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

Abstract

The derivation of different models of non linear acoustic in thermo-ellastic 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, to compare to the constant state perturbations, correctors allows to obtain the approximation results for the solutions of these models and to estimate the time during which they keep closed in the L2 norm. The KZK and NPE equations are also considered as paraxial approximations of the Kuznetsov equation, which is a model obtained only by perturbations from the Navier-Stokes/Euler system. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. In the aim to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Navier-Stokes system and the Kuznetsov equation in a half-space with periodic in time initial and boundary data) were obtained.