Time-Dependent Loss of Derivatives for Hyperbolic Operators with Non Regular Coefficients

Abstract

In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coecients in any space dimension N ≥ 1. We will suppose the coecients to be log-Zygmund continuous in time and log-Lipschitz continuous in space. Paradierential calculus with parameters will be the main tool to get energy estimates in Sobolev spaces and these estimates will present a time-dependent loss of derivatives.

A Note on Complete Hyperbolic Operators with log-Zygmund Coefficients

“…The aim of the present note is to extend the result of [7] for homogeneous second order operators to the case where also lower order terms come into play. In particular, according with the results of [10], [11] and [9], we will assume the first order coefficients to be Hölder continuous in the space variable, and the zero order coefficient to be just bounded.…”

“…Nevertheless, they had to restrict themselves to the case of space dimension N = 1 (see also [9] for the case of a complete operator). The result in the general instance N ≥ 1 was only recently proved in paper [7]. Even if the difficulty of the additional space variables seems to be just technical, the approach assumed in [7], i.e.…”

“…The result in the general instance N ≥ 1 was only recently proved in paper [7]. Even if the difficulty of the additional space variables seems to be just technical, the approach assumed in [7], i.e. of paradifferential calculus depending on parameters, actually gives a more general and complete point of view of the problem.…”

“…The present paper is, in a certain sense, the continuation of the recent work [7]: it is devoted to the study of strictly hyperbolic operators with log-Zygmund in time coefficients. In particular, we want to prove the H ∞ well-posedness of the Cauchy problem related to a complete second order operator,…”

“…The present paper is the continuation of the recent work [7], and it is devoted to strictly hyperbolic operators with non-regular coefficients. We focus here on the case of complete operators whose second order coefficients are log-Zygmund continuous in time, and we investigate the H ∞ well-posedness of the associate Cauchy problem.…”