Multidimensional stability of large-amplitude Navier-Stokes shocks

TL;DR

This paper investigates the multidimensional stability of large-amplitude Navier-Stokes shocks using a combination of asymptotic estimates and numerical Evans-function computations, finding unconditional stability across a broad parameter range.

Contribution

It provides the first successful numerical Evans function computation for multidimensional viscous shock stability and extends stability analysis to full shock amplitude range for various gas models.

Findings

Unconditional stability within the considered parameter range.

First numerical Evans function computation for multidimensional viscous shocks.

Results agree with inviscid Euler shock stability findings.

Abstract

Extending results of Humpherys-Lyng-Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier--Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients $\mu$, $\mu +\eta$, and $\nu=\kappa/c_v$ in the physical ratios predicted by statistical mechanics, with Mach number $M>1.035$. Our results indicate unconditional stability within the parameter range considered; this agrees with the results of Erpenbeck and Majda for the corresponding inviscid case of Euler shocks. Notably, this study includes the first successful numerical computation of an Evans function associated with the multidimensional stability of a viscous shock wave. The methods introduced may be used in principle to decide stability for any $\gamma$-law gas, $\gamma>1$, and arbitrary $\mu>|\eta|\ge 0$, $\nu>0$ or, indeed, for shocks of much more general models and equations, including in particular viscoelasticity, combustion, and magnetohydrodynamics (MHD).