Cauchy problem for the Kuznetsov equation

Adrien Dekkers; Anna Rozanova-Pierrat

doi:10.3934/dcds.2019012

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2019, Volume 39, Issue 1: 277-307. Doi: 10.3934/dcds.2019012

This issue Previous Article Double minimality, entropy and disjointness with all minimal systems Next Article The variational discretization of the constrained higher-order Lagrange-Poincaré equations

Cauchy problem for the Kuznetsov equation

Laboratory Mathématiques et Informatique pour la Complexité et les Systèmes, CentraleSupélec, Univérsité Paris-Saclay, Campus de Gif-sur-Yvette, Plateau de Moulon, 3 rue Joliot Curie, 91190 Gif-sur-Yvette, France

Received: November 30, 2017

Revised: May 31, 2018

Published: December 2018

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Abstract

We consider the Cauchy problem for a model of non-linear acoustic, named the Kuznetsov equation, describing a 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 inviscid 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.

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Mathematics Subject Classification: Primary: 35L15, 35L71; Secondary: 35B45.

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