References
Brezis, H., “Opérateurs Maximaux Monotones”, Amsterdam: North-Holland 1973.
Brockway, G.S., On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics, Arch.Rational Mech. Anal. 48, 213–244 (1972).
Chernoff, P. & J. Marsden, “Some Properties of Infinite Dimensional Hamiltonian Systems”, Springer Lecture Notes 425 (1974).
Choquet-Bruhat, Y. (Y. Fourès-Bruhat), Théorème d'existence pour certain systèmes d'équations aux dérivées partielles non linéaires, Acta Math. 88, 141–225 (1952).
Choquet-Bruhat, Y., “Cauchy Problem”, In: Witten, L. (Ed.): Gravitation; an introduction to current research, New York: John Wiley 1962.
Choquet-Bruhat, Y., Solutions C∞ d'équations hyperboliques non linéaires, C.R. Acad. Sci. Paris 272, 386–388 (1971) (see also Gen. Rel. Grav. 1 (1971)).
Choquet-Bruhat, Y., Problèmes mathématiques en relativité, Actes Congress Intern. Math. Tome 3, 27–32 (1970).
Choquet-Bruhat, Y., Stabilité de solutions d'équations hyperboliques non linéaires. Application à l'espace-temps de Minkowski en relativité générale, C.R. Acad. Sci. Paris 274, Ser. A (1972), 843. (See also Uspeskii Math. Nauk. XXIX (2) 176, 314–322 (1974).)
Choquet-Bruhat, Y., and L. Lamoureau-Brousse, Sur les équations de l'elasticité relativiste, C.R. Acad. Sci. Paris 276, 1317–1320 (1973).
Choquet-Bruhat, Y. and J.E. Marsden, tSolution of the local mass problem in general relativity Comm. Math. Phys. (to appear). (See also C.R. Acad. Si. 282, 609–612 (1976)).
Courant, R. & D. Hilbert, “Methods of Mathematical Physics”, Vol. II, New York: Interscience 1962.
Dionne, P., Sur les problèmes de Cauchy bien posés, J. Anal. Math. Jerusalem 10, 1–90 (1962/63).
Dorroh, J.R. & J.E. Marsden, Differentiability of nonlinear semigroups (to appear).
Duvaut, T. & J.L. Lions, “Les Inéquations en Mécanique et en Physique”, Paris: Dunod 1972.
Fichera, G., “Existence Theorems in Elasticity”, in Handbuch der Physik (ed. C. Truesdell), Vol. IVa/2, Berlin-Heidelberg-New York: Springer 1972.
Fischer, A. & J. Marsden, The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, I, Commun. Math. Phys. 28, 1–38 (1972).
Fischer, A. & J. Marsden, Linearization stability of nonlinear partial differential equations, Proc. Symp. Pure Math. 27, Part 2, 219–263 (1975).
Frankl, F., Über das Anfangswertproblem für lineare und nichtlineare hyperbolische partielle Differentialgleichungen zweiter Ordnung, Mat. Sb. 2 (44), 814–868 (1937).
Friedman, A., “Partial Differential Equations”, New York: Holt, Rinehart and Winston 1969.
Gårding, L., “Cauchy's Problem for Hyperbolic Equations”, Lecture Notes, Chicago (1957).
Gårding, L., Energy inequalities for hyperbolic systems, in “Differential Analysis”, Bombay Colloq. (1964), 209–225, Oxford: University Press 1964.
Gurtin, M.E., “The Linear Theory of Elasticity”, in Handbuch der Physik (ed. C. Truesdell), Vol. IVa/2, Berlin-Heidelberg-New York: Springer 1972.
Hawking, S.W. & G.F.R. Ellis, “The Large Scale Structure of Spacetime”, Cambridge (1973).
Hille, E. & R. Phillips, “Functional Analysis and Semi-groups”, AMS (1967).
Kato, T., “Perturbation Theory for Linear Operators”, Berlin-Heidelberg-New York: Springer 1966.
Kato, T., Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo 17, 241–258 (1970).
Kato, T., Linear evolution equations of “hyperbolic” type II, J. Math. Soc. Japan 25, 648–666 (1973).
Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58, 181–205 (1975).
Kato, T., “Quasi-linear Equations of Evolution, with Applications to Partial Differential Equations”, Springer Lecture Notes 448, 25–70 (1975).
Knops, R.J. & L.E. Payne, “Uniqueness Theorems in Linear Elasticity”, New York: Springer 1971.
Knops, R.J. & L.E. Payne, Continuous data dependence for the equations of classical elastodynamics, Proc. Camb. Phil. Soc. 66, 481–491 (1969).
Knops, R.J. & E.W. Wilkes, “Theory of Elastic Stability”, in Handbuch der Physik (ed. C. Truesdell), Vol. VIa/3, Berlin-Heidelberg-New York: Springer 1973.
Kōmura, Y., Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan 19, 473–507 (1967).
Krzyzanski, M. & J. Schauder, Quasilineare Differentialgleichungen zweiter Ordnung vom hyperbolischen Typus, Gemischte Randwertaufgaben, Studia. Math. 5, 162–189 (1934).
Lax, P., Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure and Appl. Math. 8, 615–633 (1955).
Lax, P., “Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves”, SIAM (1973).
Leray, J., “Hyperbolic Differential Equations”, Institute for Advanced Study (Notes) (1953).
Lichnerowicz, A., “Relativistic Hydrodynamics and Magnetohydrodynamics”, Amsterdam: Benjamin 1967.
Marsden, J., “Applications of Global Analysis in Mathematical Physics”, Publish or Perish (1974).
Marsden, J., The existence of non-trivial, complete, asymptotically flat spacetimes, Publ. Dept. Math. Lyon 9, 183–193 (1972).
Morrey, C.B., “Multiple Integrals in the Calculus of Variations”, Berlin-Heidelberg-New York: Springer 1966.
Palais, R., “Foundations of Global Nonlinear Analysis”, Amsterdam: Benjamin 1968.
Petrovskii, I., Über das Cauchysche Problem für lineare und nichtlineare hyperbolische partielle Differentialgleichungen, Mat. Sb. 2 (44), 814–868 (1937).
Schauder, J., Das Anfangswertproblem einer quasi-linearen hyperbolischen Differentialgleichung zweiter Ordnung in beliebiger Anzahl von unabhängigen Veränderlichen, Fund. Math. 24, 213–246 (1935).
Sobolev, S.S., “Applications of Functional Analysis in Mathematical Physics”, Translations of Math. Monographs 7, Am. Math. Soc. (1963).
Sobolev, S.S., On the theory of hyperbolic partial differential equations, Mat. Sb. 5 (47), 71–99 (1939).
Wang, C.-C. & C. Truesdell, “Introduction to Rational Elasticity”, Leyden: Noordhoff 1973.
Wheeler, L. & R.R. Nachlinger, Uniqueness theorems for finite elastodynamics, J. Elasticity 4, 27–36 (1974).
Wilcox, C.H., Wave operators and asymptotic solutions of wave propagation problems of classical physics, Arch. Rational Mech. Anal. 22, 37–78 (1966).
Yosida, K., “Functional Analysis”, Berlin-Heidelberg-New York: Springer 1971.
Author information
Authors and Affiliations
Additional information
Communicated by S. Antman
Rights and permissions
About this article
Cite this article
Hughes, T.J.R., Kato, T. & Marsden, J.E. Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 63, 273–294 (1977). https://doi.org/10.1007/BF00251584
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00251584