Stability of isentropic viscous shock profiles in the high-Mach number limit

By a combination of asymptotic ODE estimates and numerical Evans function calculations, we establish stability of viscous shock solutions of the isentropic compressible Navier–Stokes equations with $\gamma$-law pressure (i) in the limit as Mach number $M$ goes to infinity, for any $\gamma \ge 1$ (proved analytically), and (ii) for $M\ge 2,500$, $\gamma \in [1,2.5]$ (demonstrated numerically). This builds on and completes earlier studies by Matsumura–Nishihara and Barker–Humpherys–Rudd–Zumbrun establishing stability for low and intermediate Mach numbers, respectively, indicating unconditional stability, independent of shock amplitude, of viscous shock waves for $\gamma$-law gas dynamics in the range $\gamma \in [1,2.5]$. Other $\gamma$-values may be treated similarly, but have not been checked numerically. The main idea is to establish convergence of the Evans function in the high-Mach number limit to that of a pressureless, or ``infinitely compressible’', gas with additional upstream boundary condition determined by a boundary-layer analysis. Recall that low-Mach number behavior is incompressible.