Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic--parabolic systems

TL;DR

This paper proves the long-time stability of large-amplitude noncharacteristic boundary layers in hyperbolic-parabolic systems, including Navier-Stokes equations, by linking stability to an Evans function condition that is numerically checkable.

Contribution

It extends stability analysis to large-amplitude boundary layers in hyperbolic-parabolic systems, establishing a spectral condition for nonlinear stability.

Findings

Stability is equivalent to an Evans function condition.

The Evans function condition is numerically checkable.

Large-amplitude boundary layers in gas dynamics are shown to be stable.

Abstract

Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic-parabolic systems including the Navier--Stokes equations of compressible gas- and magnetohydrodynamics, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index $\gamma \ge 1$. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for compressive (``shock-type'') layers in other than the nearly-constant case. The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime.