Le problème de Cauchy pour leséquations d'ondeà coefficients non Lipschtziens; application au prolongement de solutions d'équations d'ondes non linéaires.
AbstractIn this paper we study the Cauchy problem for second order strictly hyperbolic operators of the formwhen the coefficients of the principal part are not Lipschitz continuous, but only "Log-Lipschitz" with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant version of a previous paper of the first author with N.Lerner [6]. We also give an application of the method to a continuation theorem for nonlinear wave equations where the coefficients above depend on u: the smooth solution can be extended as long as it remains Log-Lipschitz.
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.
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