On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media
If one considers a 3D acoustic beam concentrated near the \(x_1\)-axis and propagating along it, then one can consider the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation as a mathematical model. One assumes that the density \(\rho (t,x)=\rho (t, x_1 , x_2 ,x_3 )\) and the velocity \(u(t,x)\) of the fluid are slightly fluctuated (of order \(\varepsilon\)) from constant states \(\rho_0\) and \(u_0\), respectively. Let \(\nu\) be the viscosity coefficient, \(c\) be the sound speed, and let \(\gamma\) be the heat capacity ratio (a thermodynamical constant determined by specific heat capacities). One introduces a new coordinate systme \((\tau, z,y)\) by \(\tau =t-x_1 /c,\;z=\varepsilon x_1,\;y=\sqrt{\varepsilon}(x_2 ,x_3 )\). Then, denotng \(\rho =\rho_0 +\varepsilon I(\tau ,z,y)\), one can approximate the original Navier-Stokes equation (or Euler equation) by \[ cI_{\tau z} -\frac{\gamma +1}{4\rho_0} (I^2)_{\tau\tau} -\frac{\nu}{2c^2 \rho_0} I_{\tau\tau\tau}-\frac{c^2}{2} \Delta_y I=0, \] which is called the KZK equation. The author shows how one can derive this equation from the original Navier-Stokes equation (or Euler equation), by means of perturbation theory and Hilbert expansion theory. To ensure the validity of such an approximation, the author shows that the solution of the KZK equation gives an asymptotic solution of the original Navier-Stokes equation (or Euler equation). A stability result for the Navier-Stokes equation is also given.
scientific article; zbMATH DE number 5657334