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QUALITATIVE ANALYSIS OF THE KHOKHLOV–ZABOLOTSKAYA–KUZNETSOV (KZK) EQUATION

ANNA ROZANOVA-PIERRAT

Laboratoire de Physique de la Matière Condensée, École Polytechnique, 91128 Palaiseau Cedex, France

https://doi.org/10.1142/S0218202508002863Cited by:17 (Source: Crossref)

Abstract

The Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation is considered as a model of nonlinear acoustic which describes the nonlinear propagation of a finite-amplitude focused sound beam which is essentially one-directional, in the thermo-viscous medium.^1,7,8 The aim of this paper is the study of the existence, uniqueness, stability, regularity, continuous dependence on the initial value and blow-up of solution of the KZK equation in Sobolev spaces H^s of periodic on x functions and with mean value zero. Existence of shock waves for the model with zero viscosity is proved using S. Alinhac's method.² Global existence in time of the beam's propagation in viscous media is established for small enough initial data. Existence result is proved by two methods: first by the fractional step method in the particular case ℝ³ and s = 3 to justify the numerical results of Thierry Le Pollès²⁵ and second for the general case ℝⁿ and s > [n/2] + 1 by the approach used in Refs. 12 and 13 for the Kadomtsev–Petviashvili (KP) equation.

Keywords:

AMSC: 22E46, 53C35, 57S20

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