Viscous boundary layers in hyperbolic-parabolic systems with Neumann boundary conditions

TL;DR

This paper studies boundary layers in hyperbolic-parabolic systems with Neumann and mixed boundary conditions, revealing new boundary layer behaviors and providing a rigorous analysis of the small viscosity limit.

Contribution

It introduces a novel analysis of noncharacteristic boundary layers with Neumann and mixed boundary conditions in hyperbolic-parabolic systems, including a full boundary-layer expansion and error estimates.

Findings

Boundary layers with Neumann conditions can be characterized by reduced hyperbolic problems.

A formal boundary-layer expansion can be constructed with rigorous error bounds.

The small viscosity limit is characterized for these boundary conditions.

Abstract

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.