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September 2009 On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media
Anna Rozanova-Pierrat
Commun. Math. Sci. 7(3): 679-718 (September 2009).

Abstract

We consider the derivation of the Khokhlov-Zabolotskaya-Kuznetzov (KZK) equation from the nonlinear isentropic Navier-Stokes and Euler systems. The KZK equation is a mathematical model that describes the nonlinear propagation of a finite-amplitude sound pulse in a thermo-viscous medium. The derivation of the KZK equation has to date been based on the paraxial approximation of small perturbations around a given state of the Navier-Stokes system. However, this method does not guarantee that the solution of the initial Navier-Stokes system can be reconstructed from the solution of the KZK equation. We introduce a corrector function in the derivation ansatz that allows one to validate the KZK-approximation. We also give the analysis of other types of derivation. We prove the validation of the KZK-approximation for the non-viscous case, as well as for the viscous nonlinear and linear cases. The results are obtained in Sobolev spaces for functions periodic in one of the space variables and with a mean value of zero. The existence of a unique regular solution of the isentropic Navier-Stokes system in the half space with boundary conditions that are both periodic and mean value zero in time is also obtained.

Citation

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Anna Rozanova-Pierrat. "On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media." Commun. Math. Sci. 7 (3) 679 - 718, September 2009.

Information

Published: September 2009
First available in Project Euclid: 26 October 2009

zbMATH: 1186.35147
MathSciNet: MR2569029

Subjects:
Primary: 35Q, 41A65, 58J37, 76L05, 76N10, 76N99

Rights: Copyright © 2009 International Press of Boston

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Vol.7 • No. 3 • September 2009
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