On the laws of relativistic electromagneto-fluid dynamics. (English) Zbl 0308.76075

Having noticed that the restriction to strictly hyperbolic equations (as studied by Petrovskiĭ, Leray and others) is too restrictive for general purposes of mathematical physics, the author develops the notions of symmetric hyperbolic systems [cf. author, Commun. Pure Appl. Math. 7, 345–392 (1954; Zbl 0059.08902)] and convexly dependent systems [cf. the author and P. D. Lax, Proc. Natl. Acad. Sci. USA 68, 1686–1688 (1971; Zbl 0229.35061)], and applies them to the system of equations that govern relativistic electromagneto-fluid dynamics.
A recipe for rewriting the basic equations as systems of symmetric hyperbolic equations of the first order is described. It is applicable if the equations are written as a dependent (overdetermined) system of conservation laws enjoying a certain convexity property. In contrast to what happens in the general theory of hyperbolic equations developed by Leray and Ohya, the “initial class”, i.e., the suitable class to which initial functions must belong, may be of finite order (i.e., the initial functions need not be infinitely differentiable), which physically is more plausible. In the case of relativistic magnetofluid dynamics, the system of equations (resp., the forces) has to be modified (resp., have to be chosen) in order that the solution depends on the data with such a restricted sensitivity of finite order. Two such modifications result either in omitting the electric contribution to the electromagnetic forces or in allowing the fluid energy to depend on the charge density in a suitable way.
In the case of polarized electro-magneto-fluid dynamics, the dependent conservation laws, which automatically satisfy the convexity condition, are associated with a Lagrangian. This allows the deduction of a suitable form of the electromagnetic energy-momentum tensor whose expression is closely related to that of S. R. de Groot and L. G. Suttorp [see their “Foundations of electrodynamics.” Amsterdam: North-Holland (1972)].
Along the same line, the author draws enlightening conclusions as regards, on the one hand, laws involving angular momenta, and on the other, polarized fields with conductivity, where it is shown that Ohmian losses should be related to the rotational character of the flow and have an effect on the angular momentum.


76Y05 Quantum hydrodynamics and relativistic hydrodynamics
76W05 Magnetohydrodynamics and electrohydrodynamics
35L45 Initial value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
78A25 Electromagnetic theory (general)
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[1] Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bevegten Körpern pp 53– (1908) · JFM 39.0909.02
[2] Abraham, Zur Elektrodynamik bewegter Körper, Rendiconti, Circ. Mat. Palermo 28 pp 1– (1909)
[3] Ann. Phys. 44 pp 537– (1914)
[4] Dällenbach, Allgemeine kovariante Grundgleichungen des elektromagnetishen Feldes im Inneren ponderabler Materie vom Standpunkt der Elecktronentheorie, Ann. Physik 58 pp 523– (1919)
[5] Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 pp 345– (1954) · Zbl 0059.08902
[6] Leray, Systèmes hyperboliques non-stricts pp 83– (1969)
[7] Leray, Systèmes linéaires, hyperboliques non-stricts pp 104– (1964)
[8] Bruhat, Étude des équations des fluides chargés relativistes inductifs et conducteurs, Commun. Math. Phys. 3 pp 334– (1966) · Zbl 0171.46605
[9] Lichnerowicz, Relativistic Hydrodynamics and Magnetohydrodynamics (1967)
[10] de Groot, The relativistic energy-momentum tensor in polarized media, I-VII, Physica (1967)
[11] Brevik, The electro-magnetic energy momentum tensor within material media, Dan. Vid. Selsk. Math. Phys. Medd. 37 (11) (1970)
[12] Taub, Variational principles and relativistic magnetohydrodynamics pp 177– (1969)
[13] Friedrichs, A limiting process leading to the equations of relativistic and nonrelativistic fluid dynamics pp 177– (1969)
[14] Friedrichs, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci., USA 86 (8) pp 1686– (1971) · Zbl 0229.35061
[15] Fischer, The Einstein evolution equations as a first-order quasilinear, symmetric hyperbolic system, I, Comm. Math. Phys. 28 pp 1– (1972)
[16] Fischer, General relativity, partial differentiation equations and dynamical systems, Proc. Symp. Pure Math. 23 (1973)
[17] Belinfante, On the spin angular momentum of mesons, Physica 6 pp 888– (1939) · Zbl 0022.04607
[18] Belinfante, On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields, Physica 7 pp 462– (1940) · Zbl 0024.14203
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