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

Abstract

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.