Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers

TL;DR

This paper investigates the spectral stability of noncharacteristic boundary layers in the isentropic Navier-Stokes equations, using a combination of asymptotic and numerical methods, revealing stability for a range of parameters and highlighting the complexity introduced by boundary conditions.

Contribution

It extends stability analysis of shock-like boundary layers to the boundary layer context, incorporating new parameters and stability indices, and providing numerical evidence for stability across parameter ranges.

Findings

Boundary layers are stable for gamma in [1,3] across amplitudes.

Inflow boundary layers exhibit delicate stability depending on displacement and amplitude.

Use of Evans function and stability indices to determine stability in complex cases.

Abstract

Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or "shock-like", boundary layers of the isentropic compressible Navier-Stokes equations with gamma-law pressure by a combination of asymptotic ODE estimates and numerical Evans function computations. Our results indicate stability for gamma in the interval [1, 3] for all compressive boundary-layers, independent of amplitude, save for inflow layers in the characteristic limit (not treated). Expansive inflow boundary-layers have been shown to be stable for all amplitudes by Matsumura and Nishihara using energy estimates. Besides the parameter of amplitude appearing in the shock case, the boundary-layer case features an additional parameter measuring displacement of the background profile, which greatly complicates the resulting case structure. Moreover, inflow boundary layers turn out to have quite delicate stability in both large-displacement and large-amplitude limits, necessitating the additional use of a mod-two stability index studied earlier by Serre and Zumbrun in order to decide stability.