No loss of derivatives for hyperbolic operators with Zygmund-continuous coefficients in time

Ferruccio Colombini (1) , Daniele Del Santo (2) , Francesco Fanelli (3)

1 Dipartimento di Matematica
2 Dipartimento di Matematica e Geoscienze [Trieste]
3 EDPA - Équations aux dérivées partielles, analyse

In this note we prove a well-posedness result, without loss of derivatives, for strictly hyperbolic wave operators having coefficients which are Zygmund-continuous in the time variable and Lipschitz-continuous in the space variables. The proof is based on Tarama's idea of introducing a lower order corrector in the energy, in order to produce special algebraic cancellations when computing its time derivative, combined with paradifferential calculus with parameters, in order to handle the low regularity of the coefficients with respect to $x$.

Domaines: Equations aux dérivées partielles [math.AP]

Ferruccio Colombini, Daniele Del Santo, Francesco Fanelli. No loss of derivatives for hyperbolic operators with Zygmund-continuous coefficients in time. Anomalies in Partial Differential Equations, 43, Springer International Publishing, pp.127-148, 2021, Springer INdAM Series, ⟨10.1007/978-3-030-61346-4_6⟩. ⟨hal-02457392⟩