The Well-Posedness Issue in Sobolev Spaces for Hyperbolic Systems with Zygmund-Type Coefficients

Ferruccio Colombini (1) , Daniele del Santo (2) , Francesco Fanelli (3) , Guy Métivier (4)

1 Dipartimento di Matematica
2 Università degli studi di Trieste = University of Trieste
3 UCBL - Université Claude Bernard Lyon 1
4 IMB - Institut de Mathématiques de Bordeaux

In this paper we study the well-posedness of the Cauchy problem for first order hyperbolic systems with constant multiplicities and with low regularity coefficients depending just on the time variable. We consider Zygmund and log-Zygmund type assumptions, and we prove well-posedness in H ∞ respectively without loss and with finite loss of derivatives. The key to obtain the results is the construction of a suitable symmetrizer for our system, which allows us to recover energy estimates (with or without loss) for the hyperbolic operator under consideration. This can be achievied, in contrast with the classical case of systems with smooth (say Lipschitz) coefficients, by adding one step in the diagonalization process, and building the symmetrizer up to the second order.

Mots clés: hyperbolic system with constant multiplicities; Zygmund and log-Zygmund conditions; microlocal symmetrizability; energy estimates; H ∞ well-posedness

Domaines: Equations aux dérivées partielles [math.AP]

Ferruccio Colombini, Daniele del Santo, Francesco Fanelli, Guy Métivier. The Well-Posedness Issue in Sobolev Spaces for Hyperbolic Systems with Zygmund-Type Coefficients. Communications in Partial Differential Equations, 2015, 40, pp.2082-2121. ⟨10.1080/03605302.2015.1082107⟩. ⟨hal-01260632⟩