On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and
validation of the KZK-approximation for viscous and non-viscous thermo-elastic
media
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{1256562819,
author = {Rozanova-Pierrat, Anna},
title = {On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and
validation of the KZK-approximation for viscous and non-viscous thermo-elastic
media},
journal = {Commun. Math. Sci.},
volume = {7},
number = {1},
year = {2009},
pages = { 679-718},
language = {en},
url = {http://dml.mathdoc.fr/item/1256562819}
}
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., Tome 7 (2009) no. 1, pp. 679-718. http://gdmltest.u-ga.fr/item/1256562819/