Communications in Mathematical Sciences

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

Full-text: Open access

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.

Article information

Source
Commun. Math. Sci., Volume 7, Number 3 (2009), 679-718.

Dates
First available in Project Euclid: 26 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.cms/1256562819

Mathematical Reviews number (MathSciNet)
MR2569029

Zentralblatt MATH identifier
1186.35147

Subjects
Primary: 35Q 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 58J37: Perturbations; asymptotics 76N10: Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30] 76N99: None of the above, but in this section 76L05: Shock waves and blast waves [See also 35L67]

Keywords
KZK equation isentropic Navier-Stokes system entropy paraxial approximation

Citation

Rozanova-Pierrat, Anna. 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 (2009), no. 3, 679--718. https://projecteuclid.org/euclid.cms/1256562819


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