We discuss the different roles of the entropy principle in modern thermodynamics. We start with the approach of rational thermodynamics in which the entropy principle becomes a selection rule for physical constitutive equations. Then we discuss the entropy principle for selecting admissible discontinuous weak solutions and to symmetrize general systems of hyperbolic balance laws. A particular attention is given on the local and global well-posedness of the relative Cauchy problem for smooth solutions. At the end we give some recent results on closure procedure for the moments theory associated to the Boltzmann equation (Extended Thermodynamics).
Si presenta una breve rassegna dei diversi ruoli che ha il principio di entropia nella moderna termodinamica. Nell'ambito della termodinamica razionale il principio di entropia diventa un criterio di selezione per le equazioni costitutive ammissibili mentre nel caso di soluzioni deboli di sistemi iperbolici non lineari diventa un criterio di selezione dei processi fisicamente ammissibili. Inoltre tutti i sistemi iperbolici di leggi di bilancio che sono compatibili con un principio di entropia convessa sono simmetrici ed è possibile riconoscere teorie a nido mediante l'introduzione dei sottosistemi principali. Particolare attenzione è dedicata allanalisi qualitativa dimostrando che in presenza di dissipazione il problema di Cauchy è ben posto in senso globale ed esistono, per dati iniziali sufficientemente piccoli, soluzioni regolari per tutti i tempi che tendono a stati costanti di equilibrio. Infine vengono applicati questi risultati alla teoria della Termodinamica Estesa che governa i processi dei gas rarefatti.
@article{BUMI_2005_8_8B_1_1_0,
author = {Tommaso Ruggeri},
title = {The entropy principle: from continuum mechanics to hyperbolic systems of balance laws},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {8-A},
year = {2005},
pages = {1-20},
zbl = {1150.80001},
mrnumber = {2122973},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2005_8_8B_1_1_0}
}
Ruggeri, Tommaso. The entropy principle: from continuum mechanics to hyperbolic systems of balance laws. Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005) pp. 1-20. http://gdmltest.u-ga.fr/item/BUMI_2005_8_8B_1_1_0/
[1] - , The thermomechanics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13 (1963), 167-178. | MR 153153 | Zbl 0113.17802
[2] , On the entropy inequality, Arch. Rational Mech. Anal., 26 (1967), 118-141. | MR 214336 | Zbl 0163.46701
[3] - - Eds., Entropy, Series in Appl. Math. Princeton University Press, Princeton (2003). | MR 2035814 | Zbl 1187.00001
[4] , Shock Waves and Entropy, Contributions to Functional Analysis, 603-634, Ed. , New York, Academic Press (1971). | MR 393870 | Zbl 0268.35014
[5] - , Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688. | MR 285799 | Zbl 0229.35061
[6] , Hyperbolic Conservation Laws in Continuum Physics, Springer Verlag, Berlin (2000). | MR 1763936 | Zbl 1078.35001
[7] , The Riemann Problem for General System of Conservation Laws, J. Differential Equations, 18 (1975), 218-234. | MR 369939 | Zbl 0297.76057
[8] , The Admissible Solutions of Hyperbolic Conservation Laws, Memoir of AMS, 240 (1981), 78. | MR 603391 | Zbl 0446.76058
[9] - , Entropy Production and Admissibility of Shocks, Acta Math. Appl. Sin., Engl. Ser., 19, No. 1 (2003), 1-12. | MR 2053765 | Zbl 1029.35172
[10] - , Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. Henri Poincarè, 34-A (1981), 65-84. | MR 605357 | Zbl 0473.76126
[11] , Sur l'Existence et la Recherche d'Équations de Conservation Supplémentaires pour les Systèmes Hyperboliques, C.R. Acad. Sc. Paris, 278-A (1974), 909-912. Non Linear Fields and Waves. In CIME Course, Recent Mathematical Methods in Nonlinear Wave Propagation, Lecture Notes in Mathematics 1640, Ed. Springer-Verlag (1995), 103-152. | MR 342870 | Zbl 0279.35058
[12] , An interesting class of quasilinear systems, Sov. Math., Dokl.2 (1961), 947-949; translation from Dokl. Akad. Nauk SSSR, 139 (1961), 521-523. | MR 131653 | Zbl 0125.06002
[13] - , Hyperbolic Principal Subsystems: Entropy Convexity and Sub characteristic Conditions, Arch. Rat. Mech. Anal., 137 (1997), 305- 320. | MR 1463797 | Zbl 0878.35070
[14] - , On the shock structure problem for hyperbolic system of balance laws and convex entropy, Contin. Mech. Thermodyn., 10, No. 5 (1998), 285-292. | MR 1652858 | Zbl 0922.76237
[15] , Maximum of Entropy Density in Equilibrium and Minimax Principle for an Hyperbolic System of Balance Laws, Contributions to Continuum Theories, Anniversary Volume for Krzysztoff Wilmanski, editor WIAS-Report No. 18 (2000).
