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 γ-law pressure (i) in the limit as Mach number M goes to infinity, for any γ ≥ 1 (proved analytically), and (ii) for M ≥ 2, 500, γ ∈ [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 γ-law gas dynamics in the range γ ∈ [1, 2.5]. Other γ-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 boundarylayer analysis. Recall that low-Mach number behavior is incompressible.1 The result of [21] is obtained by energy estimates combining the techniques of [34] with those of [14,15]; a similar approach has been used in [30] to obtain smallamplitude zero-mass stability of Boltzmann shocks. See [37,12] for an alternative approach based on asymptotic ODE methods.
Stability of isentropic viscous shock profiles 3[34], in fact yields stability of small-amplitude shocks of general symmetric hyperbolic-parabolic systems, largely settling the problem of small-amplitude shock stability for continuum mechanical systems.However, there remains the interesting question of large-amplitude stability. The main result in this direction, following a general strategy proposed in [47], is a "refined Lyapunov theorem" 2 established by Mascia and Zumbrun [32,46] for general symmetric hyperbolicparabolic systems, stating that linearized and nonlinear L 1 ∩ H 3 → L 1 ∩ H 3 orbital stability (the standard notions of stability) are equivalent to spectral stability, or nonexistence of nonstable (nonnegative real part) eigenvalues of the linearized operator L about the wave, other than the single zero eigenvalue arising through translational invariance of the underlying equations.This reduces the problem of large-amplitude stability to the study of the associated eigenvalue equation (L−λ)u = 0, a standard analytically and numerically well-posed (boundary value) problem in ODE, which can be attacked by the large body of techniques developed for asymptotic, exact, and numerical study of ODE. In particular, there exist well-developed and efficient numerical algorithms to determine the number of unstable roots for any specific linearized operator L, independent of its origins in the PDE setting; see, e.g., [9,10,11,8,22] and references therein. In this sense, the problem of determining stability of any single wave is satisfactorily resolved, or, for that matter, of any compact family of waves. To determine stability of a family of waves across an unbounded parameter regime, how...
