  We initiate the study of noncharacteristic boundary layers in
hyperbolic-parabolic problems with Neumann boundary conditions. More generally,
we study boundary layers with mixed Dirichlet--Neumann boundary conditions
where the number of Dirichlet conditions is fewer than the number of hyperbolic
characteristic modes entering the domain, that is, the number of boundary
conditions needed to specify an outer hyperbolic solution. We have shown
previously that this situation prevents the usual WKB approximation involving
an outer solution with pure Dirichlet conditions. It also rules out the usual
maximal estimates for the linearization of the hyperbolic-parabolic problem
about the boundary layer.
  Here we show that for linear, constant-coefficient, hyperbolic-parabolic
problems one obtains a reduced hyperbolic problem satisfying Neumann or mixed
Dirichlet--Neumann rather than Dirichlet boundary conditions. When this
hyperbolic problem can be solved, a unique formal boundary-layer expansion can
be constructed. In the extreme case of pure Neumann conditions and totally
incoming characteristics, we carry out a full analysis of the quasilinear case,
obtaining a boundary-layer approximation to all orders with a rigorous error
analysis. As a corollary we characterize the small viscosity limit for this
problem. The analysis shows that although the associated linearized hyperbolic
and hyperbolic--parabolic problems do not satisfy the usual maximal estimates
for Dirichlet conditions, they do satisfy analogous versions with losses.
