In this dissertation we give a rigorous mathematical analysis of the asymptotic stability of arbitrary-amplitude noncharacteristic viscous boundary layers in dimensions d ≥ 1, motivated by physical applications such as compressible gas dynamics and magnetohydrodynamics (MHD) equations. Briefly, the dissertation addresses the following problems.

One-dimensional stability. Our first result concerns the stability of one-dimensional boundary layers for a class of symmetrizable hyperbolic-parabolic systems. We obtain the results by following the approach of detailed derivation of pointwise Green function bounds; more specifically, we build on the works of C. Mascia and K. Zumbrun in their treating the shock cases for the hyperbolic-parabolic systems and of S. Yarahmadian and K. Zumbrun in their treating the boundary layers for the strictly parabolic systems.

Multi-dimensional stability. Our second result concerns the long-time stability of multi-dimensional boundary layers of a general class of systems. Under the so-called uniform Evans stability condition, we prove the stability of the layers in dimensions d ≥ 2, following a modified version of the approach of K. Zumbrun in treating the multi-dimensional shock cases, involving estimates between various L^p spaces.

Multi-dimensional stability for systems with variable multiplicities . Our third result is to extend the existing stability results to certain MHD layers for which the constant multiplicity assumption used in previous analyses fails to hold. In addition, the removal of a technical assumption is done. We encompass the extension by employing the recent work of O. Guès, G. Métivier, M. Williams, and K. Zumbrun in the construction of Kreiss' symmetrizers.

Spectral stability of isentropic Navier-Stokes layers. Our final result concerns the verification of the uniform Evans stability condition. By making use of the Evans-function framework of K. Zumbrun and others, we verify numerically the stability of large-amplitude compressive, or "shock-like", boundary layers of the isentropic Navier-Stokes equations.