Stability criterion for small perturbations for a quasi-gasdynamic system of equations

The stability of small perturbations against a constant background is studied for a system of quasi-gasdynamic equations in an arbitrary number of space variables. It is established that, for a fixed adiabatic exponent $\gamma$, the stability is determined only by the background Mach number, and a necessary and sufficient condition for stability at any Mach number is $\gamma\le\bar\gamma$, where $\bar\gamma\approx6.2479$. The proof is based on a direct analysis of the corresponding complex characteristic numbers depending on several parameters. The multidimensional case is successfully reduced to the one-dimensional one. Then, the generalized Routh-Hurwitz criterion is applied in conjunction with analytical calculations based on Mathematica.