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The Cauchy problem for quasi-linear symmetric hyperbolic systems. (English) Zbl 0343.35056

MSC:
35L60Nonlinear first-order hyperbolic equations
35D05Existence of generalized solutions of PDE (MSC2000)
35L45First order hyperbolic systems, initial value problems
35B45A priori estimates for solutions of PDE
46E40Spaces of vector- and operator-valued functions
References:
[1]Kato, T., Linear evolution equations of ?hyperbolic? type. J. Fac. Sci. Univ. Tokyo 17, 241-258 (1970).
[2]Kato, T., Linear evolution equations of ?hyperbolic type?, II. J. Math. Soc. Japan 25, 648-666 (1973). · Zbl 0262.34048 · doi:10.2969/jmsj/02540648
[3]Friedrichs, K.O., Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7, 345-392 (1954). · Zbl 0059.08902 · doi:10.1002/cpa.3160070206
[4]Sobolev, S.L., Applications of functional analysis in mathematical physics. AMS Translations of Math. Monographs. 7, 1963.
[5]Fischer, A.E., & J.E. Marsden, The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic systems, I. Commun. Math. Phys. 28, 1-38 (1972). · Zbl 0247.35082 · doi:10.1007/BF02099369
[6]Bers, L., F. John, & M. Schechter, Partial Differential Equations. Interscience 1964.
[7]Kallman, R.R., & G.-C. Rota, On the inequality ?f??2 ? 4 ?f? ?f??. Inequalities, Vol. 2, pp. 187-192. Academic Press 1970.
[8]Hormander, L., Linear partial differential operators. Springer 1963.
[9]Massey, F.J. III, Abstract evolution equations and the mixed problem for symmetric hyperbolic systems, Trans. Amer. Math. Soc. 168, 165-188 (1972). · doi:10.1090/S0002-9947-1972-0298231-4