Stability of strong ideal-gas shock layers

TL;DR

This paper investigates the spectral stability of ideal-gas shock layers using a combination of asymptotic analysis and numerical Evans-function computations, demonstrating stability across a range of parameters and establishing a foundation for multi-dimensional extensions.

Contribution

The paper extends spectral stability analysis of shock waves to the ideal-gas case using combined asymptotic and numerical methods, providing new stability results and a framework for future multi-dimensional studies.

Findings

Spectral stability holds for shock waves across a range of gas constants and heat conduction ratios.

Evans function converges in the large-amplitude limit to a limiting shock profile's Evans function.

The family of boundary-value problems is compact and suitable for numerical investigation.

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.