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).


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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