A well-posedness result for hyperbolic operators with Zygmund coefficients

TL;DR

This paper establishes a well-posedness result for hyperbolic operators with Zygmund continuous coefficients, showing energy estimates without derivative loss, even with non-Lipschitz coefficients, in a specific initial data space.

Contribution

It proves a novel energy estimate for hyperbolic operators with Zygmund coefficients, enabling well-posedness results under less regularity than previously possible.

Findings

Energy estimate with no derivative loss for Zygmund coefficients

Well-posedness of the Cauchy problem in specific initial data space

Use of paradifferential calculus with parameters in the proof

Abstract

In this paper we prove an energy estimate with no loss of derivatives for a strictly hyperbolic operator with Zygmund continuous second order coefficients both in time and in space. In particular, this estimate implies the well-posedness for the related Cauchy problem. On the one hand, this result is quite surprising, because it allows to consider coefficients which are not Lipschitz continuous in time. On the other hand, it holds true only in the very special case of initial data in $H^{1/2}\times H^{-1/2}$. Paradifferential calculus with parameters is the main ingredient to the proof.