On asymptotic stability of noncharacteristic viscous boundary layers

TL;DR

This paper proves the long-time stability of a broader class of multi-dimensional noncharacteristic viscous boundary layers in hyperbolic-parabolic systems, including some MHD layers, by improving previous methods and assumptions.

Contribution

It extends stability results to more systems and dimensions, and removes a technical assumption on the glancing set, using Kreiss' symmetrizers for low-frequency estimates.

Findings

Stability established for larger class of systems in dimensions d≥2.

Applicable to certain magnetohydrodynamics (MHD) layers.

Introduced a new proof technique for low-frequency estimates.

Abstract

We extend our recent work with K. Zumbrun on long-time stability of multi-dimensional noncharacteristic viscous boundary layers of a class of symmetrizable hyperbolic-parabolic systems. Our main improvements are (i) to establish the stability for a larger class of systems in dimensions $d\ge 2$, yielding the result for certain magnetohydrodynamics (MHD) layers; (ii) to drop a technical assumption on the so--called glancing set which was required in previous works. We also provide a different proof of low-frequency estimates by employing the method of Kreiss' symmetrizers, replacing the one relying on detailed derivation of pointwise bounds on the resolvent kernel.