Large Viscous Boundary Layers
1. Introduction
In this paper, we study the linear and nonlinear stability of viscous boundary
layers which arise when one considers small viscosity parabolic perturbations of hy-
perbolic equations. For linear equations this problem is studied in [BBB], [Ba-Ra],
[Lio]. The semilinear case is solved in [Gul]. For quasilinear equations, a partial
answer is given in [Gr-Gu] (see also [Gi-Se] for results in one space dimension).
Indeed, the analysis in [Gr-Gu] has two parts. In the first part, approximate so-
lutions are obtained using formal expansions in series of the viscosity e. In the
second part, the authors prove the stability of this approximate solution, proving
that the exact solution is actually close to the approximate one, using a smallness
condition (as in [Gi-Se]). By an example, they also show that some condition is
needed. However, the smallness condition is not natural and does not allow large
boundary layers.
The goal of this paper is to remove this smallness assumption, replacing it
by an accurate assumption based on the analysis of an Evans function. Evans
functions have been introduced in the study of the stability of planar viscous shock
and boundary layers (see, e.g., [GZ], [ZH], [ZS], [Z], [S], [Rou], and references
therein1). They play the role of the Lopatinski determinant for constant coefficient
boundary value problems. When they vanish in the open left half plane, the problem
is strongly unstable and when they do not vanish in the closed half space, the
problem is expected to be strongly stable. Rescaling the variables, the results
of [ZH], [Z] can be used to study the linear stability of boundary layers created
by viscous perturbations of constant state solutions of hyperbolic equations on a
half space providing some estimates of the corresponding Green's function. This
indicates that assumptions on the Evans function should be the correct approach
in the study of the stability of boundary layers. This has been proved to be correct
in space dimension one [Gr-Ro] and the goal of this paper is to extend the analysis
to multidimensional problems.
The one space dimensional analysis in [Gr-Ro] is based on integrations along
characteristics for the hyperbolic equations and on pointwise estimates of the Green's
function for the parabolic part, which are then combined to yield
L1
bounds on
the Green's function for the linearized equations about the full boundary layer
expansion. In multi-dimensions, both ingredients break down, due to more com-
plicated geometry of characteristic surfaces. In particular, the known estimates of
the parabolic Green's function [Z] consist of LP bounds, p 2, and do not include
pointwise behavior. Moreover, it is known from study of the constant-coefficient
For the origins of this method in the study of reaction-diffusion equations; see, e.g., [El]-
[E4], [J], [AGJ], [PW], [K].
l
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