Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
thermodynamics; symmetric hyperbolicity; kinetic theory; light scattering
thermodynamics; symmetric hyperbolicity; kinetic theory; light scattering
Summary:
Extended thermodynamics is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear differential equations of first order. If the constitutive functions are subject to the requirements of the entropy principle, one may write them in symmetric hyperbolic form by a suitable choice of fields. \endgraf The kinetic theory of gases, or the moment theories based on the Boltzmann equation, provide an explicit example for extended thermodynamics. The theory proves its usefulness and practicality in the successful treatment of light scattering in rarefied gases. \endgraf It would seem that extended thermodynamics is worthy of the attention of mathematicians. It may offer them a non-trivial field of study concerning hyperbolic equations, if ever they get tired of the Burgers equation.
References:
[1] Boillat, G.: La propagation des ondes. Gauthier-Villars Paris French (1965). MR 0197989 | Zbl 0151.45104
[2] Boillat, G., Ruggeri, T.: Hyperbolic principal subsystems: Entropy convexity and subcharacteristic conditions. Arch. Ration. Mech. Anal. 137 (1997), 305-320. DOI 10.1007/s002050050030 | MR 1463797 | Zbl 0878.35070
[3] Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory of gases and wave velocities. Contin. Mech. Thermodyn. 9 (1997), 205-212. DOI 10.1007/s001610050066 | MR 1467331 | Zbl 0892.76075
[4] Cattaneo, C.: Sulla conduzione del calore. Atti Semin. Mat. Fis. Univ. Modena 3 (1949), 83-101 Italian. MR 0032898 | Zbl 0035.26203
[5] Friedrichs, K. O., Lax, P.: Systems of conservation equations with a convex extension. Proc. Natl. Acad. Sci. USA 68 (1971), 1686-1688. DOI 10.1073/pnas.68.8.1686 | MR 0285799 | Zbl 0229.35061
[6] Godunov, S. K.: An interesting class of quasi-linear systems. Sov. Math. Dokl. 2 (1961), 947-949. MR 0131653
[7] Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2 (1949), 331-407. DOI 10.1002/cpa.3160020403 | MR 0033674 | Zbl 0037.13104
[8] Green, W. A.: The growth of plane discontinuities propagating into a homogeneous deformed elastic material. Arch. Ration. Mech. Anal. 16 (1964), 79-88. DOI 10.1007/BF00281332 | MR 0161511
[9] Kawashima, S.: Large-time behaviour of solutions to hyperbolic-parabolic systems of conservations laws and applications. Proc. R. Soc. Edinb., Sect. A 106 (1987), 169-184. MR 0899951
[10] Liu, I-Shih: Method of Lagrange multipliers for the exploitation of the entropy principle. Arch. Ration. Mech. Anal. 46 (1972), 131-148. DOI 10.1007/BF00250688 | MR 0337164
[11] Liu, I-Shih, Müller, I.: Extended thermodynamics of classical and degenerate gases. Arch. Ration. Mech. Anal. 83 (1983), 285-332. DOI 10.1007/BF00963838 | MR 0714978
[12] Müller, I.: On the entropy inequality. Arch. Ration. Mech. Anal. 26 (1967), 118-141. DOI 10.1007/BF00285677 | MR 0214336
[13] Müller, I.: Zum Paradox der Wärmeleitungstheorie. Zeitschrift für Physik 198 (1967).
[14] Müller, I.: Zur Ausbreitungsgeschwindigkeit von Störungen in kontinuierlichen Medien. Dissertation TH Aachen (1966).
[15] Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. 2nd ed. Springer Tracts of Natural Philosophy Vol. 37. Springer New York (1998). MR 1632151
[16] Müller, I., Weiss, W., Reitebuch, D.: Extended thermodynamics---consistent in order of magnitude. Contin. Mech. Thermodyn. 15 (2003), 113-146. DOI 10.1007/s00161-002-0106-0 | MR 1970289 | Zbl 1057.80002
[17] Ruggeri, T.F.: Galilean invariance and entropy principle for systems of balance laws. Contin. Mech. Thermodyn. 1 (1989), 3-20. DOI 10.1007/BF01125883 | MR 1001434 | Zbl 0759.35039
[18] Ruggeri, T., Strumia, A.: Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A, Vol. 34 (1981), 65-84. MR 0605357 | Zbl 0473.76126
[19] Weiss, W.: Zur Hierarchie der erweiterten Thermodynamik. Dissertation TU Berlin (1990).
Search
Partner of
