Cauchy Problem for the Kuznetsov Equation

TL;DR

This paper studies the well-posedness and global existence of solutions for the Kuznetsov equation, a model in nonlinear acoustics, considering both viscous and non-viscous cases, with results on regularity, energy estimates, and blow-up behavior.

Contribution

It extends existing results by proving global existence for small initial data in the viscous case and improving well-posedness results for less regular data in the non-viscous case.

Findings

Global existence of regular solutions for small initial data in viscous case

Optimal estimates of maximal existence time depending on space dimension

Confirmation of blow-up results through L2-stability estimates

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.