COLOMBINI, F; MÉTIVIER, Guy

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COLOMBINI, F
Dipartimento di Matematica

MÉTIVIER, Guy
Institut de Mathématiques de Bordeaux [IMB]

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We consider the well-posedness of the Cauchy problem in Gevrey spaces for N × N first order weakly hyperbolic systems. The question is to know wether the general results of M.D.Bronštein [Br] and K.Kajitani [Ka2] can be improved when the coefficients depend only on time and are smooth, as it has been done for the scalar wave equation in [CJS]. The anwser is no for general systems, and yes when the system is uniformly diagonalizable: in this case we show that the Cauchy problem is well posed in all Gevrey classes G s when the coefficients are C ∞. Moreover, for 2 × 2 systems and some other special cases, we prove that the Cauchy problem is well posed in G s for s < 1 + k when the coefficients are C k , which is sharp following the counterexamples of S.Tarama [Ta1]. The main new ingredient is the construction, for all hyperbolic matrix A, of a family of approximate symmetrizers, S ε , the coefficients of which are polynomials of ε and the coefficients of A and A *. MSC Classification : 35 L 50, 35 L 45, 35 L 40.< Réduire

Mots clés en anglais

hyperbolic systems

Cauchy problem

symmetrizers

well posed- ness

Gevrey spaces

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The Cauchy problem for weakly hyperbolic systems

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Résumé en anglais

Mots clés en anglais

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