Existence and stability of noncharacteristic boundary-layers for the compressible Navier-Stokes and viscous MHD equations

TL;DR

This paper proves the existence and stability of noncharacteristic boundary layers for compressible Navier-Stokes and MHD equations, establishing conditions for their stability and describing the small viscosity limit.

Contribution

It introduces a spectral stability condition based on Evans functions that ensures existence and stability of boundary layers, including for small amplitudes and in multiple dimensions.

Findings

Existence of boundary layers under uniform Evans stability.

Equivalence of small-amplitude stability to constant layer stability.

First results on multi-dimensional boundary layers in the compressible case.

Abstract

For a general class of hyperbolic-parabolic systems including the compressible Navier-Stokes and compressible MHD equations, we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical Navier-Stokes boundary conditions. Our first main result, using the abstract framework established by the authors in the companion work \cite{GMWZ6}, is to show that existence and stability of arbitrary amplitude exact boundary-layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an Evans function (uniform Evans stability). Whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with "residual" boundary conditions. Our second is to show that uniform Evans stability for small-amplitude layers is equivalent to Evans stability of the limiting constant layer, which in turn can be checked by a linear-algebraic computation. Finally, for a class of symmetric-dissipative systems including the physical examples mentioned above, we carry out energy estimates showing that constant (and thus small-amplitude) layers always satisfy uniform Evans stability. This yields existence of small-amplitude multi-dimensional boundary layers for the compressible Navier-Stokes and MHD equations. For both equations these appear to be the first such results in the compressible case.