On the Cauchy problem for microlocally symmetrizable hyperbolic systems with log-Lipschitz coefficientsReport as inadecuate




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1 Dipartimento di Matematica 2 Dipartimento di Matematica e Geoscienze Trieste 3 EDP, Analyse ICJ - Institut Camille Jordan Villeurbanne 4 IMB - Institut de Mathématiques de Bordeaux

Abstract : The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in space problem we establish energy estimates with finite loss of derivatives, which is linearly increasing in time. This implies well-posedness in H ∞ , if the coefficients enjoy enough smoothness in x. From this result, by standard arguments i.e. extension and convexification we deduce also local existence and uniqueness. A huge part of the analysis is devoted to give an appropriate sense to the Cauchy problem, which is not evident a priori in our setting, due to the very low regularity of coefficients and solutions. 2010 Mathematics Subject Classification: 35L45 primary; 35B45, 35B65 secondary.

Keywords : hyperbolic systems microlocal symmetrizability log-Lipschitz regularity loss of derivatives global and local Cauchy problem well-posedness





Author: Ferruccio Colombini - Daniele Santo - Francesco Fanelli - Guy Métivier -

Source: https://hal.archives-ouvertes.fr/



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