Communications in Mathematical Sciences

Volume 18 (2020)

Number 8

Models of nonlinear acoustics viewed as an approximation of the Navier–Stokes and Euler compressible isentropic systems

Pages: 2075 – 2119

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n8.a1

Authors

Adrien Dekkers (Laboratory Mathématiques et Informatique pour la Complexité et les Systèmes, CentraleSupélec, Univérsité Paris-Saclay, Gif-sur-Yvette, France)

Anna Rozanova-Pierrat (Laboratory Mathématiques et Informatique pour la Complexité et les Systèmes, CentraleSupélec, Univérsité Paris-Saclay, Gif-sur-Yvette, France)

Abstract

The derivation of different models of non linear acoustic in thermo-elastic media as the Kuznetsov equation, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation and the nonlinear progressive wave equation (NPE) from an isentropic Navier–Stokes/Euler system is systematized using the Hilbert-type expansion in the corresponding perturbative and (for the KZK and NPE equations) paraxial ansatz. The use of small correctors, to compare to the constant state perturbations, allows to obtain the approximation results for the solutions of these models and to estimate the time during which they keep closed in the $L^2$ norm. In the aim to compare the solutions of the exact and approximate systems in found approximation domains a global well-posedness result for the Navier–Stokes system in a half-space with time periodic initial and boundary data was obtained.

Keywords

non-linear acoustic, approximations of the Navier–Stokes system, Kuznetsov, KZK and NPE equations

2010 Mathematics Subject Classification

35B51, 35L71, 35Q30, 35Q31

Full Text (PDF format)

Received 28 April 2019

Accepted 21 May 2020

Published 22 December 2020