[16] - , Stability of constant equilibrium state for dissipative balance laws system with a convex entropy, Quarterly of Applied Math. To appear (2003). | MR 2032577 | Zbl 1068.35067
[17] , Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. R. Soc. Edinb., Sect. A, 106 (1987), 169-194. | MR 899951 | Zbl 0653.35066
[18] - , The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, Commun. Math. Phys., 28 (1972), 1-38. | MR 309507 | Zbl 0247.35082
[19] , Compressible fluid flow and systems of conservation laws in several space variables, Springer Verlag, New York (1984). | MR 748308 | Zbl 0537.76001
[20] , Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150, No. 3 (1999), 225-279. | MR 1738119 | Zbl 0966.76079
[21] - , Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Arch. Rat. Mech. Anal., 169 (2003), 89-117. | MR 2005637 | Zbl 1037.35041
[22] - , Rational Extended Thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy 37, Springer-Verlag, New York (1998). | MR 1632151 | Zbl 0895.00005
[23] , On the kinetic theory of rarefied gases, Comm. Appl. Math., 2 (1949), 331-407. | MR 33674 | Zbl 0037.13104
[24] - , Moment equations in the kinetic theory of gases and wave velocities, Contin. Mech. Thermodyn., 9, No. 4 (1997), 205-212. | MR 1467331 | Zbl 0892.76075
[25] - , Maximum wave velocity in the moments system of a relativistic gas, Contin. Mech. Thermodyn., 11, No. 2 (1999), 107-111. | MR 1680244 | Zbl 0935.76077
[26] - , Relativistic gas: Moment equations and maximum wave velocity, J. Math. Phys., 40, No. 12 (1999), 6399-6406. | MR 1725865 | Zbl 0962.82060
[27] - , Maximum velocity for wave propagation in a relativistic rarefied gas, Contin. Mech. Thermodyn., 11, No. 5 (1999), 331-338. | MR 1723707 | Zbl 0946.76084
[28] , Die Berechnung von kontinuierlichen Stoßstrukturen in der Kinetischen Gastheorie, Habilitation thesis TU Berlin (1997).
[29] - , Light scattering and extended thermodynamics, Cont. Mech. Thermodyn., 7 (1995), 123-144. | MR 1333703
[30] , An extended moment method in radiative transfer: The matrices of mean absorption and scattering coefficients, Annals of Physics, 257, No. 2 (1997), 111-135. | MR 1460874 | Zbl 0937.76065
[31] - , Thermal conductivity and dynamic pressure in extended thermodynamics of chemically reacting mixtures of gases, Ann. Inst. Henri Poincaré, Phys. Théor., 69, No. 3 (1998), 309-334. | MR 1648986 | Zbl 0964.80007
[32] - - , Extended hydrodynamical model of carrier transport in semiconductors, SIAM J. Appl. Math., 61, No. 1 (2000), 74-101. | MR 1776388 | Zbl 0966.35076
[33] , Maximisation of the Entropy in Non-Equilibrium, J. Phys. A: Math. Gen., 20 (1987), 6505-6512. | MR 926398 | Zbl 0633.76081
[34] , Breakdown of Shock Wave Structure Solutions, Phys. Rev., 47-E, (6) (1993), 4135-4140. | MR 1377905
[35] , Continuous shock structure in extended thermodynamics, Phys. Review E, Part A, 52 (1995), 5760-5768.