Time-dependent loss of derivatives for hyperbolic operators with non regular coefficients

TL;DR

This paper investigates the Cauchy problem for hyperbolic operators with low regularity coefficients, demonstrating a time-dependent loss of derivatives using paradifferential calculus in Sobolev spaces.

Contribution

It introduces a novel approach to handle hyperbolic operators with log-Zygmund and log-Lipschitz coefficients, establishing energy estimates with derivative loss.

Findings

Established energy estimates with time-dependent derivative loss

Applied paradifferential calculus to low regularity coefficients

Extended analysis to any space dimension N≥1

Abstract

In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coefficients in any space dimension $N\geq1$. We will suppose the coefficients to be log-Zygmund continuous in time and log-Lipschitz continuous in space. Paradifferential 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.