Cauchy Problem for the Kuznetsov Equation

Abstract : We consider the Cauchy problem for a model of non-linear acoustics, named the Kuznetsov equation, describing sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the non-viscous case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a L 2-stability estimate, obtained between a regular and a less regular solutions.
Type de document :
Pré-publication, Document de travail
2017
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Contributeur : Adrien Dekkers <>
Soumis le : vendredi 24 novembre 2017 - 16:50:59
Dernière modification le : mercredi 29 novembre 2017 - 01:14:08

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  • HAL Id : hal-01648010, version 1
  • ARXIV : 1711.09789

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Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy Problem for the Kuznetsov Equation. 2017. 〈hal-01648010〉

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