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 coefficients in any space dimension N ≥ 1. 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.

Keywords:

• Energy estimates
• Hyperbolic operators
• Log-Zygmund regularity
• Non-Lipschitz coefficient
• Well-posedness

2010 Mathematics Subject Classfication:

• Primary 35L15
• Secondary 35B65, 35S50, 35B45

Acknowledgment

The work was mostly prepared during the Ph.D. thesis of the third author at SISSA and Université Paris-Est: he is grateful to both these institutions. During the final part, the third author was partially supported by Grant MTM2011-29306-C02-00, MICINN, Spain, ERC Advanced Grant FP7-246775 NUMERIWAVES, ESF Research Networking Programme OPTPDE and Grant PI2010-04 of the Basque Government.