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Fischer, Arthur E.; Marsden, Jerrold E.

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I. (English) Zbl 0247.35082

Commun. Math. Phys. 28, 1-38 (1972).
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Cited in 2 Reviews
Cited in 94 Documents

MSC:

35L60 First-order nonlinear hyperbolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35Q99 Partial differential equations of mathematical physics and other areas of application
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[1] Abraham, R.: Lectures on global analysis. Mimeographed, Princeton University.
[2] Arnowitt, R., Deser, S., Misner, C. W.: The dynamics of general relativity. In: Witten, L. (Ed.): Gravitation; an introduction to current research. New York: John Wiley 1962. · Zbl 1152.83320
[3] Fourès-Bruhat, Y.: Théorèm d’existence pour certain systèmes d’équations aux dérivées partielles non linéaires. Acta Math.88, 141–225 (1952). · Zbl 0049.19201 · doi:10.1007/BF02392131
[4] —- Cauchy problem. In: Witten, L. (Ed.): Gravitation; an introduction to current research. New York: John Wiley 1962. · Zbl 0115.43103
[5] Cantor, M.: Diffeomorphism groups over manifolds with non-compact base (preprint).
[6] Choquet-Bruhat, Y.: Espaces-temps einsteiniens généraux, chocs gravitationels. Ann. Inst. Henri Poincaré8, 327–338 (1968). · Zbl 0162.29703
[7] —- SolutionsC d’équations hyperboliques non linéares. C. R. Acad. Sci. Paris272, 386–388 (1971).
[8] —- Problèmes mathématiques en relativité. Actes Congres intern. Math. Tome3, 27–32 (1970).
[9] –C solutions of hyperbolic non-linear equations; applications in general relativity and gravitation. Gen. Rel. Grav.1 (1971).
[10] – Stabilité de solutions d’équations hyperboliques non linéares. Application à l’espace-temps de Minkowski en relativité générale. C. R. Acad. Sci. Paris274, Ser. A. (843) (1972).
[11] —- Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commun. math. Phys.14, 329–335 (1969). · Zbl 0182.59901 · doi:10.1007/BF01645389
[12] Chernoff, P.: Note on product formulas for operator semigroups. J. Funct. An.2, 238–242 (1968). · Zbl 0157.21501 · doi:10.1016/0022-1236(68)90020-7
[13] Chernoff, P., Marsden, J.: On continuity and smoothness of group actions. Bull. Am. Math. Soc.76, 1044–1049 (1970). · Zbl 0202.23202 · doi:10.1090/S0002-9904-1970-12552-6
[14] – Hamiltonian systems and quantum mechanics (in preparation).
[15] Courant, R., Hilbert, D.: Methods of mathematical physics, Vol. II. New York: Interscience 1962. · Zbl 0099.29504
[16] Dionne, P.: Sur les problèmes de Cauchy bien posés. J. Anal. Math. Jerusalem10, 1–90 (1962/63). · Zbl 0112.32301 · doi:10.1007/BF02790303
[17] Dunford, N., Schwartz, J.: Linear operators (I). New York: Interscience 1962. · Zbl 0084.10402
[18] Ebin, D.: On the space of Riemannian metrics. In: Proc. Symp. Pure Math., vol. 15. Providence, R. I.: Math. Soc. 1970. · Zbl 0205.53702
[19] —- Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math.92, 102–163 (1970). · Zbl 0211.57401 · doi:10.2307/1970699
[20] Fischer, A., Marsden, J.: The Einstein equations of evolution – A geometric approach. J. Math. Phys.13, 546–568 (1972). · Zbl 0233.58009 · doi:10.1063/1.1666014
[21] —- —- General relativity, partial differential equations and dynamical systems. In: Proc. Symp. Pure Math., vol. 23. Providence, Rhode Island: Am. Math. Soc. 1972 (to appear).
