1, 40–50. MR 1553503, DOI https://doi.org/10.1007/BF00281562
Morton E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), no. 2, 113–126. MR 1553521, DOI https://doi.org/10.1007/BF00281373
W. A. Day and Morton E. Gurtin, On the symmetry of the conductivity tensor and other restrictions in the nonlinear theory of heat conduction, Arch. Rational Mech. Anal. 33 (1969), no. 1, 26–32. MR 1553529, DOI https://doi.org/10.1007/BF00248154
Jace W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–204. MR 295683, DOI https://doi.org/10.1090/S0033-569X-1971-0295683-6
William Alan Day, The thermodynamics of simple materials with fading memory, Springer-Verlag, New York-Heidelberg, 1972. Springer Tracts in Natural Philosophy, Vol. 22. MR 0366234
Bernard D. Coleman and David R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54 (1974), 1–104. MR 395502, DOI https://doi.org/10.1007/BF00251256
François Trèves, Basic linear partial differential equations, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 62. MR 0447753
R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), no. 2, 313–332. MR 515894, DOI https://doi.org/10.1016/0022-247X%2878%2990234-2
V. L. Kolpashchikov and A. I. Shnip, Linear thermodynamic theory of heat conduction with memory, Inzh.-Fiz. Zh. 46 (1984), no. 6, 1008–1017 (Russian); English transl., J. Engrg. Phys. 46 (1984), no. 6, 732–739. MR 771210, DOI https://doi.org/10.1007/BF00825640

G. M. Troianiello, Elliptic Differential Equation and Obstacle Problems, Plenum Press, New York and London, 1987

Mauro Fabrizio and Barbara Lazzari, On asymptotic stability for linear viscoelastic fluids, Differential Integral Equations 6 (1993), no. 3, 491–505. MR 1202554
Giorgio Gentili, Dissipativity conditions and variational principles for the heat flux equation with memory, Differential Integral Equations 4 (1991), no. 5, 977–989. MR 1123347

C. Giorgi, Alcune conseguenze delle restrizioni termodinamiche per mezzi viscoelastici lineari, Quad. Sem. Mat. Univ. Brescia, 6/91

Claudio Baiocchi, Soluzioni ordinarie e generalizzate del problema di Cauchy per equazioni differenziali astratte lineari del secondo ordine in spazi di Hilbert, Ricerche Mat. 16 (1967), 27–95 (Italian). MR 226229
A. Morro and M. Fabrizio, On uniqueness in linear viscoelasticity: a family of counterexamples, Quart. Appl. Math. 45 (1987), no. 2, 321–325. MR 895102, DOI https://doi.org/10.1090/S0033-569X-1987-0895102-5

H. Grabmueller, On linear theory of heat conduction in materials with memory, Proc. Roy. Soc. Edinburgh, A-76 (1976–1977), 119–137

Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 919738