Stability of strong ideal-gas shock layers

Abstract

Extending recent results in the isentropic case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the spectral stability of shock-wave solutions of the compressible Navier--Stokes equations with ideal gas equation of state. Our main results are that, in appropriately rescaled coordinates, the Evans function associated with the linearized operator about the wave (i) converges in the large-amplitude limit to the Evans function for a limiting shock profile of the same equations, for which internal energy vanishes at one endstate; and (ii) has no unstable (positive real part) zeros outside a uniform ball $|\lambda|\le \Lambda$. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with the (nonphysical) limiting case, form a compact family of boundary-value problems that can be conveniently investigated numerically. An extensive numerical Evans-function study yields one-dimensional spectral stability, independent of amplitude, for gas constant $\gamma$ in $[1.2, 3]$ and ratio $\nu/\mu$ of heat conduction to viscosity coefficient within $[0.2,5]$ ($\gamma\approx 1.4$, $\nu/\mu\approx 1.47$ for air). Other values may be treated similarly but were not considered. The method of analysis extends also to the multi-dimensional case, a direction that we shall pursue in a future work.