Abstract
We study linear and nonlinear stability of large-amplitude multidimensional
viscous boundary layers arising through the small viscosity perturbation of a hy-
perbolic initial-boundary value problem with noncharacteristic boundary. Our
main result is to show that, provided there holds the necessary condition that all
"frozen," planar boundary layers associated with the inner layer of the profile satisfy
an appropriate Evans function condition, then the linearized equations about the
full profile are well-posed in
L2,
with sufficiently strong estimates on the solution
and its derivatives as to yield a full nonlinear stability result and thereby nonlin-
ear continuation/validation of the formal boundary layer expansion (alternatively,
short-time existence for prepared initial data). The method of analysis is by sym-
metrizers and an appropriate extension of Kreiss' analysis of hyperbolic equations.
Notable technical aspects include reduction to constant coefficients of the resolvent
equation by an extension of the Gap Lemma of Evans function theory, clarification
of the role of block structure in Kreiss-type estimates, and the use of conormal
derivative estimates in the hyperbolic-parabolic setting.
Received by the editor March 26, 2002.
2000 Mathematics Subject Classification. 35L60 (35B35).
Key words and phrases. Boundary layers, noncharacteristic hyperbolic initial-boundary-
value problems, Kreiss symmetrizers, Evans function.
G.M. thanks Indiana University for its hospitality during a visit in which this work was
partially carried out.
K.Z. thanks the University of Rennes and E.N.S. Lyon for their hospitality during two visits
in which this work was partially carried out. Thanks also to Mark Williams for his careful reading
of early versions of the manuscript and many helpful suggestions/corrections. Research of K.Z.
was partially supported under NSF grant number DMS-0070765.
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