## Communications in Mathematical Sciences

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

### On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media

#### 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