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 [S.I. Aanonsen and al., J. Acoust. Soc. Am., 75, 749-768, 1984], [M.A. Averkiou, Y.S. Lee and M.F. Hamilton, J. Acoust. Soc. Am., 94, 2876-2883, 1993], [M.A. Averkiou and M.F. Hamilton, J. Acoust. Soc. Am., 102, 2539-2548, 1997], [A. Kitkauskaite and A. Kopustinskas, available at http://www.etf.rtu.lv/Latvieshu%20lapa/pasn str/konf/p 7.pdf], [Y.S. Lee and M.F. Hamilton, J. Acoust. Soc. Am., 97, 906-917, 1995]. 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 [N.S. Bakhvalov, Ya. M. Zhileikin and E.A. Zabolotskaya, American Institute of Physics, New York, 1987, Nelineinaya teoriya zvukovih puchkov, Moscow “Nauka”, 1982]. 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 [P. Donnat, J.L. Joly, G. Metivier and J. Rauch, Indiana Univ. Math. J., 47, 1167-1241, 1998], [D. Sanchez, J. Diff. Equ., 210, 263-289, 2005], [B. Texier, Adv. Diff. Equ., 9, 1-52, 2004]. 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.