[22] Fock, V.: The theory of space, time and gravitation. New York: MacMillan 1964. · Zbl 0085.42301
[23] Friedrichs, K. O.: Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Math.7, 345–392 (1954). · Zbl 0059.08902 · doi:10.1002/cpa.3160070206
[24] – A limiting process leading to the equations of relativistic and nonrelativistic fluid dynamics. In: La Magnétohydrodynamique Classique et Relativiste, Lille 1969. Colloques Internat. CNRS, No. 184, Paris, 1970.
[25] Hawking, S.: Stable and generic properties in general relativity. Gen. Rel. Grav.1, 393–400 (1971). · Zbl 0332.53040 · doi:10.1007/BF00759218
[26] Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966. · Zbl 0148.12601
[27] —-: Integration of the equation of evolution in a Banach space. J. Math. Soc. Japan5, 208–234 (1953). · Zbl 0052.12601 · doi:10.2969/jmsj/00520208
[28] —-: Approximation theorems for evolution equations. In: Aziz, A. (Ed.): Lectures in Differential Equations. Mathematical Studies, Vol. 19. New York: Van Nostrand 1969.
[29] —-: Linear evolution equations of ”hyperbolic type”. J. Fac. Sci. Univ. Tokyo17, 241–258 (1970). · Zbl 0222.47011
[30] Lang, S.: Introduction to differentiable manifolds. New York: Interscience 1962. · Zbl 0103.15101
[31] Lax, P.: Cauchy’s problem for hyperbolic equations and the differentiability of solutions of elliptic equations. Commun. Pure Appl. Math.8, 615–633 (1955). · Zbl 0067.07502 · doi:10.1002/cpa.3160080411
[32] Leray, J.: Lectures on hyperbolic equations with variable coefficients. Princeton, Inst. for Adv. Stud. 1952. · Zbl 0046.09901
[33] Lichnerowicz, A.: Relativistic Hydrodynamics and Magnetohydrodynamics. New York: Benjamin 1967. · Zbl 0193.55401
[34] Marsden, J., Ebin, D., Fischer, A.: Diffeomorphism groups, hydrodynamics and relativity. In: Proc. Canadian Math. Congress XIII, Halifax (1971) (to appear). · Zbl 0284.58002
[35] Nirenberg, L.: On elliptic differential equations. Scuola Norm. Super. Pisa13, 115–162 (1959). · Zbl 0088.07601
[36] Palais, R.: Seminar on the Atiyah Singer index theorem. Ann. of Math. Studies, no. 57. Princeton: Princeton University Press 1965. · Zbl 0137.17002
[37] Phillips, R.: Dissipative operators and hyperbolic systems of partial differential equations. Trans. Am. Math. Soc.90, 193–254 (1959). · Zbl 0093.10001 · doi:10.1090/S0002-9947-1959-0104919-1
[38] Schulenberger, J., Wilcox, C.: Completeness of the wave operators for perturbations of uniformly propagative systems. J. Funct. An.7, 447–474 (1971). · Zbl 0223.47007 · doi:10.1016/0022-1236(71)90028-0
[39] Segal, I.: Non-linear semi-groups. Ann. Math.78, 339–364 (1963). · Zbl 0204.16004 · doi:10.2307/1970347
[40] Sobolev, S. L.: Applications of functional analysis in mathematical physics. Providence, R. I.: Am. Math. Soc. 1963. · Zbl 0123.09003
[41] Synge, J. L.: Relativity: The general theory. Amsterdam: North-Holland Publ. Co. 1960. · Zbl 0090.18504
[42] Wilcox, C.: The domain of dependence inequality for symmetric hyperbolic systems. Bull. Am. Math. Soc.70, 149–154 (1964). · Zbl 0117.31202 · doi:10.1090/S0002-9904-1964-11056-9
[43] Yosida, K.: Functional Analysis. Berlin-Heidelberg-New York: Springer 1965. · Zbl 0126.11504
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