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Propagation de l’analyticité des solutions de systèmes hyperboliques non-linéaires. (French) Zbl 0545.35063

The authors consider fully-nonlinear scalar equations of order m \[ F(x,u(x),\nabla u(x),...,\partial^{\alpha}u(x))=0\quad x\in {\mathbb{R}}^ n,\quad | \alpha | \leq m \] or first-order systems (U being a vector) \[ \sum^{n}_{i=1}A_ i(x,U(x))\partial U/\partial x_ i+B(x,U(x))U=0. \] Theorems about propagation of analytic regularity across a surface are proved under the assumptions that the surface be space-like and the linearized operators be hyperbolic. Consequences of these theorems can be formulated as follows: if the Cauchy data on an analytic hypersurface are analytic, the solution is analytic in the corresponding domain of influence of the (hyperbolic) linearized operator. These results have been extended recently to the case where the operator may contain ”elliptic factors”, and to ”pseudo-differential” systems as well (such as the Euler equation of fluid mechanics).

MSC:

35L75 Higher-order nonlinear hyperbolic equations
35A20 Analyticity in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L60 First-order nonlinear hyperbolic equations
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References:

[1] Baouendi, M.S., Goulaouic, C.: Cauchy problems for analytic pseudodifferential operators. Comm. in Partial Differential Equations1, 135-189 (1976) · Zbl 0344.35077
[2] Baouendi, M.S., Goulaouic, C.: Remarks on the abstract form of non linear Cauchy Kovalevsky theorems. Comm. in Partial Differential Equations2, 1151-1162 (1977) · Zbl 0391.35006
[3] Bardos, C.: Analyticité de la solution de l’équation d’Euler dans un ouvert de ? n . C.R. Acad. Sc. Parist 283, 255-258 (1976) · Zbl 0343.35014
[4] Bardos, C., Benachour, S., Zerner, M.: Analytiticité des solutions périodiques de l’équation d’Euler en dimension deux. C.R. Acad. Sc. Parist 282, 995-998 (1976) · Zbl 0329.35012
[5] Benachour, S.: Analyticité des solutions périodiques de l’équation d’Euler en dimension trois. C.R. Acad. Sc. Parist 283, 107-110 (1976) · Zbl 0334.35005
[6] Bernstein, S.: Démonstration du théorème de M. Hilbert sur la nature analytique des solutions des équations de type elliptique Math. Z.28, 330-348 (1928) · JFM 54.0506.02
[7] Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Scient. E.N.S., 4éme série14, 209-246 (1981) · Zbl 0495.35024
[8] Bony, J.M.: Interaction des singularités pour les équations aux dérivées partielles non linéaires; Séminaire Goulaouic-Meyer-Schwartz, Année 1981-82-Ecole Polytechnique, Paris
[9] Friedman, A.: On the regularity of the solutions of non linear elliptic and parabolic systems of partial differential equations. J. Math. Mech.7, 43-60 (1958) · Zbl 0078.27702
[10] Friedrichs, K.O.: Symetric hyperbolic linear differential equations. Comm. Pure Appl. Math.7, 345-392 (1954) · Zbl 0059.08902
[11] Hörmander, L.: Linear partial differential operators. Berlin-Heidelberg-New York: Springer 1969 · Zbl 0175.39201
[12] Hörmander, L.: Uniqueness theorem and wave front sets for solutions of linear differential equations with analytic coefficients. Comm. Pure Appl. Math.24, 671-704 (1971) · Zbl 0226.35019
[13] Bros, J., Iagolnitzer, D.: Support essentiel et structure analytique des distributions; Séminaire Goulaouic-Lions-Schwartz, Année 1974-75, Ecole Polytechnique, Paris · Zbl 0333.46029
[14] Lascar, B.: Singularité des solutions d’équations aux dérivées partielles non linéaires. C.R. Acad. Sc. Parist 287, 527-529 (1978) · Zbl 0387.35018
[15] Lax, P.D.: Non linear hyperbolic equations. Comm. on Pure Appl. Math.6, 231-258 (1953) · Zbl 0050.31705
[16] Mizohata, S.: Analyticity of solutions of hyperbolic systems with analytic coefficients. Comm. Pure Appl. Math.14, 547-559 (1961) · Zbl 0105.07203
[17] Morrey, C.B.: Multiple integrals in the calculs of variations. Berlin-Heidelberg-New York: Springer 1966 · Zbl 0142.38701
[18] Nirenberg, L.: Pseudodifferential operators. Proc. Sympos. Pure Math., vol. 16. Amer. Math. Soc., Providence R.I. 149-167 (1970)
[19] Nirenberg, L.: An abstract from of the non linear Cauchy-Kovalevsky theorem. J. Diff. Geometry6, 561-576 (1972) · Zbl 0257.35001
[20] Nishida, T.: A note on the Nirenberg’s theorem as an abstract form of the non linear Cauchy-Kovalevsky theorem in a scale of Banach spaces. J. Diff. Geometry12, 629-633 (1977) · Zbl 0368.35007
[21] Ovsjannikov, L.V.: A non linear Cauchy problem in a scale of Banach spaces. Dok. Akad. Nauk SSRR200 (1971); Sov. Math. Dokl12, 1497-1502 (1971) · Zbl 0234.35018
[22] Petrowsky, I.: Sur l’analyticité des solutions des systèmes d’équations différentielles. Ree. Math. N.S. Math. Sbornik5, (47) 3-70 (1939) · Zbl 0022.22601
[23] Rauch, J.: Singularities of solutions to semi-linear wave equations. J. Math. Pure Appl.58, 299-308 (1979) · Zbl 0388.35045
[24] Sato, M.: Hyperfunctions and partial differential equations. Proc. Int. Conf. Funct. Anal. Tokyo 91-94 (1969)
[25] Taylor, M.E.: Pseudodifferential operators; Princeton Mathematical Series, no 34, Princeton University Press
[26] Wagschall, C.: Le problème de Goursat non linéaire, Séminaire Goulaouic-Schwartz, Année 1978-79, Ecole Polytechnique, Paris
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