A well-posedness result for hyperbolic operators with Zygmund coefficients

Abstract

In this paper we prove an energy estimate with no loss of derivatives for a strictly hyperbolic operator with Zygmund continuous second order coecients 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 coecients 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

Some local questions for hyperbolic systems with non-regular time dependent coefficients

“…The special issue is that they are still suitable for well-posedness of hyperbolic Cauchy problems, with the same kind of loss which would pertain to the latter conditions: so, for pure Zygmund hypothesis one recovers well-posedness in any H s with no loss of derivatives, while log-Zygmund hypothesis entails a finite loss, which depends on time. This picture is very similar to what was already obtained in the case of scalar wave equations (2) by Tarama, see paper [19] (see also [3] and [4] and the references therein). There, the main point was to compensate the worse behaviour of the coefficients by introducing a lower order corrector in the definition of the energy, in order to produce special algebraic cancellations in the energy estimates.…”

“…On the other hand, in order to make this argument consistent, we need a general existence result for operator (4). This is provided by the following statement, which is in the same spirit of the celebrated Cauchy-Kovalevska Theorem.…”

“…Lemma 2.3. Let the operator L be defined as in (4). Then L is microlocally symmetrizable, with bounded symmetrizer S, if and only if the adjoint operator…”

“…Finally, counterxemples show that, for too irregular coefficients, the loss of derivatives can be infinite, precluding any possibility of recovering well-posedness or observability, even in a weak sense. We refer to [3]- [4] (concerning the well-posedness theory) and [10]- [21] (concerning observability and control of waves) for an overview about this topic and for further references, as well as for some recent progress in these directions.…”

A Note on Complete Hyperbolic Operators with log-Zygmund Coefficients

“…This result has been recently generalized in [8] to the case of a jk 's Zygmund continuous both in t and x (assuming also Hölder first order coefficients and bounded zero order coefficient), but only in the space H 1/2 ×H −1/2 . If an isotropic Zygmund assumption implies or not loss of regularity for general initial data in H s × H s−1 is still an open problem.…”